An intersection theorem in L-convex spaces with applications

Abstract

A new intersection theorem is obtained in L-convex spaces without linear structure. As its applications, a fixed point theorem, a maximal element theorem, a coincidence theorem, some new minimax inequalities and a saddle point theorem are given in L-convex spaces. Our results generalize many known theorems in the literature.  2005 Published by Elsevier Inc.

A minimax inequality in L-convex spaces and its applications

“…Since D is compact, by (ii) and Proposition 3.1.11 of [1], the set F(D,y) = Ux ED F(x,y) is compact in Z.…”

“…In this paper, by Corollary [1], we establish Ky Fan's section theorem and apply this section theorem to minimax inequality, maximal element theorem. As applications, we obtain some new existence theorems for abstract generalized vector equilibrium problems in L-convex spaces.…”

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.…”