Generalized Fan-Browder Fixed Point Theorem And Nash Equilibrium Problems

Both hyperconvex metric space and G-convex space are the abstract spaces without linear structure. But hyperconvex metric space and linear space can not be contained each other. Although there is not linear structure which is in linear space in hyperconvex metric space and G-convex space, there is the notion of admissible set in hyperconvex convex space and the notion of G- convexity in G- convex space. Therefore, with regard to the fixed point theorem, coincide point theorem, section theorem, largest element theorem, saddle point theorem, maximum and minimum inequality, Ky Fan inequality, match theorem, best approximation theorem etc. in Hausdorff topology linear space, there are correspondence results in hyperconvex metric space and G-convex space. In this paper, we discuss the existence of the solution and the application of a generalized Ky Fan inequality in hyperconvex metric space and G-convex space; and use the Nash equilibrium conclusion in game theory to analyze some practical economic phenomenon.The content of this paper as follow:This paper contains five parts. In section 1, the introduction section, we explained the research purpose, the sense, the overview and the main research matter of this article. In section 2, the existence of the solution of a generalized Ky Fan inequality in hyperconvex spaces is discussed by quoting the notion of strongly path transfer lower semicontinuous (SPTl.s.c). And prove a generalized Fan-Browder fixed point theorem. Then obtain a section theorem and a largest element theorem in hyperconvex spaces. In section 3, the existence of the solution of a generalized Ky Fan inequality in G-convex spaces is discussed. Then obtain a generalized Fan-Browder fixed point theorem, saddle theorem and Nash equilibrium existence theorem. In section 4, by using the Fort theorem, the stability of the set of KKM points in G-convex space is studied, and so the generic stability and the existence of one essential component of the set of KKM points in G-convex space is proved. In section 5, it's about the application of the inlinear functional analysis, especially apply the Nash equilibrium conclusion in game theory to economics and management. In this chapter, we use the Nash equilibrium conclusion in game theory to analyze the venal and bribery phenomenon in revenue process, and put forward some feasible measure to clean off the lawless dealing, and compute the Nash equilibrium point of a practical economic question with emulating data and analyze the result.