Open Access

Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces

Fixed Point Theory and Applications20102010:618767

DOI: 10.1155/2010/618767

Received: 8 August 2010

Accepted: 24 September 2010

Published: 28 September 2010

Abstract

Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that and for all . We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary by , , where and , are sequences of positive numbers satisfying , , converges strongly to a common fixed point of and ; that is, there exists such that .

1. Introduction

Let be a Banach space, and let be a nonempty subset of . We will denote by the family of nonempty bounded closed subsets of and by the family of nonempty compact convex subsets of . Let be the Hausdorff distance on , that is,
(1.1)

where is the distance from the point to the subset .

A mapping is said to be nonexpansive if
(1.2)

A point is called a fixed point of if .

A multivalued mapping is said to be nonexpansive if
(1.3)

A point is called a fixed point for a multivalued mapping if .

We use the notation standing for the set of fixed points of a mapping and standing for the set of common fixed points of and . Precisely, a point is called a common fixed point of and if .

In 2006, S. Dhompongsa et al. [1] proved a common fixed point theorem for two nonexpansive commuting mappings.

Theorem 1.1 (see [1, Theorem 4.2]).

Let be a nonempty bounded closed convex subset of a uniformly Banach space , and let , and be a nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume that and are commuting; that is, if for every such that and , there holds . Then, and have a common fixed point.

In this paper, we introduce an iterative process in a new sense, called the modified Ishikawa iteration method with respect to a pair of single-valued and multivalued nonexpansive mappings. We also establish the strong convergence theorem of a sequence from such process in a nonempty compact convex subset of a uniformly convex Banach space.

2. Preliminaries

The important property of the uniformly convex Banach space we use is the following lemma proved by Schu [2] in 1991.

Lemma 2.1 (see [2]).

Let be a uniformly convex Banach space, let be a sequence of real numbers such that for all , and let and be sequences of such that , , and for some .  Then, .

The following observation will be used in proving our results, and the proof is straightforward.

Lemma 2.2.

Let be a Banach space, and let be a nonempty closed convex subset of . Then,
(2.1)

where and is a multivalued nonexpansive mapping from into .

A fundamental principle which plays a key role in ergodic theory is the demiclosedness principle. A mapping defined on a subset of a Banach space is said to be demiclosed if any sequence in the following implication holds: and implies .

Theorem 2.3 (see [3]).

Let be a nonempty closed convex subset of a uniformly convex Banach space , and let be a nonexpansive mapping. If a sequence in converges weakly to and converges to 0 as , then .

In 1974, Ishikawa introduced the following well-known iteration.

Definition 2.4 (see [4]).

Let be a Banach space, let be a closed convex subset of , and let be a selfmap on . For , the sequence of Ishikawa iterates of is defined by
(2.2)

where and are real sequences.

A nonempty subset of is said to be proximinal if, for any , there exists an element such that = dist . We will denote by the family of nonempty proximinal bounded subsets of .

In 2005, Sastry and Babu [5] defined the Ishikawa iterative scheme for multivalued mappings as follows.

Let be a compact convex subset of a Hilbert space , and let be a multivalued mapping, and fix .
(2.3)

where , are sequences in [ ] with such that and .

They also proved the strong convergence of the above Ishikawa iterative scheme for a multivalued nonexpansive mapping with a fixed point under some certain conditions in a Hilbert space.

Recently, Panyanak [6] extended the results of Sastry and Babu [5] to a uniformly convex Banach space and also modified the above Ishikawa iterative scheme as follows.

Let be a nonempty convex subset of a uniformly convex Banach space , and let be a multivalued mapping
(2.4)

where , are sequences in [ ] with and such that and , respectively. Moreover, and such that and , respectively.

Very recently, Song and Wang [7, 8] improved the results of [5, 6] by means of the following Ishikawa iterative scheme.

Let be a multivalued mapping, where . The Ishikawa iterative scheme is defined by
(2.5)

where and such that and , respectively. Moreover, such that .

At the same period, Shahzad and Zegeye [9] modified the Ishikawa iterative scheme and extended the result of [7, Theorem 2] to a multivalued quasinonexpansive mapping as follows.

Let be a nonempty convex subset of a Banach space , and let be a multivalued mapping, where . The Ishikawa iterative scheme is defined by
(2.6)

where and .

In this paper, we introduce a new iteration method modifying the above ones and call it the modified Ishikawa iteration method.

Definition 2.5.

Let be a nonempty closed bounded convex subset of a Banach space , let be a single-valued nonexpansive mapping, and let be a multivalued nonexpansive mapping. The sequence of the modified Ishikawa iteration is defined by
(2.7)

where , , and , .

3. Main Results

We first prove the following lemmas, which play very important roles in this section.

Lemma 3.1.

Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued and a multivalued nonexpansive mapping, respectively, and satisfying for all . Let be the sequence of the modified Ishikawa iteration defined by (2.7). Then, exists for all .

Proof.

Letting and , we have
(3.1)

Since is a decreasing and bounded sequence, we can conclude that the limit of exists.

We can see how Lemma 2.1 is useful via the following lemma.

Lemma 3.2.

Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued and a multivalued nonexpansive mapping, respectively, and satisfying for all . Let be the sequence of the modified Ishikawa iteration defined by (2.7). If for some , then, .

Proof.

Let . By Lemma 3.1, we put and consider
(3.2)
Then, we have
(3.3)
Further, we have
(3.4)

By Lemma 2.1, we can conclude that .

Lemma 3.3.

Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued and a multivalued nonexpansive mapping, respectively, and satisfying for all . Let be the sequence of the modified Ishikawa iteration defined by (2.7). If , for some , then .

Proof.

Let . We put, as in Lemma 3.2, . For , we have
(3.5)
and hence
(3.6)
Therefore, since ,
(3.7)
Thus,
(3.8)
It follows that
(3.9)
Since, from (3.3), , we have
(3.10)
Recall that
(3.11)
Hence, we have
(3.12)

Using the fact that and by (3.10), we can conclude that .

The following lemma allows us to go on.

Lemma 3.4.

Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued and a multivalued nonexpansive mapping, respectively, and satisfying for all . Let be the sequence of the modified Ishikawa iteration defined by (2.7). If , , then .

Proof.

Consider
(3.13)
Then, we have
(3.14)

Hence, by Lemmas 3.2 and 3.3, .

We give the sufficient conditions which imply the existence of common fixed points for single-valued mappings and multivalued nonexpansive mappings, respectively, as follows

Theorem 3.5.

Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued and a multivalued nonexpansive mapping, respectively, and satisfying for all . Let be the sequence of the modified Ishikawa iteration defined by (2.7). If , , then for some subsequence of implies .

Proof.

Assume that . From Lemma 3.4, we have
(3.15)
Since is demiclosed at 0, we have , and hence , that is, . By Lemma 2.2 and by Lemma 3.4, we have
(3.16)

It follows that . Therefore as desired.

Hereafter, we arrive at the convergence theorem of the sequence of the modified Ishikawa iteration. We conclude this paper with the following theorem.

Theorem 3.6.

Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued and a multivalued nonexpansive mapping, respectively, and satisfying for all . Let be the sequence of the modified Ishikawa iteration defined by (2.7) with , . Then converges strongly to a common fixed point of and .

Proof.

Since is contained in which is compact, there exists a subsequence of such that converges strongly to some point , that is, . By Theorem 3.5, we have , and by Lemma 3.1, we have that exists. It must be the case in which . Therefore, converges strongly to a common fixed point of and .

Declarations

Acknowledgments

The first author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree for this research. The authors would like to express their deep gratitude to Prof. Dr. Sompong Dhompongsa whose guidance and support were valuable for the completion of the paper. This work was completed with the support of the Commission on Higher Education and The Thailand Research Fund under Grant MRG5180213.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Burapha University
(2)
Department of Mathematics, Faculty of Science, Chiang Mai University

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Copyright

© K. Sokhuma and A. Kaewkhao. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.