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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 264910, 9 pages
http://dx.doi.org/10.1155/2013/264910
Research Article

Implicit Iterative Scheme for a Countable Family of Nonexpansive Mappings in 2-Uniformly Smooth Banach Spaces

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 1 January 2013; Accepted 22 March 2013

Academic Editor: Qamrul Hasan Ansari

Copyright © 2013 Ming Tian and Xin Jin. 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

Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operator and get strong convergence under some mild assumptions. Our results improve and extend the corresponding conclusions announced by many others.

1. Introduction

Let be a real -uniformly smooth Banach space with induced norm , . Let be the dual space of . Let denote the generalized duality mapping from into given by . In our paper, we consider real 2-uniformly smooth Banach spaces, that is, , so the normalized duality mapping is . If is smooth, then is single valued. Throughout this paper, we use to denote the fixed points set of the mapping .

In what follows, we write to indicate that the sequence converges weakly to . implies that the sequence converges strongly to .

Given a nonlinear operator , it is well-known that the generalized variational inequality problem over is to find a , such that Scholars mainly proposed iterative algorithms to solve the generalized variational inequalities, and some of them focused on the existence of the solutions of generalized variational inequalities; see [1, 2] and references therein.

Variational inequalities are developed from operator equations and have been playing an essential role in management science, mechanics, and finance. As for mathematics, variational inequality problems mainly originate from partial differential equations, optimization problems; see [37] and references therein.

Definition 1. A mapping is said to be(1)-strongly accretive if for each , there exists a and , such that (2)-Lipschitzian continuous if for each , there exists a constant , such that
In particular, is called nonexpansive if ; it is said to be contractive if .
Yamada [3] introduced the hybrid steepest descent method: where is a nonexpansive mapping in Hilbert spaces. Under some appropriate conditions, Yamada [3] proved that the sequence generated by (4) converges strongly to the unique solution of the variational inequality: , for all .
Moudafi [4] introduced the classical viscosity approximation method for nonexpansive mappings and defined a sequence by where is a sequence in . Xu [6] proved that under certain appropriate conditions on , the sequence generated by (5) converges strongly to the unique solution of the variational inequality: , for all , (where ) in Hilbert spaces as well as in some Banach spaces.
Marino and Xu [7] considered the following general iterative method in Hilbert spaces: where is a strongly positive bounded linear operator. It is proved that if the sequence satisfies appropriate conditions, the sequence generated by (6) converges strongly to the unique solution of the variational inequality: , for all , where is the fixed points set of a nonexpansive mapping .
Tian [8] considered the following general iterative algorithm (GIA) in Hilbert spaces: It is proved that if the sequence satisfies appropriate conditions, the sequence generated by (7) converges strongly to the unique solution of the variational inequality: , for all .
In 2001, Soltuz [9] introduced the following backward Mann scheme iteration: where is a nonexpansive mapping and got strong convergence in Hilbert spaces.
In order to find a common fixed point of a finite family of nonexpansive mappings , where stands for , in 2001, Xu and Ori [10] introduced the following implicit process: where , and a weak convergence is obtained in real Hilbert spaces.
Ceng et al. [11] introduced an iterative algorithm to find a common fixed point of a finite family of nonexpansive semigroups in reflexive Banach spaces with a weak sequentially continuous duality mapping, which satisfy the uniformly asymptotical regularity condition: Under some appropriate conditions one the parameter sequences , , , and the sequence generated by (10) converges strongly to the approximate solution of a variational inequality problem.
In order to find a common element of the solution set of a general system of variational inequalities and the fixed-point set of the mapping , Ceng et al. [12] constructed a new relaxed extragradient iterative method: Under mild assumptions, they obtained a strong convergence theorem.
Yao et al. [13] introduced the following Halpern-type implicit iterative method where is a continuous pseudocontraction: and obtained a strong convergence theorem in Banach spaces.
Hu [14] introduced an iteration for a nonexpansive mapping in Banach spaces, which guarantee a uniformly Gêteaux differentiable norm as follows: and several strong convergent theorems are obtained.
Very recently, Jung [15] proposed an iterative process in the frame of Hilbert spaces as follows: where is a mapping defined by and is a -strictly pseudocontraction. Strong convergence theorems are established.
Motivated and inspired by Soltuz [9], Xu and Ori [10], Ceng et al. [11], Ceng et al. [12], Yao et al. [13], Hu [14], and Jung [15], we consider the following new implicit iteration in real 2-uniformly smooth Banach spaces: where and are real sequences in , is an -Lipschitzian continuous with Lipschitzian constant , is an -strongly accretive and -Lipschitzian continuous mapping with and , and is a countable family of nonexpansive mappings.
In this paper, we prove that the implicit iterative process (15) has strong convergence and find the unique solution of variational inequality: Our results improve and extend the corresponding conclusions announced by many others.

2. Preliminaries

Let . Then the norm of is said to be Gâteaux differentiable if exists for each . In this case, is said to be smooth. The norm of is called uniformly Gâteaux differentiable, if for each , is attained uniformly for . The norm of is called Fréchet differentiable, if for each , is attained uniformly for . The norm of is called uniformly Fréchet differentiable, if is attained uniformly for . It is well known that (uniformly) Fréchet differentiability of the norm of implies (uniformly) Gâteaux differentiability of the norm of . If the norm on is uniformly Gâteaux differentiable, the generalized duality mapping is single-valued and uniformly continuous on any bounded subsets of .

Let be the modulus of smoothness of defined by A Banach space is said to be uniformly smooth if as . A Banach space is said to be -uniformly smooth, if there exists a fixed constant , such that . It is well known that the is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable [16].

The so-called gauge function is defined as follows: let be a continuous strictly increasing function, such that and as . The duality mapping associated with a gauge function is defined by It is known that real 2-uniformly smooth Banach spaces have a weakly continuous duality mapping with a guage function , which is the same as the normalized duality mapping . Set , for all , then , where denotes the subdifferential in the sense of convex analysis. In fact, for , we have and

Lemma 2 (see [17]). Let be a real -uniformly smooth Banach space for some , then there exists some positive constant , such that in particular, if is a real 2-uniformly smooth Banach space, then there exists a best smooth constant , such that

Lemma 3 (see [18]). Assume that a Banach space has a weakly continuous duality mapping :(i) for all , the following inequality holds:   in particular, for all , there holds: (ii) assume that a sequence converges weakly to a point , then the following equation holds:

Lemma 4 (see [19]). Assume that is a sequence of nonnegative real numbers, such that where is a sequence in and is a sequence in , such that (a); (b) or .
Then .

Lemma 5. Let be a real 2-uniformly smooth Banach space. Let be a nonexpansive mapping over , and let be an -strongly accretive and -Lipschitzian continuous mapping with and . For and , set , and define a mapping by . Then is a contraction on ; that is, .

Proof. From , we have . Setting , we have . For each , by Lemma 2, we have
Hence, it implies that This completes the proof.

To deal with a family of mappings, we will introduce the following concept called the condition.

Definition 6 (see [20]). Let be a real Banach space, let be a nonempty subset of , and let be a countable family of mappings of with . Then is said to satisfy the condition, if for any bounded subset of , the following inequality holds:

Lemma 7 (see [20]). Let be a Banach space, let be a nonempty closed subset of , and let be a family of self-mappings of satisfying the AKTT condition. Then for each , converges strongly to a point in . Moreover, let the mapping be defined by Then for any bounded subset of , the following equality holds:

Lemma 8 (see [1]). Suppose that . Then the following inequality holds: for arbitrary positive real numbers and .

3. Main Results

In order to obtain the main results, we divide this section into 3 parts. In Proposition 9, we give the path convergence. In Proposition 10, under the demiclosed assumption and combined with Proposition 9, we find the unique solution of a variational inequality. In Theorem 11, we prove that the sequence defined by the implicit scheme (15) converges strongly to the unique solution of (16).

Throughout this paper, we assume that is a real -uniformly smooth Banach space, which guarantees a weakly continuous duality as proposed in Section 1.

Proposition 9 (the path convergence). Let be a nonexpansive mapping with , and let be an -Lipschitzian continuous mapping with Lipschitzian constant . is an -strongly accretive and -Lipschitzian continuous mapping with and . For , let , and set and . Then assume that is defined by Then converges strongly as to a fixed point of  , which is the unique solution of the variational inequality VIP:

Proof. Consider a mapping on defined by It is easy to see that is a contraction. Indeed, for any , by Lemma 5, we have Hence, by the Banach contraction mapping principle, has a unique fixed point, denoted by , which uniquely solves the fixed point equation (33).
We divided the proof into several steps.
Step 1. We show the uniqueness of the solution of the variational inequality (34). Assume that both and are solutions of the variational inequality (34), then we have Adding up (37) yields Indeed, from the given conditions , and , , we have
Thus, we conclude that . So the uniqueness of the variational inequality (35) is guaranteed.
Step 2. We show that is bounded. Taking , it follows from Lemma 5 that It follows that Hence is bounded, so are and .
Step 3. Next, we will show that has a subsequence converging strongly to .
Assume , and set . By the definition of , we have
Since is bounded, there exists a subsequence of converging weakly to as .
Set . Define a mapping by Again, is weakly continuous, by Lemma 3, and it follows that From (42), we have and we also note that so, we obtain This implies that ; that is, .
By Lemma 5, we have This implies that Since is bounded, there exists a subsequence of satisfying Since the mapping is single-valued and weakly continuous, it follows from (50) that as . Thus, there exists a subsequence, such that .
Step 4. Finally, we show that is the unique solution of variational inequality (34).
Since , we can derive that
It follows that, for any ,
Since is a nonexpansive mapping, for all , we conclude that
Now replacing in (52) with and letting , from (42), we have that , thus we can conclude that So, is a solution of (34). Hence, by uniqueness. Therefore, as . This completes the proof.

Proposition 10 (the demiclosed result). Let be a nonexpansive mapping with , and let be an -Lipschitzian continuous self-mapping on with Lipschitzian constant . is an -strongly accretive and -Lipschitzian continuous mapping with and . Assume that the net is defined as Proposition 9 which converges strongly as to . Suppose that the sequence is bounded and satisfies the condition (the so-called demiclosed property). Then the following inequality VIP holds:

Proof. Set . From the given condition and the boundness of and , it is obvious that when .
From (33) and the fact that is a nonexpansive mapping, we obtain that which implies that It follows that Taking the as and recalling (42) and the continuity of , we conclude that
On the other hand, since is a real -uniformly smooth Banach space, and is single-valued and strong-weak* uniformly continuous on , as , we have Thus, from (59) and (60), we obtain So (55) is valid. This completes the proof.

Finally, we study the following implicit iterative method process: the initial is arbitrarily selected, and the iterative algorithm is recursively defined by where the sequences and are sequences in and satisfy the following conditions: (C1), (C2).

Theorem 11. Let be a countable family of self-nonexpansive mappings on , such that . Let be an -Lipschitzian continuous self-mapping on with Lipschitzian constant . is an -strongly accretive and -Lipschitzian continuous mapping with and . Suppose that the sequences and satisfy the controlling conditions (C1)-(C2). Let , and set and . Assume that satisfies the AKTT condition. Then defined by (62) converges strongly to a common fixed point of   which equivalently solves the following variational inequality:

Proof. First we show that is well defined. Consider a mapping on defined by It is easy to see that is a contraction. Indeed, for any , by Lemma 5, we have
Hence, is a contraction. By the Banach contraction mapping principle, we conclude that has a unique fixed point, denoted by . So (62) is well defined.
Then we show that is bounded. Taking any , we have which implies that By induction, it follows that Hence is bounded, so are the and .
Next, we show that From (C1) and the definition of , we observe that By Lemma 7 and (70), we have
Let be defined by (33), from Propositions 9 and 10, and we have that converges strongly to and
As required, finally we show that . As a matter of fact, by Lemmas 5 and 8, we have which implies that It is easily to see that Thus, (C2) yields that . Applying Lemma 4 and (72) to (74), we conclude that .
This completes the proof.

Remark 12. Our result in Proposition 9 extends Theorem 3.1 of Tian [8] from real Hilbert spaces to real -uniformly smooth Banach spaces. If we set , our result in Theorem 11 extends Theorem 3.2 of Tian [8] from real Hilbert spaces to real 2-uniformly smooth Banach spaces as well as from a single nonexpansive mapping to a countable family of nonexpansive mappings.

Remark 13. In 2008, Hu [14] introduced a modified Halpern-type iteration for a single nonexpansive mapping in Banach spaces which have a uniformly Gêteaux differentiable norm as follows: Under some appropriate assumptions, he proved that the sequence defined by the iteration process (76) converges strongly to the fixed point of .

Corollary 14. If we take , , , and in (62), we extend the classical viscosity approximation [4] under a mild assumption: the contraction mapping is replaced by an -Lipschitzian continuous mapping . Our proving process needs no Banach limit and is different from the proving process given by Xu [6] in some aspects.

Remark 15. Ceng et al. [11] introduced the following iterative algorithm to find a common fixed point of a finite family of nonexpansive semigroups in reflexive Banach spaces: Under some appropriate conditions one the parameter sequences , , , and the sequence converges strongly to the approximate solution of a variational inequality problem.
If we set and , the algorithm is simplified into viscosity-form iterative schemes for a finite family of nonexpansive semigroups. Our algorithms are considered in full space and avoid the generalized projections or sunny nonexpansive retractions in Banach space. For further improving our works, in order to obtain more general results, we should take the results given by Ceng et al. in [11] into account.

Acknowledgments

Ming Tian was supported in part by the Fundamental Research Funds for the Central Universities (no. ZXH2012K001). Xin Jin was supported in part by Technology Innovation Funds of Civil Aviation University of China for Graduate (YJSCX12-18).

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