Study On Fixed Point And Best Proximity Point Problems For Nonlinear Operators In Several Kinds Of Spaces

Nonlinear operator theory is the cornerstone of nonlinear functional analysis,while the study on fixed point and best proximity points of nonlinear operators is one of the important topics of nonlinear operator theory,which has wide applications in solving nonlinear equations.In this thesis,we mainly study best proximity point,coincidence best proximity point,fixed point,common fixed point and approximate coincidence point problems in metric spaces,metric spaces with w₀-distance,Menger PM-spaces and partially ordered Menger PM-spaces for a mapping or a pair of mappings.The thesis is divided into four chapters.In Chapter 1,we briefly introduce the development of study on fixed point and best proximity point problems for operators in different kinds of spaces,and give some preliminaries.In Chapter 2,we study best proximity point and coincidence best proximity point problems for mappings in metric spaces.First of all,we introduce the concept of generalized p-proximity contraction in metric space,and prove a best proximity point theorem.Secondly,we propose the concept of MT-cyclic-noncyclic pair by combining MT-function classes and cyclic-noncyclic pair,and establish a new coincidence best proximity point theorem.In Chapter 3,we discuss the best proximity point problems for mappings in metric spaces with w₀-distance.First of all,we introduce the concepts of pproximal ?-?-?-quasi contraction and a proximal admissible mapping with respect to ? in metric space with w₀-distance,and prove a best proximity point theorem for two single-valued mappings in such spaces based on these concepts.Secondly,we give some corollaries and corresponding results,and provide an example to show the validity of the main result.Finally,we apply the main result to study the exsitence and uniqueness of solutions to a class of nonlinear integral equations.In Chapter 4,we investigate common fixed point and approximate coincidence point problems for a pair of mappings in probabilistic metric spaces.First of all,we establish a corresponding fixed point theorem for cyclic mappings in Menger PMspaces by introducing the concept of cyclic R-contractions in such spaces.Secondly,we construct new contractive conditions in partially ordered Menger PM-spaces,and prove some new common fixed point theorems and approximate coincidence point theorems for a hybrid pair of mappings(one of them is a single-valued mapping,while the other is a set-valued one).Finally,we apply the new theorems to discuss the existence of solutions to a class of a system of nonliear integral equations.