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

- K Sokhuma
^{1}and - A Kaewkhao
^{2}Email author

**2010**:618767

https://doi.org/10.1155/2010/618767

© K. Sokhuma and A. Kaewkhao. 2010

**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

where is the distance from the point to the subset .

A point is called a fixed point of if .

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.

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]).

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.

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.

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.

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.

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

Definition 2.5.

## 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.

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.

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.

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.

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.

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

## References

- Dhompongsa S, Kaewcharoen A, Kaewkhao A:
**The Domínguez-Lorenzo condition and multivalued nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(5):958–970. 10.1016/j.na.2005.05.051MathSciNetView ArticleMATHGoogle Scholar - Schu J:
**Weak and strong convergence to fixed points of asymptotically nonexpansive mappings.***Bulletin of the Australian Mathematical Society*1991,**43**(1):153–159. 10.1017/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Semicontractive and semiaccretive nonlinear mappings in Banach spaces.***Bulletin of the American Mathematical Society*1968,**74:**660–665. 10.1090/S0002-9904-1968-11983-4MathSciNetView ArticleMATHGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Sastry KPR, Babu GVR:
**Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point.***Czechoslovak Mathematical Journal*2005,**55(130)**(4):817–826.MathSciNetView ArticleMATHGoogle Scholar - Panyanak B:
**Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces.***Computers & Mathematics with Applications*2007,**54**(6):872–877. 10.1016/j.camwa.2007.03.012MathSciNetView ArticleMATHGoogle Scholar - Song Y, Wang H:
**Erratum to: "Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces" [Comput. Math. Appl. 54 (2007) 872–877].***Computers & Mathematics with Applications*2008,**55**(12):2999–3002. 10.1016/j.camwa.2007.11.042MathSciNetView ArticleMATHGoogle Scholar - Song Y, Wang H:
**Convergence of iterative algorithms for multivalued mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(4):1547–1556. 10.1016/j.na.2008.02.034MathSciNetView ArticleMATHGoogle Scholar - Shahzad N, Zegeye H:
**On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(3–4):838–844. 10.1016/j.na.2008.10.112MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.