Existence theorems of an extension for generalized strong vector quasi-equilibrium problems
© Sitthithakerngkiet and Plubtieng; licensee Springer. 2013
Received: 27 August 2013
Accepted: 21 November 2013
Published: 13 December 2013
In this paper, we study generalized strong vector quasi-equilibrium problems in topological vector spaces. Using the generalization of Fan-Browder fixed point theorem, we provide existence theorems for an extension of generalized strong vector quasi-equilibrium problems with and without monotonicity. The results in this paper generalize, extend and unify some well-known existence theorems in literature.
MSC:49J30, 49J40, 47H10, 47H17.
The minimax inequalities of Fan  are fundamental in proving many existence theorems in nonlinear analysis. Their equivalence to the equilibrium problems was introduced by Takahashi [, Lemma 1] Blum and Oettli  and Noor and Oettli . The equilibrium problem theory provides a novel and united treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization. This theory has had a great impact and influence in the development of several branches of pure and applied sciences. During this period, many results on existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, for example, [5–12]).
The first problem is called weak vector equilibrium problem (see, for instance, [13, 17, 18]) and the second one is normally called strong vector equilibrium problem (see ). However, Kazmi and Khan  called problem (1.2) the generalized system (for short, GS). Recently, many existence results extended and improved WVEP and its particular cases (see, for instance, [21–25]), but not VEP.
where is a multivalued mapping, is a nonlinear mapping and is denoted by the space of all continuous linear operators for X to Y. This above formulation is the generalization of vector variational inequalities, variational-like inequality problems and vector complementarity problems in infinite dimensional spaces studied by many authors (see [28–30] and references therein).
The main motivation of this paper is to establish some existence results for a solution to the new type of the generalized strong vector quasi-equilibrium problems GSVQEP with and without monotonicity by using the generalization of Fan-Browder fixed point theorem.
Let us recall some definitions and lemmas that are needed in the main results of this paper.
Definition 2.1 
T is said to be upper semicontinuous at if for each and each open set V in Y with , there exists an open neighborhood U of x in X such that for each .
T is said to be lower semicontinuous at if for each and each open set V in Y with , there exits an open neighborhood U of x in X such that for each .
T is said to be continuous on X if it is at the same time upper semicontinuous and lower semicontinuous on X. It is also known that is lower semicontinuous if and only if for each closed set V in Y, the set is closed in X.
T is said to be closed if the graph of T, i.e., , is a closed set in .
Definition 2.2 
- (i)A multivalued bi-operator is said to be C-strongly pseudomonotone if it satisfies
- (ii)A multivalued mapping is said to be C-convex if for all and for all ,And the mapping G is said to be generalized hemicontinuous (in short, g.h.c.) if for all and for all ,
Definition 2.3 
T is said to be hemicontinuous if, for any given and for , the mapping is continuous at 0+;
- (ii)T is said to be C-η-strongly pseudomonotone if, for any ,
η is said to be affine in the second argument if, for any and (), with and any , .
The following lemma is useful in what follows and can be found in .
T has the local intersection property;
There exits a map such that for each , is open for each and .
Subsequently, Browder  obtained in 1986 the following fixed point theorem.
Theorem 2.5 (Fan-Browder fixed point theorem)
Let X be a nonempty compact convex subset of a Hausdorff topological vector space and be a map with nonempty convex values and open fibers (i.e., for , is called the fiber of T on y). Then T has a fixed point.
The generalization of the Fan-Browder fixed point theorem was obtained by Balaj and Muresan  in 2005 as follows.
Theorem 2.6 Let X be a compact convex subset of a topological vector space and be a map with nonempty convex values having the local intersection property. Then T has a fixed point.
Lemma 2.7 
Let X be a bounded subset of E. Then the usual pairing is continuous.
3 Main theorem
In this section, we shall investigate the existence results for GSVQEP and GQVLIP with monotonicity and without monotonicity. First, we present the following lemma which is of Minty’s type for GSVQEP.
Find such that , , .
Find such that , , .
Proof (i) → (ii) It is clear by the C-strong pseudomonotonicity.
Then we have , because C is a convex cone. Since F is g.h.c. in the first argument, we have , , . It implies that , for all . This completes the proof. □
In the following theorem, we present the existence result for GSVQEP by assuming the monotonicity of the function.
Theorem 3.2 Let K be a nonempty compact convex subset of X. Let be a set-valued mapping such that for any , is a nonempty convex subset of K and for each , is open in K. Let the set be closed. Assume that is C-strongly pseudomonotone, g.h.c. in the first argument, C-convex and l.s.c. in the second argument. Then GSVQEP has a solution.
Hence, there exists such that . If , we have , which contradicts the assumptions. Then and hence . This means that and for all . This completes the proof by Lemma 3.1. □
The following example shows that GSVQEP has a solution under the condition of Theorem 3.2.
Similarly, in another case, we have F is C-convex in the second argument. Clearly, F is g.h.c. in the first argument and l.s.c. in the second argument.
Moreover, this example asserts that −0.5 is one of the solutions because if , then . Note that for all , . Therefore for all .
Now, we present an existence theorem for GSVQEP when F is not necessarily monotone.
Theorem 3.4 Let K be a nonempty compact convex subset of X, let be a set-valued mapping such that for each , is a nonempty convex subset of K, and let the set be closed. Assume that is C-convex in the second argument and for each , the set is open. Then GSVQEP has a solution.
for all . Since and it is a fixed point of S, , which is a contradiction. This completes the proof. □
If we set , then Theorem 3.2 and Theorem 3.4 are reduced to Theorem 1 and Theorem 3 in Kum and Wong , respectively. Moreover, Theorem 3.2 is a multivalued version of Theorem 2.3 in Kazmi and Khan .
Let for all , where and . As a consequence of Theorem 3.2 and using the same argument as in Kum and Wang (, Theorem 2), we have the following existence result for GQVLIP.
Corollary 3.5 Let K be a nonempty compact convex subset of X, let be a set-valued mapping such that for any , is a nonempty convex subset of K and for each , is open in K. Let the set be closed, let be affine and continuous in the first argument and hemicontinuous in the second argument, and let be a C-strongly pseudomonotone and g.h.c. with nonempty compact values where is equipped with topology of bounded convergence. Then GQVLIP has a solution.
As a consequence of Theorem 3.4, we obtain the following existence result for GQVLIP.
Corollary 3.6 Let K be a nonempty compact convex subset of X. Let be a set-valued mapping such that for each , is a nonempty convex subset of K and let the set be closed. Assume that is affine in the first argument and is a nonlinear mapping such that, for every , the set is open. Then GQVLIP has a solution.
The first author would like to thank the King Mongkut’s University of Technology North Bangkok, Thailand.
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