## Numerical and Analytical Methods for Variational Inequalities and Related Problems 2013

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# Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

**Academic Editor:**Shih-sen Chang

#### Abstract

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

#### 1. Introduction

Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction from to , where is the set of real numbers. The equilibrium problem for is to find such that The set of solutions of (1) is denoted by . Numerous problems in physics, optimization, and economics are reduced to find the solution of an equilibrium problem (e.g., see [1]). For solving an equilibrium problem, we assume that the bifunction satisfies the following conditions:(A1) for all ; (A2) is monotone; that is, for all ; (A3) for every , ; (A4) is convex and lower semicontinuous for each . Equilibrium problems have been studied extensively (see [1–6]).

Let be a mapping of into . The effective domain of is denoted by ; that is, . A multivalued mapping is said to be monotone if A monotone operator is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. For a maximal monotone operator on and , we may define a single-valued operator , which is called the resolvent of for . It is known that is firmly nonexpansive. A basic problem for maximal monotone operator is to The classical method for solving problem (3) is the proximal point algorithm. Rockafellar [7] established the weak convergence of the proximal point algorithm for maximal monotone operators. The question whether the strong convergence of the proximal point algorithm holds was answered in the negative by Güler [8]. In order to obtain the strong convergence theorem, some researchers have been devoted to modifying the proximal point algorithm (see [9, 10]). Given a positive constant , a mapping is said to be -inverse strongly monotone if Generally, monotone inclusion problem is to where the mapping is inverse strongly monotone and is maximal monotone. We write for the set of solutions of problem (5); that is, . It is well known that (see [11]). Takahashi et al. [12] constructed the following iterative sequence. Let , and let be a sequence generated by Under appropriate conditions, they proved that the sequence converges strongly to a point . Lin and Takahashi [13] introduced an iterative sequence that converges strongly to an element of , where is another maximal monotone operator.

Motivated by the above results, especially by Chuang et al. [2] and Takahashi et al. [12], we introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of [12].

#### 2. Preliminaries

Throughout this paper, let be a real Hilbert space with inner product and norm and let be a nonempty closed convex subset of . A mapping is nonexpansive if for all . The set of fixed points of is denoted by ; that is, . It is well known that is closed and convex. We denote the strong convergence and the weak convergence of to by and , respectively. For any , there exists a unique point such that is called the metric projection of onto . Note that is a nonexpansive mapping. For and , we have

Now consider inverse strongly monotone operators. For , let be an -inverse strongly monotone operator. If , then is a nonexpansive mapping. Indeed, for and , we get Therefore, the operator is a nonexpansive mapping of into .

We collect some important facts and tools.

Lemma 1 (see [1]). *Let be a nonempty closed convex subset of and let be a bifunction from to satisfying (A1)–(A4). If and , then there exists such that
*

Lemma 2 (see [4]). *Let be a nonempty closed convex subset of , and let be a bifunction from to satisfying (A1)–(A4). For , define a mapping as follows:
**
Then, the following hold: *(i)* is single valued;*(ii)* is firmly nonexpansive; that is, for any , ;*(iii)*;*(iv)* is closed and convex. *

Lemma 3 (see [6]). *Suppose that (A1)–(A4) hold. If and , then
*

Lemma 4 (see [14]). *Let be a real Hilbert space and let be a maximal monotone operator on . Then, the following holds:
**
for all and . *

Lemma 5 (see [15]). *Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .*

The following lemma is an immediate consequence of the inner product on .

Lemma 6. *For all , the inequality holds.*

Lemma 7 (demiclosedness principle [16]). *Let be a nonempty closed convex subset of and a nonexpansive mapping. Let be a point in and let be a sequence in . Suppose that and that . Then, . *

Lemma 8 (see [17]). * Let be a sequence of nonnegative real numbers satisfying , where*(i)*, ; *(ii)*. ** Then, .*

#### 3. Strong Convergence Theorems

In this section, we introduce a new iterative method for finding a common element of the set of zeros of the sum of maximal monotone operators and the set of solutions of an equilibrium problem, and we prove the strong convergence theorems in Hilbert spaces.

Theorem 9. *Let be a nonempty closed convex subset of a real Hilbert space and an -inverse strongly monotone operator of into with . Suppose that is a maximal monotone operator on and is a bifunction from to with (A1)–(A4). Assume that . Let and be sequences in and let be a sequence in . Let be a sequence generated by
**
Assume the following conditions are satisfied:*(c1)* for some ;*(c2)* and ; *(c3)*; *(c4)* and ; *(c5)* and . ** Then, the sequence converges strongly to .*

*Proof. *Note that the set is closed and convex since and are closed and convex.

With the help of Lemmas 1 and 2, we have , and, for any ,
Notice that
It follows that
By condition (c1), the sequence is bounded. Hence, there exists a positive number such that . From a simple inductive process, we have
which yields that is bounded. So are and .

We have from Lemmas 3 and 4 that
Thus, conditions (c2), (c4), and (c5) yield that
Recall that . It follows from Lemma 5 that
Consequently,

For any , according to Lemma 2, we have
which yields that
By (14), one has
which implies that
This together with (22) deduces that

Next, we prove that
where . In order to show this inequality, we can choose a subsequence of such that
Owing to the boundedness of , without loss of generality, we assume that . Now we show that . As is contained in and is a closed convex set, one has .

We firstly prove that . By (14), we have
The monotonicity of yields that
Replacing by , we obtain
It follows from (21), (27), and (A4) that
For , , set . Then, and . Thus,
Dividing by , we see that
Letting , we get
namely, .

Secondly, we prove that . Thanks to (c5), there is a subsequence of such that . Without loss of generality, we assume that . Observe that
Note that expressions (21) and (27) imply that
Therefore,
Since is nonexpansive and , Lemma 7 gives ; that is, . Thus, we have .

By (29) and the property of metric projection, we have
Inequality (28) arrives.

Finally, we prove that . Using (14) again, we obtain
Note that
It follows from Lemma 8 that converges strongly to .

The following results are direct consequences of Theorem 9.

Corollary 10. *Let be a nonempty closed convex subset of a real Hilbert space and an -inverse strongly monotone operator of into with . Suppose that is a maximal monotone operator on with . Let and be sequences in and let be a sequence in . Let be a sequence generated by
**
If conditions (c1)–(c3) and (c5) are satisfied, then the sequence converges strongly to .*

*Proof. *Letting for all and in Theorem 9, the desired result follows.

*Remark 11. *When , the algorithm in Corollary 10 corresponds to the one in Theorem 10 of Takahashi et al. [12]. Thus, Corollary 10 includes, as a special case, Theorem 10 of [12].

Corollary 12. *Let be a nonempty closed convex subset of a real Hilbert space and let be a bifunction from to with (A1)–(A4). Assume that the set is nonempty. Let and be sequences in and let be a sequence in . Let be a sequence generated by
**
If conditions (c1)–(c4) are satisfied, then the sequence converges strongly to . *

*Proof. *Putting , (i.e., the subdifferential of the indicator function of ) in Theorem 9, we get the result.

#### 4. Applications

In this section, we apply the results in the preceding section to variational inequality and optimization problems.

Now we consider the variational inequality problem. Let be a real Hilbert space and let be a proper lower semicontinuous convex function of into . Then, the subdifferential of is defined as follows: for all . Rockafellar [18] claimed that is a maximal monotone operator. Let be the indicator function of ; that is, Since is a proper lower semicontinuous convex function on , the subdifferential of is a maximal monotone operator. The resolvent of for is defined by We have where , .

Let be a nonlinear mapping. The variational inequality problem for is to find such that The set of its solutions is denoted by . It is easy to see that (for more details, see [13]). Using Corollary 10, we obtain the strong convergence theorem for the variational inequality problem.

Theorem 13. *Let be a nonempty closed convex subset of a real Hilbert space and an -inverse strongly monotone operator of into with . Assume that the set is nonempty. Let and be sequences in and let be a sequence in . Let be a sequence generated by
**
If conditions (c1)–(c3) and (c5) are satisfied, then the sequence converges strongly to .*

*Proof. *Notice that . Corollary 10 yields the conclusion.

Next we study the optimization problem where is a proper lower semicontinuous convex function of into such that is contained in . We denote by the set of solutions of problem (51). Let be a bifunction defined by It is clear that satisfies (A1)–(A4) and . Therefore, by Corollary 12, the following result is obtained.

Theorem 14. * Let be a proper lower semicontinuous convex function of into and let be a nonempty closed convex subset of such that is contained in . Suppose that the set is nonempty. Let and be sequences in and let be a sequence in . Let be a sequence generated by
**
If conditions (c1)–(c4) are satisfied, then the sequence converges strongly to . *

*Proof. *Letting in Corollary 12, we get the required result.

*Remark 15. *Letting in our results, we obtain the algorithms for minimal-norm solutions of corresponding problems.

#### Conflict of Interests

The authors declare that there is no conflict of interests.

#### Acknowledgments

The authors would like to thank the referees and editors for their valuable comments and suggestions. The work was supported by graduate funds of Beijing University of Technology (no. ykj-2013-9422).

#### References

- E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”
*The Mathematics Student*, vol. 63, no. 1–4, pp. 123–145, 1994. View at: Google Scholar | MathSciNet - C. S. Chuang, L. J. Lin, and W. Takahashi, “Halpern's type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points,”
*Journal of Global Optimization*, vol. 56, no. 4, pp. 1591–1601, 2013. View at: Publisher Site | Google Scholar - P. Sunthrayuth and P. Kumam, “A new computational technique for common solutions between systems of generalized mixed equilibrium and fixed point problems,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 230392, 17 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,”
*Journal of Nonlinear and Convex Analysis*, vol. 6, no. 1, pp. 117–136, 2005. View at: Google Scholar | MathSciNet - D. N. Qu and C. Z. Cheng, “A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixed-point problems in Banach space,”
*Fixed Point Theory and Applications*, vol. 2011, article 17, 2011. View at: Publisher Site | Google Scholar - F. Cianciaruso, G. Marino, and L. Muglia, “Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces,”
*Journal of Optimization Theory and Applications*, vol. 146, no. 2, pp. 491–509, 2010. View at: Publisher Site | Google Scholar | MathSciNet - 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: Google Scholar | MathSciNet - O. Güler, “On the convergence of the proximal point algorithm for convex minimization,”
*SIAM Journal on Control and Optimization*, vol. 29, no. 2, pp. 403–419, 1991. View at: Publisher Site | Google Scholar | MathSciNet - S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces,”
*Journal of Approximation Theory*, vol. 106, no. 2, pp. 226–240, 2000. View at: Publisher Site | Google Scholar | MathSciNet - H. K. Xu, “A regularization method for the proximal point algorithm,”
*Journal of Global Optimization*, vol. 36, no. 1, pp. 115–125, 2006. View at: Publisher Site | Google Scholar | MathSciNet - K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “On a strongly nonexpansive sequence in Hilbert spaces,”
*Journal of Nonlinear and Convex Analysis*, vol. 8, no. 3, pp. 471–489, 2007. View at: Google Scholar | MathSciNet - W. Takahashi, N. C. Wong, and J. C. Yao, “Two generalized strong convergence theorems of Halpern's type in Hilbert spaces and applications,”
*Taiwanese Journal of Mathematics*, vol. 16, no. 3, pp. 1151–1172, 2012. View at: Google Scholar | MathSciNet - L. J. Lin and W. Takahashi, “A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications,”
*Positivity*, vol. 16, no. 3, pp. 429–453, 2012. View at: Publisher Site | Google Scholar | MathSciNet - S. Takahashi, W. Takahashi, and M. Toyoda, “Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces,”
*Journal of Optimization Theory and Applications*, vol. 147, no. 1, pp. 27–41, 2010. View at: Publisher Site | Google Scholar | MathSciNet - T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2005, article 103, 2005. View at: Google Scholar | MathSciNet - H. H. Bauschke and P. L. Combettes,
*Convex Analysis and Monotone Operator Theory in Hilbert Spaces*, Springer, 2011. View at: Publisher Site | MathSciNet - H. K. Xu, “Another control condition in an iterative method for nonexpansive mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 65, no. 1, pp. 109–113, 2002. View at: Publisher Site | Google Scholar | MathSciNet - R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,”
*Pacific Journal of Mathematics*, vol. 33, no. 1, pp. 209–216, 1970. View at: Google Scholar | MathSciNet

#### Copyright

Copyright © 2013 Huan-chun Wu and Cao-zong Cheng. 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.