ON THE CONVEXITY OF THE SOLUTION SET OF SYMMETRIC VECTOR EQUILIBRIUM PROBLEMS

Abstract

In this paper, we rst introduce symmetric vector and scalar equilibrium
problems in the setting of vector spaces ( that is, without using topological structure)
and then provide sucient conditions which under them their solution sets are equal.
After that we are going to study convexity of the solution sets of the symmetric vector
equilibrium problems for set-valued mappings in the setting of topological vector spaces.
Finally, an existence theorem for a solution of the symmetric vector equilibrium problem
for set-valued mappings is presented which our method is dierent from [37]. The results of
this note extend the corresponding results given in [9, 28, 33, 36, 37] from locally convex
spaces to topological vector spaces without assuming boundedness or monotonicity on
the maps. Moreover, the coercivity condition presented in this article, in order to relax
compactness, is more general than that given in [37]. Also one can consider the existence
result obtained in this article as the set valued version of the existence result presented in
[18, 19] and [17, 22] (by omitting the compactness on the sets).