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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 852760, 21 pages
http://dx.doi.org/10.1155/2013/852760
Research Article

Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros of -Accretive Operators in Banach Spaces

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 1 July 2013; Accepted 28 July 2013

Academic Editor: Chi-Ming Chen

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

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of an -accretive operator in the setting of real uniformly convex and 2-uniformly smooth Banach space . Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the set of zeros of in . 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 literature.

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 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 to indicate the strong convergence. A mapping is said to be

(i) accretive if for each there exists such that

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

(iii) -inverse strongly accretive if for each there exists such that for some;

(iv) -strictly pseudocontractive [1] (see also [2]) 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, [35]. 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 strictly convex. 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 [6], 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 .

Let be an operator with domain and range in is said to be accretive if for each and there exists such that An accretive operator is said to be -accretive if for all . Denote by the resolvent of ; that is, for each , It is known that is a nonexpansive mapping from to which will be assumed convex. In 2008, Chen and Zhu [7] derived the following strong convergence theorems for viscosity approximation methods for accretive operators in a uniformly smooth Banach space .

Theorem  CZ1. Let be a uniformly smooth Banach space. Suppose that is an -accretive operator in such that is convex and is a fixed contractive map. For each , is defined byThen as , converges strongly to a zero of .

Theorem  CZ2. Let be a uniformly smooth Banach space. Suppose that is an -accretive operator in such that is convex and is a fixed contractive map. The sequence is defined bywhere and satisfy the following conditions: (i),, and ,(ii) for all and .

Then as , converges strongly to a zero of .

In the meantime, Ceng et al. [8] derived some strong convergence theorems of composite iterative schemes for zeros of -accretive operators in uniformly smooth Banach spaces. Furthermore, motivated by strong convergence results for hybrid steepest-descent methods in [9, 10], Ceng et al. [11] established some strong convergence theorems for hybrid steepest-descent methods for nonexpansive and -accretive operators in a uniformly smooth Banach space . Subsequently, Ceng et al. [12] introduced hybrid viscosity approximation method for finding zeros of -accretive operators, which combine viscosity approximation method with hybrid steepest-descent method, and obtained the following strong convergence theorems.

Theorem  CASY1. Let be a uniformly smooth Banach space, let be an -accretive operator in with , and let be a contractive map. Assume that is -strongly accretive and -strictly pseudocontractive with . For each and each integer , let be defined bywhere for some and with . Then as converges strongly to a zero of , which is a unique solution of the variational inequality problem (VIP)

Theorem  CASY2. Let be a uniformly smooth Banach space, let be an -accretive operator in with , and let be a contractive map. Assume that is -strongly accretive and -strictly pseudocontractive with . Given sequences in in , and in for some , suppose that there hold the following conditions: (i) and ,(ii),(iii) for some ,(iv), , , and .

Then for any given point , the sequence generated byconverges strongly to a zero of , which is a unique solution of the VIP as above.

On the other hand, Cai and Bu [13] 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 (16) is denoted by GSVI . In particular, if , a real Hilbert space, then GSVI (16) reduces to the following GSVI of finding such that where and are two positive constants. The set of solutions of problem (17) is still denoted by GSVI . In particular, if , then problem (17) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [14]. Further, if additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding such that The solution set of the VIP (18) is denoted by VI . Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [15]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the condition that the involved operator must be strongly monotone and Lipschitz continuous.

Recently, Ceng et al. [16] transformed problem (17) into a fixed point problem in the following way.

Lemma 1 (see [16]). For given is a solution of problem (17) 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 mapping is -inverse strongly monotone for , then the mapping is nonexpansive provided for .

In 1976, Korpelevič [17] proposed an iterative algorithm for solving the VIP (18) in Euclidean space : with being a given number, which is known as the extragradient method (see also [18]). The literature on the VIP is vast, and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 13, 1932] and references therein, to name but a few.

In particular, whenever is still a real smooth Banach space, , and , then GSVI (16) reduces to the variational inequality problem (VIP) of finding such that which was considered by Aoyama et al. [33]. Note that VIP (21) is connected with the fixed point problem for nonlinear mapping (see, e.g., [34]), the problem of finding a zero point of a nonlinear operator (see, e.g., [35]), and so on. It is clear that VIP (21) extends VIP (18) from Hilbert spaces to Banach spaces.

In order to find a solution of VIP (21), Aoyama et al. [33] introduced the following iterative scheme for an accretive operator : where is a sunny nonexpansive retraction from onto . Then they proved a weak convergence theorem.

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (16) which contains VIP (21) as a special case. Very recently, Cai and Bu [13] constructed an iterative algorithm for solving GSVI (16) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space .

Lemma 2 (see [36]). Let be a 2-uniformly smooth Banach space. Then where is the 2-uniformly smooth constant of and is the normalized duality mapping from into .

Define the mapping as follows: The fixed point set of is denoted by . Then their strong convergence theorem on the proposed method is stated as follows.

Theorem  CB (see [13, Theorem 3.1]). Let be a nonempty closed 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 with for . Let be a contraction of into itself with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping defined by (24). For arbitrarily given , let be the sequence generated bySuppose that and are two sequences in satisfying the following conditions: (i) and ,(ii).

Assume that for any bounded subset of , let be a mapping of into defined by for all , and suppose that . Then converges strongly to , which solves the VIP:

It is easy to see that the iterative scheme in Theorem CB is essentially equivalent to the following two-step iterative scheme:

For the convenience of implementing the argument techniques in [16], the authors [13] have used the following inequality in a real smooth and uniform convex Banach space .

Proposition 3 (see [37]). 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 .

Let be a nonempty closed convex subset of a real uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be a contraction with coefficient . Motivated and inspired by the research going on in this area, we introduce and analyze modified hybrid steepest-descent methods for the GSVI (16), with solutions being also zeros of an -accretive operator in . Here the modified hybrid steepest-descent methods are based on Korpelevich’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI (16) and the set of zeros of in . 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 literature [12, 13, 16, 32].

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 4 can be found in [38]. Lemma 5 is an immediate consequence of the subdifferential inequality of the function .

Lemma 4. Letbe 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 5. In a real smooth Banach space , there holds the inequality where is the normalized duality mapping.

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 ofinto 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 6 (see [39]). 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),(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 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 . The following lemma follows easily from Lemma 6.

Lemma 7. 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 (16) if and only if , where .

In terms of Lemma 7, 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 8 is the resolvent identity which can be found in [40], and Lemma 9 can be found in [41].

Lemma 8. For , there holds the identity

Lemma 9. Assume that . Then for all .

Lemma 10 (see [42]). Let be a nonempty closed 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.

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 11 (see [43]). 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 .

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

Lemma 12 (see [34]). Let be a uniformly smooth Banach space. Let be a nonempty closed convex subset of , let be a nonexpansive mapping with , and let . Then the net defined by converges strongly to a point in . If one defines a mapping by for all , then solves the VIP:

The following proposition will be used frequently throughout this paper.

Proposition 13 (see [11]). Let be a real smooth Banach space and let be a mapping.(i)If is -strictly pseudocontractive, then is Lipschitzian with constant .(ii)If is -strongly accretive and -strictly pseudocontractive with , then is contractive with constant .(iii)If is -strongly accretive and -strictly pseudocontractive with , then, for any fixed number , is contractive with constant .

3. Main Results

In this section, we introduce our modified hybrid steepest-descent schemes and show the strong convergence theorems. We will need the following useful lemmas in the sequel.

Lemma 14 (see [13, Lemma 2.8]). Let be a nonempty closed 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 15 (see [13, Lemma 2.9]). Let be a nonempty closed 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 nonexpansive.

Let be a real smooth Banach space and let be an -accretive operator in such that is convex. Let be a sunny nonexpansive retraction from onto . Let be -strongly accretive and -strictly pseudocontractive with . Let the mapping be -inverse strongly accretive for and let be a contractive map with coefficient . In this section, we will consider the problem of finding a point , which is a unique solution of the VIP: where is the fixed point set of the mapping with for . For each and each integer , we choose a number arbitrarily and then consider the following mapping defined as Then, is a contractive map. Indeed, utilizing Proposition 13(iii) and Lemma 15, we have for all and hence is contractive due to . By Banach’s Contraction Mapping Principle, there exists a unique fixed point of   in ; that is,

Theorem 16. Let be a uniformly convex and 2-uniformly smooth Banach space and let be an -accretive operator in such that is convex. Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive for , let be a contractive map with coefficient , and let be -strongly accretive and -strictly pseudocontractive with . Assume that , where is the fixed point set of the mapping with for . For each and each integer , let be defined by (47), where for some and with . Then as converges strongly to , which is a unique solution of the VIP (44).

Proof. First let us show that for some is bounded. Indeed, since with , there exists some such that for all .
Take . Then utilizing Proposition 13, 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 . Taking into account , we get From Lemma 14 we have From the last two inequalities, we obtain which, together with (50), implies that So it immediately follows that Since for , we have Utilizing Proposition 3 and Lemma 6, we have which implies that In the same way, we derive which implies that Substituting (57) for (59), we get which, together with (50), implies that So it immediately follows that Hence, from (55) we conclude that Utilizing the properties of and , we get which leads to That is,
Note that for all and that is bounded and so are , , and . Hence, we have as . Also, observe that This, together with (66) and (67), implies that Utilizing the nonexpansivity of , we obtain from (67) and (69) that Since for all , utilizing Lemma 9 we have
For any integer , for simplicity put for all . 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 (70) and (71) we obtain