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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 832548, 8 pages
http://dx.doi.org/10.1155/2013/832548
Research Article

Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces

1Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 27 March 2013; Accepted 11 May 2013

Academic Editor: Ru Dong Chen

Copyright © 2013 H. Zegeye and N. Shahzad. 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 iterative process which converges strongly to solutions of a certain variational inequity problem for -inverse strongly accretive mappings in the set of common fixed points of finite family of strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Let be a real normed linear space with dual . For , we denote by the generalized duality mapping from to defined by where denotes the duality pairing. In particular, is called the normalized duality map. It is well known (see e.g., [1]) that is single valued if is smooth and that In the sequel, we will denote the single-valued generalized map by .

A mapping with domain and range in is called -strongly accretive if there exist and such that is called-inverse strongly accretive if there exist and such that Let be a nonempty, closed, and convex subset of and, let be a nonlinear mapping. The variational inequality problem is to for some . The set of solutions of variational inequality problem is denoted by . If , a real Hilbert space, the variational inequality problem reduces to which was introduced and studied by Stampacchia [2].

Variational inequality theory has emerged as an important tool in studying a wide class of related problems arising in mathematical, physical, regional, engineering, and nonlinear optimization sciences (see, for instance, [312]).

In 1976, Korpelevič [4] introduced the following well-known extragradient method: where is the metric projection from onto its subset , for some , and is an accretive operator. He proved that the sequence converges to a solution of the variational inequality (6).

Furthermore, Noor [6] proved that the iterative scheme, given by where is an accretive operator, converges to a solution of the variational inequality (6).

We note that the above algorithms give strong convergence to a solution of the variational inequality (6). However, both algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces.

In 2006, Aoyama et al. [13] introduced and studied the following iterative algorithm in a uniformly convex and 2-uniformly smooth Banach spaces possessing weakly sequentially continuous duality mapping: where is a sunny nonexpansive retraction from onto a closed and convex , is an -inverse strongly accretive mapping and and subsets of real numbers, satisfy certain conditions. They proved that the sequence in (9) converges weakly to a point .

Recently, Yao et al. [8] introduced and considered the following iterative method for -strongly accretive mappings in a uniformly convex and 2-uniformly smooth Banach space possessing weakly sequentially continuous duality mapping: where is a sunny nonexpansive retraction from onto . They proved that the sequence defined by (10) converges strongly to provided that real sequences , , , and satisfy certain conditions.

Let be a nonempty subset of a real Banach space . A mapping is called -strictly pseudocontractive of Browder-Petryshyn type [14] if for all there exist and such that is called Lipschitz if there exists such that If in (12), then is called contraction, while is said to be nonexpansive if .

If , a real Hilbert space, then (11) is equivalent to the inequality and we can assume also that , so that . A point is a fixed point of if , and we denote by the set of fixed points of ; that is, .

In 2001, Yamada [7] introduced a hybrid steepest descent method which relates solutions of variational inequality problems with fixed point of mappings in Hilbert spaces. He proved that if is nonexpansive self-map on and is an -strongly accretive mapping from into satisfying certain conditions, then the sequence defined by converges strongly to the unique solution of the variational inequality The above results naturally bring us to the following question.

Question. Could we produce an iterative scheme which approximates a solution of variational inequality (5) for -inverse strongly accretive mappings in Banach spaces?

In this paper, motivated by Yao et al. [8] and Yamada [7], it is our purpose to introduce an iterative scheme which converges strongly to a solution of the variational inequality (5) for -inverse strongly accretive mapping in the set of common fixed points of finite family of strictly pseudocontractive mappings in a uniformly convex and -uniformly smooth Banach space possessing weakly sequentially continuous duality mapping. Our results complement or improve the results of Yao et al. [8], Aoyama et al. [13], and some authors.

2. Preliminaries

Let be a real Banach space. The modulus of smoothness of is the function defined by . If for all , then is said to be smooth. If there exists a constant and a real number , such that , then is said to be -uniformly smooth.

If is a real -uniformly smooth Banach space, then by [15], the following geometric inequality holds: for all and some real constant .

It is well known (see e.g., [16]) that The Banach space is said to be uniformly convex if, given , there exists , such that, for all with , and , . It is well known that , , and Sobolev spaces , are uniformly convex.

Let be closed convex and a mapping of onto . Then, is said to be sunny if for all and . A mapping of into is said to be a retraction if . If a mapping is a retraction, then for every , range of . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto , and it is said to be a nonexpansive retract of if there exists a nonexpansive retraction of onto . If , the metric projection is a sunny nonexpansive retraction from to any closed convex subset of . Moreover, if is a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space and is a nonexpansive mapping of into itself with , then the set is a sunny nonexpansive retract of .

In what follows, we will make use of the following lemmas.

Lemma 1 (see, e.g., [17]). Let be a smooth Banach space, and let be a nonempty subset of . Let be a retraction, and let be the normalized duality map on . Then, the following are equivalent:(i)is sunny nonexpansive,(ii) for all and .

Lemma 2 (see [18]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , and . Then, .

Lemma 3 (see [13]). Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto , and let be an accretive operator of into . Then, for all , where .

Lemma 4 (see [17]). Let be a nonempty bounded closed convex subset of a uniformly convex Banach space , and let be nonexpansive mapping of into itself. If is a sequence of such that weakly and strongly, then is a fixed point of .

Lemma 5 (see [19]). Let be a real Banach space. Then, for any given , the following inequality holds:

Lemma 6 (see [20]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let , be a family of nonexpansive mappings such that . Let be real numbers in such that , and let . Then, is nonexpansive, and .

Lemma 7 (see [21]). Let be a nonempty, closed and convex subset of a real uniformly convex and smooth Banach space . Let , , be -strictly pseudocontractive mappings such that . Let with . Then is -strictly pseudocontractive with and .

Lemma 8 (see [22]). Let be a nonempty closed and convex subset of a real -uniformly smooth Banach space for . Let be a -strictly pseudocontractive mapping. Then, for , where is the Lipchitz constant of and is a constant in (16), the mapping is nonexpansive, and .

Lemma 9. Let be a nonempty closed and convex subset of a a real -uniformly smooth Banach space for . Let be an -inverse strongly accretive mapping. Then, for , the mapping is nonexpansive.

Proof. Now, using inequality (16), we get that The proof is complete.

Lemma 10 (see [10]). Let be a uniformly convex Banach space, and let be a closed ball of . Then, there exists a continuous strictly increasing convex function with such that for , with .

Lemma 11 (see [5]). Let be sequences of real numbers such that there exists a subsequence of such that , for all . Then, there exists a nondecreasing sequence such that , and the following properties are satisfied by all (sufficiently large) numbers : In fact, .

3. Main Results

Let be a nonempty closed convex subset of a real uniformly convex and -uniformly smooth Banach space . Let , for , be a -strictly pseudocontractive mappings, and let be an -inverse strongly accretive mapping. Then, in what follows, we will study the variational inequality and the following iteration process: where , for , such that , , for , and , and is the real number in (16). In addition, we assume and as real sequences satisfying the following control conditions: (i) , (ii) , , .

We now prove our main theorem.

Theorem 12. Let be a nonempty closed convex subset of a real uniformly convex and -uniformly smooth Banach space possessing weakly sequentially continuous duality mapping. Let , for , be -strictly pseudocontractive mappings, and let be -inverse strongly accretive mapping. Let be a sequence defined by (25). Assume that , where . Then, converges strongly to , where is a sunny nonexpansive retraction of onto , which is a solution of the variational inequality (24).

Proof. By Lemmas 7 and 8 we have that is nonexpansive. In addition, by Lemma 9 we get that is nonexpansive. Let and, let . Then from (25), Lemmas 8 and 9 we have that
Thus, from (25) and (27), we get that
Therefore, by induction, which implies that and hence , , and are bounded. Furthermore, from (25), we obtain that And, hence, from (25) and (31), we have that for some . Thus, using the properties of , , , (32), and Lemma 2, we obtain that , as , which implies from (31) that , as . Again from (25), we have that , as . Consequently,
Now, we prove that converges strongly to the point . Let , and let . Then, since , we have that Furthermore, from (25), Lemma 5, and Lemma 10, we get that which implies that
Now, following the method of proof of Lemma 3.2 of Maingé [5], we consider two cases.
Case 1. Suppose that there exists such that is decreasing for all . Then, we get that is convergent. Thus, from (36) and the fact that , as , we have that which implies that
In addition, since is bounded subset of a reflexive space , we can choose a subsequence of such that and . This implies from (34) and (33) that. Then, from (39) and Lemma 4, we have that . Moreover, from (39) and Lemma 4, we have that , and by Lemma 3, we get , and hence . Therefore, using the fact that has a weakly sequentially continuous duality mapping and Lemma 1, we immediately obtain that Then, it follows from (37), (40), and Lemma 2 that , as . Consequently, .
Case 2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 11, there exists a nondecreasing sequence such that , and for all . Now, from (36) and the fact that , we get that and , as . Thus, like in Case 1, we obtain that Moreover, from (37), we have that which implies from (42) and (44) that Now, since , we obtain that and using (43), we get that . This together with (44) implies that , as . But , for all ; thus, we obtain that . Therefore, from both cases, we can conclude that converges strongly to , which is a solution of the variational inequality (24), and the proof is complete.

If in Theorem 12, we consider that , we get the following corollary.

Corollary 13. Let be a nonempty closed convex subset of a real uniformly convex and -uniformly smooth Banach space possessing weakly sequentially continuous duality mapping. Let be a -strictly pseudocontractive mapping, and let be an -inverse strongly accretive mapping. Let be a sequence defined by (25), where . Assume that . Then, converges strongly to which is a solution of the variational inequality

If in Theorem 12, we assume that , for , are nonexpansive, we get the following corollary.

Corollary 14. Let be a nonempty closed convex subset of a real uniformly convex and -uniformly smooth Banach space possessing weakly sequentially continuous duality mapping. Let , for , be nonexpansive mappings, and let be an -inverse strongly accretive mapping. For , let the sequence be generated iteratively by where for , , , are as in (24). Assume that , where . Then, converges strongly to , which is a solution of the variational inequality problem (24).

Proof. Lemma 6 and the method of proof of Theorem 12 provide the required assertion.

If in Corollary 14, we consider that , we get the following corollary.

Corollary 15. Let be a nonempty closed convex subset of a real uniformly convex and -uniformly smooth Banach space possessing weakly sequentially continuous duality mapping. Let be a nonexpansive mapping, and let be an -inverse strongly accretive mapping. For , let the sequence be generated iteratively by Assume that . Then, converges strongly to , which is a solution of the variational inequality problem

If in Corollary 14, we assume that , we obtain the following corollary.

Corollary 16. Let be a nonempty closed convex subset of a real uniformly convex and -uniformly smooth Banach space possessing weakly sequentially continuous duality mapping. Let be an -inverse strongly accretive mapping. For , let the sequence be generated iteratively by Assume that . Then, converges strongly to , where is a sunny nonexpansive retraction of onto .

If in Theorem 12, we assume that is an -strongly accretive and -Lipschitzian continuous mapping, we obtain the following corollary.

Corollary 17. Let be a nonempty closed convex subset of a real uniformly convex and -uniformly smooth Banach space possessing weakly sequentially continuous duality mapping. Let , for , be -strictly pseudocontractive mappings, and Let be an -strongly accretive and -Lipschitzian continuous mapping. Let be a sequence defined by (25) for . Assume that , where . Then, converges strongly to , which is a solution of the variational inequality problem

Proof. We note that if is an -strongly accretive and -Lipschitzian continuous mapping of into , then we have that and hence, is an -inverse strongly accretive mapping with . Thus, the conclusion follows from Theorem 12.

If , a real Hilbert space, then is a uniformly convex and -uniformly smooth Banach space for possessing weakly sequentially continuous duality mapping. In this case, we have that , projection mapping from onto . Thus, we have the following corollary.

Corollary 18. Let be a nonempty closed convex subset of a real Hilbert space . Let , for , be -strictly pseudocontractive mappings, and let be an -inverse strongly accretive mapping. For , let the sequence be generated iteratively by where , for , such that , , for , and . Assume that , where . Then, converges strongly to , which is a solution of the variational inequality

Remark 19. Theorem 12 complements Theorem 3.2 of Yao et al. [8] in more general Banach spaces for -inverse strongly accretive mappings. Moreover, Theorem 12 improves Theorem 3.1 of Aoyama et al. [13] and Theorem 3.7 of Saejung et al. [23] in the sense that our convergence is strong in the set of common fixed points of finite family of strictly pseudocontractive mappings.

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