A Generalized KKM Theorem And Its Applications

Nonlinear analysis mainly studies on the nonlinear questions in some fields. The discussion of existence is an important aspect of nonlinear analysis. The theorem of KKM and its skill has an important role in the discussion of the existence. This paper mainly focuses on hot questions in nonlinear functional analysis, and generalizes some results in previous studies. This paper has two main parts:1. Under a set-valued mapping condition, the previous KKM theorem is given in H-metric space. A general KKM theorem (Non-empty intersection theorem), in this paper, are given in H-metric space with two extreme conditions. Some applications are obtained on the basis of KKM theorem. Fixed point theorems in H-metric space, minimax inequalities, existence theorems of maximal element, section theorems, existence theorems for abstract generalized vector balance equilibrium point, and existence theorems of upper and lower bounds of the balance. The former results were given under one extreme conditions. The paper obtains the further results by changing one extreme condition into two extreme conditions.2.The above-mentioned theory can be extended to the G-convex metric space by giving the generalized KKM theorem in G-convex metric space. Some applications in G-convex metric space are gotten based on KKM theorem. New fixed point theorems in the G-convex metric space, minimax inequalities, new maximal element existence theorems, new section theorems which are mapping in H-metric space can be obtained under some conditions. The advanced results are obtained by changing conditions in H-metric space.