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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 761864, 21 pages
http://dx.doi.org/10.1155/2013/761864
Research Article

Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 30 April 2013; Accepted 6 June 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Lu-Chuan Ceng and Ching-Feng Wen. 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 composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

1. Introduction

Let be a real Banach space whose dual space is denoted by . The normalized duality mapping is defined by where denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that is nonempty for each . Let be a nonempty, closed, and convex subset of . A mapping is called nonexpansive if for every . The set of fixed points of is denoted by . We use the notation to indicate the weak convergence and the one to indicate the strong convergence. A mapping is said to be accretive if for each , there exists such that It is said to be -strongly accretive if for each , there exists such that for some . The mapping is called -inverse strongly-accretive if for each , there exists such that for some and is said to be -strictly pseudocontractive if for each , there exists such that for some .

Let denote the unite sphere of . A Banach space is said to be uniformly convex if for each , there exists such that for all , It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space is said to be smooth if the limit exists for all ; in this case, is also said to have a Gâteaux differentiable norm. is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for . Moreover, it is said to be uniformly smooth if this limit is attained uniformly for . The norm of is said to be the Fréchet differential if for each , this limit is attained uniformly for . In addition, we define a function called the modulus of smoothness of as follows: It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all . As pointed out in [1], no Banach space is -uniformly smooth for . In addition, it is also known that is single-valued if and only if is smooth, whereas if is uniformly smooth, then the mapping is norm-to-norm uniformly continuous on bounded subsets of . If has a uniformly Gâteaux differentiable norm, then the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of .

Very recently, Cai and Bu [2] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space , which involves finding such that where is a nonempty, closed, and convex subset of , , and are two nonlinear mappings, and and are two positive constants. Here the set of solutions of GSVI (9) is denoted by GSVI. In particular, if , a real Hilbert space, then GSVI (9) reduces to the following GSVI of finding such that which and are two positive constants. The set of solutions of problem (10) is still denoted by GSVI. It is clear that the problem (10) covers as special case the classical variational inequality problem (VIP) of finding such that The solution set of the VIP (11) is denoted by .

Recently, Ceng et al. [3] transformed problem (10) into a fixed point problem in the following way.

Lemma 1 (see [3]). For given is a solution of problem (10) if and only if is a fixed point of the mapping defined by where and is the the projection of onto .

In particular, if the mappings is -inverse strongly monotone for , then the mapping is nonexpansive provided for .

Let be a nonempty, closed, and convex subset of a real smooth Banach space . Let be a sunny nonexpansive retraction from onto , and let be a contraction with coefficient . In this paper we introduce composite implicit and explicit iterative algorithms for solving GSVI (9) and the common fixed point problem of an infinite family of nonexpansive mappings of into itself. These composite iterative algorithms are based on Korpelevich’s extragradient method [4] and viscosity approximation method [5]. Let the mapping be defined by We first propose a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space : where is -inverse-strongly accretive with for and , , , and are the sequences in such that for all . It is proven that under appropriate conditions, converges strongly to , which solves the following VIP: On the other hand, we also propose another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gateaux differentiable norm: where is -strictly pseudocontractive and -strongly accretive with for and , , , and are the sequences in such that for all . It is proven that under mild conditions, also converges strongly to , which solves the VIP (15). The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 2 can be found in [6]. Lemma 3 is an immediate consequence of the subdifferential inequality of the function .

Lemma 2. Let be a sequence of nonnegative real numbers satisfying where , , and satisfy the conditions:(i) and ,(ii),(iii), , and .Then .

Lemma 3. In a smooth Banach space , there holds the inequality

Lemma 4 (see [7]). Let and be bounded sequences in a Banach space , and let be a sequence in which satisfies the following condition: Suppose that , , and . Then .

Let be a subset of , and let be a mapping of into . Then is said to be sunny if whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for every where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . The following lemma concerns the sunny nonexpansive retraction.

Lemma 5 (see [8]). Let be a nonempty, closed, and convex subset of a real smooth Banach space . Let be a nonempty subset of . Let be a retraction of onto . Then the following are equivalent:(i) is sunny and nonexpansive;(ii), ;(iii), , .

It is well known that if a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto ; that is, . If is a nonempty, closed, and convex subset of a strictly convex and uniformly smooth Banach space and if is a nonexpansive mapping with the fixed point set , then the set is a sunny nonexpansive retract of .

Lemma 6 (see [9]). Given a number . A real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function , , such that for all and such that and .

Lemma 7 (see [10]). Let be a nonempty, closed, and convex subset of a Banach space . Let , be a sequence of mappings of into itself. Suppose that . Then for each , converges strongly to some point of . Moreover, let be a mapping of into itself defined by for all . Then .

Let be a nonempty, closed, and convex subset of a Banach space , and let be a nonexpansive mapping with . As previous, let be the set of all contractions on . For and , let be the unique fixed point of the contraction on ; that is,

Lemma 8 (see [11, 12]). Let be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let be a nonempty, closed, and convex subset of , let be a nonexpansive mapping with , and . Then the net defined by converges strongly to a point in . If we define a mapping by , , then solves the VIP:

Lemma 9 (see [13]). Let be a nonempty, closed, and convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose that is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for is defined well, nonexpansive, and holds.

3. Implicit Iterative Schemes

In this section, we introduce our implicit iterative schemes and show the strong convergence theorems. We will use the following useful lemmas in the sequel.

Lemma 10 (see [2, Lemma 2.8]). Let be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space . Let the mapping be -inverse-strongly accretive. Then, one has for , where . In particular, if , then is nonexpansive for .

Lemma 11 (see [2, Lemma 2.9]). Let be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive for . Let be the mapping defined by If   for , then is nonexpensive.

Lemma 12 (see [2, Lemma 2.10]). Let be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be two nonlinear mappings. For given is a solution of GSVI (9) if and only if where .

Remark 13. By Lemma 12, we observe that which implies that is a fixed point of the mapping .

We now state and prove our first result on the implicit iterative scheme.

Theorem 14. Let be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive for . Let be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping . For arbitrarily given , let be the sequence generated by where for and , , , and are the sequences in such that , . Suppose that the following conditions hold:(i) and ,(ii) and ,(iii).Assume that for any bounded subset of , and let be a mapping of into itself defined by for all . Suppose that . Then converges strongly to , which solves the following VIP:

Proof. Take a fixed arbitrarily. Then by Lemma 12, we know that and for all . Moreover, by Lemma 11, we have which hence implies that Thus, from (27), we have It immediately follows that is bounded, and so are the sequences , , and due to (30) and the nonexpansivity of .
Let us show that as . As a matter of fact, from (27), we have Simple calculations show that It follows that which hence yields Now, we write , , where . It follows that for all , This together with (35) implies that where for some . So, from , condition (iii), and the assumption on , it immediately follows that In terms of condition (ii) and Lemma 4, we get Hence we obtain
Next we show that as .
For simplicity, put , , and . Then . From Lemma 10, we have Substituting (41) into (42), we obtain According to Lemma 3, we have from (27) which hence yields This together with (43) and the convexity of , we have where for some . So, it follows that Since for , from conditions (i), (ii), and (40), we obtain Utilizing [14, Proposition 1] and Lemma 5, we have which implies that In the same way, we derive which implies that Substituting (50) into (52), we get From (46) and (53), we have which implies that Utilizing conditions (i), (ii), from (40) and (48), we have Utilizing the properties of and , we deduce that From (57), we obtain That is, On the other hand, since and are bounded, by Lemma 6, there exists a continuous strictly increasing function , such that for which together with (30) implies that It immediately follows that According to condition (ii), we get Since , , and , we conclude that Utilizing the property of , we have We note that So, That is, We observe that Thus, from (59)–(68), we obtain that By (70) and Lemma 7, we have In terms of (59) and (71), we have Define a mapping , where is a constant. Then by Lemma 9, we have that . We observe that From (59) and (72), we obtain
Now, we claim that where with being the fixed point of the contraction Then solves the fixed point equation . Thus we have By Lemma 3, we conclude that