Inclusion and Intersection Theorems With Applications in Equilibrium Theory in G-Convex Spaces

Abstract

Abstract. In this paper we obtain a very general theorem of ρ-compatibility for three multivalued mappings, one of them from the class B. More exactly, we show that given a G-convex space Y , two topological spaces X and Z, a (binary) relation ρ on 2 Z and three mappings P : X Z, Q : Y Z and T ∈ B(Y, X) satisfying a set of conditions we can find (e x, e y) ∈ X × Y such that e x ∈ T (e y) and P (e x)ρ Q(e y). Two particular cases of this general result will be then used to establish existence theorems for the solutions of some general equilibrium problems.

An intersection theorem for set-valued mappings

“…In 1961 [8], Ky Fan extended the famous Knaster-Kuratowski-Mazurkiewicz (simply, KKM) theorem to arbitrary topological vector spaces obtaining a remarkable intersection theorem for the family of values of a set-valued mapping. Since then many intersection theorems have appeared (see [1], [5], [7], [13], [15], [17]). The main theorem of our paper fits into this interesting group of results.…”