On the symmetric vector quasi-equilibrium problems

Abstract

In this paper we, using a particular technique, consider the symmetric vector quasi-equilibrium problems in the Hausdorff topological vector space. As applications of our existence theorem, a coincidence point theorem and the existence of vector optimization problem for a pair of vector-valued mappings are obtained. Moreover, we answer an open question raised by Fu in [J.Y. Fu, Symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl. 285 (2003) 708-713].

On the convexity of the solution set of symmetric vector equilibrium problems

“…then (2.1) reduces to a single-valued symmetric vector equilibrium problem considered in [17][18][19].…”

“…On the other hand, the symmetric vector equilibrium problem is a generalization of the equilibrium problem which has been studied by many authors. A main topic of the current research is to establish existence theorems for a solution of the symmetric vector equilibrium problem(see, for examples, [17][18][19]23]) and some important properties of the solution set to the symmetric vector equilibrium problem.…”

Existence and Well-Posedness for Symmetric Vector Quasi-Equilibrium Problems

“…In this paper, we will introduce existence and wellposed theorem of (SVQEP) for discontinuous vector-valued mapping, which extend the corresponding result in [12] in metric space. Then, by using the conditions of the existence theorem of the solutions to (SVQEP) in [14], we obtain sufficient conditions for the generalized Levitin-Polyak wellposedness of (SVQEP), which improve the result of [16,Theorem 4.1].…”

“…(SVQEP) is a generalization of the equilibrium problem, proposed by Blum and Oettli [13], and a unified model of several problems, for example, vector optimization problems, problems of vector Nash equilibria, vector variational inequalities, and vector complementarity problems. Farajzadeh [14] considered existence theorem of the solution of symmetric vector quasi-equilibrium problems in the Hausdorff topological vector space and answered the open question raised by Fu [12]. In [15], Li et al obtained existence results for two classes of generalized vector quasiequilibrium problems.…”

On the existence and stability of solutions to stochastic equilibrium problems

“…Therefore, properties of solutions to equilibrium problem and its related-problems are worthwhile topics to discuss, and they have been studied by many mathematicians all over the world (see, e.g. [30][31][32][33] and the references therein). To the best of our knowledge, conditions for the existence and stability of solutions to stochastic equilibrium problems have not been under investigation so far despite of their importance in practice.…”