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]).
Recently, the equilibrium problem has been extensively generalized to the vector mappings (see [6–8, 10–16]). Let X and Y be real topological vector spaces and K be a nonempty subset of X. Let C be a closed and convex cone in Y with , where intC denotes the topological interior of C. For a vector-value function , at least two different vector equilibrium problems are the following problems:
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.
For a more general form of vector equilibrium problem, we let be a multivalued map with nonempty values where denotes the family of subsets of K. Then we consider the following problem: find such that
It is known that a vector quasi-equilibrium problem (for short, VQEP) was introduced by Ansari et al. . If the mapping F is replaced by a multivalued map, saying with Y being a topological vector space, VEP can be generalized in the following way: find such that
It is called generalized vector equilibrium problem (for short, GVEP) and it has been studied by many authors; see, for example, [14, 15, 21, 22, 26] and references therein. In 2003, Ansari and Flores-Bazás  introduced the generalized vector quasi-equilibrium problem (for short, GVQEP): find such that
which is a general form of GVEP; more examples can also be found in [13–15, 26]. In another way, Kum and Wong  considered the multivalued generalized system (for short, MGS): find such that
Throughout this paper, unless otherwise specified, we assume that X and Y are Hausdorff topological vector spaces, K is a nonempty convex subset of X and C is a pointed closed convex cone in Y with . For a given multivalued bi-operator such that for each , where denotes the family of subsets of Y, the new type of generalized strong vector quasi-equilibrium problem (for short, GSVQEP) is the problem to find such that
where is a multivalued map with nonempty values. If we set for all , then the GSVQEP reduces to the following generalized quasi-variational like inequality problem (for short, GQVLIP): find such that
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.
Let X and Y be two topological vector spaces, and let be a set-valued mapping.
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 .
T is said to be hemicontinuous if, for any given and for , the mapping is continuous at 0+;
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 .
Lemma 2.4LetXbe a topological space andYbe a set. Letbe a map with nonempty values. Then the following are equivalent:
Thas the local intersection property;
There exits a mapsuch thatfor each , is open for eachand .
Subsequently, Browder  obtained in 1986 the following fixed point theorem.
Theorem 2.5 (Fan-Browder fixed point theorem)
LetXbe a nonempty compact convex subset of a Hausdorff topological vector space andbe a map with nonempty convex values and open fibers (i.e., for , is called the fiber ofTony). ThenThas a fixed point.
The generalization of the Fan-Browder fixed point theorem was obtained by Balaj and Muresan  in 2005 as follows.
Theorem 2.6LetXbe a compact convex subset of a topological vector space andbe a map with nonempty convex values having the local intersection property. ThenThas a fixed point.
LetXbe a bounded subset ofE. Then the usual pairingis 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.
Lemma 3.1LetKbe a nonempty and convex subset ofX, letbe a set-valued mapping such that for any , is a nonempty convex subset ofKand letbe g.h.c. in the first argument, C-convex in the second argument andC-strongly pseudomonotone. Then the following problems are equivalent:
Findsuch that , , .
Findsuch that , , .
Proof (i) → (ii) It is clear by the C-strong pseudomonotonicity.
(ii) → (i) Let . For any and , we set . By the assumption (ii) and the convexity of , we conclude that
Since F is C-convex in the second argument, we have
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 3.2LetKbe a nonempty compact convex subset ofX. Letbe a set-valued mapping such that for any , is a nonempty convex subset ofKand for each , is open inK. Let the setbe closed. Assume thatisC-strongly pseudomonotone, g.h.c. in the first argument, C-convex and l.s.c. in the second argument. Then GSVQEP has a solution.
Proof For any , we define the set-valued mapping by
and for any , we denoted the complement of by . For each , we define multivalued maps by
Clearly, and are nonempty sets for all , and by the C-strong pseudomonotonicity of F, we have for all . We claim that is convex. Let and . Since F is C-convex in the second argument, we have
Then we have is convex and so is convex by the convexity of . Next, we will show that is open in K for each . Since F is l.s.c. in the second argument and by the definition of , we have closed and so is open in K. By assumption, we obtain that
is open in K. It is easy to see that the mapping H has no fixed point because , . From the contrapositive of the generalization of the Fan-Browder fixed point theorem and Lemma 2.4, we have
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.
Example 3.3 Let , and . Define the mapping and by
respectively. By the definition of A, see Figure 1, we have the set which is closed and for each , is open in K.
We see that F is C-strongly pseudomonotone. Indeed, if , then we only consider the case , so . That is,
Let and . If , then
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 3.4LetKbe a nonempty compact convex subset ofX, letbe a set-valued mapping such that for each , is a nonempty convex subset ofK, and let the setbe closed. Assume thatisC-convex in the second argument and for each , the setis open. Then GSVQEP has a solution.
Proof We proceed with the contrary statements, that is, for each , or there exists such that
For every , we define the sets and as follows:
By the assumption, we have the set is open in K and we see that is an open cover of K. Since K is compact, there exists a finite subcover such that . By a partition of unity, there exists a family of real-valued continuous functions subordinate to such that for all , and and for each , . Let . Then C is a simplex of a finite dimensional space. Define a mapping by
Hence, we have S is continuous since is continuous for each i. From Brouwer’s fixed point theorem, there exists such that . We define a set-valued mapping by
Now, we note that for any , . Since F is C-convex in the second argument, it follows from (3.1), (3.2) and (3.3) that we have
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 3.5LetKbe a nonempty compact convex subset ofX, letbe a set-valued mapping such that for any , is a nonempty convex subset ofKand for each , is open inK. Let the setbe closed, letbe affine and continuous in the first argument and hemicontinuous in the second argument, and letbe aC-strongly pseudomonotone and g.h.c. with nonempty compact values whereis 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 3.6LetKbe a nonempty compact convex subset ofX. Letbe a set-valued mapping such that for each , is a nonempty convex subset ofKand let the setbe closed. Assume thatis affine in the first argument andis a nonlinear mapping such that, for every , the setis open. Then GQVLIP has a solution.
Fan K: Minimax theorems. Proc. Natl. Acad. Sci. USA 1953, 39: 42–47. 10.1073/pnas.39.1.42
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between all authors. SP designed theorems and methods of the proof and interpreted the results. KS proved the theorems, interpreted the results and wrote the paper. All authors read and approved the final manuscript.
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Sitthithakerngkiet, K., Plubtieng, S. Existence theorems of an extension for generalized strong vector quasi-equilibrium problems.
Fixed Point Theory Appl2013, 342 (2013). https://doi.org/10.1186/1687-1812-2013-342