Vector Equilibrium Problems Under Asymptotic Analysis

摘要

Given a closed convex set K in R~n; a vector function F : K x K → R~m; a closed convex (not necessarily pointed) cone P(x) in R~m with non-empty interior, int P(x) ≠ Φ, various existence results to the problem find x ∈ K such that F(x, y) is not an element of -int P(x) any y ∈ K, under P(x)-convexity/lower semicontinuity of F(x, ·) 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 nonempty 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) = R_+~m 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.