Fan-KKM Theorem Research Articles

Existence Results for Strong Mixed Vector Equilibrium Problem for Multivalued Mappings

We consider a strong mixed vector equilibrium problem in topological vector spaces. Using generalized Fan-Browder fixed point theorem (Takahashi 1976) and generalized pseudomonotonicity for multivalued mappings, we provide some existence results for strong mixed vector equilibrium problem without using KKM-Fan theorem. The results in this paper generalize, improve, extend, and unify some existence results in the literature. Some special cases are discussed and an example is constructed.

Extended Mixed Vector Equilibrium Problems

We study extended mixed vector equilibrium problems, namely, extended weak mixed vector equilibrium problem and extended strong mixed vector equilibrium problem in Hausdorff topological vector spaces. Using generalized KKM-Fan theorem (Ben-El-Mechaiekh et al.; 2005), some existence results for both problems are proved in noncompact domain.

Existence of Solutions of New Generalized Mixed Vector Variational-Like Inequalities in Reflexive Banach Spaces

In this paper, we extend the concept of monotonicity for a vector set-valued mapping to semimonotonicity for a vector set-valued mapping. Then, we prove solvability results for a class of new generalized mixed vector variational-like inequalities by applying the Fan-KKM theorem and Nadler's result. On the other hand, we introduce the concepts of complete semicontinuity and strong semicontinuity for vector multivalued mappings. Moreover, by using the Brouwer fixed point theorem, we prove the solvability for the class of generalized vector variational-like inequalities without monotonicity assumption. Using this result, we obtain a theorem and corollary that improve and extend some known results.

Existence Results for Nonsmooth Vector Quasi-Variational-Like Inequalities

We introduce nonsmooth vector quasi-variational-like inequalities (NVQVLI) by means of a bifunction. We establish some existence results for solutions of these inequalities by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in Al-Homidan et al. (2012), one can easily derive the relations among these inequalities and a vector quasi-optimization problem. Hence, the existence results for a solution of a vector quasi-optimization problem can be derived by using our results. The results of this paper extend several known results in the literature.

Ordered Variational Inequalities and Ordered Complementarity Problems in Banach Lattices

We introduce the concepts of ordered variational inequalities and ordered complementarity problems with both domain and range in Banach lattices. Then we apply the Fan-KKM theorem and KKM mappings to study the solvability of these problems.

Existence of solutions of a new system of generalized variational inequalities in Banach spaces

In this article, we consider the solutions of the system of generalized variational inequality problems in Banach spaces. By employing the generalized projection operator, the well-known Fan's KKM theorem and Kakutani-Fan-Glicksberg fixed point theorem, we establish some new existence theorems of solutions for two classes of generalized set-valued variational inequalities in reflexive Banach spaces under some suitable conditions.

Vector mixed quasi complementarity problems in Banach spaces

In this paper, we introduce a new class of set-valued vector implicit quasi complementarity problems with corresponding set-valued implicit quasi variational inequality problems. By means of the Fan–KKM theorem, we investigate the nonemptiness and compactness of solution sets of these problems. Our work generalizes and improves some results appeared recently in the literature.

Generalized vector implicit quasi complementarity problems

In this paper, we introduce and study a generalized class of vector implicit quasi complementarity problem and the corresponding vector implicit quasi variational inequality problem. By using Fan-KKM theorem, we derive existence of solutions of generalized vector implicit quasi variational inequalities without any monotonicity assumption and establish the equivalence between those problems in Banach spaces.

Generalized vector complementarity-type problems in topological vector spaces

In this paper, we introduced the generalized vector variational inequality-type problem and the generalized vector complementarity-type problem in the setting of topological vector space. By utilizing a modified version of the Fan–KKM theorem, we investigated the nonemptyness and compactness of solution sets of these problems without the demipseudomonotonicity assumption. Further, we prove that solution sets of both the problems are equivalent to each other under some suitable conditions. The results of this paper generalize and improve several results that appeared recently in the literature.

Existence theorems for generalized vector variational inequalities with a variable ordering relation

In this paper we study the solvability of the generalized vector variational inequality problem, the GVVI problem, with a variable ordering relation in reflexive Banach spaces. The existence results of strong solutions of GVVIs for monotone multifunctions are established with the use of the KKM-Fan theorem. We also investigate the GVVI problems without monotonicity assumptions and obtain the corresponding results of weak solutions by applying the Brouwer fixed point theorem. These results are also the extension and improvement of some recent results in the literature.

Existence of solutions of vector variational inequalities and vector complementarity problems

In this paper, we consider vector variational inequality and vector F-complementarity problems in the setting of topological vector spaces. We extend the concept of upper sign continuity for vector-valued functions and provide some existence results for solutions of vector variational inequalities and vector F-complementarity problems. Moreover, the nonemptyness and compactness of solution sets of these problems are investigated under suitable assumptions. We use a version of Fan-KKM theorem and Dobrowolski's fixed point theorem to establish our results. The results of this paper generalize and improve several results recently appeared in the literature.

Existence of solutions of generalized variational inequalities in reflexive Banach spaces

In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E ∗ , a multifunction T : K → 2 E ∗ , and a vector b ∈ E ∗ , find x ̄ ∈ K such that there exists u ̄ ∈ T ( x ̄ ) satisfying 〈 u ̄ − b , y − x ̄ 〉 ≥ 0 , for all y ∈ K . By using generalized projection and the well-known Fan–KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.

Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.

Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.

Generalized vector quasi-equilibrium problems with set-valued mappings

This paper is devoted to study a new class of generalized vector quasi-equilibrium problems with set-valued mappings. By means of the Fan–KKM Theorem and lower semicontinuity with respect to cone order of the set-valued mapping, we obtain an existence result for this class of generalized vector quasi-equilibrium problems with set-valued mappings. Our result extents and improves some previous results.

Generalized Vector Variational-Like Inequalities

Using a generalized Fan's KKM theorem, some existence results for generalized vector variational-like inequalities in noncompact settings are established. Some applications to vector optimization problems are given. The results presented in this paper extend and unify corresponding results of other authors.

On the existence of Nash equilibriums for infinite matrix games

The fundamental theorem of game theory states that for every matrix game with finite strategies there exists at least one optimal strategy. It is known that the fundamental theorem of game theory does not hold, in general, in infinite matrix games. In this paper, we provide a characteristic of Nash equilibriums for ∞ × ∞ matrix games and prove an existence theorem of optimal strategies by using the Fan–KKM theorem. We give some applications of these theorems.

A Fuzzy Extension of Generalized Implicit Vector Variational-like Inequalities

In this paper, we study the existence of solutions to the generalized implicit vector variational like inequality problems with fuzzy mappings in Hausdorff topological vector spaces using KKM-Fan Theorem.

Vector [formula omitted]-implicit complementarity problems with corresponding variational inequality problems

In this work, we present some new existence theorems for solutions for a new class of generalized vector F -implicit complementarity problems and the corresponding generalized vector F -implicit variational inequality problems by using the Fan–KKM Theorem under some suitable assumptions without monotonicity, and establish the equivalence between those problems in Banach spaces.

Generalized Vector Variational Inequalities in Topological Vector Spaces

The solvability of the generalized weak vector implicit variational inequality problem, generalized strong vector implicit variational inequality problem and generalized vector variational inequality problem are proved by using a generalized Fan's KKM theorem. Our results extend and unify corresponding results of other authors.