ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 576108, 13 pagesdoi:10.5402/2011/576108

Research Article

Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces

Yekini Shehu

Department of Mathematics, University of Nigeria, Nsukka, Nigeria

Received 20 April 2011; Accepted 9 June 2011

Academic Editors: A. Levy and O. Miyagaki

Copyright © 2011 Yekini Shehu. 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.

Abstract.

We introduce a new iterative process to approximate a common fixed point of a finite family of multivalued maps in a uniformly convex real Banach space and establish strong convergence theorems for the proposed process. Furthermore, strong convergence theorems for finite family of quasi-nonexpansive multivalued maps are obtained. Our results extend important recent results.

1. Introduction

Let be a nonempty, closed, and convex subset of a real Hilbert space . The set is called proximinal if for each , there exists such that , where . Let , and denote the families of nonempty, closed and bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of , respectively. The Hausdorff metric on is defined by for . A single-valued map is called nonexpansive if for all . A multivalued map is said to be nonexpansive if for all . An element is called a fixed point of (resp., if (resp., ). The set of fixed points of is denoted by . A multivalued map is said to be quasi-nonexpansive if for all and for all .

A is said to satisfy Condition (I) if there is a nondecreasing function with for such that for all .

The fixed point theory of multivalued nonexpansive mappings is much more complicated and difficult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multivalued mappings. The first results in this direction were established by Markin [1] in Hilbert spaces and by Browder [2] for spaces having weakly continuous duality mapping. Dozo [3] generalized these results to a Banach space satisfying Opial's condition.

In 1974, by using Edelstein's method of asymptotic centers, Lim [4] obtained a fixed point theorem for a multivalued nonexpansive self-mapping in a uniformly convex Banach space.

Theorem 1.1 (Lim [4]). Let be a nonempty, closed convex, and bounded subset of a uniformly convex Banach space and a multivalued nonexpansive mapping. Then, has a fixed point.

In 1990, Kirk and Massa [5] gave an extension of Lim's theorem proving the existence of a fixed point in a Banach space for which the asymptotic center of a bounded sequence in a closed bounded convex subset is nonempty and compact.

Theorem 1.2 (Kirk and Massa [5]). Let be a nonempty, closed convex, and bounded subset of a Banach space and a multivalued nonexpansive mapping. Suppose that the asymptotic center in of each bounded sequence of is nonempty and compact. Then, has a fixed point.

Banach contraction mapping principle was extended nicely multivalued mappings by Nadler [6] in 1969. (Below is stated in a Banach space setting).

Theorem 1.3 (Nadler [6]). Let be a nonempty closed subset of a Banach space and a multivalued contraction. Then, has a fixed point.

In 1953, Mann [7] introduced the following iterative scheme to approximate a fixed point of a nonexpansive mapping in a Hilbert space : where the initial point is taken arbitrarily in and is a sequence in . However, we note that Mann's iteration has only weak convergence; see, for example, [8].

In 2005, Sastry and Babu [9] proved that the Mann and Ishikawa iteration schemes for a multivalued map with a fixed point converge to a fixed point of under certain conditions. They also claimed that the fixed point may be different from .

In 2007, Panyanak [10] extended the results of Sastry and Babu to uniformly convex Banach spaces and proved the following theorems.

Theorem 1.4 (Panyanak [10]). Let be a uniformly convex Banach space, a nonempty closed bounded convex subset of , and a multivalued nonexpansive mapping that satisfies condition . Assume that (i) and (ii). Suppose that a nonempty proximinal subset of . Then, the Mann iterates defined by , where such that and such that , converges strongly to a fixed point of .

Theorem 1.5 (Panyanak [10]). Let be a uniformly convex Banach space, a nonempty compact convex subset of , and a multivalued nonexpansive mapping with a fixed point . Assume that (i) ; (ii) and (iii) . Then, the Ishikawa iterates defined by , such that , and such that converges strongly to a fixed point of .

Later, Song and Wang [11] noted there was a gap in the proofs of Theorem 1.5 above and of [9, Theorem 5]. They further solved/revised the gap and also gave the affirmative answer Panyanak [10] question using the Ishikawa iterative scheme. In the main results, the domain of is still compact, which is a strong condition (see [11, Theorem 1]) and satisfies condition (see [11, Theorem 1]).

Recently, Shahzad and Zegeye [12] proved the following theorems for quasi-nonexpansive multivalued map and multivalued map in uniformly convex Banach space.

Theorem 1.6 (Shahzad and Zegeye [12]). Let be a uniformly convex real Banach space and a nonempty, closed and convex subset of . Let be a quasi-nonexpansive multivalued map with for which , for all. Let be a sequence defined iteratively by , , and . Assume that satisfies condition (I) and . Then, converges strongly to a fixed point of .

Theorem 1.7 (Shahzad and Zegeye [12]). Let be a uniformly convex real Banach space and a nonempty, closed, and convex subset of . Let be a multivalued map with such that is nonexpansive. Let be a sequence defined iteratively by , , and . Assume that satisfies condition (I) and . Then, converges strongly to a fixed point of .

More recently, Abbas et al. [13] introduced the following one-step iterative process to compute common fixed points of two multivalued nonexpansive mappings. Using (1.11), Abbas et al. [13] proved weak and strong convergence theorems for approximation of common fixed point of two multivalued nonexpansive mappings in Banach spaces.

Motivated by the ongoing research and the above mentioned results, we introduce a new iterative scheme for approximation of common fixed points of finite family of multivalued maps in a real Banach space. Furthermore, we prove strong convergence theorems for approximation of common fixed points of finite family of multivalued maps in a uniformly convex real Banach space. Next, we prove a necessary and sufficient condition for strong convergence of our new iterative process to a common fixed point of finite family of multivalued maps. Finally, we introduce a new iterative scheme and prove strong convergence theorems for finite family of quasi-nonexpansive multivalued maps in a uniformly convex real Banach space. Our results extend the results of Sastry and Babu [9], Panyanak [10], Shahzad and Zegeye [12], and Song and Wang [11].

2. Preliminaries

Let be Banach space and . The modulus of convexity of is the function defined by is uniformly convex if for any , there exists a such that if with and , then . Equivalently, is uniformly convex if and only if for all .

A family is said to satisfy Condition (II) if there is a nondecreasing function with for such that for all and .

The mapping is called hemicompact if for any sequence in such that as , there exists a subsequence of such that . We note that if is compact, then every multivalued mapping is hemicompact.

Let be a nonempty, closed, and convex subset of a real Banach space . Let be a multimap and Then, is nonempty and compact for every . Furthermore, we observe that if is a fixed point of .

A mapping is -nonexpansive ([14]) if for all and with , there exists with such that

It is known that -nonexpansiveness is different from nonexpansiveness for multimaps. There are some -nonexpansive multimaps which are not nonexpansive and some nonexpansive multimaps which are not -nonexpansive ([15, 16]).

By the definition of Hausdorff metric, we obtain that if a multimap is -nonexpansive, then is nonexpansive.

Throughout this paper, we write to indicate that the sequence converges strongly to .

Also, this following lemma will be used in the sequel.

Lemma 2.1 (Schu [17]). Suppose that is a uniformly convex Banach space and for all positive integers . Also, suppose that and are two sequences of such that and hold for some . Then, .

3. Main Results

We now introduce the following iteration scheme. Let be a real normed space and a nonempty subset of . Let be multivalued maps of into with such that are nonexpansive and a sequence in such that for all . Let be a sequence defined iteratively by where .

Lemma 3.1. Let be a real normed space and a nonempty subset of . Let be multivalued maps of into with such that are nonexpansive. Let a sequence in such that for all . Let be a sequence defined iteratively by (3.1). Then,

Proof. Let . Then, from (3.1), we have the following estimates: Thus, exists. Let for some . Then, Since is nonexpansive mapping and , we have for each . Taking on both sides, we obtain Next, Using (3.5), (3.8), and Lemma 2.1, we obtain This yields Thus, . Furthermore, Using (3.11) and Lemma 2.1, we obtain This yields Thus, . So, Then, In a similar way, we can show that This completes the proof.

Theorem 3.2. Let be a uniformly convex real Banach space and a nonempty, closed, and convex subset of . Let be multivalued maps of into with such that are nonexpansive and satisfying condition (II). Let a sequence in such that for all . Let be a sequence defined iteratively by (3.1). Then, converges strongly to a common fixed point of .

Proof. Since satisfies condition (II), we have that as . Thus, there is a subsequence of and a sequence such that for all . By Lemma 3.1, we obtain We now show that is a Cauchy sequence in . Observe that This shows that is a Cauchy sequence in , and thus converges to . Since and as , it follows that , and thus , and converges strongly to . Since exists, it follows that converges strongly to . This completes the proof.

Corollary 3.3. Let be a uniformly convex real Banach space and a nonempty, closed, and convex subset of . Let be -nonexpansive multimaps of into with and satisfying condition (II). Let a sequence in such that for all . Let be a sequence defined iteratively by (3.1). Then, converges strongly to a common fixed point of .

Theorem 3.4. Let be a uniformly convex real Banach space and a nonempty, closed and convex subset of . Let be multivalued maps of into with such that are nonexpansive and is hemicompact and continuous for each . Let a sequence in such that for all . Let be a sequence defined iteratively by (3.1). Then, converges strongly to a common fixed point of .

Proof. Since for all and is hemicompact for each , there is a subsequence of such that as for some . Since is continuous for each , we have . As a result, we have that for all, and so, . Since exists, it follows that converges strongly to . This completes the proof.

Corollary 3.5. Let be a uniformly convex real Banach space and a nonempty, closed, and convex subset of . Let be -nonexpansive multimaps of into with and is hemicompact and continuous for each . Let a sequence in such that for all . Let be a sequence defined iteratively by (3.1). Then, converges strongly to a common fixed point of .

Theorem 3.6. Let be a uniformly convex real Banach space and a nonempty compact convex subset of . Let be multivalued maps of into with such that are nonexpansive. Let a sequence in such that for all . Let be a sequence defined iteratively by (3.1). Then, converges strongly to a common fixed point of .

Proof. From the compactness of , there exists a subsequence of such that for some . Thus, Hence, . Now, on taking in place of , we get that exists. This completes the proof.

The following result gives a necessary and sufficient condition for strong convergence of the sequence in (3.1) to a common fixed point of .

Theorem 3.7. Let be a nonempty, closed, and convex subset of a real Banach space . Let be multivalued maps of into with such that are nonexpansive. Let a sequence in such that for all . Let be a sequence defined iteratively by (3.1). Then, converges strongly to a common fixed point of if and only if .

Proof. The necessity is obvious. Conversely, suppose that . By (3.3), we have This gives Hence, exists. By hypothesis, , so we must have .Next, we show that is a Cauchy sequence in . Let be given, and since , there exists such that for all , we have In particular, so that there must exist a such that Now, for , we have Hence, is a Cauchy sequence in a closed subset of a Banach space , and therefore, it must converge in . Let . Now, for each , we obtain gives that which implies that . Consequently, .

All the results we have obtained so far can be established for finite family of quasi-nonexpansive multivalued maps. Let be quasi-nonexpansive multivalued maps of into such that for which , for all . Let a sequence in such that for all . Let be a sequence defined iteratively by where , . Thus, we obtain the following theorems using iterative process (3.28).

Theorem 3.8. Let be a uniformly convex real Banach space and a nonempty, closed, and convex subset of . Let be quasi-nonexpansive multivalued maps of into such that for which , for all and satisfying condition (II). Let a sequence in such that for all . Let be a sequence defined iteratively by (3.28). Then, converges strongly to a common fixed point of .

Theorem 3.9. Let be a uniformly convex real Banach space and a nonempty, closed, and convex subset of . Let be quasi-nonexpansive multivalued maps of into such that for which , for all and is hemicompact and continuous for each . Let a sequence in such that for all . Let be a sequence defined iteratively by (3.28). Then, converges strongly to a common fixed point of .

Theorem 3.10. Let be a uniformly convex real Banach space and a nonempty compact convex subset of . Let be quasi-nonexpansive multivalued maps of into such that for which , for all . a sequence in such that for all . Let be a sequence defined iteratively by (3.28). Then, converges strongly to a common fixed point of .

Corollary 3.11 (Abbas et al. [13]). Let be a uniformly convex real Banach space satisfying Opial's condition. Let be a nonempty, closed, and convex of . Let be multivalued nonexpansive mappings of into such that . Let , and be sequence in satisfying . Let be a sequence defined iteratively by where such that and whenever is a fixed point of any one of mappings and . Then, converges weakly to a common fixed point of .

Corollary 3.12 (Abbas et al. [13]). Let be a real Banach space and a nonempty, closed, and convex subset of . Let be multivalued nonexpansive mappings of into such that . Let and be sequence in satisfying . Let be a sequence defined iteratively by where , such that and whenever is a fixed point of any one of mappings and . Then, converges strongly to a common fixed point of if and only if .

Remark 3.13. Our results extend the results of Sastry and Babu [9], Panyanak [10], and Song and Wang [11] from approximation of a fixed point of a single multivaued nonexpansive mapping to approximation of common fixed point of a finite family of quasi-nonexpansive multivaued mappings.

Remark 3.14. Our results extend the results of Shahzad and Zegeye [12] from approximation of a fixed point of a single quasi-nonexpansive multivaued mapping and single multivalued map to approximation of common fixed point of a finite family of quasi-nonexpansive multivaued mappings and a finite family of multivalued maps.

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