Abstract and Applied Analysis

Volume 2012 (2012), Article ID 493862, 11 pages

http://dx.doi.org/10.1155/2012/493862

## Modified Noor's Extragradient Method for Solving Generalized Variational Inequalities in Banach Spaces

^{1}Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China^{2}Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{3}Department 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.

#### Abstract

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.

#### 1. Introduction

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 [1]. 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 [24] 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 [10] 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. [25] which is connected with the fixed point problem for nonlinear mapping. For solving the aforementioned generalized variational inequality (1.1), Aoyama et al. [25] introduced an iterative algorithm: where is a sunny nonexpansive retraction from onto , and and are two real number sequences. Aoyama et al. [25] obtained on the aforementioned method (1.6) for solving variational inequality (1.1). Motivated by (1.6), Yao and Maruster [26] presented a modification of (1.5): Yao and Maruster [26] proved that (1.7) converges strongly to the solution of the generalized variational inequality (1.1). Yao et al. [27] 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.

#### 2. Preliminaries

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 [28]). *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 [29]). *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 [25]). *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 [30]). *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 [31]). *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.

#### 4. Conclusion

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 [10] 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.

#### Acknowledgment

The authors thank the referees for useful comments and suggestions.

#### References

- G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,”
*Comptes Rendus de l'Academie des Sciences*, vol. 258, pp. 4413–4416, 1964. View at Google Scholar · View at Zentralblatt MATH - Y. Censor, A. N. Iusem, and S. A. Zenios, “An interior point method with Bregman functions for the variational inequality problem with paramonotone operators,”
*Mathematical Programming A*, vol. 81, no. 3, pp. 373–400, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Glowinski,
*Numerical Methods for Nonlinear Variational Problems*, Springer-Verlag, New York, NY, USA, 1984. - A. N. Iusem, “An iterative algorithm for the variational inequality problem,”
*Computational and Applied Mathematics*, vol. 13, no. 2, pp. 103–114, 1994. View at Google Scholar · View at Zentralblatt MATH - P. Jaillet, D. Lamberton, and B. Lapeyre, “Variational inequalities and the pricing of American options,”
*Acta Applicandae Mathematicae*, vol. 21, no. 3, pp. 263–289, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor,
*On Variational Inequalities*, Brunel University, London, UK, 1975. - M. A. Noor, “General variational inequalities,”
*Applied Mathematics Letters*, vol. 1, no. 2, pp. 119–122, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Wiener-Hopf equations and variational inequalities,”
*Journal of Optimization Theory and Applications*, vol. 79, no. 1, pp. 197–206, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “New approximation schemes for general variational inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “A class of new iterative methods for general mixed variational inequalities,”
*Mathematical and Computer Modelling*, vol. 31, no. 13, pp. 11–19, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Some developments in general variational inequalities,”
*Applied Mathematics and Computation*, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Extended general variational inequalities,”
*Applied Mathematics Letters*, vol. 22, no. 2, pp. 182–186, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Some aspects of extended general variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 303569, 16 pages, 2012. View at Publisher · View at Google Scholar - M. A. Noor and E. A. Al-Said, “Wiener-Hopf equations technique for quasimonotone variational inequalities,”
*Journal of Optimization Theory and Applications*, vol. 103, no. 3, pp. 705–714, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Aslam Noor and Z. Huang, “Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings,”
*Applied Mathematics and Computation*, vol. 191, no. 2, pp. 504–510, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,”
*Applied Mathematics and Information Sciences*, vol. 6, no. 1, pp. 65–69, 2012. View at Google Scholar - H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,”
*Journal of Optimization Theory and Applications*, vol. 119, no. 1, pp. 185–201, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in
*Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications*, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473–504, North-Holland, Amsterdam, Holland, 2001. View at Google Scholar · View at Zentralblatt MATH - J. C. Yao, “Variational inequalities with generalized monotone operators,”
*Mathematics of Operations Research*, vol. 19, no. 3, pp. 691–705, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,”
*Journal of Global Optimization*. In press. View at Publisher · View at Google Scholar - Y. Yao, M. A. Noor, and Y.-C Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar - Y. Yao, M. A. Noor, and Y.-C Liou, “Iterative algorithms for general multi-valued variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 768272, 10 pages, 2012. View at Publisher · View at Google Scholar - E. Zeidler,
*Nonlinear Functional Analysis and its Applications. III: Variational Methods and Applications*, Springer-Verlag, New York, NY, USA, 1985. - G. M. Korpelevich, “An extragradient method for finding saddle points and for other problems,”
*Ekonomika i Matematicheskie Metody*, vol. 12, no. 4, pp. 747–756, 1976. View at Google Scholar - K. Aoyama, H. Iiduka, and W. Takahashi, “Weak convergence of an iterative sequence for accretive operators in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2006, Article ID 35390, 13 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao and S. Maruster, “Strong convergence of an iterative algorithm for variational inequalities in Banach spaces,”
*Mathematical and Computer Modelling*, vol. 54, no. 1-2, pp. 325–329, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y. C. Liou, C. L. Li, and H. T. Lin, “Extended extragradient methods for generalized variational inequalities,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 237083, 14 pages, 2012. View at Publisher · View at Google Scholar - H. K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. E. Bruck Jr., “Nonexpansive retracts of Banach spaces,”
*Bulletin of the American Mathematical Society*, vol. 76, pp. 384–386, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” in
*Nonlinear Functional Analysis (Proceedings of Symposia in Pure Mathematics, Vol. XVIII, Part 2, Chicago, Ill., 1968)*, pp. 1–308, American Mathematical Society, Providence, RI, USA, 1976. View at Google Scholar · View at Zentralblatt MATH - H. K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH