Journal of Applied Mathematics

Volume 2012 (2012), Article ID 137932, 11 pages

http://dx.doi.org/10.1155/2012/137932

## Projection Algorithms for Variational Inclusions

^{1}School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, China^{2}Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China^{3}Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan

Received 7 October 2011; Accepted 22 October 2011

Academic Editor: Yonghong Yao

Copyright © 2012 Youli Yu 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

We present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality.

#### 1. Introduction

Let be a real Hilbert space. Let be a single-valued nonlinear mapping and be a set-valued mapping. Now we concern the following variational inclusion, which is to find a point such that where is the zero vector in . The set of solutions of problem (1.1) is denoted by . If , then problem (1.1) becomes the generalized equation introduced by Robinson [1]. If , then problem (1.1) becomes the inclusion problem introduced by Rockafellar [2]. It is known that (1.1) provides a convenient framework for the unified study of optimal solutions in many optimization-related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, game theory, and so forth. Also various types of variational inclusions problems have been extended and generalized. Recently, Zhang et al. [3] introduced a new iterative scheme for finding a common element of the set of solutions to the problem (1.1) and the set of fixed points of nonexpansive mappings in Hilbert spaces. Peng et al. [4] introduced another iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. For some related works, see [5–28] and the references therein.

Inspired and motivated by the works in the literature, in this paper, we present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality.

#### 2. Preliminaries

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of . Recall that a mapping is said to be -inverse strongly monotone if there exists a constant such that ,for all . A mapping is strongly positive on if there exists a constant such that for all .

For any , there exists a unique nearest point in , denoted by , such that Such a is called the metric projection of onto . We know that is nonexpansive. Further, for and ,

A set-valued mapping is called monotone if, for all , and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if, for , for every implies .

Let the set-valued mapping be maximal monotone. We define the resolvent operator associated with and as follows: where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse strongly monotone, and that a solution of problem (1.1) is a fixed point of the operator for all , see for instance [29].

Lemma 2.1 (see [30]). *Let be a maximal monotone mapping and be a Lipschitz-continuous mapping. Then the mapping is maximal monotone.*

Lemma 2.2 (see [8]). *Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .*

Lemma 2.3 (see [31]). *Assume is a sequence of nonnegative real numbers such that , where is a sequence in and is a sequence such that*(1)*;*(2)* or .**Then .*

#### 3. Main Result

In this section, we will prove our main result. First, we give some assumptions on the operators and the parameters. Subsequently, we introduce our iterative algorithm for finding solutions of the variational inclusion (1.1). Finally, we will show that the proposed algorithm has strong convergence.

In the sequel, we will assume that(A1) is a nonempty closed convex subset of a real Hilbert space ;(A2) is a strongly positive bounded linear operator with coefficient , is a maximal monotone mapping and is an -inverse strongly monotone mapping;(A3) is a constant satisfying .

Now we introduce the following iteration algorithm.

*Algorithm 3.1. *For given arbitrarily, compute the sequence as follows:
where and are two real sequences in .

Now we study the strong convergence of the algorithm (3.1)

Theorem 3.2. *Suppose that . Assume the following conditions are satisfied:*(i)*;*(ii)*;*(iii)*.**Then the sequence generated by (3.1) converges strongly to which solves the following variational inequality:
*

*Proof. *Take . It is clear that
We divide our proofs into the following five steps:(1)the sequence is bounded.(2).(3).(4) where .(5).

*Proof of (1.1). *Since is -inverse strongly monotone, we have
It is clear that if , then is nonexpansive. Set . It follows that
Since is linear bounded self-adjoint operator on , then
Observe that
that is to say is positive. It follows that
From (3.1), we deduce that
Therefore, is bounded.

*Proof of (3.1). *Set for all . Then, we have
Note that
Substituting (3.11) into (3.10), we get
Therefore,
This together with Lemma 2.2 imply that
Hence,

*Proof of (3.4). *From (3.4), we get
By (3.1), we obtain
where is some constant satisfying . From (3.16) and (3.17), we have
Thus,
which imply that

*Proof of (3.10). *Since is 1-inverse strongly monotone, we have
which implies that
Substitute (3.22) into (3.17) to get
Then we derive
So, we have

We note that is a contraction. As a matter of fact,
for all . Hence has a unique fixed point, say . That is . This implies that for all . Next, we prove that

First, we note that there exists a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that .

Next, we show that . In fact, since is -inverse strongly monotone, is Lipschitz-continuous monotone mapping. It follows from Lemma 2.1 that is maximal monotone. Let , that is, . Again since , we have , that is, . By virtue of the maximal monotonicity of , we have
and so
It follows from , and that
It follows from the maximal monotonicity of that , that is, . Therefore, . It follows that

*Proof of (3.11). *First, we note that , then for all , we have . Thus,
that is,
So,
where and . It is easy to see that and . Hence, by Lemma 2.3, we conclude that the sequence converges strongly to . This completes the proof.

#### 4. Conclusion

The results proved in this paper may be extended for multivalued variational inclusions and related optimization problems.

#### Acknowledgment

This research was partially supported by Youth Foundation of Taizhou University (2011QN11).

#### References

- S. M. Robinson, “Generalized equations and their solutions. I: basic theory,”
*Mathematical Programming Study*, no. 10, pp. 128–141, 1979. View at Google Scholar · View at Zentralblatt MATH - R. T. Rockafellar, “Monotone operators and the proximal point algorithm,”
*SIAM Journal on Control and Optimization*, vol. 14, no. 5, pp. 877–898, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,”
*Applied Mathematics and Mechanics*, vol. 29, no. 5, pp. 571–581, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. W. Peng, Y. Wang, D. S. Shyu, and J. C. Yao, “Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 720371, 15 pages, 2008. View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Equivalence of variational inclusions with resolvent equations,”
*Nonlinear Analysis*, vol. 41, no. 7-8, pp. 963–970, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor and T. M. Rassias, “Projection methods for monotone variational inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 237, no. 2, pp. 405–412, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor and Z. Huang, “Some resolvent iterative methods for variational inclusions and nonexpansive mappings,”
*Applied Mathematics and Computation*, vol. 194, no. 1, pp. 267–275, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, K. I. Noor, and E. Al-Said, “Some resolvent methods for general variational
inclusions,”
*Journal of King Saud University—Science*, vol. 23, no. 1, pp. 53–61, 2011. View at Google Scholar - M. A. Noor and K. I. Noor, “Sensitivity analysis for quasi-variational inclusions,”
*Journal of Mathematical Analysis and Applications*, vol. 236, no. 2, pp. 290–299, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Three-step iterative algorithms for multivalued quasi variational inclusions,”
*Journal of Mathematical Analysis and Applications*, vol. 255, no. 2, pp. 589–604, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Generalized set-valued variational inclusions and resolvent equations,”
*Journal of Mathematical Analysis and Applications*, vol. 228, no. 1, pp. 206–220, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y. J. Cho, and Y. C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,”
*European Journal of Operational Research*, vol. 212, no. 2, pp. 242–250, 2011. View at Publisher · View at Google Scholar - Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in
Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article ID 79, 2011. View at Publisher · View at Google Scholar - Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general
errors,”
*Optimization Letters*. In press. View at Publisher · View at Google Scholar - Y. Yao, Y. C. Liou, and C. P. Chen, “Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings,”
*Taiwanese Journal of Mathematics*, vol. 15, pp. 1979–1998, 2011. View at Google Scholar - Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,”
*Mathematical & Computer Modelling*. In press. - S. Adly, “Perturbed algorithms and sensitivity analysis for a general class of variational inclusions,”
*Journal of Mathematical Analysis and Applications*, vol. 201, no. 2, pp. 609–630, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Chang, “Set-valued variational inclusions in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 248, no. 2, pp. 438–454, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Chang, “Existence and approximation of solutions of set-valued variational inclusions in Banach spaces,”
*Nonlinear Analysis*, vol. 47, pp. 583–594, 2001. View at Google Scholar - X. P. Ding, “Perturbed Ishikawa type iterative algorithm for generalized quasivariational inclusions,”
*Applied Mathematics and Computation*, vol. 141, no. 2-3, pp. 359–373, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. J. Huang, “Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions,”
*Computers & Mathematics with Applications*, vol. 35, no. 10, pp. 1–7, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. J. Lin, “Variational inclusions problems with applications to Ekeland's variational principle, fixed point and optimization problems,”
*Journal of Global Optimization*, vol. 39, no. 4, pp. 509–527, 2007. 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, “General variational inequalities,”
*Applied Mathematics Letters*, vol. 1, no. 2, pp. 119–121, 1988. 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. - Y. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modied 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, Y. C. Liou, C. L. Li, and H. T. Lin, “Extended extra-gradient methods for generalized variational inequalities,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 237083, 14 pages, 2012. View at Publisher · View at Google Scholar - B. Lemaire, “Which fixed point does the iteration method select?” in
*Recent Advances in Optimization*, vol. 452 of*Lecture Notes in Economics and Mathematical Systems*, Springer, Berlin, Germany, 1997. View at Google Scholar · View at Zentralblatt MATH - H. Brezis,
*Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert*, North-Holland Publishing, Amsterdam, The Netherlands, 1973. - 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