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Best proximity and fixed point results for cyclic multivalued mappings under a generalized contractive condition

Manuel De la Sen1*, Shyam Lal Singh2, Madjid Eshaghi Gordji3, Asier Ibeas4 and Ravi P Agarwal56

Author Affiliations

1 Institute of Research and Development of Processes, University of Basque Country, Campus of Leioa (Bizkaia) - Aptdo. 644 - Bilbao, Bilbao, 48080, Spain

2 Pt. L. M. S. Government Autonomous Postgraduate College, Rishikesh, 249201, India

3 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

4 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, Barcelona, 08193, Spain

5 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX, 78363-8202, USA

6 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

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Fixed Point Theory and Applications 2013, 2013:324  doi:10.1186/1687-1812-2013-324


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/324


Received:26 July 2013
Accepted:5 November 2013
Published:27 November 2013

© 2013 Sen et al.; 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 is devoted to investigating the existence of fixed points and best proximity points of multivalued cyclic self-mappings in metric spaces under a generalized contractive condition involving Hausdorff distances. Some background results for cyclic self-mappings or for multivalued self-mappings in metric fixed point theory are extended to cyclic multivalued self-mappings. An example concerned with the global stability of a time-varying discrete-time system is also discussed by applying some of the results obtained in this paper. Such an example includes the analysis with numerical simulations of two particular cases which are focused on switched discrete-time control and integrate the associate theory in the context of multivalued mappings.

MSC: 47H10, 55M20, 54H25.

Keywords:
best proximity points; cyclic self-mappings; fixed points; metric space; multi-control; multivalued self-mappings; uniform convex Banach space

1 Introduction

Important attention is being devoted to investigation of fixed point theory for single-valued and multivalued mappings concerning some relevant properties like, for instance, stability of the iterations, fixed points of contractive and nonexpansive self-mappings and the existence of either common or coupled fixed points of several multivalued mappings or operators. See, for instance, [1-24] and references therein. Related problems concerning the computational aspects of iterative calculations and best approximations based on fixed point theory have been also investigated. See, for instance, [21-23,25,26] and some references therein. On the other hand, a fixed point result for partial metric spaces and partially ordered metric spaces can be found in [27-30] and [4,15,31,32], respectively, and references therein.

This paper is devoted to the investigation of some properties of fixed point and best proximity point results for multivalued cyclic self-mappings under a general contractive-type condition based on the Hausdorff metric between subsets of a metric space [1-3,33,34]. This includes, as a particular case, contractive single-valued self-mappings [1-3,25,33-36], and similar problems for cyclic (strictly contractive or not) self-mappings [35-37] as well. Some previous results on multivalued contractions are retaken by generalizing the contractive condition and extended to cyclic multivalued self-mappings by extending the results of Đorić and Lazović in [1] (then being extended in [6] concerning results on common fixed points of a pair of multivalued maps in a complete metric space) which are based on previous Suzuki et al. and Ćirić’s results for single-valued self-mappings in some background literature papers. See, for instance, [2,3,33,34] and references therein. Through this paper, we consider a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M1">View MathML</a> and a multivalued 2-cyclic self-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M2">View MathML</a> (being simply referred to as a multivalued cyclic self-mapping in the sequel), where A and B are nonempty closed subsets of X, so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M3">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M4">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M5">View MathML</a>. Let us consider the subset of the set of real numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M6">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M7">View MathML</a>, let the symbols ‘∨’ and ‘∧’ denote the logic disjunction (‘or’) and conjunction (‘and’), and define the functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M8">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M9">View MathML</a> as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M10">View MathML</a>

(1.1)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M11">View MathML</a>

(1.2a)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M12">View MathML</a>

(1.2b)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M13">View MathML</a>

(1.3)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M14">View MathML</a> for some real constants <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M15">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M16">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M17">View MathML</a>

(1.4)

Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M18">View MathML</a> is non-increasing since all its partial derivatives with respect to K, α, β exist and are non-positive; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M19">View MathML</a> and note also that Δ is the union of the subsets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M20">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M21">View MathML</a>.

A general contractive condition is then proposed and discussed based on the Hausdorff metric on subsets of a vector space and the constraints (1.1)-(1.4). For this purpose, some preparatory concepts are needed. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M22">View MathML</a> be a family of all nonempty and closed subsets of the vector space X. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M23">View MathML</a>, then we can define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M24">View MathML</a> as the generalized hyperspace of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a> equipped with the Hausdorff metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M26">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M27">View MathML</a>

(1.5)

with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M28">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M29">View MathML</a> being nonempty sets. The gap between the nonempty sets A and B is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M30">View MathML</a>

The proposed general contractive condition to be then discussed is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M31">View MathML</a>

(1.6)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a> is a multivalued cyclic self-mapping on the subset <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M33">View MathML</a> of X, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M34">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M35">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a> is a complete metric space including the case that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M37">View MathML</a> is a Banach space with a norm-induced metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M38">View MathML</a>, so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M39">View MathML</a> is a complete metric space, is used, subject to (1.1)-(1.4), in the main result Theorem 2.1 below. In this context, Tx is the image set through T of any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M40">View MathML</a> which is in B, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M41">View MathML</a> (respectively, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M42">View MathML</a>) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43">View MathML</a> (respectively, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M44">View MathML</a>). It is inspired by that proposed in [34] for single-valued self-mappings while it generalizes that proposed and discussed in [1] for multivalued self-mappings which is based on the Hausdorff generalized metric.

See Figure 1 with the plots of the various involved sets Δ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M45">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M21">View MathML</a> and some of their relevant subsets in the contractive condition subject to (1.1)-(1.4).

thumbnailFigure 1. The setsΔand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M47">View MathML</a>,<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M48">View MathML</a>.

Note that the proposed contractive condition, in fact, considers the worst case, given by the maximum of (1.1), of such a contractive condition of [1], reflected in (1.2a), with one based on a Kannan-type contractive condition associated with the choice of possible distinct values for the constants α and β, which is reflected in (1.2b) subject to (1.3)-(1.4). In particular, the choice <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M49">View MathML</a> gives a Kannan-type contractive condition in (1.2b). Note the importance of Kannan-type contractions for single-valued mappings in the sense that a metric space is complete if and only if each Kannan contraction has a unique fixed point [27,38,39]. The incorporation of (1.2b), (1.3)-(1.4) to (1.1) to build the general contractive condition allows an obvious direct generalization of the usual contractive condition, based on the Banach principle combined with a Kannan-type constraint, since both of them do not imply each other. In this context, note, for instance, that the simple scalar single-valued sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M50">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M51">View MathML</a>, with initial condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M52">View MathML</a>, is a strict contraction if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M53">View MathML</a>. However, it is not a Kannan contraction for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M53">View MathML</a>. This is easily seen as follows. Check the Kannan condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M55">View MathML</a> for the self-mapping T on R defining the sequence solution and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M56">View MathML</a>, for instance, for points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M57">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M58">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M59">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M60">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M61">View MathML</a>. Then the Kannan contractive test is subject to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M62">View MathML</a>, which is not fulfilled for given nonzero sufficiently small values of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M63">View MathML</a> and any real <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M64">View MathML</a>. It is possible also to check in a similar way a failure of the generalized Kannan-extended contractive condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M65">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M66">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M67">View MathML</a> for given nonzero sufficiently small values of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M68">View MathML</a>.

In the current approach, a combination of distinct contractive conditions for the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M69">View MathML</a> pairs of values belonging to some relevant sets constructed from the subsets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M70">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M71">View MathML</a> of Δ is itself combined with the two point-to-point possibilities of combinations of the comparisons <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M72">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M73">View MathML</a>. The various constraints are used to prove the convergence of the iterated sequences constructed with cyclic self-mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a> to best proximity points. On the other hand, the use of ωD in the contractive condition, instead of the distance in-between subsets, allows via the choice of some real constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M75">View MathML</a> to deal with problems where the achievement of limits of sequences at best proximity points is not of particular interest but just their limits superior belonging to certain subsets of the relevant sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M76">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M77">View MathML</a> containing the best proximity points. In this case, the permanence of the relevant sequences after a finite time in subsets of the sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M78">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M79">View MathML</a> after a finite number of steps is guaranteed. The standard analysis can be used for the particular case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M80">View MathML</a>. The performed study in the manuscript seems to be also promising for its extension to the study of single-valued and multivalued proximal contraction mappings in-between subsets of metric spaces because of the close formal relation between cyclic self-mappings and proximal mappings. See, for instance, [40] and references therein.

2 Main results

The first main result follows.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a>be a complete metric space, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a>be, in general, a multivalued cyclic self-mapping, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M83">View MathML</a>are nonempty, closed and subject to the contractive constraint

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M84">View MathML</a>

(2.1)

subject to (1.1)-(1.4), for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M85">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M15">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M87">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M88">View MathML</a>. Assume also that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M89">View MathML</a>

(2.2)

Then the following properties hold:

(i) There is a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M33">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M92">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M93">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M94">View MathML</a>

IfAandBare bounded sets which intersect, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M95">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a>is a Cauchy sequence having its limit in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97">View MathML</a>, with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M98">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99">View MathML</a>for any given<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100">View MathML</a>.

IfAandBare not bounded, then the above property still holds if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M101">View MathML</a>. Furthermore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M102">View MathML</a>exists if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M103">View MathML</a>for any given<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100">View MathML</a>with the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a>being constructed in such a way that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M106">View MathML</a>.

If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107">View MathML</a>, then the sequence of sets<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M108">View MathML</a>converges to a subset<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M109">View MathML</a>of best proximity points inA (in the sense that any point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M110">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111">View MathML</a>) and the sequence of sets<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M112">View MathML</a>converges to a subset<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M113">View MathML</a>of best proximity points inBwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M114">View MathML</a>.

If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M115">View MathML</a>, i.e., if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M117">View MathML</a>, and any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a>being iteratively generated as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M98">View MathML</a>, for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M120">View MathML</a>, is a Cauchy sequence which converges to a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M121">View MathML</a>of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M122">View MathML</a>.

(ii) Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116">View MathML</a>, thatAandBare convex, and that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M124">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M125">View MathML</a>are fixed points of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M122">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M127">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M128">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M129">View MathML</a>, that is, the image sets of any fixed points are identical.

(iii) Consider a uniformly convex Banach space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M130">View MathML</a>, so that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M131">View MathML</a>is a complete metric space for the norm-induced metric<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M132">View MathML</a>, and letAandBbe nonempty, disjoint, convex and closed subsets ofXwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M122">View MathML</a>satisfying the contractive conditions (2.1)-(2.2) with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M134">View MathML</a>.

Then a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135">View MathML</a>built so that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M136">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M137">View MathML</a>is a Cauchy sequence inAif<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M138">View MathML</a>and a Cauchy sequence inBif<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M139">View MathML</a>so that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M140">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M141">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M142">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M141">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M145">View MathML</a>, then the sequences of sets<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M146">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M147">View MathML</a>converge to unique best proximity points<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M148">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M149">View MathML</a>inAandB, respectively.

Proof The proof is organized by firstly splitting it into two parts, namely, the situations: (a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M150">View MathML</a> defined in (1.2a), or (b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M151">View MathML</a>, defined in (1.2b), gives the maximum for M, defined in (1.1); and then in five distinct cases concerning (1.3), subject to (1.4), as follows.

(a) Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M152">View MathML</a>. Take, with no loss in generality, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154">View MathML</a> and note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M155">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M156">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M157">View MathML</a>, and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154">View MathML</a>, then it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M159">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M160">View MathML</a>, then one gets from the definition of Hausdorff metric (1.5) and the contractive condition (2.1), which holds for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M161">View MathML</a>, that for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M163">View MathML</a>

(2.3)

since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M15">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M165">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M166">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M167">View MathML</a>. Also, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M168">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M169">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M170">View MathML</a>. Thus, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M172">View MathML</a> so that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M173">View MathML</a>

(2.4)

since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M174">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M167">View MathML</a>.

(b) Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M176">View MathML</a> so that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M177">View MathML</a>

(2.5)

This implies also that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M178">View MathML</a> and again (2.4) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M154">View MathML</a>. As a result,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M180">View MathML</a>

By interchanging the roles of the sets A and B, one also gets by proceeding in a similar way:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M181">View MathML</a>

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M182">View MathML</a>

(2.6)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M183">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M184">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M185">View MathML</a>. Note that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a> is cyclic, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M187">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43">View MathML</a> and conversely.

Now, construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M33">View MathML</a> as follows: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M191">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M145">View MathML</a>, …, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M193">View MathML</a>, …, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M194">View MathML</a> which satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M195">View MathML</a>

(2.7)

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M196">View MathML</a>. On the other hand,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M197">View MathML</a>

(2.8)

so that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M198">View MathML</a>

(2.9)

and we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a> is a Cauchy sequence if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M200">View MathML</a> (i.e., if A and B intersect provided that they are bounded or simply if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M201">View MathML</a>) since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M202">View MathML</a>, which has a limit z in X, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M203">View MathML</a> is complete, which is also in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97">View MathML</a> which is nonempty and closed since A and B are both nonempty and closed since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M205">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M35">View MathML</a>. On the other hand, for any distance <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M207">View MathML</a> between A and B,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M208">View MathML</a>

(2.10)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M209">View MathML</a>

(2.11)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M210">View MathML</a>

(2.12)

Note that the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M211">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M212">View MathML</a> are bounded if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M213">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M214">View MathML</a> are such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M215">View MathML</a>, which is always guaranteed if A and B are bounded. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M216">View MathML</a>, then one gets from the above relations that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M217">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M218">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M219">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M220">View MathML</a>. Thus, any sequences of sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135">View MathML</a> contain the best proximity points of A and B, respectively, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107">View MathML</a> and, conversely, of B and A if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M139">View MathML</a> and converge to them. This follows by contradiction since, if not, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M225">View MathML</a>, there is some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M226">View MathML</a>, some subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M227">View MathML</a> of natural numbers with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M228">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M229">View MathML</a>, and some related subsequences of real numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M230">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M231">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M232">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M233">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M234">View MathML</a> is impossible.

Now, assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M115">View MathML</a> and consider separately the various cases in (1.3)-(1.4), by using the contractive condition (2.1), subject to (1.1)-(1.4), to prove that there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97">View MathML</a> to which all sequences converge by using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M238">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a> being a Cauchy sequence since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a> is complete and A and B are nonempty and closed.

Case 1: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M241">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M242">View MathML</a>.

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M243">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M244">View MathML</a>. Thus, the contradiction <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M245">View MathML</a> holds if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M244">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M247">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M250">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M247">View MathML</a> since Tz is closed. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M252">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M253">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M254">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M252">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M258">View MathML</a>. The proof that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M260">View MathML</a> is similar since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M261">View MathML</a> from the definitions of the sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M262">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M263">View MathML</a>, and the fact that distances have the symmetry property.

Case 2: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M264">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M265">View MathML</a>.

Then the contractive condition becomes <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M266">View MathML</a>. Then either <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M269">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M270">View MathML</a>. But the second possibility is impossible since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M271">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M272">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> since Tz is closed.

Case 3: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M274">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M275">View MathML</a>.

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M276">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M277">View MathML</a>, which implies for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M279">View MathML</a> that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M280">View MathML</a>, equivalently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M281">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M282">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M284">View MathML</a> is impossible. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> since Tz is closed.

Case 4: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M241">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M287">View MathML</a>.

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M288">View MathML</a>, which is a contradiction for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> since Tz is closed.

Case 5: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M291">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M292">View MathML</a>.

Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M293">View MathML</a>

which is a contradiction if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M248">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M236">View MathML</a> since Tz is closed. A combined result of Cases 1-5 is that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M296">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100">View MathML</a>. Now, assume again that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116">View MathML</a> and that there are two distinct fixed points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M299">View MathML</a> necessary located in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97">View MathML</a> to which the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M301">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M302">View MathML</a> converge to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M303">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M304">View MathML</a>, respectively, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M305">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M306">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M308">View MathML</a>. Assume also that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M309">View MathML</a>. One gets from the contractive condition (2.1), subject to (1.1)-(1.4), that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M310">View MathML</a>

Thus, construct sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M311">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M312">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M313">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M314">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M315">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M316">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M318">View MathML</a> which is nonempty, closed and convex, for any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M319">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M320">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M322">View MathML</a> are in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M324">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M325">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M326">View MathML</a>) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M327">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M328">View MathML</a>) as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M330">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M331">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M332">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M97">View MathML</a> contradicting the hypothesis that such sets are distinct. Properties (i)-(ii) have been proven.

Property (iii) is proven by using, in addition, [[35], Lemma 3.8], one gets

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M334">View MathML</a>

for any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M100">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M98">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M338">View MathML</a> is a uniformly convex Banach space, A and B are nonempty and disjoint closed subsets of X and A is convex. Note that Lemma 3.8 of [35] and its given proof remain fully valid for multivalued cyclic self-maps since only metric properties were used in its proof. It turns out that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221">View MathML</a> is a Cauchy sequence, then bounded, with a limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M340">View MathML</a> in A, which is also a best proximity point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M341">View MathML</a> in A since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M342">View MathML</a>

and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135">View MathML</a> converges to some point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M344">View MathML</a>, which is also a best proximity point in B (then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M345">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M346">View MathML</a>), since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M338">View MathML</a> is a uniformly convex Banach space and A and B are nonempty closed and convex subsets of X. In the same way, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M348">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M349">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221">View MathML</a> are bounded sequences since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135">View MathML</a> is bounded and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M352">View MathML</a>. Also, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M139">View MathML</a> and B is convex, then the above result holds with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M354">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M355">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M356">View MathML</a>. Now, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M357">View MathML</a>, the reformulated five cases in the proof of Property (i) would lead to contradictions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M358">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M359">View MathML</a> or if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M360">View MathML</a>. From Proposition 3.2 of [35], there are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M148">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M149">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M363">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a> is cyclic satisfying the contractive conditions (2.1)-(2.2), where A and B are nonempty and closed subsets of a complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M365">View MathML</a>, with convergent subsequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135">View MathML</a> in both A and B, respectively, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M368">View MathML</a> and in B and A, respectively, for any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M369">View MathML</a>. Assume that some given sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221">View MathML</a> in A is generated from some given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M107">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M372">View MathML</a>, which converges to the best proximity point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M373">View MathML</a> in A of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M374">View MathML</a>. Assume also that there is some sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M375">View MathML</a>, distinct from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M135">View MathML</a>, in A generated from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M377">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M378">View MathML</a> which converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M379">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M380">View MathML</a> is a best proximity point in B of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a>. Consider the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a> obtained by using the norm-induced metric in the Banach space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M383">View MathML</a> so that both spaces can be mutually identified to each other. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M384">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M43">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M386">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M387">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M388">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M340">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M390">View MathML</a> are best proximity points of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a> in A and B and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M392">View MathML</a> is the closure of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M393">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M394">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M149">View MathML</a> and then any sequence converges to best proximity points.

It is now proven by contradiction that the best proximity points in A and B are unique. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M396">View MathML</a> are two distinct best proximity points of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a> in A. Then there are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M398">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M399">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M400">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M401">View MathML</a> so that, one gets

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M402">View MathML</a>

which leads to the contradiction <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M403">View MathML</a>, and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M404">View MathML</a>. Hence Property (iii) has been proven. □

A special case of Theorem 2.1 is stated and proven in the subsequent result.

Corollary 2.2Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M405">View MathML</a>is a uniform Banach space with associate norm-induced metric<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M406">View MathML</a>, and letAandBbe nonempty closed and convex subsets ofX. Assume also that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M407">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M408">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M409">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M134">View MathML</a>in the contractive condition (2.1). If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M411">View MathML</a>, then there are<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M412">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M413">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M414">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M415">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M416">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M417">View MathML</a>are, respectively, best proximity points of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a>inAandB, respectively, and simultaneously, fixed points of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M419">View MathML</a>, respectively. In addition, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M116">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M421">View MathML</a>is a fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a>. The result also holds if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M423">View MathML</a> (and, in particular, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M424">View MathML</a>).

Proof Assume, with no loss in generality, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M425">View MathML</a>. Take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M426">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M427">View MathML</a> by noting that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M428">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a> a multivalued cyclic self-mapping. □

Remark 2.3 Note that the particular case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M430">View MathML</a> in the contractive condition (2.1) is useful to investigate multivalued cyclic Kannan self-mappings which are contractive with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M431">View MathML</a> and some of their generalizations [33,34].

The following result follows directly from Theorem 2.1 and Corollary 2.2 without proof.

Corollary 2.4Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M32">View MathML</a>is a single-valued cyclic self-mapping whereAandBare nonempty closed subsets ofXwhere<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a>is a complete metric space. Then Theorem 2.1 and Corollary 2.2 still hold mutatis-mutandis for a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M434">View MathML</a>ifAandBare convex and intersect and best proximity points are<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M435">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M436">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M437">View MathML</a>, if, in addition<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M405">View MathML</a>is a uniformly convex Banach space.

Remark 2.5 The results of this section can be extended mutatis-mutandis to multivalued <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M439">View MathML</a>-cyclic self-maps <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M440">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M441">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M442">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M443">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M444">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a> being a complete metric space. See [2,3,36,37] and references therein for some background results for single-valued cyclic s-self-mappings.

3 Example of application to time-varying discrete-time dynamic systems

3.1 Multi-control discrete-time linear dynamic system

The problems of stability in differential equations, difference equations and related dynamic systems are closely related to fixed point theory of single-valued functions since stable equilibrium points are fixed points [41-44]. Also, fixed point theory of a class of cyclic self-mappings has been recently applied to differential and difference impulsive equations in a stability context study [44]. On the other hand, some typical applications of multivalued maps can be located in the framework of dynamic programming techniques for optimal control of dynamic systems [26,45]. Several tentative controls are tested to obtain the one which minimizes a suitable cost function on a certain ahead time-interval. One of them is selected as the optimal one. Đorić and Lazović discussed in [1] a dynamic programming application in the continuous-time domain of contractive multivalued-self maps under the theoretical results of their paper. Switches among distinct parameterizations of a dynamic system and the associate stabilization problem have been discussed in the literature. Also, switching processes among different estimators of unknown systems according to the optimization or suboptimization of some appropriate loss function have been described so as to improve the estimation error. See, for instance, [41-43] and included references. On the other hand, fixed point theory has been shown to be useful to discuss the stability of iterative sequences and, in general, for the analysis of the stability of discrete dynamic systems. See, for instance, [46] and references therein. We now discuss a linear time-varying discrete control problem under several tentative controls at each stage with the purpose of selecting the control sequence which guarantees a prescribed stability degree of the feedback system. The problem is stated in such a way that the tentative state-trajectory solution is formally stated as a multivalued function generating several point-to-point iterated sequences and one of them is being selected. In particular, each current state generates a set of tentative ones at the next sampling time which belongs to the image set of the current sampled state. The convergence to fixed points or to best proximity points, if the trajectory solution sequence has a cyclic nature, describes the convergence either to equilibrium points or to a limit cycle of the solution. This second case occurs when the mapping defining the state-trajectory solution is cyclic and the subsets on whose union such a mapping is defined do not intersect. Consider the discrete time-varying control system:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M446">View MathML</a>

(3.1)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M447">View MathML</a> is the state vector sequence for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M448">View MathML</a> under some nonzero initial state <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M449">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M450">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M451">View MathML</a> and all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a> is the linear time-varying control where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M453">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a> is a sequence of control gain matrices in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M455">View MathML</a> which is chosen from an admissible set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M456">View MathML</a> of cardinal <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M457">View MathML</a> values for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>. System (3.1) is said to be an uncontrolled (or open-loop) system if the control sequence is identically zero [26]. The controlled (or closed-loop) system for any time-varying control being generated by a state-feedback control law under a gain matrix sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M459">View MathML</a> results to be

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M460">View MathML</a>

(3.2)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M461">View MathML</a> is a sequence of matrices in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M462">View MathML</a> of closed-loop dynamics. The stabilization via linear state-feedback of (3.2) and its links to fixed point theory via Theorem 2.1 are now discussed. For any sequence of natural numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M463">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>, the following relation is obtained from (3.2):

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M465">View MathML</a>

(3.3)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M466">View MathML</a>

(3.4)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M467">View MathML</a>

(3.5)

where the superscript ‘T’ denotes matrix transposition. Note that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M468">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M470">View MathML</a> is the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M471">View MathML</a> controllability matrix of (3.1) on the sequence of samples <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M472">View MathML</a>. Any prefixed state is reachable in any given prefixed number of samples from a null initial condition by some linear time-invariant state-feedback control in at most p samples if and only if (3.1) is reachable, that is, if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M473">View MathML</a>

(3.6)

for any sequence of integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M475">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M476">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M478">View MathML</a> is uniformly bounded. It is controllable to the origin if and only if it is reachable, that is, (3.6) holds and, furthermore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M479">View MathML</a> are all non-singular for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>. It is well known that if the dynamic system (3.1) is controllable to the origin, then it is also stabilizable in the sense that some linear time-varying state-feedback control sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M481">View MathML</a> is such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M482">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M483">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M484">View MathML</a>. The controllability assumption can be weakened while keeping the stabilizability property as follows.

Proposition 3.1Assume that (3.1) is stabilizable (which is guaranteed if it is controllable to the origin).

Then the following properties hold:

(i) There is a sequence of control gain matrices<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M485">View MathML</a>such that all the matrices in the subsequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M486">View MathML</a>are convergent matrices with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M487">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477">View MathML</a>being some existing sequence with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M489">View MathML</a>being a uniformly bounded sequence for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M61">View MathML</a>.

(ii) The subsequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M491">View MathML</a>of states of the closed-loop system (3.2) converges to zero as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M492">View MathML</a>. As a result, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a>of states of the closed-loop system also converges to zero as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111">View MathML</a>.

Proof One gets from (3.2) that if such a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M495">View MathML</a> of finite natural numbers exists for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M498">View MathML</a>

(3.7)

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111">View MathML</a> for some existing sequence of stabilizing controller gains <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M500">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M501">View MathML</a>, and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M502">View MathML</a> are all convergent matrices, i.e., with all their eigenvalues being of modulus less than one. Note that, since system (3.1) is stabilizable, then such a sequence of nonnegative integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M503">View MathML</a> always exists since it exists with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M504">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M476">View MathML</a>. Now, it follows from (3.7) for any vector-induced matrix norm that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M506">View MathML</a>

(3.8)

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111">View MathML</a> for any integer <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M508">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M509">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M111">View MathML</a>, and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M511">View MathML</a>

(3.9)

since the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512">View MathML</a> is uniformly bounded. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M90">View MathML</a> converges to zero for any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M484">View MathML</a>. □

Proposition 3.1 is linked to Theorem 2.1 of Section 2 in the subsequent result.

Theorem 3.2The following properties hold:

(i) Assume that system (3.1) is stabilizable and a linear time-varying feedback control<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M515">View MathML</a>is used where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M516">View MathML</a>for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>. Assume also that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M518">View MathML</a>for some sequence of nonnegative integers<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474">View MathML</a>, such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512">View MathML</a>is uniformly bounded, for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M477">View MathML</a>, there is a controller gain<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M523">View MathML</a>for some integer<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M524">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M525">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>such that any matrix in the subsequence of matrices<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M527">View MathML</a>is convergent for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M528">View MathML</a>for some uniformly bounded sequence of samples<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M529">View MathML</a>and some set of upper-bounded positive integer numbers<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M530">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>.

(ii) If, in addition, the elements of the subsequence of pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M532">View MathML</a>are all controllable for some sequence of nonnegative integers<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M495">View MathML</a>, with the sequence of natural numbers<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M534">View MathML</a>being uniformly bounded, then the closed-loop system can be exponentially stabilized via time-varying linear control with prescribed stability degree.

Outline of proof Property (i) follows directly from Proposition 3.1. Property (ii) follows since all the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M535">View MathML</a> being controllable implies that the matrices of the closed-loop dynamics satisfy at the subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474">View MathML</a> of samples the following matrix relation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M537">View MathML</a>

(3.10)

where the superscript ‘<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M538">View MathML</a>’ stands for the ith row vector of matrix, ‘≈’ stands for matrix similarity, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M539">View MathML</a> denotes the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M540">View MathML</a> identity matrix and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M541">View MathML</a> denotes some prefixed p-real row vector by the appropriate choice of the real controller matrix <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M542">View MathML</a>, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M543">View MathML</a> is controllable, towards the achievement of a suitable closed-loop stability degree. Note that the closed-loop matrix of dynamics at the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M544">View MathML</a>-sample is similar by a similarity transformation to its companion block partitioned form in (3.10). Thus, both matrices have the same characteristic monic polynomial, thus the same characteristic roots which are also the prefixed eigenvalues of the closed-loop dynamics given by (3.10), which can be arbitrarily fixed via <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M542">View MathML</a> such that its non-leading real coefficients are the components of the real row vector <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M541">View MathML</a>. Thus, the sequence of closed-loop matrices can be chosen with the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M547">View MathML</a> having a stability degree <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M548">View MathML</a> such that the stability degree of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M549">View MathML</a>. This follows since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M550">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512">View MathML</a> is uniformly bounded. Thus, the time-varying closed-loop system is exponentially stable with prescribed stability degree ρ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M552">View MathML</a> for any integer <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M553">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>. □

The stability degree is defined by the modulus of the dominant eigenvalue of the matrix of dynamics if the dominant eigenvalue is simple and such a number is a strict upper-bound of the stability degree, otherwise. At samples which are not in the subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474">View MathML</a>, the controller gains may be chosen arbitrarily. The exponential stabilization of the closed-loop system is now related to Theorem 2.1 as follows. Assume that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M556">View MathML</a> of sets of matrices <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M557">View MathML</a> contains at least a stabilizing matrix such that Theorem 3.2(ii) holds via stabilization with such stabilizing matrices.

Then Theorem 2.1 is applicable to some subset <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M558">View MathML</a> being a nonempty bounded set about <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M559">View MathML</a> such that the initial condition of (3.1) satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M560">View MathML</a> of with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M115">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M562">View MathML</a>. Take the distance function equal to the Euclidean norm so that we can consider the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M563">View MathML</a> to be identical to the Banach space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M37">View MathML</a>. Re-denote the sequence of points in A as the states <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M565">View MathML</a> (the replacement is made following the notation of Theorem 2.1), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M566">View MathML</a> . If Theorem 3.2(ii) holds, then there is some bounded <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M567">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M568">View MathML</a> defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M569">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M99">View MathML</a> is a contractive mapping which defines the state trajectory solution at the points of the sampling subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M474">View MathML</a>. Take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M572">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M573">View MathML</a> in Theorem 2.1. Note that if the stabilizing matrix is chosen within the sequence of matrices, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M574">View MathML</a> is single-valued. If all the matrices in the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M575">View MathML</a> are tested, then the multivalued map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M568">View MathML</a> is defined as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M577">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M578">View MathML</a> satisfies the Hausdorff particular contractive condition of Theorem 2.1. Also, one of the points of the sequence of sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M579">View MathML</a> satisfies the point-to-point contractive particular condition of Theorem 2.1, by virtue of such a theorem, according to the constraints

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M580">View MathML</a>

(3.11)

obtained from the stabilizing control matrix sequence, and, furthermore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M581">View MathML</a> for all samples given by the integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M582">View MathML</a>. Note that the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M583">View MathML</a>-matrix norm of any real matrix M of any order satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M584">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M585">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M586">View MathML</a> stand, respectively, for the maximum and minimum eigenvalues of the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M587">View MathML</a>-matrix with all its eigenvalues being real. A weak result is obtained with the particular case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M588">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M589">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M590">View MathML</a> in the contractive condition of Theorem 2.1. In this case, we have a multivalued contractive Kannan self-mapping. In both cases, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M591">View MathML</a> is a fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M592">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M28">View MathML</a> which is also a stable equilibrium point of the closed-loop dynamic system. Now assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M594">View MathML</a>, that is, the uncontrolled system (3.1) is scalar subject to a scalar control with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M595">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M596">View MathML</a> for some given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M597">View MathML</a>. Take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M598">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M599">View MathML</a>, and note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M600">View MathML</a>. The tentative controller gains used are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M601">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M602">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M603">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M604">View MathML</a> for the bounded sets of integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M605">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>, where the nonnegative real sequences of sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M607">View MathML</a> are uniformly bounded and contain a strictly decreasing positive real sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M608">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M609">View MathML</a> and some existing difference sequence of integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M512">View MathML</a> being uniformly upper-bounded for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a>.

The formalism of Section 2 is applicable to bounded sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M612">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M613">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M614">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M615">View MathML</a>, then a particular case of the above result follows for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M616">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M617">View MathML</a>, then the closed-loop state-trajectory solutions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M618">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M221">View MathML</a> converge to the best proximity points ε and −ε, respectively, if the initial condition is in A and, conversely, if it is in B under the sequence of stabilizing matrices <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M620">View MathML</a>.

3.2 Numerical example: a vector-valued discrete-time dynamic system with multiple parameterizations

A numerical simulation of the above-presented example (3.1) is given now. Consider the discrete-time dynamic switched system described by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M621">View MathML</a>

(3.12)

with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M622">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M623">View MathML</a> being the so-called switching function, which selects one of the dynamic systems subscripted by 1, 2 or 3 which parameterize the time-varying system (3.12) at each discrete-time instant (or sample), n. This dynamic system is a simplified version of an automobile roll dynamics enhancement control system given in [15]. The switching function is assumed, for simulation purposes, to be the 1-sample periodic (cyclic) sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M624">View MathML</a> . The following state-feedback gains are considered:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M625">View MathML</a>

(3.13)

The control design problem can be formulated as how to select the appropriate feedback gain at each sample, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M626">View MathML</a>, from the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M627">View MathML</a> in order to guarantee the asymptotic stability of the closed-loop. For this purpose, a dynamic optimization procedure can be used. Therefore, n, one considers the multivalued map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M628">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M629">View MathML</a> for each sample <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M452">View MathML</a> from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M631">View MathML</a> to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M631">View MathML</a>. The Banach space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M633">View MathML</a> can be identified with the metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M634">View MathML</a> by taking the distance <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M635">View MathML</a> to be the Euclidean norm. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M636">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M637">View MathML</a> so that it is direct to apply the formalism and results of Section 2. The multivalued composite map represents the set of reachable states starting from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321">View MathML</a> for all potential feedback gains <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M627">View MathML</a> at each sample. Figure 2 displays graphically this concept. The starting point is depicted with a circle. The application of the multivalued map T to this point produces the three points (each one corresponding to one of the feedback matrices <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M641">View MathML</a>), labeled as first iteration in Figure 2. A second application of T generates three more points from each previous one, providing nine new points, which are depicted in Figure 2 as the second iteration. This procedure can be continued to generate the complete set of reachable states from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321">View MathML</a>. The ‘plus’ symbols are used to represent the image for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M643">View MathML</a>, dots are used for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M644">View MathML</a>, while squares are used to represent the image for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M645">View MathML</a>.

thumbnailFigure 2. Graphical representation of two iterations of the multivalued mapT.

The control algorithm generates all the images of the multivalued map T and then chooses the gain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M626">View MathML</a> in such a way that the null vector, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M647">View MathML</a>, is a fixed point of the multivalued map T. In this example, the choice <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M648">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M649">View MathML</a> allows stabilizing asymptotically the system. Then, according to Proposition 3.1, all the states are bounded and the norm of the state converges to zero asymptotically as Figures 3 and 4 show.

thumbnailFigure 3. Evolution of the states with initial condition<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M650">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M651">View MathML</a>.

thumbnailFigure 4. Evolution of the norm of the state,<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M652">View MathML</a>.

In addition, it can be verified that the following matrices are convergent as Proposition 3.1(i) states:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M653">View MathML</a>

while the eigenvalues of the matrix product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M654">View MathML</a>, which describes the evolution of the discrete dynamics, converge asymptotically to zero as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M655">View MathML</a>.

3.3 Numerical example: a scalar discrete-time dynamic system with multiple parameterization

Now consider the controlled single-input single-output (SISO) dynamic system of the control sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M656">View MathML</a> given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M657">View MathML</a>

(3.14)

with the state-feedback control law <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M658">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M659">View MathML</a>. This system defines the multivalued map

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M660">View MathML</a>

(3.15)

Note that (3.14) is a 2-cyclic self-mapping with non-disjoint semi-closed sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M661">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M662">View MathML</a>) of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M663">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M664">View MathML</a> whose intersection is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M665">View MathML</a>. Consider the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M25">View MathML</a> which is also a Banach space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M383">View MathML</a> if the defined distance is the Euclidean norm. Thus, from each value of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M321">View MathML</a>, (3.12) generates an image set of dimension 3, the points being labeled as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M669">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M670">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M671">View MathML</a>. If the iteration process goes on, then each one of these values generates three more ones as depicted in Figure 5.

thumbnailFigure 5. Several iterations of the multivalued map (3.14).

The multivalued map generates three images at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M672">View MathML</a> from the starting value at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M673">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M674">View MathML</a>. Then at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M675">View MathML</a> three more values are obtained from each previous one. However, note that only four different values are obtained at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M675">View MathML</a> and five at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M677">View MathML</a>. Thus, the image of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M678">View MathML</a> possesses repeated values. Moreover, note that as the number of iterations increases, there are a larger number of points approaching zero since the use of the stabilizing gain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M643">View MathML</a> forces some of the previously obtained points to approach zero. The particular numerical values for the first iterations showed in Figure 5 are as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M680">View MathML</a>

where the subscript denotes the sample while the superscript denotes the sequence of gains used to reach the point from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M673">View MathML</a>. System (3.11) is asymptotically stabilizable provided that at least one of the following conditions for the sequence of feedback gains <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M682">View MathML</a> is met:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M683">View MathML</a> contains <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M644">View MathML</a> at least once,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M683">View MathML</a> contains <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M643">View MathML</a> an infinite (countable) number of times.

For instance, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M648">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M649">View MathML</a> satisfies the above condition (ii) and, according to Proposition 3.1, the norm of the state will converge to zero as shown in Figure 6.

thumbnailFigure 6. Evolution of the norm of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/324/mathml/M689">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors participated actively in the ellaboration of the whole paper. All authors read and approved the final manuscript.

Acknowledgements

The first and fourth authors (M De la Sen and A Ibeas) are 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. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. The second author (SL Singh) acknowledges the support by the UGC New Delhi under Emeritus Fellowship. The authors are also grateful to the referees for their valuable comments which helped to improve the manuscript.

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