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Convergence theorems for mixed type asymptotically nonexpansive mappings

Weiping Guo1, Yeol Je Cho2* and Wei Guo3

Author Affiliations

1 School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, Jiangsu, 215009, P.R. China

2 Department of Mathematics Education and the RINS College of Education, Gyeongsang National University, Chinju, 660-701, Korea

3 Department of Aerospace Engineering and Mechanics, University of Minnesota, Twin Cities, Minneapolis, MN, 55455, USA

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Fixed Point Theory and Applications 2012, 2012:224  doi:10.1186/1687-1812-2012-224

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/224


Received:27 April 2012
Accepted:16 November 2012
Published:11 December 2012

© 2012 Guo et al.; licensee Springer

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

Abstract

In this paper, we introduce a new two-step iterative scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong and weak convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

Keywords:
mixed type asymptotically nonexpansive mapping; strong and weak convergence; common fixed point; uniformly convex Banach space

1 Introduction

Let K be a nonempty subset of a real normed linear space E. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M1">View MathML</a> is said to be asymptotically nonexpansive if there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M2">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M3">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M4">View MathML</a>

(1.1)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>.

In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self-mappings, and proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space E and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.

Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive self-mappings in Banach spaces (see [2-16]), which extend and improve the result of Goebel and Kirk in several ways.

Recently, Chidume et al.[10] introduced the concept of asymptotically nonexpansive nonself-mappings, which is a generalization of an asymptotically nonexpansive self-mapping, as follows.

Definition 1.1[10]

Let K be a nonempty subset of a real normed linear space E. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7">View MathML</a> be a nonexpansive retraction of E onto K. A nonself-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M8">View MathML</a> is said to be asymptotically nonexpansive if there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M9">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M10">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M12">View MathML</a>

(1.2)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>.

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.

In 2003, also, Chidume et al.[10] studied the following iteration scheme:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M15">View MathML</a>

(1.3)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17">View MathML</a> is a sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M18">View MathML</a> and P is a nonexpansive retraction of E onto K, and proved some strong and weak convergence theorems for an asymptotically nonexpansive nonself-mapping.

In 2006, Wang [11] generalized the iteration process (1.3) as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M19">View MathML</a>

(1.4)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M21">View MathML</a> are two asymptotically nonexpansive nonself-mappings and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23">View MathML</a> are real sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M24">View MathML</a>, and proved some strong and weak convergence theorems for two asymptotically nonexpansive nonself-mappings. Recently, Guo and Guo [12] proved some new weak convergence theorems for the iteration process (1.4).

The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and to prove some strong and weak convergence theorems for the new iteration scheme in uniformly convex Banach spaces.

2 Preliminaries

Let E be a real Banach space, K be a nonempty closed convex subset of E and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7">View MathML</a> be a nonexpansive retraction of E onto K. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M26">View MathML</a> be two asymptotically nonexpansive self-mappings and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M27">View MathML</a> be two asymptotically nonexpansive nonself-mappings. Then we define the new iteration scheme of mixed type as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M28">View MathML</a>

(2.1)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23">View MathML</a> are two sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M24">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a> are the identity mappings, then the iterative scheme (2.1) reduces to the sequence (1.4).

We denote the set of common fixed points of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M39">View MathML</a> and denote the distance between a point z and a set A in E by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M40">View MathML</a>.

Now, we recall some well-known concepts and results.

Let E be a real Banach space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M41">View MathML</a> be the dual space of E and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M42">View MathML</a> be the normalized duality mapping defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M43">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M44">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M45">View MathML</a> denotes duality pairing between E and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M41">View MathML</a>. A single-valued normalized duality mapping is denoted by j.

A subset K of a real Banach space E is called a retract of E[10] if there exists a continuous mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M48">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M49">View MathML</a>. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M50">View MathML</a> is called a retraction if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M51">View MathML</a>. It follows that if a mapping P is a retraction, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M52">View MathML</a> for all y in the range of P.

A Banach space E is said to satisfy Opial’s condition[17] if, for any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> of E, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54">View MathML</a> weakly as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11">View MathML</a> implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M56">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M57">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M58">View MathML</a>.

A Banach space E is said to have a Fréchet differentiable norm[18] if, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M59">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M60">View MathML</a>

exists and is attained uniformly in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M61">View MathML</a>.

A Banach space E is said to have the Kadec-Klee property[19] if for every sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> in E, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54">View MathML</a> weakly and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M64">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54">View MathML</a> strongly.

Let K be a nonempty closed subset of a real Banach space E. A nonself-mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M8">View MathML</a> is said to be semi-compact[11] if, for any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> in K such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M68">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11">View MathML</a>, there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70">View MathML</a> converges strongly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M73">View MathML</a>.

Lemma 2.1[15]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M74">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M75">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M76">View MathML</a>be three nonnegative sequences satisfying the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M77">View MathML</a>

for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M78">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M79">View MathML</a>is some nonnegative integer, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M80">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M81">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M82">View MathML</a>exists.

Lemma 2.2[8]

LetEbe a real uniformly convex Banach space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M83">View MathML</a>for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>. Also, suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M86">View MathML</a>are two sequences ofEsuch that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M87">View MathML</a>

hold for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M88">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M89">View MathML</a>.

Lemma 2.3[10]

LetEbe a real uniformly convex Banach space, Kbe a nonempty closed convex subset ofEand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M8">View MathML</a>be an asymptotically nonexpansive mapping with a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M9">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M10">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M11">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M94">View MathML</a>is demiclosed at zero, i.e., if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M54">View MathML</a>weakly and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M96">View MathML</a>strongly, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M97">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M98">View MathML</a>is the set of fixed points ofT.

Lemma 2.4[16]

LetXbe a uniformly convex Banach space andCbe a convex subset ofX. Then there exists a strictly increasing continuous convex function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M99">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M100">View MathML</a>such that, for each mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M101">View MathML</a>with a Lipschitz constant<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M102">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M103">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M104">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M105">View MathML</a>.

Lemma 2.5[16]

LetXbe a uniformly convex Banach space such that its dual space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M106">View MathML</a>has the Kadec-Klee property. Suppose<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>is a bounded sequence and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M108">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M109">View MathML</a>

exists for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M110">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M111">View MathML</a>denotes the set of all weak subsequential limits of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M113">View MathML</a>.

3 Strong convergence theorems

In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.

Lemma 3.1LetEbe a real uniformly convex Banach space andKbe a nonempty closed convex subset ofE. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M114">View MathML</a>be two asymptotically nonexpansive self-mappings with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M115">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M116">View MathML</a>be two asymptotically nonexpansive nonself-mappings with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M117">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M118">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M119">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>, respectively, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M121">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>be the sequence defined by (2.1), where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23">View MathML</a>are two real sequences in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M24">View MathML</a>. Then

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M126">View MathML</a>exists for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a>;

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M128">View MathML</a>exists.

Proof (1) Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M129">View MathML</a>. For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a>, it follows from (2.1) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M131">View MathML</a>

(3.1)

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M132">View MathML</a>

(3.2)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M118">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M119">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M136">View MathML</a>. It follows from Lemma 2.1 that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M126">View MathML</a> exists.

(2) Taking the infimum over all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a> in (3.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M139">View MathML</a>

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>. It follows from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M136">View MathML</a> and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □

Lemma 3.2LetEbe a real uniformly convex Banach space andKbe a nonempty closed convex subset ofE. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M114">View MathML</a>be two asymptotically nonexpansive self-mappings with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M115">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M116">View MathML</a>be two asymptotically nonexpansive nonself-mappings with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M117">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M118">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M119">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>, respectively, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M121">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>be the sequence defined by (2.1) and the following conditions hold:

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M17">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M23">View MathML</a>are two real sequences in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M153">View MathML</a>for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M154">View MathML</a>;

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M155">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>.

Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M158">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>.

Proof Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M129">View MathML</a>. For any given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M126">View MathML</a> exists by Lemma 3.1. Now, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M163">View MathML</a>. It follows from (3.2) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M136">View MathML</a> that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M165">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M166">View MathML</a>

Taking lim sup on both sides in (3.1), we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M167">View MathML</a> and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M168">View MathML</a>

Using Lemma 2.2, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M169">View MathML</a>

(3.3)

By the condition (b), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M170">View MathML</a>

and so, from (3.3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M171">View MathML</a>

(3.4)

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M172">View MathML</a>

Taking lim inf on both sides in the inequality above, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M173">View MathML</a>

by (3.4) and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M174">View MathML</a>

Using (3.1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M175">View MathML</a>

In addition, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M176">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M177">View MathML</a>

It follows from Lemma 2.2 that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M178">View MathML</a>

(3.5)

Now, we prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M179">View MathML</a>

Indeed, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M180">View MathML</a> by the condition (b). It follows from (3.5) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M181">View MathML</a>

(3.6)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M182">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M7">View MathML</a> is a nonexpansive retraction of E onto K, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M184">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M185">View MathML</a>

(3.7)

Furthermore, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M186">View MathML</a>

Thus it follows from (3.5), (3.6) and (3.7) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M187">View MathML</a>

(3.8)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M188">View MathML</a> by the condition (b) and

Using (3.3) and (3.8), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M190">View MathML</a>

(3.9)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M191">View MathML</a>

(3.10)

It follows from

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M192">View MathML</a>

and (3.3) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M193">View MathML</a>

(3.11)

In addition, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M194">View MathML</a>

Using (3.3) and (3.11), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M195">View MathML</a>

(3.12)

Thus, using (3.9), (3.10) and the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M196">View MathML</a>

we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M197">View MathML</a>. It follows from (3.6) and the inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M198">View MathML</a>

that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M199">View MathML</a>

(3.13)

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M200">View MathML</a>

from (3.8), (3.11) and (3.13), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M201">View MathML</a>

(3.14)

Again, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M202">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M203">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a> are two asymptotically nonexpansive nonself-mappings, we have

(3.15)

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>. It follows from (3.12), (3.14) and (3.15) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M209">View MathML</a>

(3.16)

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M211">View MathML</a>

Using (3.4), (3.8) and (3.12), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M212">View MathML</a>

(3.17)

In addition, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M213">View MathML</a>

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214">View MathML</a>. Thus it follows from (3.6), (3.10), (3.16) and (3.17) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M215">View MathML</a>

Finally, we prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M216">View MathML</a>

In fact, by the condition (b), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M217">View MathML</a>

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214">View MathML</a>. Thus it follows from (3.5), (3.6), (3.9) and (3.10) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M219">View MathML</a>

This completes the proof. □

Now, we find two mappings, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M220">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M221">View MathML</a>, satisfying the condition (b) in Lemma 3.2 as follows.

Example 3.1[20]

Let ℝ be the real line with the usual norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M222">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M223">View MathML</a>. Define two mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M224">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M225">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M226">View MathML</a>

Now, we show that T is nonexpansive. In fact, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M227">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M228">View MathML</a>, then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M229">View MathML</a>

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M230">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M231">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M232">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M233">View MathML</a>, then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M234">View MathML</a>

This implies that T is nonexpansive and so T is an asymptotically nonexpansive mapping with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M235">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>. Similarly, we can show that S is an asymptotically nonexpansive mapping with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M237">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>.

Next, we show that two mappings S, T satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:

Case 1. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M227">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M240">View MathML</a>

Case 2. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M228">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M242">View MathML</a>

Case 3. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M232">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M244">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M245">View MathML</a>

Case 4. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M230">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M231">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M248">View MathML</a>

Therefore, the condition (b) in Lemma 3.2 is satisfied.

Theorem 3.1Under the assumptions of Lemma 3.2, if one of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>is completely continuous, then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>defined by (2.1) converges strongly to a common fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>.

Proof Without loss of generality, we can assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a> is completely continuous. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> is bounded by Lemma 3.1, there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M260">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M261">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M260">View MathML</a> converges strongly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M263">View MathML</a>. Moreover, we know that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M264">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M265">View MathML</a>

by Lemma 3.2, which imply that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M266">View MathML</a>

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M267">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M268">View MathML</a>. Thus, by the continuity of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M273">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M274">View MathML</a>

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>. Thus it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M276">View MathML</a>. Furthermore, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M277">View MathML</a> exists by Lemma 3.1, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M278">View MathML</a>. This completes the proof. □

Theorem 3.2Under the assumptions of Lemma 3.2, if one of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>is semi-compact, then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>defined by (2.1) converges strongly to a common fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M158">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214">View MathML</a> by Lemma 3.2 and one of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a> is semi-compact, there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M70">View MathML</a> converges strongly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M297">View MathML</a>. Moreover, by the continuity of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M302">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M303">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>. Thus it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M276">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M277">View MathML</a> exists by Lemma 3.1, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M278">View MathML</a>. This completes the proof. □

Theorem 3.3Under the assumptions of Lemma 3.2, if there exists a nondecreasing function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M308">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M309">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M310">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M311">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M312">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M49">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M39">View MathML</a>, then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>defined by (2.1) converges strongly to a common fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M158">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M214">View MathML</a> by Lemma 3.2, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M322">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M308">View MathML</a> is a nondecreasing function satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M309">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M310">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M311">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M128">View MathML</a> exists by Lemma 3.1, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M328">View MathML</a>.

Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> is a Cauchy sequence in K. In fact, from (3.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M330">View MathML</a>

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M332">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a>. For any m, n, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M334">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M335">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M336">View MathML</a>. Thus, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M338">View MathML</a>

Taking the infimum over all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M340">View MathML</a>

Thus it follows from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M328">View MathML</a> that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> is a Cauchy sequence. Since K is a closed subset of E, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges strongly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M297">View MathML</a>. It is easy to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M345">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M346">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M347">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M348">View MathML</a> are all closed and so F is a closed subset of K. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M328">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M350">View MathML</a>, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges strongly to a common fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>. This completes the proof. □

4 Weak convergence theorems

In this section, we prove weak convergence theorems for the iterative scheme defined by (2.1) in uniformly convex Banach spaces.

Lemma 4.1Under the assumptions of Lemma 3.1, for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M356">View MathML</a>, the limit

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M357">View MathML</a>

exists for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M358">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>is the sequence defined by (2.1).

Proof Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M360">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M361">View MathML</a> and, from Lemma 3.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M362">View MathML</a> exists. Thus it remains to prove Lemma 4.1 for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M363">View MathML</a>.

Define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M364">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M365">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M49">View MathML</a>. It is easy to prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M367">View MathML</a>

(4.1)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M332">View MathML</a>. Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M370">View MathML</a>, it follows from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M371">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M372">View MathML</a> that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M373">View MathML</a>. Setting

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M374">View MathML</a>

(4.2)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M375">View MathML</a>, from (4.1) and (4.2), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M376">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M5">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M378">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M379">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M127">View MathML</a>. Let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M381">View MathML</a>

(4.3)

Then, using (4.3) and Lemma 2.4, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M382">View MathML</a>

It follows from Lemma 3.1 and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M383">View MathML</a> that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M384">View MathML</a> uniformly for all m. Observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M385">View MathML</a>

Thus we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M386">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M387">View MathML</a> exists for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M363">View MathML</a>. This completes the proof. □

Lemma 4.2Under the assumptions of Lemma 3.1, ifEhas a Fréchet differentiable norm, then, for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M389">View MathML</a>, the limit

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M390">View MathML</a>

exists, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>is the sequence defined by (2.1). Furthermore, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M111">View MathML</a>denotes the set of all weak subsequential limits of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M394">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M395">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M396">View MathML</a>.

Proof This follows basically as in the proof of Lemma 3.2 of [12] using Lemma 4.1 instead of Lemma 3.1 of [12]. □

Theorem 4.1Under the assumptions of Lemma 3.2, ifEhas a Fréchet differentiable norm, then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>defined by (2.1) converges weakly to a common fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>.

Proof Since E is a uniformly convex Banach space and the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> is bounded by Lemma 3.1, there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M403">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> which converges weakly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M405">View MathML</a>. By Lemma 3.2, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M406">View MathML</a>

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M120">View MathML</a>. It follows from Lemma 2.3 that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M408">View MathML</a>.

Now, we prove that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges weakly to q. Suppose that there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410">View MathML</a> converges weakly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M413">View MathML</a>. Then, by the same method given above, we can also prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M414">View MathML</a>. So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M415">View MathML</a>. It follows from Lemma 4.2 that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M416">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M417">View MathML</a>, which shows that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges weakly to q. This completes the proof. □

Theorem 4.2Under the assumptions of Lemma 3.2, if the dual space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M41">View MathML</a>ofEhas the Kadec-Klee property, then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>defined by (2.1) converges weakly to a common fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>.

Proof Using the same method given in Theorem 4.1, we can prove that there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M403">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> which converges weakly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M408">View MathML</a>.

Now, we prove that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges weakly to q. Suppose that there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410">View MathML</a> converges weakly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M297">View MathML</a>. Then, as for q, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M350">View MathML</a>. It follows from Lemma 4.1 that the limit

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M434">View MathML</a>

exists for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M358">View MathML</a>. Again, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M436">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M437">View MathML</a> by Lemma 2.5. This shows that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges weakly to q. This completes the proof. □

Theorem 4.3Under the assumptions of Lemma 3.2, ifEsatisfies Opial’s condition, then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a>defined by (2.1) converges weakly to a common fixed point of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M34">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M37">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M38">View MathML</a>.

Proof Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M403">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> which converges weakly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M408">View MathML</a>.

Now, we prove that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges weakly to q. Suppose that there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M410">View MathML</a> converges weakly to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M451">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M452">View MathML</a>. Then, as for q, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M453">View MathML</a>. Using Lemma 3.1, we have the following two limits exist:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M454">View MathML</a>

Thus, by Opial’s condition, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M455">View MathML</a>

which is a contradiction and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M456">View MathML</a>. This shows that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/224/mathml/M53">View MathML</a> converges weakly to q. This completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

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