About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 374268, 5 pages
http://dx.doi.org/10.1155/2013/374268
Research Article

Mixed Equilibrium Problems with Weakly Relaxed α-Monotone Bifunction in Banach Spaces

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand

Received 28 May 2013; Accepted 1 August 2013

Academic Editor: Sompong Dhompongsa

Copyright © 2013 Wutiphol Sintunavarat. 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 introduce the class of mixed equilibrium problems with the weakly relaxed α-monotone bi-function in Banach spaces. Using the KKM technique, we obtain the existence of solutions for mixed equilibrium problem with weakly relaxed α-monotone bi-function in Banach spaces. The results presented in this paper extend and improve the corresponding results in the existing literature.

1. Introduction

Let be a nonempty subset of a real reflexive Banach space . Let be a real valued function and let be an equilibrium bi-function; that is, , for all . Then the mixed equilibrium problem (for short, (MEP)) is to find such that In particular, if , this problem reduces to the classical equilibrium problem (for short, (EP)), which is to find such that

Equilibrium problems and mixed equilibrium problems play an important role in many fields, such as economics, physics, mechanics, and engineering sciences. Also, the equilibrium problems and mixed equilibrium problems include many mathematical problems as particular cases for example, mathematical programming problems, complementary problems, variational inequality problems, Nash equilibrium problems in noncooperative games, minimax inequality problems, and fixed point problems. Because of their wide applicability, equilibrium problems and mixed equilibrium problems have been generalized in various directions for the past several years.

The monotonicity and generalized monotonicity play an important role in the study of equilibrium problems and mixed equilibrium problems. In recent years, a substantial number of papers on existence results for solving equilibrium problems and mixed equilibrium problems based on different generalization of monotonicity such as pseudomonotonicity, quasimonotonicity, relaxed monotonicity, semimonotonicity, -monotonicity, and -pseudomonotonicity, see [117], appeared.

In 1990, Karamardian and Schaible [14] introduced various kinds of generalized monotone mappings. Afterward, several researcher in [24] extended the idea of Karamardian and Schaible [14] for bi-functions to study equilibrium problems.

In 2003, Fang and Huang [10] considered two classes of the variational-like inequalities with the relaxed monotone and relaxed semimonotone mappings. They obtained the existence solutions of variational-like inequalities with relaxed monotone and relaxed semimonotone mappings in Banach spaces using the KKM technique. Later, Bai et al. [1] introduced a new concept of relaxed pseudomonotone mappings and obtained the solutions for the variational-like inequalities. Afterward, Mahato and Nahak [18] defined the weakly relaxed pseudomonotone bi-function to study the equilibrium problems.

Recently, Mahato and Nahak [19] introduce the concept of the relaxed -monotonicity for bi-functions. They also obtained the existence of solutions for mixed equilibrium problems with the relaxed -monotone bi-function in reflexive Banach spaces, by using the KKM technique.

The purpose of this paper is to introduce the class of weakly relaxed -monotone bi-functions which contain the class of relaxed -monotone bi-functions. The existence of solutions for mixed equilibrium problems with bi-function in such class is given. Our results in this paper extend and improve the results of Mahato and Nahak [19] and many results in the literature.

2. Preliminaries

In this paper, unless otherwise specified, let be a nonempty closed convex subset of a real reflexive Banach space . The following definitions and lemma will be useful in our paper.

Definition 1. A real valued function defined on a convex subset of is said to be hemicontinuous if , for each .

Definition 2. Let be a set-valued mapping. Then is said to be KKM mapping if for any finite subset of , we have , where denotes the convex hull of .

Remark 3. Let . If is KKM mapping and for all , then also is KKM mapping.

Lemma 4 (see [20]). Let be a nonempty subset of a Hausdorff topological vector space and let be a KKM mapping. If is closed in for all and compact for some , then .

Definition 5. Let be a Banach space. A function is lower semicontinuous at if for any sequence in such that converges to .

Definition 6. Let be a Banach space. A function is weakly upper semicontinuous at if for any sequence in such that converges to weakly.

Definition 7 (see [19]). A bi-function is said to be a relaxed -monotone if there exists a function with for all and such that where is a real constant.

Remark 8. If then from (5), it follows that is monotone; that is, Therefore, the monotonicity implies relaxed -monotonicity. However, the converse of previous statement is not true in general, which is shown by the following examples.

Example 9. Let , and let bi-function be defined by for all . Then when . So is not monotone.
However, it easy to see that is a relaxed -monotone with . In fact,

Example 10. Let . For , is a reflexive Banach space. Let , which is nonempty, closed, and convex subset of . Define by Then, for , that is, is not monotone mapping.
But, if we choose by , then is a relaxed -monotone.

3. Mixed Equilibrium Problems with Weakly Relaxed -Monotone Bi-Function

In this section, we introduce the new class of bi-functions. Using KKM technique, we study and prove the existence of solutions for mixed equilibrium problem with bi-function in such class in Banach spaces.

Definition 11. A bi-function is said to be a weakly relaxed -monotone if there exists a function with for all and such that

Remark 12. We obtain that the relaxed -monotonicity implies weakly relaxed -monotonicity. So the class of relaxed -monotone bi-functions is a subclass of weakly relaxed -monotone bi-function class.

Next, we discuss the existence solution of the (MEP) (1), using the concept of the weakly relaxed -monotonicity.

Theorem 13. Suppose is a weakly relaxed -monotone which is hemicontinuous in the first argument, and convex in the second argument let be a convex function. Then, the (MEP) and the following problem are equivalent:

Proof. Suppose that the (MEP) (1) has a solution. So there exists such that Since is, weakly relaxed -monotone, we have and then Therefore, is a solution of problem (15).
Conversely, suppose is a solution of problem (15) and is any point in . For , we let . Since is convex, we obtain that . From (15), we have By the convexity of in the second argument, we have that is, The convexity of implies that and thus From (19), (21), and (23), we have and so Since is hemicontinuous in the first argument and taking we get From (12), we get is indeterminate form. Using L’ Hopital’s rule, we obtain that By property (13) of weakly relaxed -monotone , we have and then Therefore, is a solution of (MEP).

Theorem 14. Let be a nonempty bounded closed convex subset of a real reflexive Banach space . Suppose that is a weakly relaxed -monotone and hemicontinuous in the first argument; let be a convex and lower semicontinuous function. Assume that (a)for fixed , the mapping is convex and lower semicontinuous; (b) is weakly upper semicontinuous. Then the problem (MEP) has a solution.

Proof. Consider the set valued mapping defined by for all .
It is easy to see that solves the problem (MEP) that is if and only if . Thus it is sufficient to prove that .
Next, we show that is a KKM mapping. Assuming this contrary, then there exists such that . This implies that there exists such that , where , , and , but .
From the construction of , we have By the convexity of in the second variable and the convexity of and (32), we obtain that which is a contradiction. Therefore, is a KKM mapping.
Now we define another set valued mapping such that for all .
Next, we will prove that for all . For each , let ; then From the weakly relaxed -monotonicity of , we get
This implies that and hence for all . So is also a KKM mapping.
By assumption, and are convex lower semicontinuous functions, where fixed . Then it is easy to see that they are both weakly lower semicontinuous. From the definition of and the weakly upper semicontinuity of , we get is weakly closed for all .
Since is closed bounded and convex, it is also weakly compact, and then is weakly compact in for each . From Lemma 4 and Theorem 13, we obtain that So there exists , such that then the problem (MEP) has a solution. This completes the proof.

It easy to see that the relaxed -monotonicity implies the weakly relaxed -monotonicity. So Theorem 14 can be deduced to the following corollary.

Corollary 15. Let be a nonempty bounded closed convex subset of a real reflexive Banach space . Suppose that is relaxed -monotone and hemicontinuous in the first argument; let be a convex and lower semicontinuous function. Assume that (a)for fixed , the mapping is convex and lower semicontinuous; (b) is weakly upper semicontinuous. Then the problem (MEP) has a solution.

Next, we study and prove result for the case of is unbounded set.

Theorem 16. Let be a nonempty unbounded closed convex subset of a real reflexive Banach space . Suppose that is a weakly relaxed -monotone and hemicontinuous in the first argument; let be a convex and lower semicontinuous function. Assume that (a)for fixed , the mapping is convex and lower semicontinuous; (b) is weakly upper semicontinuous. (c) satisfied the weakly coercivity condition, that is, there exists such that whenever and is large enough.
Then the problem (MEP) has a solution.

Proof. For , define .
Consider the problem: find such that Since is bounded, by Theorem 14, we get that the problem (40) has at least one solution .
For in the weakly coercivity condition (c), we choose . From (40), we have Since , we have . If , we may choose large enough so that by the weakly coercivity condition (c), we get which contradicts (41). Therefore, we must have such that .
For each , we can choose small enough such that . From (40), for each , we have This implies that Therefore, the problem (MEP) has a solution. This completes the proof.

Corollary 17. Let be a nonempty unbounded closed convex subset of a real reflexive Banach space . Suppose that is relaxed -monotone and hemicontinuous in the first argument; let be a convex and lower semicontinuous function. Assume that (a)for fixed , the mapping is convex and lower semicontinuous; (b) is weakly upper semicontinuous;(c) satisfied the weakly coercivity condition; that is, there exists such that whenever and large enough.
Then the problem (MEP) has a solution.

Remark 18. Theorems 13, 14, and 16 are improving the results of Fang [10] from the corresponding results of variational-like inequality problems to equilibrium problems. Also, these results are extensions of the main results of Mahato and Nahak [19].

References

  1. M. R. Bai, S. Z. Zhou, and G. Y. Ni, “Variational-like inequalities with relaxed η-α pseudomonotone mappings in Banach spaces,” Applied Mathematics Letters, vol. 19, no. 6, pp. 547–554, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Bianchi and S. Schaible, “Generalized monotone bifunctions and equilibrium problems,” Journal of Optimization Theory and Applications, vol. 90, no. 1, pp. 31–43, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Bianchi and S. Schaible, “Equilibrium problems under generalized convexity and generalized monotonicity,” Journal of Global Optimization, vol. 30, no. 2-3, pp. 121–134, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Bianchi and R. Pini, “A note on equilibrium problems with properly quasimonotone bifunctions,” Journal of Global Optimization, vol. 20, no. 1, pp. 67–76, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. L. C. Ceng and J. C. Yao, “On generalized variational-like inequalities with generalized monotone multivalued mappings,” Applied Mathematics Letters, vol. 22, no. 3, pp. 428–434, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Q. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. W. Cottle and J. C. Yao, “Pseudomonotone complementarity problems in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 75, no. 2, pp. 281–295, 1992. View at Publisher · View at Google Scholar
  8. X. P. Ding, “Existence and algorithm of solutions for generalized mixed implicit quasi-variational inequalities,” Applied Mathematics and Computation, vol. 113, no. 1, pp. 67–80, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K. Fan, “Some properties of convex sets related to fixed point theorems,” Mathematische Annalen, vol. 266, no. 4, pp. 519–537, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  12. R. Glowinski, J. L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, The Netherlands, 1981. View at MathSciNet
  13. D. Goeleven and D. Motreanu, “Eigenvalue and dynamic problems for variational and hemivariational inequalities,” Communications on Applied Nonlinear Analysis, vol. 3, no. 4, pp. 1–21, 1996. View at Zentralblatt MATH · View at MathSciNet
  14. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. U. Verma, “On monotone nonlinear variational inequality problems,” Commentationes Mathematicae Universitatis Carolinae, vol. 39, no. 1, pp. 91–98, 1998. View at Zentralblatt MATH · View at MathSciNet
  17. X. Q. Yang and G. Y. Chen, “A class of nonconvex functions and pre-variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 359–373, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. N. K. Mahato and C. Nahak, “Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces,” Journal of Applied Mathematics and Computing, vol. 40, no. 1-2, pp. 499–509, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. N. K. Mahato and C. Nahak, “Mixed equilibrium problems with relaxed α-monotone mapping in Banach spaces,” Rendiconti del Circolo Matematico di Palermo, 2013. View at Publisher · View at Google Scholar
  20. K. Fan, “A generalization of Tychonoff's fixed point theorem,” Mathematische Annalen, vol. 142, pp. 305–310, 1961. View at MathSciNet