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Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Buthinah A Bin Dehaish1* and WM Kozlowski2

Author Affiliations

1 Department of Mathematics, King Abdulaziz University, P.O. Box 53909, Jeddah, 21593, Saudi Arabia

2 School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

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Fixed Point Theory and Applications 2012, 2012:118  doi:10.1186/1687-1812-2012-118


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/118


Received:4 April 2012
Accepted:2 July 2012
Published:20 July 2012

© 2012 Dehaish and Kozlowski; 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

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a> be a uniformly convex modular function space with a strong Opial property. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2">View MathML</a> be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.

MSC: 47H09, 47H10.

Keywords:
fixed point; nonexpansive mapping; fixed point iteration process; Mann process; Ishikawa process; modular function space; Orlicz space; Opial property; uniform convexity

1 Introduction

In 2008, Kirk and Xu [21] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2">View MathML</a>, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M5">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M6">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M7">View MathML</a>. Their main result (Theorem 3.5) states that every asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, bounded and convex subset C of a uniformly convex Banach space X has a fixed point. As pointed out by Kirk and Xu, asymptotic pointwise mappings seem to be a natural generalization of nonexpansive mappings. The conditions on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M8">View MathML</a> can be for instance expressed in terms of the derivatives of iterations of T for differentiable T. In 2009 these results were generalized by Hussain and Khamsi to metric spaces, [9].

In 2011, Khamsi and Kozlowski [18] extended their result proving the existence of fixed points of asymptotic pointwise ρ-nonexpansive mappings acting in modular function spaces. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwise ρ-nonexpansive mapping. This paper aims at filling this gap.

Let us recall that modular function spaces are natural generalization of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others, see the book by Kozlowski [24] for an extensive list of examples and special cases. There exists an extensive literature on the topic of the fixed point theory in modular function spaces, see, e.g., [3-5,8,13,14,17-20,24] and the papers referenced there.

It is well known that the fixed point construction iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful numerical methods for solving various nonlinear equations and variational problems, often of great importance for applications in various areas of pure and applied science. There exists an extensive literature on the subject of iterative fixed point construction processes for asymptotically nonexpansive mappings in Hilbert, Banach and metric spaces, see, e.g., [1,2,6,7,9,12,16,30-36,38-42] and the works referred there. Kozlowski proved convergence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces [25] and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces [26]. Recently, weak and strong convergence of such processes to common fixed points of semigroups of mappings in Banach spaces has been demonstrated by Kozlowski and Sims [28].

We would like to emphasize that all convergence theorems proved in this paper define constructive algorithms that can be actually implemented. When dealing with specific applications of these theorems, one should take into consideration how additional properties of the mappings, sets and modulars involved can influence the actual implementation of the algorithms defined in this paper.

The paper is organized as follows:

(a) Section 2 provides necessary preliminary material on modular function spaces.

(b) Section 3 introduces the asymptotic pointwise nonexpansive mappings and related notions.

(c) Section 4 deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems.

(d) Section 5 utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process.

(e) Section 6 establishes the almost everywhere convergence theorem for generalized Ishikawa process.

(f) Section 7 provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact set C.

2 Preliminaries

Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M9">View MathML</a> be a δ-ring of subsets of Ω such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M10">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M11">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M12">View MathML</a>. Let us assume that there exists an increasing sequence of sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M13">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M14">View MathML</a>. By ℰ we denote the linear space of all simple functions with supports from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M9">View MathML</a>. By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M16">View MathML</a> we will denote the space of all extended measurable functions, i.e., all functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M17">View MathML</a> such that there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M18">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M19">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M20">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M21">View MathML</a>. By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M22">View MathML</a> we denote the characteristic function of the set A.

Definition 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M23">View MathML</a> be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M24">View MathML</a>;

(ii) ρ is monotone, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M25">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M21">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M27">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M28">View MathML</a>;

(iii) ρ is orthogonally subadditive, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M29">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M32">View MathML</a>;

(iv) ρ has the Fatou property, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M33">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M21">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M35">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M32">View MathML</a>;

(v) ρ is order continuous in ℰ, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M38">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M39">View MathML</a>.

Similarly, as in the case of measure spaces, we say that a set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M40">View MathML</a> is ρ-null if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M41">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M42">View MathML</a>. We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual, we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind we define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M43">View MathML</a>

(2.1)

where each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M44">View MathML</a> is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists we will write ℳ instead of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M45">View MathML</a>.

Definition 2.2 Let ρ be a regular function pseudomodular.

(1) We say that ρ is a regular convex function semimodular if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M46">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M48">View MathML</a>ρ-a.e.;

(2) We say that ρ is a regular convex function modular if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M49">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M48">View MathML</a>ρ-a.e.;

The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.

Let us denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M51">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M52">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M53">View MathML</a>. It is easy to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M54">View MathML</a> is a function pseudomodular in the sense of Def.2.1.1 in [24] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [22-24].

Remark 2.1 We limit ourselves to convex function modulars in this paper. However, omitting convexity in Definition 2.1 or replacing it by s-convexity would lead to the definition of nonconvex or s-convex regular function pseudomodulars, semimodulars and modulars as in [24].

Definition 2.3[22-24]

Let ρ be a convex function modular.

(a) A modular function space is the vector space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M55">View MathML</a>, or briefly <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>, defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M57">View MathML</a>

(b) The following formula defines a norm in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a> (frequently called Luxemurg norm):

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M59">View MathML</a>

In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.

Theorem 2.1[22-24]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>.

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M62">View MathML</a>is complete and the norm<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M63">View MathML</a>is monotone w.r.t. the natural order in ℳ.

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M64">View MathML</a>if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M65">View MathML</a>for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47">View MathML</a>.

(3) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M65">View MathML</a>for an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47">View MathML</a>then there exists a subsequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M69">View MathML</a>of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M71">View MathML</a>ρ-a.e.

(4) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70">View MathML</a>converges uniformly tofon a set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M73">View MathML</a>then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M74">View MathML</a>for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M47">View MathML</a>.

(5) Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M76">View MathML</a>ρ-a.e. There exists a nondecreasing sequence of sets<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M77">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M78">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70">View MathML</a>converges uniformly tofon every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M80">View MathML</a> (Egoroff theorem).

(6) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M81">View MathML</a>whenever<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M82">View MathML</a>ρ-a.e. (Note: this property is equivalent to the Fatou property.)

(7) Defining<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M83">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M84">View MathML</a>we have:

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M85">View MathML</a>,

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M86">View MathML</a>has the Lebesgue property, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M87">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M88">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M90">View MathML</a>.

(c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M86">View MathML</a>is the closure of ℰ (in the sense of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M92">View MathML</a>).

The following definition plays an important role in the theory of modular function spaces.

Definition 2.4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. We say that ρ has the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>-property if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M95">View MathML</a>

whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M90">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M97">View MathML</a>.

Theorem 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. The following conditions are equivalent:

(a) ρhas<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>,

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M100">View MathML</a>is a linear subspace of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>,

(c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M102">View MathML</a>,

(d) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M103">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M104">View MathML</a>,

(e) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M65">View MathML</a>for an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M88">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M64">View MathML</a>, i.e., the modular convergence is equivalent to the norm convergence.

We will also use another type of convergence which is situated between norm and modular convergence. It is defined, among other important terms, in the following definition.

Definition 2.5 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>.

(a) We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70">View MathML</a> is ρ-convergent to f and write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M110">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M111">View MathML</a>.

(b) A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70">View MathML</a> where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M113">View MathML</a> is called ρ-Cauchy if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M114">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M115">View MathML</a>.

(c) A set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116">View MathML</a> is called ρ-closed if for any sequence of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M117">View MathML</a>, the convergence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M118">View MathML</a> implies that f belongs to B.

(d) A set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116">View MathML</a> is called ρ-bounded if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M120">View MathML</a>.

(e) A set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116">View MathML</a> is called strongly ρ-bounded if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M122">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M123">View MathML</a>.

(f) A set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M116">View MathML</a> is called ρ-compact if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M125">View MathML</a> in C there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M126">View MathML</a> and an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M127">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M128">View MathML</a>.

(g) A set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M129">View MathML</a> is called ρ-a.e. closed if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70">View MathML</a> in C which ρ-a.e. converges to some f, then we must have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M127">View MathML</a>.

(h) A set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M129">View MathML</a> is called ρ-a.e. compact if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M70">View MathML</a> in C, there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M126">View MathML</a> which ρ-a.e. converges to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M127">View MathML</a>.

(i) Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>. The ρ-distance between f and C is defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M138">View MathML</a>

Let us note that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M76">View MathML</a> does not imply in general <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M140">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M141">View MathML</a>. Using Theorem 2.1, it is not difficult to prove the following:

Proposition 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>.

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>isρ-complete,

(ii) ρ-balls<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M144">View MathML</a>areρ-closed andρ-a.e. closed.

Let us compare different types of compactness introduced in Definition 2.5.

Proposition 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. The following relationships hold for sets<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M146">View MathML</a>:

(i) IfCisρ-compact, thenCisρ-a.e. compact.

(ii) IfCis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M147">View MathML</a>-compact, thenCisρ-compact.

(iii) Ifρsatisfies<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M147">View MathML</a>-compactness andρ-compactness are equivalent in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>.

Proof

(i) follows from Theorem 2.1 part (3).

(ii) follows from Theorem 2.1 part (2).

(iii) follows from (2.2) and from Theorem 2.2 part (e).

 □

3 Asymptotic pointwise nonexpansive mappings

Let us recall the modular definitions of asymptotic pointwise nonexpansive mappings and associated notions, [18].

Definition 3.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a> be nonempty and ρ-closed. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M153">View MathML</a> is called an asymptotic pointwise mapping if there exists a sequence of mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M154">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M155">View MathML</a>

(i) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M156">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136">View MathML</a> and every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M158">View MathML</a>, then T is called ρ-nonexpansive or shortly nonexpansive.

(ii) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M159">View MathML</a> converges pointwise to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M160">View MathML</a>, then T is called asymptotic pointwise contraction.

(iii) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M161">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136">View MathML</a>, then T is called asymptotic pointwise nonexpansive.

(iv) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M163">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M136">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M165">View MathML</a>, then T is called strongly asymptotic pointwise contraction.

Denoting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M166">View MathML</a>, we note that without loss of generality we can assume that T is asymptotically pointwise nonexpansive if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M167">View MathML</a>

(3.1)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M168">View MathML</a>

(3.2)

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M169">View MathML</a>. In view of (3.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M170">View MathML</a>

(3.3)

The above notation will be consistently used throughout this paper.

By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M171">View MathML</a> we will denote the class of all asymptotic pointwise nonexpansive mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2">View MathML</a>.

In this paper, we will impose some restrictions on the behavior of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M173">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M174">View MathML</a>. This type of assumptions is typical for controlling the convergence of iterative processes for asymptotically nonexpansive mappings, see, e.g., [25].

Definition 3.2 Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M175">View MathML</a> as a class of all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M176">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M177">View MathML</a>

(3.4)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M178">View MathML</a>

(3.5)

We recall the following concepts related to the modular uniform convexity introduced in [18]:

Definition 3.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. We define the following uniform convexity type properties of the function modular ρ: Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M180">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M181">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a>. Define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M183">View MathML</a>

Let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M184">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M185">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M186">View MathML</a>. We will use the following notational convention: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M187">View MathML</a>.

Definition 3.4 We say that ρ satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M188">View MathML</a> if for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M181">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M191">View MathML</a>. Note that for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M181">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M193">View MathML</a>, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a> small enough. We say that ρ satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a> if for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M196">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M197">View MathML</a> there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M198">View MathML</a> depending only on s and ε such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M199">View MathML</a>

We will need the following result whose proof is elementary. Note that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M200">View MathML</a>, this result follows directly from Definition 3.4.

Lemma 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M180">View MathML</a>. Then for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M204">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a>there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M206">View MathML</a>depending only onsandεsuch that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M207">View MathML</a>

The notion of bounded away sequences of real numbers will be used extensively throughout this paper.

Definition 3.5 A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M208">View MathML</a> is called bounded away from 0 if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M209">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M210">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M211">View MathML</a>. Similarly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M208">View MathML</a> is called bounded away from 1 if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M214">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M158">View MathML</a>.

We will need the following generalization of Lemma 4.1 from [18] and being a modular equivalent of a norm property in uniformly convex Banach spaces, see, e.g., [36].

Lemma 3.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M218">View MathML</a>be bounded away from 0 and 1. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M219">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M220">View MathML</a>

(3.6)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M221">View MathML</a>

(3.7)

then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M222">View MathML</a>

Proof Assume to the contrary that this is not the case and fix an arbitrary <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M223">View MathML</a>. Passing to a subsequence if necessary, we may assume that there exists an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M225">View MathML</a>

(3.8)

while

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M226">View MathML</a>

(3.9)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M227">View MathML</a> is bounded away from 0 and 1 there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M228">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M229">View MathML</a> for all natural n. Passing to a subsequence if necessary, we can assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M230">View MathML</a>. For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M231">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M232">View MathML</a>, let us define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M233">View MathML</a>. Observe that the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M234">View MathML</a> is a convex function. Hence that the function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M235">View MathML</a>

(3.10)

is also convex on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M236">View MathML</a>, and consequently, it is a continuous function on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M237">View MathML</a>. Noting that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M238">View MathML</a>

(3.11)

we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M239">View MathML</a> is a continuous function of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M240">View MathML</a>. Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M241">View MathML</a>

(3.12)

By (3.8) and (3.9)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M242">View MathML</a>

(3.13)

By (3.12) the left-hand side of (3.13) tends to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M243">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M244">View MathML</a> while the right-hand side tends to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M245">View MathML</a> in view of (3.7). Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M246">View MathML</a>

(3.14)

By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a> and by Lemma 3.1, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M248">View MathML</a> satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M249">View MathML</a>

(3.15)

Combining (3.14) with (3.15) we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M250">View MathML</a>

(3.16)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M251">View MathML</a> we get a contradiction which completes the proof. □

Let us introduce a notion of a ρ-type, a powerful technical tool which will be used in the proofs of our fixed point results.

Definition 3.6 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M252">View MathML</a> be convex and ρ-bounded. A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M253">View MathML</a> is called a ρ-type (or shortly a type) if there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M254">View MathML</a> of elements of K such that for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M255">View MathML</a> there holds

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M256">View MathML</a>

Note that τ is convex provided ρ is convex. A typical method of proof for the fixed point theorems in Banach and metric spaces is to construct a fixed point by finding an element on which a specific type function attains its minimum. To be able to proceed with this method, one has to know that such an element indeed exists. This will be the subject of Lemma 3.3 below. First, let us recall the definition of the Opial property and the strong Opial property in modular function spaces, [15,17].

Definition 3.7 We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a> satisfies the ρ-a.e. Opial property if for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M258">View MathML</a> which is ρ-a.e. convergent to 0 such that there exists a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M122">View MathML</a> for which

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M260">View MathML</a>

(3.17)

the following inequality holds for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M261">View MathML</a> not equal to 0

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M262">View MathML</a>

(3.18)

Definition 3.8 We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a> satisfies the ρ-a.e. strong Opial property if for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M258">View MathML</a> which is ρ-a.e. convergent to 0 such that there exists a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M122">View MathML</a> for which

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M266">View MathML</a>

(3.19)

the following equality holds for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M261">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M268">View MathML</a>

(3.20)

Remark 3.1 Note that the ρ-a.e. Strong Opial property implies ρ-a.e. Opial property [15].

Remark 3.2 Also, note that, by virtue of Theorem 2.1 in [15], every convex, orthogonally additive function modular ρ has the ρ-a.e. strong Opial property. Let us recall that ρ is called orthogonally additive if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M269">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M270">View MathML</a>. Therefore, all Orlicz and Musielak-Orlicz spaces must have the strong Opial property.

Note that the Opial property in the norm sense does not necessarily hold for several classical Banach function spaces. For instance, the norm Opial property does not hold for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M271">View MathML</a> spaces for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M272">View MathML</a> while the modular strong Opial property holds in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M271">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M274">View MathML</a>.

Lemma 3.3[27]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>has theρ-a.e. strong Opial property. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M277">View MathML</a>be a nonempty, stronglyρ-bounded andρ-a.e. compact convex set. Then anyρ-type defined in C attains its minimum inC.

Let us finish this section with the fundamental fixed point existence theorem which will be used in many places in the current paper.

Theorem 3.1[18]

Assume<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>. LetCbe aρ-closedρ-bounded convex nonempty subset. Then any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M280">View MathML</a>asymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed points<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M281">View MathML</a>isρ-closed.

4 Demiclosedness Principle

The following modular version of the Demiclosedness Principle will be used in the proof of our convergence Theorem 5.1. Our proof the Demiclosedness Principle uses the parallelogram inequality valid in the modular spaces with the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a> property (see Lemma 4.2 in [18]). We start with a technical result which will be used in the proof of Theorem 4.1.

Lemma 4.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be a convex set, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a>is aρ-approximate fixed point sequence for T, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288">View MathML</a>, then for every fixed<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M289">View MathML</a>there holds

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M290">View MathML</a>

(4.1)

as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288">View MathML</a>.

Proof It follows from 3.5 that there exists a finite constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M292">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M293">View MathML</a>

(4.2)

Using the convexity of ρ and the ρ-nonexpansiveness of T, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M294">View MathML</a>

(4.3)

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288">View MathML</a>. □

Corollary 4.1If, under the hypothesis of Lemma 4.1, ρsatisfies additionally the<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>condition, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M297">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288">View MathML</a>.

The version of the Demiclosedness Principle used in this paper (Theorem 4.1) requires the uniform continuity of the function modular ρ in the sense of the following definition (see, e.g., [17]).

Definition 4.1 We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a> is uniformly continuous if to every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M301">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M302">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M303">View MathML</a>

(4.4)

provided <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M304">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M305">View MathML</a>.

Let us mention that the uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [37] or in sequence Orlicz spaces [11] the uniform continuity of the Orlicz modular is equivalent to the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>-type condition.

Theorem 4.1Demiclosedness Principle. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. Assume that:

(1) ρis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308">View MathML</a>,

(2) ρhas strong Opial property,

(3) ρhas<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>property and is uniformly continuous.

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be a nonempty, convex, stronglyρ-bounded andρ-closed, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M312">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M313">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M314">View MathML</a>ρ-a.e. and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M315">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316">View MathML</a>.

Proof Let us recall that by definition of uniform continuity of ρ to every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M301">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M302">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M320">View MathML</a>

(4.5)

provided <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M304">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M305">View MathML</a>. Fix any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M323">View MathML</a>. Noting that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M324">View MathML</a> due to the strong ρ-boundedness of C and that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M325">View MathML</a> by Corollary (4.1), it follows from (4.5) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M326">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M327">View MathML</a> that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M328">View MathML</a>

(4.6)

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M244">View MathML</a>. Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M330">View MathML</a>

(4.7)

Define the ρ-type φ by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M331">View MathML</a>

(4.8)

By (4.7) we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M332">View MathML</a>

(4.9)

Hence, for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M333">View MathML</a> there holds

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M334">View MathML</a>

(4.10)

Using (4.10) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M335">View MathML</a> and by passing with m to infinity, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M336">View MathML</a>

(4.11)

Since ρ satisfies the strong Opial property, it also satisfies the Opial property. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M337">View MathML</a>-a.e., it follows via the Opial property that for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M338">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M339">View MathML</a>

(4.12)

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M340">View MathML</a>

(4.13)

Combining (4.11) with (4.13), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M341">View MathML</a>

(4.14)

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M342">View MathML</a>

(4.15)

We claim that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M343">View MathML</a>

(4.16)

Assume to the contrary that (4.16) does not hold, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M344">View MathML</a>

(4.17)

By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>, it follows from (4.17) that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M346">View MathML</a> does not tend to zero. By passing to a subsequence if necessary, we can assume that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M347">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M348">View MathML</a>

(4.18)

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M289">View MathML</a>, which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M350">View MathML</a>

(4.19)

for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M351">View MathML</a>. Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M352">View MathML</a>

(4.20)

for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M351">View MathML</a>. Applying the modular parallelogram inequality valid in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308">View MathML</a> modular function spaces, see Lemma 4.2 in [18],

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M355">View MathML</a>

(4.21)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M356">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M357">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M358">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M359">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M360">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M361">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M362">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M363">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M364">View MathML</a>

(4.22)

Note that by (4.13)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M365">View MathML</a>

(4.23)

Combining (4.22) with (4.23), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M366">View MathML</a>

(4.24)

which implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M367">View MathML</a>

(4.25)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M368">View MathML</a> and applying (4.15), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M369">View MathML</a>

(4.26)

Using the properties of Ψ, we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M370">View MathML</a> tends to zero itself, which contradicts our assumption (4.17). Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M371">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M368">View MathML</a>. Clearly, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M373">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M368">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M375">View MathML</a> while <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M376">View MathML</a> by ρ-continuity of T. By the uniqueness of the ρ-limit, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M377">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M378">View MathML</a>. □

5 Convergence of generalized Mann iteration process

The following elementary, easy to prove, lemma will be used in this paper.

Lemma 5.1[2]

Suppose<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M379">View MathML</a>is a bounded sequence of real numbers and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M380">View MathML</a>is a doubly-index sequence of real numbers which satisfy

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M381">View MathML</a>

for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M382">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M379">View MathML</a>converges to an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M384">View MathML</a>.

Following Mann [29], let us start with the definition of the generalized Mann iteration process.

Definition 5.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a> be an increasing sequence of natural numbers. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a> be bounded away from 0 and 1. The generalized Mann iteration process generated by the mapping T, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M388">View MathML</a>, and the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a> denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390">View MathML</a> is defined by the following iterative formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M391">View MathML</a>

(5.1)

Definition 5.2 We say that a generalized Mann iteration process <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M392">View MathML</a> is well defined if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M393">View MathML</a>

(5.2)

Remark 5.1 Observe that by the definition of asymptotic pointwise nonexpansiveness, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M394">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M395">View MathML</a>. Hence we can always select a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M396">View MathML</a> such that (5.2) holds. In other words, by a suitable choice of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a>, we can always make <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M392">View MathML</a> well defined.

The following result provides an important technique which will be used in this paper.

Lemma 5.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be aρ-closed, ρ-bounded and convex set. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403">View MathML</a>. Assume that a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M404">View MathML</a>is bounded away from 0 and 1. Letwbe a fixed point ofTand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390">View MathML</a>be a generalized Mann process. Then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M406">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M407">View MathML</a>

(5.3)

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408">View MathML</a>. Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M409">View MathML</a>

it follows that for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M410">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M411">View MathML</a>

(5.4)

Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M412">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M413">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M414">View MathML</a>. Observe that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M416">View MathML</a>. By Lemma 5.1, there exists an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M417">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M418">View MathML</a> as claimed. □

The next result will be essential for proving the convergence theorems for iterative process.

Lemma 5.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be aρ-closed, ρ-bounded and convex set, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M422">View MathML</a>. Assume that a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M218">View MathML</a>is bounded away from 0 and 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M425">View MathML</a>be a generalized Mann iteration process. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M426">View MathML</a>

(5.5)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M427">View MathML</a>

(5.6)

Proof By Theorem 3.1, T has at least one fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M428">View MathML</a>. In view of Lemma 5.2, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M406">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M430">View MathML</a>

(5.7)

Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M431">View MathML</a>

(5.8)

and that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M432">View MathML</a>

(5.9)

Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M433">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M434">View MathML</a>, and note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M435">View MathML</a> by (5.7), and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M436">View MathML</a> by (5.8). Observe also that

(5.10)

Hence, it follows from Lemma 3.2 that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M438">View MathML</a>

(5.11)

which by the construction of the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a> is equivalent to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M440">View MathML</a>

(5.12)

as claimed. □

In the next lemma, we prove that under suitable assumption the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a> becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to recall the following notions.

Definition 5.3 A strictly increasing sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M442">View MathML</a> is called quasi-periodic if the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M443">View MathML</a> is bounded, or equivalently, if there exists a number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M444">View MathML</a> such that any block of p consecutive natural numbers must contain a term of the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M445">View MathML</a>. The smallest of such numbers p will be called a quasi-period of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M445">View MathML</a>.

Lemma 5.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be aρ-closed, ρ-bounded and convex set, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M451">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a>be bounded away from 0 and 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403">View MathML</a>be such that the generalized Mann process<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390">View MathML</a>is well defined. If, in addition, the set of indices<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M455">View MathML</a>is quasi-periodic, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a>is an approximate fixed point sequence, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M457">View MathML</a>

(5.13)

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M413">View MathML</a> be a quasi-period of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459">View MathML</a>. Observe that it is enough to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M461">View MathML</a> through <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459">View MathML</a>. Indeed, let us fix <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M182">View MathML</a>. From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M461">View MathML</a> through <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459">View MathML</a> it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M467">View MathML</a>

(5.14)

for sufficiently large k. By the quasi-periodicity of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459">View MathML</a>, to every positive integer k, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M469">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M470">View MathML</a>. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M471">View MathML</a> (the proof for the other case is identical). Since T is ρ-Lipschitzian with the constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M472">View MathML</a>, there exist a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M473">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M474">View MathML</a>

(5.15)

Note that by (5.6) and by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M476">View MathML</a> for k sufficiently large. This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M477">View MathML</a>

(5.16)

and therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M478">View MathML</a>

(5.17)

which demonstrates that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M479">View MathML</a>

(5.18)

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288">View MathML</a>. By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a> again, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M482">View MathML</a>.

To prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M287">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M461">View MathML</a> through <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M459">View MathML</a>, observe that, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M486">View MathML</a> for such k, there holds

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M487">View MathML</a>

(5.19)

which tends to zero in view of (5.5), (5.6) and (5.2). □

The next theorem is the main result of this section.

Theorem 5.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. Assume that:

(1) ρis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308">View MathML</a>,

(2) ρhas Strong Opial Property,

(3) ρhas<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>property and is uniformly continuous.

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be nonempty, ρ-a.e. compact, convex, stronglyρ-bounded andρ-closed, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M422">View MathML</a>. Assume that a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M218">View MathML</a>is bounded away from 0 and 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M425">View MathML</a>be a well-defined generalized Mann iteration process. Assume, in addition, that the set of indices<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M496">View MathML</a>is quasi-periodic. Then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M314">View MathML</a>ρ-a.e.

Proof Observe that by Theorem 4.1 in [18], the set of fixed points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M499">View MathML</a> is nonempty, convex and ρ-closed. Note also that by Lemma 3.1 in [27], it follows from the strong Opial property of ρ that any ρ-type attains its minimum in C. By Lemma 5.4, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a> is an approximate fixed point sequence, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M501">View MathML</a>

(5.20)

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M288">View MathML</a>. Consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M503">View MathML</a>, two ρ-a.e. cluster points of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a>. There exits then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M505">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M506">View MathML</a> subsequences of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M508">View MathML</a>ρ-a.e. and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M509">View MathML</a>ρ-a.e. By Theorem 4.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M510">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M511">View MathML</a>. By Lemma 5.2, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M512">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M513">View MathML</a>

(5.21)

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M514">View MathML</a>. Assume to the contrary that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M515">View MathML</a>. Then, by the strong Opial property, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M516">View MathML</a>

(5.22)

The contradiction implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M514">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a> has at most one ρ-a.e. cluster point. Since, C is ρ-a.e. compact it follows that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a> has exactly one ρ-a.e. cluster point, which means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M520">View MathML</a>ρ-a.e. Using Theorem 4.1 again, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316">View MathML</a> as claimed. □

Remark 5.2 It is easy to see that we can always construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a> with the quasi-periodic properties specified in the assumptions of Theorem 5.1. When constructing concrete implementations of this algorithm, the difficulty will be to ensure that the constructed sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a> is not “too sparse” in the sense that the generalized Mann process <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390">View MathML</a> remains well defined. The similar quasi-periodic type assumptions are common in the asymptotic fixed point theory, see, e.g., [2,25,28].

6 Convergence of generalized Ishikawa iteration process

The two-step Ishikawa iteration process is a generalization of the one-step Mann process. The Ishikawa iteration process, [10], provides more flexibility in defining the algorithm parameters, which is important from the numerical implementation perspective.

Definition 6.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a> be an increasing sequence of natural numbers. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a> be bounded away from 0 and 1, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528">View MathML</a> be bounded away from 1. The generalized Ishikawa iteration process generated by the mapping T, the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M388">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M530">View MathML</a>, and the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a> denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532">View MathML</a> is defined by the following iterative formula:

(6.1)

Definition 6.2 We say that a generalized Ishikawa iteration process <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M534">View MathML</a> is well defined if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M535">View MathML</a>

(6.2)

Remark 6.1 Observe that, by the definition of asymptotic pointwise nonexpansiveness, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M394">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M395">View MathML</a>. Hence we can always select a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M396">View MathML</a> such that (6.2) holds. In other words, by a suitable choice of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a>, we can always make <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M540">View MathML</a> well defined.

Lemma 6.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be aρ-closed, ρ-bounded and convex set. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a>be bounded away from 0 and 1, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528">View MathML</a>be bounded away from 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532">View MathML</a>be a generalized Ishikawa process. There exists then an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M406">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M551">View MathML</a>.

Proof Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M552">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M553">View MathML</a>

(6.3)

It is easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M554">View MathML</a> and that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M555">View MathML</a>. Moreover, a straight calculation shows that each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M556">View MathML</a> satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M557">View MathML</a>

(6.4)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M558">View MathML</a>

(6.5)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M559">View MathML</a>

(6.6)

Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M560">View MathML</a>, which follows directly from the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M561">View MathML</a> and from (6.5). Using (6.5) and the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M562">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M563">View MathML</a>

(6.7)

Fix any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M564">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M565">View MathML</a>, it follows that there exists a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M566">View MathML</a> such that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M567">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M568">View MathML</a>. Therefore, using the same argument as in the proof of Lemma 5.2, we deduce that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M567">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M570">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M571">View MathML</a>

(6.8)

Arguing like in the proof of Lemma 5.2, we conclude that there exists an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M572">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M573">View MathML</a>. □

Lemma 6.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be aρ-closed, ρ-bounded and convex set. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a>be bounded away from 0 and 1, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528">View MathML</a>be bounded away from 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532">View MathML</a>be a generalized Ishikawa process. Define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M582">View MathML</a>

(6.9)

Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M583">View MathML</a>

(6.10)

or equivalently

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M584">View MathML</a>

(6.11)

Proof By Theorem 3.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M585">View MathML</a>. Let us fix <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408">View MathML</a>. By Lemma 6.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M587">View MathML</a> exists. Let us denote it by r. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M408">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M590">View MathML</a> by Lemma 6.1, we have the following:

(6.12)

Note that

(6.13)

Applying Lemma 3.2 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M593">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M594">View MathML</a>, we obtain the desired equality <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M595">View MathML</a>, while (6.11) follows from (6.10) via the construction formulas for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M596">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M597">View MathML</a>. □

Lemma 6.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be aρ-closed, ρ-bounded and convex set. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M451">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M403">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M404">View MathML</a>be bounded away from 0 and 1, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M605">View MathML</a>be bounded away from 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532">View MathML</a>be a well-defined generalized Ishikawa process. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M607">View MathML</a>

(6.14)

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M608">View MathML</a>. Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M609">View MathML</a>

(6.15)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528">View MathML</a> is bounded away from 1, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M611">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M612">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M613">View MathML</a>. Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M614">View MathML</a>

(6.16)

The right-hand side of this inequality tends to zero because <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M615">View MathML</a> by Lemma 6.2 and ρ satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>. □

Lemma 6.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>be<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be aρ-closed, ρ-bounded and convex set, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M451">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a>be bounded away from 0 and 1 and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528">View MathML</a>be bounded away from 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M624">View MathML</a>be such that the generalized Ishikawa process<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532">View MathML</a>is well defined. If, in addition, the set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M455">View MathML</a>is quasi-periodic, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M627">View MathML</a>is an approximate fixed point sequence, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M628">View MathML</a>

(6.17)

Proof The proof is analogous to that of Lemma 5.4 with (6.11) used instead of (5.6) and (6.14) replacing (5.5). □

Theorem 6.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>. Assume that

(1) ρis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M308">View MathML</a>,

(2) ρhas Strong Opial Property,

(3) ρhas<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>property and is uniformly continuous.

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M137">View MathML</a>be nonempty, ρ-a.e. compact, convex, stronglyρ-bounded andρ-closed, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M422">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a>be bounded away from 0 and 1, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528">View MathML</a>be bounded away from 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a>be such that the generalized Ishikawa process<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M532">View MathML</a>is well defined. If, in addition, the set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M455">View MathML</a>is quasi-periodic, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a>generated by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M641">View MathML</a>convergesρ-a.e. to a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316">View MathML</a>.

Proof The proof is analogous to that of Theorem 5.1 with Lemma 5.4 replaced by Lemma 6.4, and Lemma 5.2 replaced by Lemma 6.1. □

7 Strong convergence

It is interesting that, provided C is ρ-compact, both generalized Mann and Ishikawa processes converge strongly to a fixed point of T even without assuming the Opial property.

Theorem 7.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M60">View MathML</a>satisfy conditions<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M195">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M146">View MathML</a>be aρ-compact, ρ-bounded and convex set, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M285">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M387">View MathML</a>be bounded away from 0 and 1, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M528">View MathML</a>be bounded away from 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M386">View MathML</a>be such that the generalized Mann process<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390">View MathML</a> (resp. Ishikawa process<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M641">View MathML</a>) is well defined. Then there exists a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316">View MathML</a>such that then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a>generated by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M390">View MathML</a> (resp. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M641">View MathML</a>) converges strongly to a fixed point ofT, that is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M657">View MathML</a>

(7.1)

Proof By the ρ-compactness of C, we can select a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M658">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M286">View MathML</a> such that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M313">View MathML</a> with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M661">View MathML</a>

(7.2)

Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M662">View MathML</a>

(7.3)

which tends to zero by Lemma 5.3 (resp. Lemma 6.4) and by (7.2). By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a> it follows from (7.3) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M664">View MathML</a>

(7.4)

Observe that by the convexity of ρ and by ρ-nonexpansiveness of T, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M665">View MathML</a>

(7.5)

which tends to zero by (7.4) and by Lemma 5.3 (resp. Lemma 6.4). Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M666">View MathML</a> which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M316">View MathML</a>. Applying Lemma 5.2 (resp. Lemma 6.1), we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M668">View MathML</a> exists. By (7.4) this limit must be equal to zero which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M669">View MathML</a>

(7.6)

 □

Remark 7.1 Observe that in view of the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M94">View MathML</a> assumption, the ρ-compactness of the set C assumed in Theorem 7.1 is equivalent to the compactness in the sense of the norm defined by ρ.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors equally participated in all stages of preparations of this article. Both authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.

References

  1. Bose, SC: Weak convergence to the fixed point of an asymptotically nonexpansive. Proc. Am. Math. Soc.. 68, 305–308 (1978). Publisher Full Text OpenURL

  2. Bruck, R, Kuczumow, T, Reich, S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math.. 65(2), 169–179 (1993)

  3. Dominguez-Benavides, T, Khamsi, MA, Samadi, S: Uniformly Lipschitzian mappings in modular function spaces. Nonlinear Anal.. 46, 267–278 (2001). Publisher Full Text OpenURL

  4. Dominguez-Benavides, T, Khamsi, MA, Samadi, S: Asymptotically regular mappings in modular function spaces. Sci. Math. Jpn.. 53, 295–304 (2001)

  5. Dominguez-Benavides, T, Khamsi, MA, Samadi, S: Asymptotically nonexpansive mappings in modular function spaces. J. Math. Anal. Appl.. 265(2), 249–263 (2002). Publisher Full Text OpenURL

  6. Fukhar-ud-din, H, Khan, AR: Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. Comput. Math. Appl.. 53, 1349–1360 (2009)

  7. Gornicki, J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carol.. 30, 249–252 (1989)

  8. Hajji, A, Hanebaly, E: Perturbed integral equations in modular function spaces. Electron. J. Qual. Theory Differ. Equ.. 20, 1–7 http://www.math.u-szeged.hu/ejqtde/ (2003)

  9. Hussain, N, Khamsi, MA: On asymptotic pointwise contractions in metric spaces. Nonlinear Anal.. 71(10), 4423–4429 (2009). Publisher Full Text OpenURL

  10. Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc.. 44, 147–150 (1974). Publisher Full Text OpenURL

  11. Kaminska, A: On uniform convexity of Orlicz spaces. Indag. Math.. 44(1), 27–36 (1982)

  12. Khan, AR: On modified Noor iterations for asymptotically nonexpansive mappings. Bull. Belg. Math. Soc. Simon Stevin. 17, 127–140 (2010)

  13. Khamsi, MA: Nonlinear semigroups in modular function spaces. Math. Jpn.. 37(2), 1–9 (1992)

  14. Khamsi, MA: Fixed point theory in modular function spaces. Proceedings of the Workshop on Recent Advances on Metric Fixed Point Theory Held in Sevilla, September, 1995. 31–35 (1995)

  15. Khamsi, MA: A convexity property in Modular function spaces. Math. Jpn.. 44(2), 269–279 (1996)

  16. Khamsi, MA: On asymptotically nonexpansive mappings in hyperconvex metric spaces. Proc. Am. Math. Soc.. 132, 365–373 (2004). Publisher Full Text OpenURL

  17. Khamsi, MA, Kozlowski, WM: On asymptotic pointwise contractions in modular function spaces. Nonlinear Anal.. 73, 2957–2967 (2010). Publisher Full Text OpenURL

  18. Khamsi, MA, Kozlowski, WM: On asymptotic pointwise nonexpansive mappings in modular function spaces. J. Math. Anal. Appl.. 380(2), 697–708 (2011). Publisher Full Text OpenURL

  19. Khamsi, MA, Kozlowski, WM, Reich, S: Fixed point theory in modular function spaces. Nonlinear Anal.. 14, 935–953 (1990). Publisher Full Text OpenURL

  20. Khamsi, MA, Kozlowski, WM, Shutao, C: Some geometrical properties and fixed point theorems in Orlicz spaces. J. Math. Anal. Appl.. 155(2), 393–412 (1991). PubMed Abstract | Publisher Full Text OpenURL

  21. Kirk, WA, Xu, HK: Asymptotic pointwise contractions. Nonlinear Anal.. 69, 4706–4712 (2008). Publisher Full Text OpenURL

  22. Kozlowski, WM: Notes on modular function spaces I. Ann. Soc. Math. Pol., 1 Comment. Math.. 28, 91–104 (1988)

  23. Kozlowski, WM: Notes on modular function spaces II. Ann. Soc. Math. Pol., 1 Comment. Math.. 28, 105–120 (1988)

  24. Kozlowski, WM: Modular Function Spaces, Dekker, New York (1988)

  25. Kozlowski, WM: Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces. J. Math. Anal. Appl.. 377, 43–52 (2011). Publisher Full Text OpenURL

  26. Kozlowski, WM: Common fixed points for semigroups of pointwise Lipschitzian mappings in Banach spaces. Bull. Aust. Math. Soc.. 84, 353–361 (2011). Publisher Full Text OpenURL

  27. Kozlowski, WM: On the existence of common fixed points for semigroups of nonlinear mappings in modular function spaces. Ann. Soc. Math. Pol., 1 Comment. Math.. 51(1), 81–98 (2011)

  28. Kozlowski, WM, Sims, B: On the convergence of iteration processes for semigroups of nonlinear mappings in Banach spaces (to appear)

  29. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc.. 4, 506–510 (1953). Publisher Full Text OpenURL

  30. Nanjaras, B, Panyanak, B: Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl.. 2010, Article ID 268780 (2010)

  31. Noor, MA, Xu, B: Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl.. 267, 444–453 (2002). Publisher Full Text OpenURL

  32. Passty, GB: Construction of fixed points for asymptotically nonexpansive mappings. Proc. Am. Math. Soc.. 84, 212–216 (1982). Publisher Full Text OpenURL

  33. Rhoades, BE: Fixed point iterations for certain nonlinear mappings. J. Math. Anal. Appl.. 183, 118–120 (1994). Publisher Full Text OpenURL

  34. Samanta, SK: Fixed point theorems in a Banach space satisfying Opial’s condition. J. Indian Math. Soc.. 45, 251–258 (1981)

  35. Schu, J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl.. 158, 407–413 (1991). Publisher Full Text OpenURL

  36. Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.. 43, 153–159 (1991). Publisher Full Text OpenURL

  37. Shutao, C: Geometry of Orlicz Spaces (1996)

  38. Tan, K-K, Xu, H-K: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc.. 114, 399–404 (1992). Publisher Full Text OpenURL

  39. Tan, K-K, Xu, H-K: A nonlinear ergodic theorem for asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.. 45, 25–36 (1992). Publisher Full Text OpenURL

  40. Tan, K-K, Xu, H-K: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl.. 178, 301–308 (1993). Publisher Full Text OpenURL

  41. Tan, K-K, Xu, H-K: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Am. Math. Soc.. 122, 733–739 (1994). Publisher Full Text OpenURL

  42. Xu, H-K: Existence and convergence for fixed points of asymptotically nonexpansive type. Nonlinear Anal.. 16, 1139–1146 (1991). Publisher Full Text OpenURL