Vector Equilibrium Problems Under Asymptotic Analysis

Abstract

Given a closed convex set K in Rn; a vector function F:K×K → Rm; a closed convex (not necessarily pointed) cone P(x) in \Rm with non-empty interior, PP(x) ≠ O, various existence results to the problem find \bar x∈K such that F(\bar x,y) \not∈-{ int} P(\bar x) \forall y∈K, under P(x)-convexity/lower semicontinuity of F(x,c) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=\Rm+ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.