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Journal of Applied Mathematics
Volume 2012, Article ID 235120, 11 pages
http://dx.doi.org/10.1155/2012/235120
Review Article

Three-Step Fixed Point Iteration for Generalized Multivalued Mapping in Banach Spaces

Department of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou 404000, China

Received 19 September 2011; Revised 2 December 2011; Accepted 8 December 2011

Academic Editor: Nazim I. Mahmudov

Copyright © 2012 Zhanfei Zuo. 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

The convergence of three-step fixed point iterative processes for generalized multivalued nonexpansive mapping was considered in this paper. Under some different conditions, the sequences of three-step fixed point iterates strongly or weakly converge to a fixed point of the generalized multivalued nonexpansive mapping. Our results extend and improve some recent results.

1. Introduction

Let be a Banach space and a nonempty subset of . The set is called proximinal if for each , there exists an element such that , where . Let denote the family of nonempty closed bounded subsets, nonempty compact subsets, nonempty proximinal bounded subsets of , and the set of fixed points, respectively. A multivalued mapping is said to be nonexpansive (quasi-nonexpansive) if where denotes the Hausdorff metric on defined by A point is called a fixed point of if . Since Banach's Contraction Mapping Principle was extended nicely to multivalued mappings by Nadler in 1969 (see [1]), many authors have studied the fixed point theory for multivalued mappings (e.g., see [2]). For single-valued nonexpansive mappings, Mann [3] and Ishikawa [4], respectively, introduced a new iteration procedure for approximating its fixed point in a Banach space as follows: where and are sequences in . Obviously, Mann iteration is a special case of Ishikawa iteration. Recently Song and Wang in [5, 6] introduce the following algorithms for multivalued nonexpansive mapping: where such that and , where and for and . They show some strong convergence results of the above iterates for multivalued nonexpansive mapping under some appropriate conditions. However, the iteration scheme constructed by Song and Wang involves the following estimates, which are not easy to be computed and the scheme is more time consuming. It is observed that Song and Wang [6] did not use the above estimates in their proofs and the assumption on , namely, for any is quite strong. It is noted that the domain of is compact, which is a strong condition. The aim of this paper is to construct an three iteration scheme for a generalized multivalued mappings, which removes the restriction of , namely, for any and also relax compactness of the domain of . The generalized multivalued mappings was introduced in [7], if where is induced by the norm. Obviously, the condition is weaker than nonexpansiveness and stronger than quasinonexpansiveness, furthermore, there are some examples of a generalized nonexpansive multivalued mapping which is not a nonexpansive multivalued mapping (see [7, 8]).

Let be a generalized nonexpansive multivalued mapping and . The three-step mean multivalued iterative scheme is defined by , where , and are appropriate sequence in , furthermore . If or , then iterative scheme (1.8) reduces to the Ishikawa and Mann multivalued iterative scheme. In fact let or or , we also have the other three algorithms.

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. The following definition was introduced in [9].

Definition 1.1. A multivalued mapping is said to satisfy Condition (A) if there is a nondecreasing function with for such that where   is the fixed point set of the multivalued mapping . From now on, stands for the fixed point set of the multivalued mapping  .

2. Preliminaries

A Banach space is said to be satisfy Opial's condition [10] if, for any sequence in , implies the following inequality: for all with . It is known that Hilbert spaces and have the Opial's condition.

Lemma 2.1 (see [7, 11]). Let , and be sequence in uniformly convex Banach space . Suppose that , and are sequence in with , and . If and  , then .

Lemma 2.2 (see [7, 11]). Let be a uniformly convex Banach space and . Then there exists a continuous strictly increasing convex function with such that for all and with .

3. Main Results

Lemma 3.1. Let be a real Banach space and be a nonempty convex subset of be a generalized multivalued nonexpansive mapping with such that is nonexpansive. Let be a sequence in defined by (1.8), then one has the following conclusion:

Proof. Let , then . Since is quasi-nonexpansive, thus we obtain similarly , then we have Then is a decreasing sequence and hence exists for any .

Lemma 3.2. Let be a uniformly convex Banach space and be a nonempty convex subset of be a generalized multivalued nonexpansive mapping with such that is nonexpansive. Let be a sequence in defined by (1.8), if the coefficient satisfy one of the following control conditions:(i) and one of the following holds: (a) and ,(b) and ,(c) and ,(d); (ii) and ;(iii); (iv) and ;then we have .

Proof. By Lemma 3.1, we know that exists for any , then it follows that , and are all bounded. We may assume that these sequences belong to where . Note that for any fixed point and is quasi-nonexpansive. By Lemma 2.2, we get and therefore we have Then Since exists for any , it follows from (3.6) that . From is continuous strictly increasing with and , then Using a similarly method together with inequalities (3.7) and , then Similarly, from (3.8) and , we have , since , then , thus we get (iii). In the sequence we prove (i) (a). From iterative scheme (1.8), we have To show that , it suffices to show that there exist a subsequence of such that  . If  , it follows from (3.9) that Since exists for any , we have From is continuous strictly increasing with and , we have This together with (3.10), (3.12), (3.15) gives Since , we have . On the other hand, if , then we may extract a subsequence of so that . This together with (i) (a) and (3.10), (3.12) gives By Double Extract Subsequence Principle, we obtain the result.
If and , we will prove (ii), Since , then This together with (3.11), (3.18), we obtain the result.
We will prove (i) (b), let . By Lemma 3.1, we let for some . From iterative scheme (1.8), we know From Lemma 3.1, we have known that and , then From (3.20) and Lemma 2.1, we have Notice that Since , we have , therefore .
We will prove (i) (c). From iterative scheme (1.8) and Lemma 3.1, we have which implies Notice that and exists. Hence from (3.25) we have Therefore, from iterative scheme (1.8) we have From Lemma 2.1, we have Notice that Since , then .
By (3.27) and Lemma 2.1, we can similarly prove (i) (d).
Finally, we will prove (iv). From iterative scheme (1.8) and Lemma 3.1, we have which implies Notice that Hence we have Thus, we have By Lemma 2.1  and , we have  .

Theorem 3.3. Let be a uniformly convex Banach space and be a nonempty convex subset of , be a generalized multivalued nonexpansive mapping with such that is nonexpansive. Let be a sequence in defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and satisfies Condition (A) with respect to the sequence , then converges strongly to a fixed point of .

Proof. By Lemma 3.2, we have . Since satisfies Condition (A) with respect to . Then Thus, we get . The remainder of the proof is the same as in [6, Theorem 2.4], we omit it.

Theorem 3.4. Let be a uniformly convex Banach space and be a nonempty convex subset of , be a generalized multivalued nonexpansive mapping with such that is nonexpansive. Let be a sequence in defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and is hemicompact, then converges strongly to a fixed point of .

Proof. By Lemma 3.2, we have . Since is hemicompact, then there exist a subsequence of such that for some . Thus, Hence, is a fixed point of . Now on take on in place of , we get that exists. It follows that as . This completes the proof.

Theorem 3.5. Let and be the same as in Lemma 3.2. If be a nonempty weakly compact convex subset of a Banach space and satisfies Opial's condition, then converges weakly to a fixed point of .

Proof. The proof of the Theorem is the same as in [6, Theorem 2.5], we omit it.

Remark 3.6. From the definition of iterative scheme (1.8), Theorems 3.3, 3.4, and 3.5 extend some results in [6, 12], and also give some new results are different from the [5]. In fact, we can present an example of a multivalued map for which is nonexpansive. A multivalued map is -nonexpansive [13] if for all and with , there exists with such that It is clear that if is -nonexpansive, then is nonexpansive. It is known that -nonexpansiveness is different from nonexpansiveness for multivalued maps. Let and be defined by for [14]. Then for and thus it is nonexpansive. Note that is -nonexpansive but not nonexpansive (see [14]).

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