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On best proximity points for pseudocontractions in the intermediate sense for non-cyclic and cyclic self-mappings in metric spaces

Manuel De la Sen

Author Affiliations

Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, 48080, Spain

Fixed Point Theory and Applications 2013, 2013:146  doi:10.1186/1687-1812-2013-146

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/146


Received:17 September 2012
Accepted:17 May 2013
Published:5 June 2013

© 2013 De la Sen; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

1 Introduction

Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated in the framework of Hilbert spaces have received a certain attention in the last years concerning their convergence properties and the existence of fixed points. See, for instance, [1-4] and references therein. Results about the existence of a fixed point are discussed in those papers. On the other hand, important attention has been paid during the last decades to the study of the convergence properties of distances in cyclic contractive self-mappings on p subsets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M1">View MathML</a> of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>, or a Banach space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M3">View MathML</a>. The cyclic self-mappings under study have been of standard contractive or weakly contractive types and of Meir-Keeler type. The convergence of sequences to fixed points and best proximity points of the involved sets has been investigated in the last years. See, for instance, [5-20] and references therein. It has to be noticed that every nonexpansive mapping [21,22] is a 0-strict pseudocontraction and also that strict pseudocontractions in the intermediate sense are asymptotically nonexpansive [2]. The uniqueness of the best proximity points to which all the sequences of iterations converge is proven in [6] for the extension of the contractive principle for cyclic self-mappings in either uniformly convex Banach spaces (then being strictly convex and reflexive [23]) or in reflexive Banach spaces [13]. The p subsets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M4">View MathML</a> of the metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>, or the Banach space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M3">View MathML</a>, where the cyclic self-mappings are defined, are supposed to be nonempty, convex and closed. If the involved subsets have nonempty intersections, then all best proximity points coincide, with a unique fixed point being allocated in the intersection of all the subsets, and framework can be simply given on complete metric spaces. The research in [6] is centered on the case of the 2-cyclic self-mapping being defined on the union of two subsets of the metric space. Those results are extended in [7] for Meir-Keeler cyclic contraction maps and, in general, with the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M7">View MathML</a>-cyclic self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M8">View MathML</a> defined on any number of subsets of the metric space with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M9">View MathML</a>. Other recent research which has been performed in the field of cyclic maps is related to the introduction and discussion of the so-called cyclic representation of a set M, as the union of a set of nonempty sets as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M10">View MathML</a>, with respect to an operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M11">View MathML</a>[14]. Subsequently, cyclic representations have been used in [15] to investigate operators from M to M which are cyclic φ-contractions, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M12">View MathML</a> is a given comparison function, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M13">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a> is a metric space. The above cyclic representation has also been used in [16] to prove the existence of a fixed point for a self-mapping defined on a complete metric space which satisfies a cyclic weak φ-contraction. In [18], a characterization of best proximity points is studied for individual and pairs of non-self-mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M15">View MathML</a>, where A and B are nonempty subsets of a metric space. The existence of common fixed points of self-mappings is investigated in [24] for a class of nonlinear integral equations, while fixed point theory is investigated in locally convex spaces and non-convex sets in [25-28]. More recently, the existence and uniqueness of best proximity points of more general cyclic contractions have been investigated in [29,30] and a study of best proximity points for generalized proximal contractions, a concept referred to non-self-mappings, has been proposed and reported in detail in [31]. Also, the study and characterization of best proximity points for cyclic weaker Meir-Keeler contractions have been performed in [32] and recent contributions on the study of best proximity and proximal points can be found in [33-38] and references therein. In general, best proximity points do not fulfill the usual ‘best proximity’ condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M16">View MathML</a> under this framework. However, best proximity points are proven to jointly globally optimize the mappings from x to the distances <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M17">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M18">View MathML</a>. Furthermore, a class of cyclic φ-contractions, which contains the cyclic contraction maps as a subclass, has been proposed in [18] in order to investigate the convergence and existence results of best proximity points in reflexive Banach spaces completing previous related results in [6]. Also, the existence and uniqueness of best proximity points of cyclic φ-contractive self-mappings in reflexive Banach spaces have been investigated in [19]. This paper is devoted to the convergence properties and the existence of fixed points of a generalized version of pseudocontractive, strict pseudocontractive and asymptotically pseudocontractive in the intermediate sense in the more general framework of metric spaces. The case of 2-cyclic pseudocontractive self-mappings is also considered. The combination of constants defining the contraction may be different on each of the subsets and only the product of all the constants is requested to be less than unity. It is assumed that the considered self-mapping can perform a number of iterations on each of the subsets before transferring its image to the next adjacent subset of the 2-cyclic self-mapping. The existence of a unique closed finite limiting sequence on any sequence of iterations from any initial point in the union of the subsets is proven if X is a uniformly convex Banach space and all the subsets of X are nonempty, convex and closed. Such a limiting sequence is of size <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M19">View MathML</a> (with the inequality being strict if there is at least one iteration with image in the same subset as its domain), where p of its elements (all of them if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M20">View MathML</a>) are best proximity points between adjacent subsets. In the case that all the subsets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M21">View MathML</a> intersect, the above limit sequence reduces to a unique fixed point allocated within the intersection of all such subsets.

2 Asymptotic contractions and pseudocontractions in the intermediate sense in metric spaces

If H is a real Hilbert space with an inner product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M22">View MathML</a> and a norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M23">View MathML</a> and A is a nonempty closed convex subset of H, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is said to be an asymptotically β-strictly pseudocontractive self-mapping in the intermediate sense for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M25">View MathML</a> if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M26">View MathML</a>

(2.1)

for some sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M27">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M28">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>[1-4,23]. Such a concept was firstly introduced in [1]. If (2.1) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M30">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is said to be an asymptotically pseudocontractive self-mapping in the intermediate sense. Finally, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M32">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically β-strictly contractive in the intermediate sense, respectively, asymptotically contractive in the intermediate sense if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M30">View MathML</a>. If (2.1) is changed to the stronger condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M36">View MathML</a>

(2.2)

then the above concepts translate into <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> being an asymptotically β-strictly pseudocontractive self-mapping, an asymptotically pseudocontractive self-mapping and asymptotically contractive one, respectively. Note that (2.1) is equivalent to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M38">View MathML</a>

(2.3)

or, equivalently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M39">View MathML</a>

(2.4)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M40">View MathML</a>

(2.5)

Note that the high-right-hand-side term <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M41">View MathML</a> of (2.3) is expanded as follows for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M42">View MathML</a>:

(2.6)

The objective of this paper is to discuss the various pseudocontractive in the intermediate sense concepts in the framework of metric spaces endowed with a homogeneous and translation-invariant metric and also to generalize them to the β-parameter to eventually be replaced with a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M44">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M45">View MathML</a>. Now, if instead of a real Hilbert space H endowed with an inner product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M22">View MathML</a> and a norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M23">View MathML</a>, we deal with any generic Banach space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M48">View MathML</a>, then its norm induces a homogeneous and translation invariant metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M49">View MathML</a> defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M50">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a> so that (2.6) takes the form

(2.7)

Define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M53">View MathML</a>

(2.8)

which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M54">View MathML</a>

(2.9)

The following result holds related to the discussion (2.7)-(2.9) in metric spaces.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M55">View MathML</a>be a metric space and consider a self-mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>. Assume that the following constraint holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M57">View MathML</a>

(2.10)

with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M58">View MathML</a>

(2.11)

for some parameterizing bounded real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M59">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M60">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M61">View MathML</a>of general terms<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M62">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M63">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M64">View MathML</a>satisfying the following constraints:

(2.12)

with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M66">View MathML</a>and, furthermore, the following condition is satisfied:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M67">View MathML</a>

(2.13)

if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M68">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70">View MathML</a>.

Then the following properties hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M71">View MathML</a>for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72">View MathML</a>so that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>is asymptotically nonexpansive.

(ii) Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>be complete, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M75">View MathML</a>be, in addition, a translation-invariant homogeneous norm and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M76">View MathML</a>, with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77">View MathML</a>being the metric-induced norm from<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M78">View MathML</a>, be a uniformly convex Banach space. Assume also that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>is continuous. Then any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M80">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M81">View MathML</a>is bounded and convergent to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M82">View MathML</a>, being in general dependent onx, in some nonempty bounded, closed and convex subsetCofA, whereAis any nonempty bounded subset ofX. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M83">View MathML</a>is bounded; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M84">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M85">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M87">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M88">View MathML</a>is a fixed point of the restricted self-mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>. Furthermore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M91">View MathML</a>

(2.14)

Proof Consider two possibilities for the constraint (2.10), subject to (2.11), to hold for each given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M92">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93">View MathML</a> as follows:

(A) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M94">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M96">View MathML</a>. Then one gets from (2.10)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M97">View MathML</a>

(2.15)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M100">View MathML</a>

(2.16)

which holds from (2.12)-(2.13) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M101">View MathML</a> since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M102">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M103">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> in (2.13) is equivalent to (2.16). Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M105">View MathML</a> is ensured either with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M106">View MathML</a> or with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M107">View MathML</a> if

(2.17)

However, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M109">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M110">View MathML</a> has to be excluded because of the unboundedness or nonnegativity of the second right-hand-side term of (2.15).

(B) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M111">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M96">View MathML</a>. Then one gets from (2.10)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M114">View MathML</a>

(2.18)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M115">View MathML</a>

(2.19)

which holds from (2.12) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M116">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M117">View MathML</a>, and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M118">View MathML</a>

(2.20)

Thus, (2.15)-(2.16), with the second option in the logic disjunction being true if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M119">View MathML</a> together with (2.18)-(2.20), are equivalent to (2.12)-(2.13) by taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M120">View MathML</a> to be either <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M121">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M122">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M96">View MathML</a>. It then follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M124">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M125">View MathML</a> from (2.15)-(2.19) since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M126">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M127">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M125">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a> is asymptotically nonexpansive. Thus, Property (i) has been proven. Property (ii) is proven as follows. Consider the metric-induced norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77">View MathML</a> equivalent to the translation-invariant homogeneous metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M132">View MathML</a>. Such a norm exists since the metric is homogeneous and translation-invariant so that norm and metric are formally equivalent. Rename <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M133">View MathML</a> and define a sequence of subsets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M134">View MathML</a> of X. From Property (i), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M135">View MathML</a> is bounded; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M136">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M137">View MathML</a> is finite, since it is bounded for any finite <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93">View MathML</a> and, furthermore, it has a finite limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>. Thus, all the collections of subsets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M140">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M141">View MathML</a> are bounded since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142">View MathML</a> is bounded. Define the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M143">View MathML</a> which is nonempty bounded, closed and convex by construction. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M144">View MathML</a> is complete, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M145">View MathML</a> is a uniformly convex Banach space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a> is asymptotically nonexpansive from Property (i), then it has a fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M147">View MathML</a>[1,23]. Since the restricted self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a> is also continuous, one gets from Property (i)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M149">View MathML</a>

(2.21)

Then any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M150">View MathML</a> is convergent (otherwise, the above limit would not exist contradicting Property (i)), and then bounded in C; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>. This also implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M152">View MathML</a> is bounded; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M154">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M85">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M157">View MathML</a>. This implies also <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M158">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M159">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M161">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a> which is then a fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a> (otherwise, the above property <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M164">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M157">View MathML</a> would be contradicted). Hence, Property (ii) is proven. □

First of all, note that Property (ii) of Theorem 2.1 applies to a uniformly convex space which is also a complete metric space. Since the metric is homogeneous and translation-invariant, a norm can be induced by such a metric. Alternatively, the property could be established on any uniformly convex Banach space by taking a norm-induced metric which always exists. Conceptually similar arguments are used in later parallel results throughout the paper. Note that the proof of Theorem 2.1(i) has two parts: Case (A) refers to an asymptotically nonexpansive self-mapping which is contractive for any number of finite iteration steps and Case (B) refers to an asymptotically nonexpansive self-mapping which is allowed to be expansive for a finite number of iteration steps. It has to be pointed out concerning such a Theorem 2.1(ii) that the given conditions guarantee the existence of at least a fixed point but not its uniqueness. Therefore, the proof is outlined with the existence of a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M167">View MathML</a> for any nonempty, bounded and closed subset A of X. Note that the set C, being in general dependent on the initial set A, is bounded, convex and closed by construction while any taken nonempty set of initial conditions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M168">View MathML</a> is not required to be convex. However, the property that all the sequences converge to fixed points opens two potential possibilities depending on particular extra restrictions on the self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a>, namely: (1) the fixed point is not unique so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M170">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> (and any A in X) so that some set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M172">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M173">View MathML</a> contains more than one point. In other words, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M174">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a> has not been proven although it is true that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M177">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a>; (2) there is only a fixed point in X. The following result extends Theorem 2.1 for a modification of the asymptotically nonexpansive condition (2.10).

Theorem 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M55">View MathML</a>be a metric space and consider the self-mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>. Assume that the constraint below holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M181">View MathML</a>

(2.22)

with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M182">View MathML</a>

(2.23)

for some parameterizing real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M183">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M184">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M185">View MathML</a>satisfying, for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M187">View MathML</a>

(2.24)

Then the following properties hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M188">View MathML</a>so that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>is asymptotically nonexpansive, and then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M190">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69">View MathML</a>if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M192">View MathML</a>

(2.25)

and the following limit exists:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M193">View MathML</a>

(2.26)

(ii) Property (ii) of Theorem 2.1 if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>is complete and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M195">View MathML</a>is a uniformly convex Banach space under the metric-induced norm<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77">View MathML</a>.

Sketch of the proof Property (i) follows in the same way as the proof of Property (i) of Theorem 2.1 for Case (B). Using proving arguments similar to those used to prove Theorem 2.1, one proves Property (ii). □

The relevant part in Theorem 2.1 being of usefulness concerning the asymptotic pseudocontractions in the intermediate sense and the asymptotic strict contractions in the intermediate sense relies on Case (B) in the proof of Property (i) with the sequence of constants <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M197">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M200">View MathML</a>; as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69">View MathML</a>. The concepts of an asymptotic pseudocontraction and an asymptotic strict pseudocontraction in the intermediate sense motivated in Theorem 2.1 by (2.7)-(2.9), under the asymptotically nonexpansive constraints (2.10) subject to (2.11) and in Theorem 2.2 by (2.22) subject to (2.23) are revisited as follows in the context of metric spaces.

Definition 2.3 Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a> is a complete metric space with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M49">View MathML</a> being a homogeneous translation-invariant metric. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically β-strictly pseudocontractive in the intermediate sense if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M206">View MathML</a>

(2.27)

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M207">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a> and some real sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M209">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M210">View MathML</a> being, in general, dependent on the initial points, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M211">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M212">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M213">View MathML</a>

(2.28)

Definition 2.4<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically pseudocontractive in the intermediate sense if (2.30) holds with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M215">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M216">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M217">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M220">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> and the remaining conditions as in Definition 2.3 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M222">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M223">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M212">View MathML</a>.

Definition 2.5<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically β-strictly contractive in the intermediate sense if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M227">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M228">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M229">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M230">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M231">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70">View MathML</a>, in Definition 2.3 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M233">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M212">View MathML</a>.

Definition 2.6<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically contractive in the intermediate sense if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M237">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M238">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M239">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M240">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M241">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M242">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> in Definition 2.3 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M244">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M245">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M246">View MathML</a>.

Remark 2.7 Note that Definitions 2.3-2.5 lead to direct interpretations of their role in the convergence properties under the constraint (2.22), subject to (2.23), by noting the following:

(1) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically β-strictly pseudocontractive in the intermediate sense (Definition 2.3), then the real sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M248">View MathML</a> of asymptotically nonexpansive constants has a general term <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M249">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a>, and it converges to a limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M251">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M252">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a> from (2.22) since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M256">View MathML</a> from (2.27). Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is trivially asymptotically nonexpansive as expected.

(2) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically pseudocontractive in the intermediate sense (Definition 2.4), then the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M259">View MathML</a> of asymptotically nonexpansive constants has the general term: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M260">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a>, and it converges to a limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M262">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M28">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is also trivially asymptotically nonexpansive as expected. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M267">View MathML</a>, note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M268">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M269">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93">View MathML</a>, while <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M271">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M272">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M274">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a> from (2.22)-(2.23).

(3) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically β-strictly contractive in the intermediate sense (Definition 2.5), then the sequence of asymptotically contractive constants is defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M278">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M280">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M282">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M283">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M285">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is an asymptotically strict contraction as expected since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M274">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a> from (2.22)-(2.23). Note that the asymptotic convergence rate is arbitrarily fast as α and β are arbitrarily close to zero, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M290">View MathML</a> becomes also arbitrarily close to zero, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M291">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M292">View MathML</a>.

(4) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically contractive in the intermediate sense (Definition 2.6), then the sequence of asymptotically contractive constants is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M294">View MathML</a>

with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M242">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M296">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M298">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M299">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M300">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M301">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is an asymptotically strict contraction as expected since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M303">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a> from (2.23). Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M306">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M299">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M308">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M309">View MathML</a>. Note also that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M310">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M311">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M312">View MathML</a>, while <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M313">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M311">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M315">View MathML</a>. In the first case, the convergence to fixed points (see Theorem 2.8 below) is guaranteed to be asymptotically faster if the self-mapping is asymptotically β-strictly contractive in the intermediate sense than if it is just asymptotically contractive in the intermediate sense if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M316">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93">View MathML</a>. Note also that if the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M209">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M319">View MathML</a> are identical in both cases, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M320">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M321">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M316">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M323">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M325">View MathML</a>.

(5) The above considerations could also be applied to Theorem 2.1 for the case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M326">View MathML</a> (Case (B) in the proof of Property (i)) being asymptotically nonexpansive for the asymptotically nonexpansive condition (2.10) subject to (2.11).

The subsequent result, being supported by Theorem 2.2, relies on the concepts of asymptotically contractive and pseudocontractive self-mappings in the intermediate sense. Therefore, it is assumed that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M327">View MathML</a>.

Theorem 2.8Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>be a complete metric space endowed with a homogeneous translation-invariant metric<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M329">View MathML</a>and consider the self-mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M331">View MathML</a>is a uniformly convex Banach space endowed with a metric-induced norm<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77">View MathML</a>from the metric<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M78">View MathML</a>. Assume that the asymptotically nonexpansive condition (2.22), subject to (2.23), holds for some parameterizing real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M334">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M335">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M336">View MathML</a>satisfying, for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M338">View MathML</a>

(2.29)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M341">View MathML</a>for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M72">View MathML</a>satisfying the conditions

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M343">View MathML</a>

(2.30)

Furthermore, the following properties hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a>is asymptoticallyβ-strictly pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M345">View MathML</a>and any given nonempty, bounded and closed subset<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M168">View MathML</a>of initial conditions if (2.29) hold with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M347">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M348">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M349">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M28">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M351">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a>has a fixed point for any such setCif<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>is continuous.

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a>is asymptotically pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M358">View MathML</a>and any given nonempty, bounded and closed subset<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M168">View MathML</a>of initial conditions if (2.29) hold with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M360">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M361">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M362">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M365">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M89">View MathML</a>has a fixed point for any such setCif<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>is continuous.

(iii) If (2.29) hold with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M372">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M373">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M374">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M231">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>is asymptoticallyβ-strictly contractive in the intermediate sense. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>has a unique fixed point.

(iv) If (2.29) hold with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M226">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M381">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M382">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M383">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M386">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>is asymptotically strictly contractive in the intermediate sense. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a>has a unique fixed point.

Proof (i) It follows from Definition 2.3 and the fact that Theorem 2.2 holds under the particular nonexpansive condition (2.22), subject to (2.23), so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M24">View MathML</a> is asymptotically nonexpansive (see Remark 2.7(1)). Property (ii) follows in a similar way from Definition 2.4 (see Remark 2.7(2)). Properties (iii)-(iv) follow from Theorem 2.2 and Definitions 2.5-2.6 implying also that the asymptotically nonexpansive self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a> is also a strict contraction, then continuous with a unique fixed point, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M392">View MathML</a> (see Remark 2.7(3)) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M299">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M394">View MathML</a> (see Remark 2.7(4)), respectively. (The above properties could also be got from Theorem 2.1 for Case (B) of the proof of Theorem 2.1(ii) - see Remark 2.7(5).) □

Example 2.9 Consider the time-varying pth order nonlinear discrete dynamic system

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M395">View MathML</a>

(2.31)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M396">View MathML</a> for some given nonempty bounded set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M397">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M398">View MathML</a> is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M399">View MathML</a> matrix sequence of elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M400">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M401">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M402">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M403">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M405">View MathML</a> defines the state-sequence trajectory solution <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M406">View MathML</a>. Equation (2.13) requires the consistency constraint <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M407">View MathML</a> to calculate <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M408">View MathML</a>. However, other discrete systems being dealt with in the same way as, for instance, that obtained by replacing <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M409">View MathML</a> in (2.31) with the initial condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M410">View MathML</a> (and appropriate ad hoc re-definition of the mapping which generates the trajectory solution from given initial conditions) do not require such a consistency constraint. The dynamic system (2.31) is asymptotically linear if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M411">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M413">View MathML</a>. Note that for the Euclidean distance (and norm), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M414">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M415">View MathML</a>. Assume that the squared spectral norm of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M416">View MathML</a> is upper-bounded by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M417">View MathML</a> for some parameterizing scalar sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M418">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M419">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M319">View MathML</a> which can be dependent, in a more general case, on the state <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M421">View MathML</a>. This holds, for instance, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M422">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M423">View MathML</a> is a real positive sequence satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M424">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M425">View MathML</a> both being potentially dependent on the state as the rest of the parameterizing sequences. Since the spectral norm equalizes the spectral radius if the matrix is symmetric, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M426">View MathML</a> can be taken exactly as the spectral radius of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M416">View MathML</a> in such a case, i.e., it equalizes the absolute value of its dominant eigenvalue. We have to check the condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M428">View MathML</a>

(2.32)

provided, for instance, that the distance is the Euclidean distance, induced by the Euclidean norm, then both being coincident, and provided also that we take the metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M429">View MathML</a> which holds, in particular, if

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M430">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M431">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M432">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M433">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M434">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M435">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M436">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M437">View MathML</a>, as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404">View MathML</a>. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M440">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M434">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M442">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M445">View MathML</a> is asymptotically nonexpansive being also an asymptotic strict β-pseudocontraction in the intermediate sense. This also implies that (2.31) is globally stable as it is proven as follows. Assume the contrary so that there is an infinite subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M446">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M447">View MathML</a> which is unbounded, and then there is also an infinite subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M448">View MathML</a> which is strictly increasing. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M449">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M450">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404">View MathML</a>, one has that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M453">View MathML</a>, any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M454">View MathML</a> and some sufficiently large <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M455">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M456">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M457">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M458">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M459">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M460">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M461">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M462">View MathML</a>. Now, take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M463">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M464">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M465">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M466">View MathML</a> and any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M467">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M468">View MathML</a>, then stability holds trivially. Assume not, and there are unbounded solutions. Thus, take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M469">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M470">View MathML</a> for any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M471">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M472">View MathML</a> and some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M473">View MathML</a>. Note that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M448">View MathML</a> is a strictly increasing real sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M475">View MathML</a> implying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M476">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M477">View MathML</a>, which leads to a contradiction to the inequality <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M478">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M479">View MathML</a> for some sufficiently large <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M480">View MathML</a>, then for some sufficiently large M, if such a strictly increasing sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M448">View MathML</a> exists. Hence, there is no such sequence, and then no unbounded sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M446">View MathML</a> for any initial condition in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142">View MathML</a>. As a result, for any initial condition in any given subset <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M397">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M485">View MathML</a> (even if it is unbounded), any solution sequence of (2.31) is bounded, and then (2.31) is globally stable. The above reasoning implies that there is an infinite collection of numerable nonempty bounded closed sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M486">View MathML</a>, which are not necessarily connected, such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M487">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404">View MathML</a> and any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M489">View MathML</a>. Assume that the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142">View MathML</a> of initial conditions is bounded, convex and closed and consider the collection of convex envelopes <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M491">View MathML</a>, define constructively the closure convex set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M492">View MathML</a> which is trivially bounded, convex and closed. Note that it is not guaranteed that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M493">View MathML</a> is either open or closed since there is a union of infinitely many closed sets involved. Note also that the convex hull of all the convex envelopes of the collection of sets is involved to ensure that A is convex since the union of convex sets is not necessarily convex (so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M494">View MathML</a> is not guaranteed to be convex while A is convex). Consider now the self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M495">View MathML</a> which defines exactly the same solution as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M405">View MathML</a> for initial conditions in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M397">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M498">View MathML</a> is identified with the restricted self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M499">View MathML</a> from a nonempty bounded, convex and closed set to itself. Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M500">View MathML</a> for the Euclidean distance is a convex metric space which is also complete since it is finite dimensional. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M501">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M502">View MathML</a> are both continuous, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M495">View MathML</a> is also continuous and has a fixed point in A from Theorem 2.8(i).

(b) If the self-mapping is asymptotically pseudocontractive in the intermediate sense, then the above conclusions still hold with the modification <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M504">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M505">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M415">View MathML</a>. From Remark 2.7(2), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M508">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M509">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M510">View MathML</a>. Thus the convergence is guaranteed to be faster for an asymptotic β-strict pseudocontraction in the intermediate sense than for an asymptotic pseudocontraction in the intermediate sense with a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M511">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M512">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a> with the remaining parameters and parametrical sequences being identical in both cases. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M514">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M515">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M516">View MathML</a> are both continuous, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M517">View MathML</a> is continuous and has a fixed point in A from Theorem 2.8(ii).

(c) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a> is asymptotically β-strictly contractive in the intermediate sense, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M519">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M404">View MathML</a> so that it is asymptotically strictly contractive and has a unique fixed point from Theorem 2.8(iii).

(d) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a> is asymptotically contractive in the intermediate sense, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M522">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M141">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M56">View MathML</a> is an asymptotic strict contraction and has a unique fixed point from Theorem 2.8(iv).

Remark 2.10 Note that conditions like (2.32) can be tested on dynamic systems being different from (2.31) by redefining, in an appropriate way, the self-mapping which generates the solution sequence from given initial conditions. This allows to investigate the asymptotic properties of the self-mapping, the convergence of the solution to fixed points, then the system stability, etc. in a unified way for different dynamic systems. Close considerations can be discussed for different dynamic systems and convergence of the solutions generated by the different cyclic self-mappings defined on the union of several subsets to the best proximity points of each of the involved subsets.

3 Asymptotic contractions and pseudocontractions of cyclic self-mappings in the intermediate sense

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M525">View MathML</a> be nonempty subsets of X. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526">View MathML</a> is a cyclic self-mapping if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M528">View MathML</a>. Assume that the asymptotically nonexpansive condition (2.10), subject to (2.11), is modified as follows:

(3.1)

(3.2)

with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M531">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>, and that the asymptotically nonexpansive condition (2.22), subject to (2.23), is modified as follows:

(3.3)

(3.4)

with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M531">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M69">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M539">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M540">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M541">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M542">View MathML</a> and Theorems 2.1, 2.2 and 2.8 hold with the replacement <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M543">View MathML</a>. Then if A and B are closed and convex, then there is a unique fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M545">View MathML</a>. In the following, we consider the case that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M546">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M547">View MathML</a>. The subsequent result based on Theorems 2.1, 2.2 and 2.8 holds.

Theorem 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>be a metric space and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526">View MathML</a>be a cyclic self-mapping, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M528">View MathML</a>, whereAandBare nonempty subsets ofX. Define the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M552">View MathML</a>of asymptotically nonexpansive iteration-dependent constants as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M554">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>provided that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>satisfies the constraint (3.1), subject to (3.2), and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M557">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M558">View MathML</a>

(3.7)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M561">View MathML</a>) and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M563">View MathML</a>) provided that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>satisfies the constraint (3.3) subject to (3.4) provided that the parameterizing bounded real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M565">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M60">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M61">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M568">View MathML</a>of general terms<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M569">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M570">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M246">View MathML</a>fulfill the following constraints:

(3.8)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M573">View MathML</a>and assuming that the following limits exist:

(3.9)

Then, the following properties hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>satisfies (3.3) subject to (3.4)-(3.9); <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M576">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M577">View MathML</a>

so that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is a cyclic asymptotically nonexpansive self-mapping. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a>is a best proximity point ofAand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580">View MathML</a>is a best proximity point ofB, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M581">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M582">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M583">View MathML</a>, which are best proximity points ofAandB (not being necessarily identical toxandy), respectively if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M584">View MathML</a>is continuous.

(ii) Property (i) also holds if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>satisfies (3.1) subject to (3.2), (3.7), (3.8)-(3.9) and (3.5b) provided that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M586">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M587">View MathML</a>.

Proof The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)

(3.10)

and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M589">View MathML</a>

(3.11)

since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M590">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M593">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M594">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M595">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M597">View MathML</a>. Note that (3.8) implies that there is no division by zero in (3.11). Now, assume that (3.10) holds with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M598">View MathML</a>. From (3.8) and (3.2), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M599">View MathML</a>, equivalently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M600">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M601">View MathML</a>, which contradicts (3.5a) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M110">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M598">View MathML</a> in (3.5a) under (3.7) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M604">View MathML</a> and, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M605">View MathML</a> from (3.6), there is no division by zero on the right-hand side of (3.10) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M598">View MathML</a>.

Also, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a> is continuous, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M608">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M609">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M611">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M612">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M613">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M593">View MathML</a>. This proves Properties (i)-(ii). □

Remark 3.2 Note that Theorem 3.1 does not guarantee the convergence of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M616">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M617">View MathML</a> to best proximity points if the initial points for the iterations <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580">View MathML</a> are not best proximity points if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a> is not contractive.

The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M621">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a> subject to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M623">View MathML</a>.

Theorem 3.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>be a metric space and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526">View MathML</a>be a cyclic self-mapping which satisfies the asymptotically nonexpansive constraint (3.1), subject to (3.2), whereAandBare nonempty subsets ofX. Let the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M626">View MathML</a>of asymptotically nonexpansive iteration-dependent constants be defined by a general term<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M627">View MathML</a>under the constraints<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M628">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M629">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M630">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M631">View MathML</a>. Then the subsequent properties hold:

(i) The following limits exist:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M632">View MathML</a>

(3.12)

(ii) Assume, furthermore, that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M2">View MathML</a>is complete, AandBare closed and convex and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M634">View MathML</a>is translation-invariant and homogeneous and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M635">View MathML</a>is uniformly convex where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77">View MathML</a>is the metric-induced norm. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M637">View MathML</a>

(3.13)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M638">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M639">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M641">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M642">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M644">View MathML</a>, wherezandTzare unique best proximity points inAandB, respectively. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M541">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M646">View MathML</a>is the unique fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>.

Proof Note from (3.9), under (3.6) and (3.7), that there is no division by zero on the right-hand side of (3.10) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M648">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M649">View MathML</a>. Then one has from (3.1)-(3.2), (3.5a), (3.6) and (3.7) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M650">View MathML</a>

(3.14)

There are several possible cases as follows.

Case A: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M651">View MathML</a> is non-increasing. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M652">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M654">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M655">View MathML</a>, one gets (3.12).

Case B: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M651">View MathML</a> is non-decreasing. Then either <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M657">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M659">View MathML</a> or it is unbounded. Then it has a subsequence which diverges, from which a strictly increasing subsequence can be taken. But this contradicts <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M660">View MathML</a> following from (3.14) subject to the given parametrical constraints. Thus, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M661">View MathML</a> is non-decreasing, it cannot have a strictly increasing subsequence so that it is bounded and has a finite limit as in Case A.

Case C: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M651">View MathML</a> has an oscillating subsequence. It is proven that such a subsequence is finite. Assume not, then if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M663">View MathML</a>, there is an integer sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M664">View MathML</a> of general term subject to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M665">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M666">View MathML</a>

but the above expression is equivalent, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M667">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M668">View MathML</a> which are in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M669">View MathML</a>, but not jointly in either A or B, to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M670">View MathML</a>

which contradicts <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M671">View MathML</a> since both sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M672">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M673">View MathML</a> are bounded; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M654">View MathML</a>. Then there is no infinite oscillating sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M675">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M676">View MathML</a> so that there is a finite limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M677">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M678">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M679">View MathML</a>. Now, proceed by contradiction by assuming the existence of some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M680">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M681">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M683">View MathML</a>. Thus, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M684">View MathML</a>, there is some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M685">View MathML</a> such that there are two consecutive nonzero elements of a nonzero real sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M686">View MathML</a>, which can depend on x and y, which satisfy <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M687">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M688">View MathML</a>

(3.15)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M683">View MathML</a>. Otherwise, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M690">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M691">View MathML</a> and any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M692">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M693">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M694">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M696">View MathML</a>. One gets, by combining (3.14) and (3.15), that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M697">View MathML</a>

(3.16)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M698">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M699">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M700">View MathML</a>, and some nonnegative real sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M701">View MathML</a> which converges to zero since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M702">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M704">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M705">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M706">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M683">View MathML</a>. The relations (3.16) contradict <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M709">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M710">View MathML</a> is positive <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a> (and it does not converge to zero) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M712','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M712">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M713">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>. Thus, one concludes that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M686">View MathML</a> converges to zero, and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M716">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M717">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M717">View MathML</a>. This leads to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M719">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720">View MathML</a> by taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M721">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M563">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562">View MathML</a>. Property (i) has been proven.

Now, Property (ii) is proven. It is first proven that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M726">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720">View MathML</a> if the metric is translation-invariant and homogeneous so that it induces a norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77">View MathML</a> if A and B are nonempty, closed and convex subsets of X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M729">View MathML</a> is a uniformly convex Banach space. Assume not and take such a norm to yield <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M730">View MathML</a>. Then if A is nonempty, closed and convex and B is nonempty and closed and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M732">View MathML</a>. It is known that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M733','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M733">View MathML</a> from Theorem 3.1(i) for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M734">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M735">View MathML</a> is a uniformly convex Banach space for the metric-induced norm (being equivalent to the translation-invariant homogeneous metric), we have the following property for the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M736','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M736">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M737">View MathML</a> satisfying for some strictly increasing nonnegative sequence of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M738">View MathML</a> and any nonnegative sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M739','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M739">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M740','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M740">View MathML</a> satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M741">View MathML</a> and any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M742">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a> that

(3.17)

(3.18)

(3.19)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>, which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M749','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M749">View MathML</a>

(3.20)

which has to be valid for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M750','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M750">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>. Now, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M752','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M752">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M753','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M753">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M755">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M756','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M756">View MathML</a>, which is a contradiction to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M757','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M757">View MathML</a> being strictly increasing, then contradicting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M758">View MathML</a> being a uniformly convex Banach space, unless <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M759','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M759">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M761','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M761">View MathML</a> converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M762">View MathML</a>. Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M764">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>, (3.15) for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M766','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M766">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70">View MathML</a> implies the existence of the first zero limit in (3.13). The existence of the second zero limit in (3.13) is proven in the same way since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M768">View MathML</a>. Since those limits are zero, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M769">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M770','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M770">View MathML</a> are Cauchy sequences in A converging to a best proximity point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M771','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M771">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a>. Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M773','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M773">View MathML</a> is necessarily the unique best proximity point in A since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M769">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M775','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M775">View MathML</a> converge to the same point. Otherwise, the first limit of (3.13) would not exist if the sequences do not converge, then a contradiction holds to a proven result, and also Property (i) would not be true, since (3.12) would not hold, if the limit of the sequence would not be a best proximity point in A, then a contradiction holds to another proven result. In the same way, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M776','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M776">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M777','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M777">View MathML</a> converge to a unique best proximity point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M778','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M778">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M779','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M779">View MathML</a>. Now, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M780','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M780">View MathML</a>. Assume not. Then since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M781','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M781">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M782">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M783','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M783">View MathML</a>, one has <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M784','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M784">View MathML</a>. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M785','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M785">View MathML</a> so that since A and B are convex,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M786','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M786">View MathML</a>

which is a contradiction. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M780','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M780">View MathML</a> is the unique best proximity of B. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M788','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M788">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M789">View MathML</a> is the unique fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M790">View MathML</a> which coincides with the unique best proximity point in A and B. □

Remark 3.4 Theorem 3.3 is known for strictly contractive cyclic self-mappings [20] satisfying the contractive condition (3.1) in the case that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M791','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M791">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M792','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M792">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M793','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M793">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M794','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M794">View MathML</a>[5-7].

It is now assumed that the cyclic self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526">View MathML</a> is asymptotically nonexpansive while not being strictly contractive for any finite number of iterations. The concepts of cyclic pseudocontractions and a strict contraction in the intermediate sense play an important role in the obtained results.

Theorem 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M3">View MathML</a>be a uniformly convex Banach space endowed with a metric-induced norm<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M77">View MathML</a>from a translation-invariant homogeneous metric<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M78">View MathML</a>, whereAand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M799">View MathML</a>are nonempty, closed and convex subsets ofXand assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>is a cyclic self-mapping. Define the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M801">View MathML</a>of asymptotically nonexpansive iteration-dependent constants as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M554">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>provided that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>satisfies the constraint (3.1), subject to (3.2); and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M806','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M806">View MathML</a>

(3.22)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M561">View MathML</a>) and for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M811','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M811">View MathML</a>) provided that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>satisfies the constraint (3.3), subject to (3.4), provided that the parameterizing bounded real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M813','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M813">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M814','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M814">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M815','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M815">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M816','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M816">View MathML</a>of general terms<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M817','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M817">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M245">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M246">View MathML</a>fulfill the following constraints:

(3.23)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M573">View MathML</a>and assuming that the following limits exist:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M822','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M822">View MathML</a>

(3.24)

Then the following properties hold:

(i) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>satisfies (3.3) subject to (3.20)-(3.24); <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M824','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M824">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M825','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M825">View MathML</a>

so that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is asymptotically nonexpansive. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M779','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M779">View MathML</a>is a best proximity point ofAand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580">View MathML</a>is a best proximity point ofB, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M829','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M829">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M830','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M830">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M831','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M831">View MathML</a>which are best proximity points ofAandB (not being necessarily identical toxandy), respectively, if furthermore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is continuous.

(ii) Property (i) also holds if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>satisfies (3.1) subject to (3.2), (3.22), (3.23)-(3.24) and (3.5b) with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M834','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M834">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M835','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M835">View MathML</a>.

(iii) Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is asymptoticallyβ-strictly pseudocontractive in the intermediate sense so that (3.21a)-(3.21b) holds with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M837','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M837">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M838','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M838">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M839','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M839">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M840','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M840">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M220">View MathML</a>, as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M844','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M844">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M584">View MathML</a>is asymptotically nonexpansive and Property (i) holds.

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is asymptotically pseudocontractive in the intermediate sense if (3.22) holds with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M847','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M847">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M848','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M848">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M849','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M849">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M850','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M850">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M218">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M351">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M51">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M630">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is asymptotically nonexpansive and Property (i) holds.

(v) If the conditions of Property (iv) are modified as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M857','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M857">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M858','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M858">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M859','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M859">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M861','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M861">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M863','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M863">View MathML</a>in (3.22), then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is asymptoticallyβ-strictly contractive in the intermediate sense. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>has a unique best proximity pointzinAand a unique best proximity pointTzinBto which the sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M866','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M866">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M867','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M867">View MathML</a>converge; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M562">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M870','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M870">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M871','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M871">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>.

(vi) If (3.4) is modified by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M873','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M873">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M874','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M874">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M875','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M875">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M240">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M208">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M231">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M219">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M33">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a>is asymptotically-strictly contractive in the intermediate sense. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a>has a unique best proximity point inAand a unique best proximity point inBto which the sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M866','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M866">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M884','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M884">View MathML</a>converge as in Property (v).

Proof The second condition of (2.18) now becomes under (3.1)-(3.2), or (3.3)-(3.4), and (3.23)-(3.24)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M885','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M885">View MathML</a>

since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M594">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M887','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M887">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M70">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M889','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M889">View MathML</a>. Also, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M575">View MathML</a> is continuous, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M891','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M891">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M609">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M893','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M893">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M612">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M895','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M895">View MathML</a> since A and B are closed and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M896','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M896">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M897','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M897">View MathML</a>. This proves Properties (i)-(ii). To prove Property (iii), note that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M898','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M898">View MathML</a> is asymptotically β-strictly pseudocontractive in the intermediate sense under (3.21a)-(3.21b)-(3.23) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M899','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M899">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M630">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M901','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M901">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> and (3.22) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M351">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M905','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M905">View MathML</a> is asymptotically nonexpansive and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M906','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M906">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M29">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M908','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M908">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580">View MathML</a> are best proximity points. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M911','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M911">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M824','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M824">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M609">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M893','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M893">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M612">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M916','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M916">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a> is continuous. Then Property (i) holds. Property (iv) is proven in a similar way as (iii) since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M556">View MathML</a> is again asymptotically nonexpansive. Properties (v)-(vi) follow since in both cases <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M526">View MathML</a> becomes a cyclic strictly contractive self-mapping for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M920','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M920">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M921','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M921">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M922','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M922">View MathML</a> and some finite <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M923','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M923">View MathML</a> in Theorem 3.3, Eq. (3.14). Thus, it is a direct proof that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M924','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M924">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M925','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M925">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M926','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M926">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M927','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M927">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M527">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M528">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M932','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M932">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M720">View MathML</a>. Furthermore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M934','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M934">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M935','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M935">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M644">View MathML</a> and there are unique best proximity points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M773','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M773">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M939','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M939">View MathML</a>. The convergence of the iterations to unique best proximity points follows using similar arguments as those used in the proof of Theorem 3.3(ii) based on the uniform convexity of the complete metric space and the fact that the subsets A and B are nonempty, convex and closed. □

Remark 3.6 Note that the existence of Theorem 3.5 of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M171">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M580">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M942','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M942">View MathML</a> is guaranteed if A is nonempty, bounded, closed and convex and B is nonempty closed and convex is also guaranteed if A is compact and B is approximately compact with respect to A, i.e., if every sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M943','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M943">View MathML</a>, such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M944','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M944">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M563">View MathML</a>, has a convergent subsequence [6,7,31].

Example 3.7 Consider the time-varying scalar controlled discrete dynamic system:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M946','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M946">View MathML</a>

(3.25)

under the feedback control sequence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M947','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M947">View MathML</a>

(3.26)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M948','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M948">View MathML</a>

(3.27)

so that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M949','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M949">View MathML</a>

(3.28)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M950','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M950">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M951','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M951">View MathML</a> for some given nonempty bounded set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M142">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M953','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M953">View MathML</a> is the control sequence. The above model can describe discrete-time dynamic systems under time-varying sampling periods or under a time-varying parameterization in general [39]. Assume that the suitable controlled solution (3.28) is of the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M954','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M954">View MathML</a>

Then

(3.29)

(3.30)

The identities (3.30) allow the feedback generation of the control sequence (3.26) from its previous values and previous solution values as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M957','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M957">View MathML</a>

(3.31)

for given parameterizing scalar sequences which can be dependent on the state <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M421">View MathML</a> (see Example 2.9). We are now defining a cyclic self-map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M959','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M959">View MathML</a> so that the solution belongs alternately to positive (respectively, nonnegative) and negative (respectively, nonpositive) real intervals <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M960','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M960">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M961','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M961">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M962','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M962">View MathML</a> (respectively, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M963','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M963">View MathML</a>), that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M964','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M964">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M965','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M965">View MathML</a>. For such an objective, consider the scalar bounded sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M966','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M966">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M967','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M967">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M968','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M968">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M969','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M969">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M970','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M970">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M971','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M971">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M972','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M972">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M99">View MathML</a> which satisfy

(3.32a)

(3.32b)

Note that by using the Euclidean distance and norm on R, it is possible to apply the theoretical formalism to the expressions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M976','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M976">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M977','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M977">View MathML</a> to prove convergence to the best proximity points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M978','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M978">View MathML</a> to which the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M979','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M979">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M980','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M980">View MathML</a> converge, respectively if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M981','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M981">View MathML</a> and conversely if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M982','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M982">View MathML</a>. Assume that:

(1) The constraints (3.32a)-(3.32b) hold;

(2) The parametrical constraints of the various parts (a) to (d) of Example 2.9 hold with the replacements and its appropriate replacements of the constraints <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M983','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M983">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M984','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M984">View MathML</a>;

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M418">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M419">View MathML</a> are redefined for this example from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M987','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M987">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M988','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M988">View MathML</a>, respectively, from (3.32a)-(3.32b).

From Theorem 3.5, the various properties of Example 2.9 hold also for this example if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M542">View MathML</a> so that the cyclic self-map is such that it alternates the values of the solution sequence between <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M990','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M990">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M991','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M991">View MathML</a>. The unique fixed point to which the solution converges is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M992','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M992">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M993','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M993">View MathML</a>, then the corresponding results are modified by convergence to each of the unique best proximity points to which the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M979','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M979">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M995','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M995">View MathML</a> converge; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M996','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/146/mathml/M996">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author is very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651, and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. He is also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07 and to the referees for their useful comments.

References

  1. Sahu, DR, Xu, HK, Yao, JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal., Theory Methods Appl.. 70, 3502–3511 (2009). Publisher Full Text OpenURL

  2. Qin, X, Kim, JK, Wang, T: On the convergence of implicit iterative processes for asymptotically pseudocontractive mappings in the intermediate sense. Abstr. Appl. Anal.. 2011, Article ID 468716. doi:10.1155/2011/468716 (2011)

  3. Ceng, LC, Petrusel, A, Yao, JC: Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense. Taiwan. J. Math.. 15(2), 587–606 (2011)

  4. Duan, P, Zhao, J: Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense. Fixed Point Theory Appl.. 2011, Article ID 13. doi:10.1186/1687-1812-2011-13 (2011)

  5. Kirk, WA, Srinivasan, PS, Veeramani, P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory. 4(1), 79–89 (2003)

  6. Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl.. 323, 1001–1006 (2006). Publisher Full Text OpenURL

  7. Karpagam, S, Agrawal, S: Best proximity point theorems for p-cyclic Meir-Keeler contractions. Fixed Point Theory Appl.. 2009, Article ID 197308. doi:10.1155/2009/197308 (2009)

  8. Di Bari, C, Suzuki, T, Vetro, C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal., Theory Methods Appl.. 69(11), 3790–3794 (2008). Publisher Full Text OpenURL

  9. De la Sen, M: Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings. Fixed Point Theory Appl.. 2010, Article ID 572057. doi:10.1155/2010/572057 (2010)

  10. De la Sen, M: Some combined relations between contractive mappings, Kannan mappings reasonable expansive mappings and T-stability. Fixed Point Theory Appl.. 2009, Article ID 815637. doi:10.1155/2009/815637 (2009)

  11. Suzuki, T: Some notes on Meir-Keeler contractions and L-functions. Bull. Kyushu Inst. Technol.. 53, 12–13 (2006)

  12. Derafshpour, M, Rezapour, S, Shahzad, N: On the existence of best proximity points of cyclic contractions. Adv. Dyn. Syst. Appl.. 6(1), 33–40 (2011)

  13. Rezapour, S, Derafshpour, M, Shahzad, N: Best proximity points of cyclic φ-contractions on reflexive Banach spaces. Fixed Point Theory Appl.. 2010, Article ID 946178. doi:10.1155/2010/946178 (2010)

  14. Al-Thagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal., Theory Methods Appl.. 70(10), 3665–3671 (2009). Publisher Full Text OpenURL

  15. Rus, IA: Cyclic representations and fixed points. Ann. T. Popoviciu Semin. Funct. Equ. Approx. Convexity. 3, 171–178 (2005)

  16. Pacurar, M, Rus, IA: Fixed point theory for cyclic φ-contraction. Nonlinear Anal., Theory Methods Appl.. 72(3-4), 1181–1187 (2010). Publisher Full Text OpenURL

  17. Karapinar, E: Fixed point theory for cyclic weak ϕ-contraction. Appl. Math. Lett.. 24, 822–825 (2011). Publisher Full Text OpenURL

  18. Shazhad, N, Sadiq Basha, S, Jeyaraj, R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl.. 148(1), 69–78 (2011). Publisher Full Text OpenURL

  19. Vetro, C: Best proximity points: convergence and existence theorems for p-cyclic mappings. Nonlinear Anal., Theory Methods Appl.. 73(7), 2283–2291 (2010). Publisher Full Text OpenURL

  20. De la Sen, M: On a general contractive condition for cyclic self-mappings. J. Appl. Math.. 2011, Article ID 542941. doi:10.1155/2011/542941 (2011)

  21. Yao, YH, Liu, YC, Chen, CP: Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings. Taiwan. J. Math.. 15(5), 1979–1998 (2011)

  22. Yao, YH, Chen, RD: Regularized algorithms for hierarchical fixed-point problems. Nonlinear Anal., Theory Methods Appl.. 74(17), 6826–6834 (2011). Publisher Full Text OpenURL

  23. Qin, X, Kang, SM, Agarwal, RP: On the convergence of an implicit iterative process for generalized asymptotically quasi-nonexpansive mappings. Fixed Point Theory Appl.. 2010, Article ID 714860. doi:10.1155/2010/714860 (2010)

  24. Pathak, HK, Khan, MS, Tiwari, R: A common fixed point theorem and its application to nonlinear integral equations. Comput. Math. Appl.. 53(6), 961–971 doi:10.1016/j.camwa.2006.08.046 (2007)

    doi:10.1016/j.camwa.2006.08.046

    Publisher Full Text OpenURL

  25. Khan, MS, Nashine, HK: On invariant approximation for noncommutative mappings in locally convex spaces. J. Comput. Anal. Appl.. 10(1), 7–15 (2008). PubMed Abstract OpenURL

  26. Nashine, HK, Khan, MS: An application of fixed point theorem to best approximation in locally convex space. Appl. Math. Lett.. 23(2), 121–127 doi:10.1016/j.aml.2009.06.025 (2010)

    doi:10.1016/j.aml.2009.06.025

    Publisher Full Text OpenURL

  27. Nashine, HK, Khan, MS: Common fixed points versus invariant approximation in nonconvex sets. Appl. Math. E-Notes. 9, 72–79 (2009)

  28. Pathak, HK, Tiwari, R, Khan, MS: A common fixed point theorem satisfying integral type implicit relations. Appl. Math. E-Notes. 7, 222–228 (2007)

  29. De la Sen, M, Agarwal, RP: Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type. Fixed Point Theory Appl.. 2012, Article ID 102. doi:10.1186/1687-1812-2011-102 (2012)

  30. De la Sen, M, Agarwal, RP: Some fixed point-type results for a class of extended cyclic self-mappings with a more general contractive condition. Fixed Point Theory Appl.. 2011, Article ID 59. doi:10.1186/1687-1812-2011-59 (2011)

  31. Basha, SS, Shahzad, N: Best proximity point theorems for generalized proximal contractions. Fixed Point Theory Appl.. 2012, Article ID 42. doi:10.1186/1687-1812-2012-42 (2012)

  32. Chen, CM, Lin, CJ: Best periodic proximity point theorems for cyclic weaker Meir-Keeler contractions. J. Appl. Math.. 2012, Article ID 856974. doi:10.1155/2012/856974 (2012)

  33. Caballero, J, Harjani, J, Sadarangani, K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl.. 2012, Article ID 231 (2012)

  34. Mongkolkeha, C, Cho, YJ, Kuman, P: Best proximity points for generalized proximal C-contraction mappings in metric spaces with partial orders. J. Inequal. Appl.. 2013, Article ID 94 (2013)

  35. Karapinar, E: Best proximity points of Kannan type cyclic weak ϕ-contractions in ordered metric spaces. An. Stiint. Univ. Ovidius Constanta, Ser. Mat.. 20(3), 51–63 (2012)

  36. Karapinar, E: Best proximity points of cyclic mappings. Appl. Math. Lett.. 25(11), 1761–1766 (2012). Publisher Full Text OpenURL

  37. Karapinar, E, Erhan, IM: Best proximity points on different type contractions. Appl. Math. Inf. Sci.. 5(3), 558–569 (2011)

  38. Raj, VS: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal., Theory Methods Appl.. 74(14), 4804–4808 (2011). Publisher Full Text OpenURL

  39. De la Sen, M: Application of the nonperiodic sampling to the identifiability and model-matching problems in dynamic systems. Int. J. Syst. Sci.. 14(4), 367–383 (1983). Publisher Full Text OpenURL