KKM Theorems And Their Applications And Abstract Generalized Vector Equilibrium Problem

This paper mainly further studies several focus problems in nonlinear functional analysis, unites and generalizes some known results in recent literature.At first, we introduce a class of generalized S-R-KKM type mapping in G-convex space, and establish generalized S-R-KKM type nonempty intersection theorem under the noncompact setting of G-convex space. As for application, some new minimax inequalities, saddle point theorem and existence theorem of maximal elements are proved in G-convex spaces; second, by using the generalized R-KKM mapping and generalized R-KKM theorems in [13], some new existence theorem of maximal elements, existence theorem of equilibrium point for the abstract generalized vector equilibrium problem and existence theorem of solutions for equilibrium problem with lower and upper bounds are obtained in topological spaces.At the same time, we especially study a class of abstract generalized vector equilibrium problem (in short, AGVEP) in G-convex space, using the generalized S-R-KKM type theorem, we establish some new existence theorems of equilibrium points for the AGVEP in noncompact G-convex spaces.Then, we obtain some new continuous selection theorems in G-convex spaces. Through the use of it, some new coincidence theorems are proved in noncompact G-convex spaces or in product G-convex spaces.