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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 945985, 18 pages
http://dx.doi.org/10.1155/2014/945985
Research Article

Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 4 August 2013; Revised 4 December 2013; Accepted 20 December 2013; Published 17 February 2014

Academic Editor: Hichem Ben-El-Mechaiekh

Copyright © 2014 Lu-Chuan Ceng 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

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings in X, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.

1. Introduction

Let be a real Banach space whose dual space is denoted by . Let denote the unit 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 strictly convex. 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 convex subset of a real Banach space . A mapping is said to be -Lipschitzian if there exists a constant such that for all . In particular, if , then is said to be nonexpansive. 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

(i)  accretive if, for each , there exists such that where is the normalized duality mapping of ,

(ii)  -inverse-strongly accretive if, for each , there exists such that for some ,

(iii)  pseudocontractive if, for each , there exists such that

(iv)  -strongly pseudocontractive if, for each , there exists such that for some ,

(v)  -strictly pseudocontractive if, for each , there exists such that for some .

It is worth emphasizing that the definition of the inverse-strongly accretive mapping is based on that of the inverse-strongly monotone mapping, which was studied by so many authors; see, for example, [17].

A Banach space is said to be smooth if the limit exists for all ; in this case, is also said to have a Gateaux differentiable norm. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for ; in this case, is also said to have a uniformly Frechet differentiable norm. The norm of is said to be the Frechet differential if, for each , this limit is attained uniformly for . In the meantime, 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 [8], 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 .

In a real smooth Banach space , we say that an operator is strongly positive (see [9]), if there exists a constant with the property where is the identity mapping.

Proposition CB [see [9, Lemma 2.5]]
Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a continuous pseudocontractive mapping with and let be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient and Lipschitzian constant . Let be a strongly positive linear bounded operator with coefficient . Assume that and . Let be defined by Then, as converges strongly to some fixed point of such that is the unique solution in to the VIP:

On the other hand, Cai and Bu [10] 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 , are two nonlinear mappings, and and are two positive constants. Here the set of solutions of GSVI (13) is denoted by . Very recently, Cai and Bu [10] constructed an iterative algorithm for solving GSVI (13) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and -uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a -uniformly smooth Banach space .

Lemma 1 (see [11]). Let be a -uniformly smooth Banach space. Then, there exists a best smooth constant such that where is the normalized duality mapping from into .

The authors [10] have used the following inequality in a real smooth and uniform convex Banach space .

Proposition 2 (see [12]). Let be a real smooth and uniform convex Banach space and let . Then, there exists a strictly increasing, continuous, and convex function , such that where .

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 3 can be found in [13]. Lemma 4 is an immediate consequence of the subdifferential inequality of the function .

Lemma 3. Let be a sequence of nonnegative real numbers such that where and are sequences of real numbers satisfying the following conditions:(i)  and  ;(ii)either or .Then, .

Lemma 4. In a smooth Banach space , there holds the inequality where is the normalized duality mapping of .

Let be a mean if is a continuous linear functional on satisfying . Then, we know that is a mean on if and only if for every . According to time and circumstances, we use instead of . A mean on is called a Banach limit if and only if for every . We know that, if is a Banach limit, then for every . So, if , , and (resp., ), as , we have

Further, it is well known that there holds the following result.

Lemma 5 (see [14]). Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a bounded sequence of ; let be a mean on and let . Then, if and only if where is the normalized duality mapping of .

Lemma 6 (see [9, Lemma 2.6]). Let be a nonempty closed convex subset of a real Banach space which has uniformly Gateaux differentiable norm. Let be a continuous pseudocontractive mapping with and let be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient and Lipschitzian constant . Let be a -strongly positive linear bounded operator with coefficient . Assume that and that converges strongly to as , where is defined by . Suppose that is bounded and that . Then, .

Lemma 7. Let be a nonempty closed convex subset of a real smooth Banach space . Let be an -strongly accretive and -strictly pseudocontractive with . Then, is nonexpansive and is Lipschitz continuous with constant . Further, for any fixed is contractive with coefficient .

Proof. From the -strictly pseudocontractivity and -strongly accretivity of , we have, for all ,
which implies that Because , we know that is nonexpansive. Also note that Now, take a fixed arbitrarily. Observe that, for all , Because , we know that is contractive with coefficient .

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 8 (see [15]). Let be a nonempty closed 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), for all ;(iii), for all , .

It is well known that, if is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto ; that is, . If is a nonempty closed 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 9. Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be nonlinear mappings. For given , is a solution of GSVI (13) if and only if , where .

Proof. We can rewrite GSVI (13) as which is obviously equivalent to because of Lemma 8. This completes the proof.

In terms of Lemma 9, define the mapping as follows: Then, we observe that which implies that is a fixed point of the mapping . Throughout this paper, the set of fixed points of the mapping is denoted by .

Lemma 10 (see [16]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then, a mapping on defined by for is well-defined, nonexpansive and holds.

Lemma 11 (see [17]). Let be a nonempty closed 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, .

3. GSVI with Hierarchical Fixed Point Problem Constraint for a Nonexpansive Mapping

In this section, we introduce our hybrid implicit viscosity scheme for solving the GSVI (13) with hierarchical fixed point problem constraint for a nonexpansive mapping and show the strong convergence theorem. First, we list several useful and helpful lemmas.

Lemma 12 (see [10, Lemma 2.8]). Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let the mapping be -inverse-strongly accretive. Then, one has for , where . In particular, if (where is the best constant of as in Lemma 1), then is nonexpansive for .

Lemma 13 (see [10, Lemma 2.9]). Let be a nonempty closed convex subset of a real -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 nonexpansive.

Lemma 14 (see [18]). Let be a Banach space, a nonempty closed and convex subset of , and a continuous and strong pseudocontraction. Then, has a unique fixed point in .

Lemma 15 (see [19]). Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and . Then, .

We now state and prove our first result.

Theorem 16. Let be a nonempty closed convex subset of a uniformly convex and -uniformly smooth Banach space such that . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive for . Let be a nonexpansive mapping such that where is the fixed point set of the mapping with for . Let be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient and Lipschitzian constant , let be -strongly accretive and -strictly pseudocontractive with , and let be a -strongly positive linear bounded operator with . Let be defined by where with . Then, as converges strongly to a point , which is the unique solution in to the VIP,

Proof. First, let us show that the net is defined well. As a matter of fact, define the mapping as follows: We may assume, without loss of generality, that . Utilizing Lemmas 7, 13, and 15, we have Hence, it is known that is a continuous and strongly pseudocontractive mapping with pseudocontractive coefficient Thus, by Lemma 14, we deduce that there exists a unique fixed point in , denoted by , which uniquely solves the fixed point equation
Let us show the uniqueness of the solution of VIP (36). Suppose that both and are solutions to VIP (36). Then, we have Adding up the above two inequalities, we obtain Note that Consequently, we have , and the uniqueness is proved.
Next, let us show that, for some , is bounded. Indeed, since with , there exists some such that for all . Take a fixed arbitrarily. Utilizing Lemma 7, we have and, hence, for all , Thus, this implies that is bounded and so are , , and .
Let us show that as .
Indeed, for simplicity, we put , , , and . Then, it is clear that and . Hence, from (43), it follows that From Lemma 12, we have From the last two inequalities, we obtain which together with (45) implies that So, it immediately follows that Since , for , we have Utilizing Proposition 2 and Lemma 8, we have that there exists such that which implies that In the same way, we derive that there exists : which implies that Substituting (52) for (54), we get which together with (45) implies that So, it immediately follows that Hence, from (50), we conclude that Utilizing the properties of and , we get which leads to That is,
Note that is bounded and so are , , and . Hence, we have as . Also, observe that This together with (61) and (62) implies that Utilizing the nonexpansivity of , we obtain which together with (62) and (64) implies that Now, let be a sequence in that converges to as , and define a function on by where is a Banach limit. Define the set and the mapping where is a constant in . Then, by Lemma 10, we know that . We observe that So, from (64) and (66), we obtain Since is a uniformly smooth Banach space, is a nonempty bounded closed convex subset of ; for more details, see [14]. We claim that is also invariant under the nonexpansive mapping . Indeed, noticing (71), we have, for , Since every nonempty closed bounded convex subset of a uniformly smooth Banach space has the fixed point property for nonexpansive mappings and is a nonexpansive mapping of , has a fixed point in , say . Utilizing Lemma 5, we get Putting , we have Since , we get It follows that Since , from (74) and the boundedness of sequences , , it follows that Therefore, for the sequence in , there exists a subsequence which is still denoted by that converges strongly to some fixed point of .
Now, we claim that such a is the unique solution in to the VIP (36).
Indeed, from (35), it follows that for all which hence implies that Since as and , we obtain from the last inequality that Utilizing the well-known Minty-type Lemma, we get So, is a solution in to the VIP (36).
In order to prove that the net converges strongly to as , suppose that there exists another subsequence such that as ; then we also have due to (71). Repeating the same argument as above, we know that is another solution in to the VIP (36). In terms of the uniqueness of solutions in to the VIP (36), we immediately get . This completes the proof.

Remark 17. It is worth emphasizing that, in the assertion of Theorem 16, “as converges strongly to a point ,” this depends on no one of the mappings , , and . Indeed, although is defined by in the proof of Theorem 16, it can be readily seen that is first found out as a fixed point of the nonexpansive self-mapping of . This shows that depends on no one of the mappings , , and .

Remark 18. Theorem 16 improves, extends, supplements, and develops Cai and Bu [9, Lemma 2.5] in the following aspects.

(i)  The GSVI (13) with hierarchical fixed point problem constraint for a nonexpansive mapping is more general and more subtle than the problem in Cai and Bu [9, Lemma 2.5] because our problem is to find a point , which is the unique solution in to the VIP:

(ii)  The iterative scheme in [9, Lemma 2.5] is extended to develop the iterative scheme in Theorem 16 by virtue of hybrid steepest-descent method. The iterative scheme in Theorem 16 is more advantageous and more flexible than the iterative scheme of [9, Lemma 2.5] because our iterative scheme involves solving two problems: the GSVI (13) and the fixed point problem of a nonexpansive mapping