Journal of Applied MathematicsVolume 2012 (2012), Article ID 720192, 14 pagesdoi:10.1155/2012/720192

Research Article

Implicit and Explicit Iterations with Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces

Jiancai Huang,¹ Shenghua Wang,² and Yeol Je Cho³

¹School of Control and Computer Engineering, North China Electric Power University, Baoding 071003, China²School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China³Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 31 December 2011; Accepted 28 January 2012

Academic Editor: Rudong Chen

Copyright © 2012 Jiancai Huang et al. 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 an implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.

1. Introduction

Let be a nonempty subset of a Banach space and be a mapping. We call nonexpansive if for all . The set of all fixed points of is denoted by , that is, .

One parameter family is said to a semigroup of nonexpansive mappings or nonexpansive semigroup on if the following conditions are satisfied: (1) for all ; (2) for all ; (3)for each , for all ; (4)for each , the mapping from , where denotes the set of all nonnegative reals, into is continuous.

We denote by the set of all common fixed points of semigroup , that is, and by the set of natural numbers.

Now, we recall some recent work on nonexpansive semigroup in literatures. In [1], Shioji and Takahashi introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space: where and . Under the certain conditions on and , they proved that the sequence defined by (1.1) converges strongly to an element in .

In [2], Suzuki introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space: where and . Under the conditions that , he proved that defined by (1.2) converges strongly to an element of . Later on, Xu [3] extended the iteration (1.2) to a uniformly convex Banach space that admits a weakly sequentially continuous duality mapping. Song and Xu [4] also extended the iteration (1.2) to a reflexive and strictly convex Banach space.

In 2007, Chen and He [5] studied the following implicit and explicit viscosity approximation processes for a nonexpansive semigroup in a reflexive Banach space admitting a weakly sequentially continuous duality mapping: where is a contraction, and . They proved the strong convergence for the above iterations under some certain conditions on the control sequences.

Recently, Chen et al. [6] introduced the following implicit and explicit iterations for nonexpansive semigroups in a reflexive Banach space admitting a weakly sequentially continuous duality mapping: where is a contraction, and . They proved that defined by (1.4) and (1.5) converges strongly to an element of , which is the unique solution of the following variation inequality problem:

For more convergence theorems on implicit and explicit iterations for nonexpansive semigroups, refer to [7–13].

In this paper, we introduce an implicit and explicit iterative process by a generalized contraction for a finite family of nonexpansive semigroups in a Banach space. Then we prove the strong convergence for the iterations and our results extend the corresponding ones of Suzuki [2], Xu [3], Chen and He [5], and Chen et al. [6].

2. Preliminaries

Let be a Banach space and the duality space of . We denote the normalized mapping from to by defined by where denotes the generalized duality pairing. For any with and , it is well known that the following inequality holds:

The dual mapping is called weakly sequentially continuous if is single valued, and , where denotes the weak convergence, then weakly star converges to [14–16]. A Banach space is called to satisfy Opial’s condition [17] if for any sequence in , , It is known that if admits a weakly sequentially continuous duality mapping , then is smooth and satisfies Opial’s condition [14].

A function is said to be an -function if , for any , and for every and , there exists such that , for all . This implies that for all .

Let be a mapping. is said to be a -contraction if there exists a -function such that for all with . Obviously, if for all , where , then is a contraction. is called a Meir-Keeler-type mapping if for each , there exists such that for all , if , then .

In this paper, we always assume that is continuous, strictly increasing and , where , is strictly increasing and onto.

The following lemmas will be used in next section.

Lemma 2.1 (see [18]). Let be a metric space and be a mapping. The following assertions are equivalent: (i) is a Meir-Keeler-type mapping;(ii)there exists an -function such that is a -contraction.

Lemma 2.2 (see [19]). Let be a Banach space and be a convex subset of . Let be a nonexpansive mapping and be a -contraction. Then the following assertions hold: (i) is a -contraction on and has a unique fixed point in ;(ii)for each , the mapping is of Meir-Keeler-type and it has a unique fixed point in .

Lemma 2.3 (see [20]). Let be a Banach space and be a convex subset of . Let be a Meir-Keeler-type contraction. Then for each there exists such that, for each with .

Lemma 2.4 (see [21]). Let C be a closed convex subset of a strictly convex Banach space E. Let be a nonexpansive mapping for each , where is some integer. Suppose that is nonempty. Let be a sequence of positive numbers with . Then the mapping defined by is well defined, nonexpansive and holds.

Lemma 2.5 (see [22]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i);(ii);(iii) or .Then .

3. Main Results

In this section, by a generalized contraction mapping we mean a Meir-Keeler-type mapping or - contraction. In the rest of the paper we suppose that from the definition of the -contraction is continuous, strictly increasing and is strictly increasing and onto, where , for all . As a consequence, we have the is a bijection on .

Theorem 3.1. Let be a nonempty closed convex subset of a reflexive Banach space which admits a weakly sequentially continuous duality mapping from into . For every , let be a semigroup of nonexpansive mappings on such that and be a generalized contraction on . Let and be the sequences satisfying and . Let be a sequence generated by Then converges strongly to a point , which is the unique solution to the following variational inequality:

Proof. First, we show that the sequence generated by (3.1) is well defined. For every and , let and define by where . Since is nonexpansive, is nonexpansive. By Lemma 2.2 we see that is a Meir-Keeler-type contraction for each . Hence, each has a unique fixed point, denoted as , which uniquely solves the fixed point equation (3.3). Hence generated by (3.1) is well defined.Now we prove that generated by (3.1) is bounded. For any , we have Using (3.4), we get and hence which implies that Hence This shows that is bounded, and so are , and .Since is reflexivity and is bounded, there exists a subsequence such that for some as . Now we prove that . For any fixed , we have By hypothesis on , , we have Further, from (3.9) we get Since admits a weakly sequentially duality mapping, we see that satisfies Opial’s condition. Thus if , we have This contradicts (3.11). So .In (3.5), replacing with and with , we see that which implies that Now we prove that is relatively sequentially compact. Since is weakly sequentially continuous, we have which implies that If , then is relatively sequentially compact. If , we have . Since is continuous, . By the definition of , we conclude that , which implies that is relatively sequentially compact.Next, we prove that is the solution to (3.2). Indeed, for any , we have Therefore, Since and is weakly sequentially continuous, we have This shows that is the solution of the variational inequality (3.2).Finally, we prove that is the unique solution of the variational inequality (3.2). Assume that with is another solution of (3.2). Then there exists such that . By Lemma 2.3 there exists such that . Since both and are the solution of (3.2), we have Adding the above inequalities, we get which is a contradiction. Therefore, we must have , which implies that is the unique solution of (3.2).In a similar way it can be shown that each cluster point of sequence is equal to . Therefore, the entire sequence converges strongly to . This completes the proof.

If letting for all in Theorem 3.1, then we get the following.

Corollary 3.2. Let be a nonempty closed convex subset of a reflexive Banach space which admits a weakly sequentially continuous duality mapping from into . For every ( ), let be a semigroup of nonexpansive mappings on such that and be a generalized contraction on . Let and be sequences satisfying . Let be a sequence generated by Then converges strongly to a point , which is the unique solution to the following variational inequality:

Theorem 3.3. Let be a nonempty closed convex subset of a reflexive and strictly convex Banach space which admits a weakly sequentially continuous duality mapping from into . For every , let be a semigroup of nonexpansive mappings on such that and be a generalized contraction on . Let and be the sequences satisfying . Let be a sequence generated Then converges strongly to a point , which is the unique solution of variational inequality (3.2).

Proof. Let and . Now we show by induction that It is obvious that (3.25) holds for . Suppose that (3.25) holds for some , where . Observe that Now, by using (3.24) and (3.26), we have By induction we conclude that (3.25) holds for all . Therefore, is bounded and so are , , .For each and , define the mapping , where . Then we rewrite the sequence (3.24) to Obviously, each is nonexpansive. Since is bounded and is reflexive, we may assume that some subsequence of converges weakly to . Next we show that . Put , , and for each . Fix . By (3.28) we have So, for all , we have Since has a weakly sequentially continuous duality mapping satisfying Opials’ condition, this implies . By Lemma 2.4, we have for each . Therefore, . In view of the variational inequality (3.2) and the assumption that duality mapping is weakly sequentially continuous, we conclude that Finally, we prove that as . Suppose that . Then there exists and subsequence of such that for all . Put , and . By Lemma 2.3 one has for all . Now, from (2.2) and (3.28) we have It follows that where is a constant.Let and . It follows from (3.33) that It is easy to see that , and (noting (3.28)) Using Lemma 2.5, we conclude that as . It is a contradiction. Therefore, as . This completes the proof.

If letting for all in Theorem 3.3, then we get the following.

Corollary 3.4. Let be a nonempty closed convex subset of a reflexive and strictly convex Banach space which admits a weakly sequentially continuous duality mapping from into . For every , let be a semigroup of nonexpansive mappings on such that and be a generalized contraction on . Let and be sequences satisfying . Let be a sequence generated Then converges strongly to a point , which is the unique solution of variational inequality (3.2).

Remark 3.5. Theorem 3.1 and Corollary 3.2 extend the corresponding ones of Suzuki [2], Xu [3], and Chen and He [5] from one nonexpansive semigroup to a finite family of nonexpansive semigroups. But Theorem 3.3 and Corollary 3.4 are not the extension of Theorem 3.2 of Chen and He [5] since Banach space in Theorem 3.3 and Corollary 3.4 is required to be strictly convex. But if letting in Theorem 3.3 and Corollary 3.4, we can remove the restriction on strict convexity and hence they extend Theorem 3.2 of Chen and He [5] from a contraction to a generalized contraction.

Remark 3.6. Our Theorem 3.1 extends and improves Theorems 3.2 and 4.2 of Song and Xu [4] from a nonexpansive semigroup to a finite family of nonexpansive semigroups and a contraction to a generalized contraction. Our conditions on the control sequences are different with ones of Song and Xu [4].

Acknowledgment

This work 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: 2011-0021821).

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