Fixed Point Theorems In P-spaces And Applications

The aim of the paper is to investigate fixed point theorems in P-spaces. We firstly give some generalizations of classical fixed point theorems and some equivalent forms of Ky-Fan inequalities in P-spaces. As applications, We obtain e —Nash equilibrium theorems in P-spaces.It consists of three parts.In Part one, we introduce some basic notions and results, including compact spaces, par-compact spaces, partition of unity theorem, simplexes, KKM lemma, semi-continuity of set-valued maps and some famous fixed point theorems with their interconnections.In Part two, we introduce the concept of P-spaces without imposing any explicitly stated linear structure and some notions in P-spaces including half convex sets, convex sets, convex functions and quasi-convex functions.In Part three, we study fixed point theorems in P-spaces. We firstly generalize classical fixed point theorems in P-spaces. Then we obtain Ky Fan's minimax inequality, Ky Fan's section theorem and variation inequality in P-spaces. As applications, we prove the existence of ε—Nash equilibrium point in P-spaces.