## Recent Developments on Sequence Spaces and Compact Operators with Applications

View this Special IssueResearch Article | Open Access

# Generalized Equilibrium Problem with Mixed Relaxed Monotonicity

**Academic Editor:**M. Mursaleen

#### Abstract

We extend the concept of relaxed -monotonicity to mixed relaxed --monotonicity. The concept of mixed relaxed --monotonicity is more general than many existing concepts of monotonicities. Finally, we apply this concept and well known KKM-theory to obtain the solution of generalized equilibrium problem.

#### 1. Introduction

Generalized monotonicities provide a way of finding parameter moves that yield monotonicity of model solutions and allow studying the monotonicity of functions or subset of variables. In recent past, many researchers have proposed many important generalizations of monotonicity such as pseudomonotonicity, relaxed monotonicity, relaxed --monotonicity, quasimonotonicity, and semimonotonicity; see [1–3]. Karamardian and Schaible [4] introduced various kinds of monotone mappings which in the case of gradient mappings are related to generalized convex functions. For more details, we refer to [5–7].

Many problems of practical interest in optimization, economics, and engineering involve equilibrium in their description. The techniques involved in the study of equilibrium problems are applicable to a variety of diverse areas and proved to be productive and innovative. Blum and Oettli [8] and Noor and Oettli [9] have shown that the mathematical programming problem can be viewed as special realization of abstract equilibrium problems.

Inspired and motivated by the recent development of equilibrium problems and their solutions methods, in this paper, we extend the concept of relaxed -monotonicity to mixed relaxed --monotonicity. Finally, this concept is applied with KKM-theory to solve a generalized equilibrium problem. The results of this paper can be viewed as generalization of many known results; see [10–13].

#### 2. Preliminaries

Let be a nonempty subset of real Banach space . Let be a real-valued function and let be an equilibrium function; that is, , for all . We consider the following generalized equilibrium problem: find such that Problem (1) has been studied by many authors in different settings; see, for instance, [14].

If , then the problem (1) reduces to the classical equilibrium problem, that is, to find such that Problem (2) was introduced and studied by Blum and Oettli [8].

We need the following definition and results in the sequel.

*Definition 1. *A real-valued function defined on a convex subset of is said to be hemicontinuous if

*Definition 2. *Let be a multivalued mapping. The is said to be a KKM-mapping if, for any finite subset of , , where denotes the convex hull.

Lemma 3 (see [15]). *Let be a nonempty subset of a topological vector space and let be a KKM-mapping. If is closed in for all and compact for at least one , then .*

*Definition 4. *Let be a Banach space. A mapping is said to be lower semicontinuous at , if
for any sequence of such that .

*Definition 5. *Let be a Banach space. A mapping is said to be weakly upper semicontinuous at , if
for any sequence of such that .

Now, we extend the definition of relaxed -monotonicity [11] to mixed relaxed --monotonicity.

*Definition 6. *A mapping is said to be mixed relaxed --monotone, if there exist mappings with , for all and , such that
where
and is a constant.

If , then Definition 6 reduces to the definition of generalized relaxed -monotone; that is,
where
If , then Definition 6 reduces to the definition of generalized relaxed -monotone; that is,
where

If both , then Definition 6 coincides with the definition of monotonicity; that is,

*Definition 7. *A mapping is said to be -diagonally convex if, for any finite subset of and with and , one has

#### 3. Existence of Solution for Generalized Equilibrium Problem

We establish this section with the discussion of existence of solution for generalized equilibrium problem by using mixed relaxed --monotonicity.

Theorem 8. *Suppose is mixed relaxed --monotone, hemicontinuous in the first argument and convex in the second argument with , for all . Let be convex in the second argument. Then, generalized equilibrium problem (1) is equivalent to the following problem.**Find such that
**
where and is a constant.*

*Proof. *Suppose that the generalized equilibrium problem (1) admits a solution; that is, there exists such that
Since is mixed relaxed --monotone, we have
Adding on both sides of (17), we have
Hence, is a solution of problem (14).

Conversely, suppose that is a solution of problem (14); that is,
Let , , and ; then clearly as is convex. Thus from (17), we have
Since is convex in the second argument, we have
which implies that

Also as is convex in the second argument, we have

Adding (22) and (24), we have
It follows that
Since is hemicontinuous in the first argument, taking , we have
that is, we have
Hence is a solution of generalized equilibrium problem (1).

Theorem 9. *Let be a nonempty bounded closed convex subset of a real Banach space . Let be a mixed relaxed --monotone, hemicontinuous in the first argument, convex in the second argument with , -diagonally convex, and lower semicontinuous. Let be convex in the second argument, -diagonally convex, and lower semicontinuous; is weakly upper semicontinuous and is weakly upper semicontinuous in the second argument. Then the mixed equilibrium problem (1) admits a solution.*

*Proof. *Consider a multivalued mapping such that
We show that ; that is, is a solution of generalized equilibrium problem (1).

Our claim is that is a KKM-mapping. Suppose to contrary that is is not a KKM-mapping; then there exists a finite subset of and with such that
It follows that
Also we have
which contradicts the -diagonal convexity of and . Hence is a KKM-mapping.

Now consider another multivalued mapping such that
We will show that , . For any given , let ; then
It follows from the mixed relaxed --monotonicity of that
that is, . Thus and consequently is also KKM-mapping.

Since and both are convex in the second argument and lower semicontinuous, thus they both are weakly lower semicontinuous. From weakly upper semicontinuity of , weakly upper semicontinuity of in the second argument, and the construction of , it is accessible to see that is weakly closed for all . Since is closed, bounded, and convex, it is weakly compact and consequently is weakly compact in for all . Therefore, from Lemma 3 and Theorem 8, we have
that is, there exists such that
Thus, the generalized equilibrium problem (1) admits a solution.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors express their sincere thanks to the referees for the careful and detailed reading of the paper and the very helpful suggestions that improved the paper substantially. The authors also acknowledge that this research was part of the research project and was partially supported by Universiti Putra Malaysia under ERGS 1-2013/5527179.

#### References

- M. Bai, S. Zhou, and G. Ni, “Variational-like inequalities with relaxed $\eta $-$\alpha $ pseudomonotone mappings in Banach spaces,”
*Applied Mathematics Letters*, vol. 19, no. 6, pp. 547–554, 2006. View at: Publisher Site | Google Scholar | MathSciNet - Y. Chen, “On the semi-monotone operator theory and applications,”
*Journal of Mathematical Analysis and Applications*, vol. 231, no. 1, pp. 177–192, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. P. Fang and N. J. Huang, “Variational-like inequalities with generalized monotone mappings in Banach spaces,”
*Journal of Optimization Theory and Applications*, vol. 118, no. 2, pp. 327–338, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Karamardian and S. Schaible, “Seven kinds of monotone maps,”
*Journal of Optimization Theory and Applications*, vol. 66, no. 1, pp. 37–46, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Ellaia and A. Hassouni, “Characterization of nonsmooth functions through their generalized gradients,”
*Optimization*, vol. 22, no. 3, pp. 401–416, 1991. View at: Publisher Site | Google Scholar | MathSciNet - S. Karamardian, S. Schaible, and J. Crouzeix, “Characterizations of generalized monotone maps,”
*Journal of Optimization Theory and Applications*, vol. 76, no. 3, pp. 399–413, 1993. View at: Publisher Site | Google Scholar | MathSciNet - S. Komlósi, “Generalized monotonicity and generalized convexity,”
*Journal of Optimization Theory and Applications*, vol. 84, no. 2, pp. 361–376, 1995. View at: Publisher Site | Google Scholar | MathSciNet - 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 - M. A. Noor and W. Oettli, “On general nonlinear complementarity problems and quasi-equilibria,”
*Le Matematiche*, vol. 49, no. 2, pp. 313–331, 1994. View at: Google Scholar | MathSciNet - S. Chang, B. S. Lee, and Y. Chen, “Variational inequalities for monotone operators in nonreflexive Banach spaces,”
*Applied Mathematics Letters*, vol. 8, no. 6, pp. 29–34, 1995. View at: Publisher Site | Google Scholar | MathSciNet - N. K. Mahato and C. Nahak, “Equilibrium problems with generalized relaxed monotonicities in Banach spaces,”
*Opsearch*, vol. 51, no. 2, pp. 257–269, 2014. View at: Publisher Site | Google Scholar | MathSciNet - R. U. Verma, “On monotone nonlinear variational inequality problems,”
*Commentationes Mathematicae Universitatis Carolinae*, vol. 39, no. 1, pp. 91–98, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. U. Verma, “On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators,”
*Journal of Mathematical Analysis and Applications*, vol. 213, no. 1, pp. 387–392, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. A. Noor, “Auxiliary principle technique for equilibrium problems,”
*Journal of Optimization Theory and Applications*, vol. 122, no. 2, pp. 371–386, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Fan, “A generalization of Tychonoff's fixed point theorem,”
*Mathematische Annalen*, vol. 142, pp. 305–310, 1961. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Haider Abbas Rizvi 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.