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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 493862, 11 pages
Modified Noor's Extragradient Method for Solving Generalized Variational Inequalities in Banach Spaces
1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
3Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
Received 9 January 2012; Accepted 20 January 2012
Academic Editor: Khalida Inayat Noor
Copyright © 2012 Shunhou Fan 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.
Motivated and inspired by Korpelevich's and Noor's extragradient methods, we suggest an extragradient method by using the sunny nonexpansive retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.
In the present paper, we focus on the following generalized variational inequality: where is a nonempty closed convex subset of a real Banach space , is a nonlinear mapping, and is the normalized duality mapping defined by We use to denote the solution set of (1.1). It is clear that (1.1) is reduced to the following variational inequality in Hilbert spaces: which was introduced and studied by Stampacchia . Variational inequalities are being used as mathematical programming tools and models to study a wide class of unrelated problems arising in mathematical, physical, regional, engineering, and nonlinear optimization sciences. See, for instance, [2–23]. In order to solve (1.3), especially, Korpelevich  introduced the following well-known extragradient method: where is the metric projection from onto its subset , and is a monotone operator. He showed that the sequence converges to some solution of the above variational inequality (1.3). Noor  further suggested and analyzed the following new iterative methods: which is known as the modified Noor's extragradient method. We would like to point out that this algorithm (1.5) is quite different from the method of Koperlevich. However, these two algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces.
The generalized variational inequality (1.1) was introduced by Aoyama et al.  which is connected with the fixed point problem for nonlinear mapping. For solving the aforementioned generalized variational inequality (1.1), Aoyama et al.  introduced an iterative algorithm: where is a sunny nonexpansive retraction from onto , and and are two real number sequences. Aoyama et al.  obtained on the aforementioned method (1.6) for solving variational inequality (1.1). Motivated by (1.6), Yao and Maruster  presented a modification of (1.5): Yao and Maruster  proved that (1.7) converges strongly to the solution of the generalized variational inequality (1.1). Yao et al.  further considered the following extended extragradient method for solving (1.1):
In this paper, motivated and inspired by Korpelevich's and Noor's extragradient methods, (1.7) and (1.8), we suggest a modified Noor's extragradient method via the sunny nonexpansive retraction for solving the variational inequalities (1.1) in Banach spaces.
Let be a nonempty closed convex subset of a real Banach space . Recall that a mapping of into is said to be accretive if there exists such that for all . A mapping of into is said to be -strongly accretive if, for , for all . A mapping of into is said to be -inverse-strongly accretive if, for , for all .
Let . A Banach space is said to uniformly convex if, for each , there exists such that for any , 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 . It is also said to be uniformly smooth if the limit (2.5) is attained uniformly for . The norm of is said to be Fréchet differentiable if, for each , the limit (2.5) is attained uniformly for . And 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 .
We need the following lemmas for proof of our main results.
Lemma 2.1 (see ). Let be a given real number with and let be a -uniformly smooth Banach space. Then for all , where is the -uniformly smoothness constant of and is the generalized duality mapping from into defined by
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 . One knows the following lemma concerning sunny nonexpansive retraction.
Lemma 2.2 (see ). Let be a closed convex subset of a smooth Banach space , let be a nonempty subset of , and let be a retraction from onto . Then is sunny and nonexpansive if and only if for all and .
Remark 2.3. (1) It is well known that if is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto .
(2) Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space and let be a nonexpansive mapping of into itself with . Then the set is a sunny nonexpansive retract of .
The following lemma characterized the set of solution of (1.1) by using sunny nonexpansive retractions.
Lemma 2.4 (see ). 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 2.5 (see ). 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 2.6 (see ). 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 .
3. Main Results
In this section, we will state and prove our main result.
Theorem 3.1. Let be a uniformly convex and 2-uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto and let be an -strongly accretive and -Lipschitz continuous mapping with . For given , let the sequence be generated iteratively by
where and are two sequences in and is a constant in for some with . Assume that the following conditions hold:(a) and ;(b).
Then defined by (3.1) converges strongly to , where is a sunny nonexpansive retraction of onto .
Proof. First, we note that must be -inverse-strongly accretive mapping. Take . By using Lemmas 2.1 and 2.4, we easily obtain the following facts.(1) for all ; in particular,
(2)If , then is nonexpansive and for all
Indeed, from Lemma 2.1, we have From (3.1), we have By (3.1) and (3.5), we have Therefore, is bounded. We observe that and hence By Lemma 2.6, we obtain From (3.1), we also have By (3.1) and (3.10), we obtain Therefore, we have Since , and , we obtain It follows that Since is -strongly accretive, we deduce which implies that that is, Next, we show that To show (3.18), since is bounded, we can choose that a sequence of converges weakly to such that We first prove . It follows that By Lemma 2.5 and (3.20), we have ; it follows from Lemma 2.4 that .
Now, from (3.19) and Lemma 2.2, we have Since and for all , we can deduce from Lemma 2.2 that Therefore, we have which implies that
Finally, by Lemma 2.6 and (3.24), we conclude that converges strongly to . This completes the proof.
Variational inequality theory provides a simple, natural, and unified framework for a general treatment of unrelated problems. These activities have motivated to generalize and extend the variational inequalities and related optimization problems in several directions using new and novel techniques. A well-known method to solve the VI is the following gradient method: This method requires some monotonicity properties of . However, we remark that there is no chance of relaxing the assumption on to plain monotonicity. To overcome this weakness of the method, Korpelevich proposed a so-called Korpelevich's method which has been extensively extended and studied. Noor  especially, suggested another method referred as Noor's method which is different from Korpelevich's method. However, these two algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces. In the present paper, we suggested a modified Noor's method which has strong convergence in Banach spaces. We hope that the ideas and technique of this paper may stimulate further research in this field.
The authors thank the referees for useful comments and suggestions.
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