Fixed Point Theorem Of Generalized ?-?-Contractive Mapping And Its Application

Nonlinear functional analysis is one of the important branches of modern mathematics.It mainly discusses the continuity,boundedness,differentiability and complete continuity of nonlinear operators,and introduces the implicit function theorem and inverse function theorem,Some topological degree theories,such as Brouwer degree of continuous image in finite dimension space,Leray Schauder degree of complete continuous field in infinite dimension Banach space and topological degree of condensed field,are established.It is effective to study the positive solution of nonlinear operator equation by combining the semi order method,topological degree theory and fixed point index theory In this paper,we mainly discuss the fixed point theorems of generalized ?-?-contractive mapping in different metric spaces,generalize and improve many known results.This paper is divided into four parts.In Chapter one,we introduce the achievements of generalized ?-?-contractive mapping that researchers have made.We give the background of three kinds of metric spaces and describe the main work of this paper.In chapter2,we discuss the concept of fuzzy metric space and establish the fixed point theorem of generalized ?-?-contractive mapping in this space.In Chapter 3,we mainly discuss the fixed point theorem of generalized ?-?-contractive mapping on the quasi metric space.In Chapter 4,we analyze the fixed point theorem of generalized ?-?-contractive mapping in S-metric space,We prove the existence and uniqueness of the fixed point by using a simpler method than the previous literature.