Some Studies On The Existence Of Fixed Points Of A Class Of Nonlinear Operators In Metric Spaces

The studies on operator equations and fixed point problems play important roles in establishing the existence and uniqueness of solutions of various types of equations.A large number of differential-integral equations in abstract spaces can ultimately be attributed to the studies of nonlinear operator equations or fixed point problems.As a generalization of metric spaces,the concept of cone metric space was introduced in 2007,and to study the existence of fixed points of some nonlinear operators fulfilling a certain contractive conditions has been one of the hot topics in the researches of fixed point theory.This thesis mainly studies the problems of existence of fixed points for nonlinear operators under a certain conditions by spectral radius and iterative methods;moreover,this thesis also deals with the calculations on topological degrees of semi-closed 1-set-contractive operators in M-PN-spaces.As an application,the existence of a solution of an integral equation is studied.The thesis is divided into four chapters as follows:The first chapter is an introduction in which we review the historical background and current situation of current studies on cone metric spaces.The second chapter is to deal with the existence of fixed points of some contractive mappings on cone metric spaces over Banach algebra and some new results are obtained.As an application,the existence of a solution of a nonlinear integral equation is considered.In the third chapter,we firstly introduce the notion of topological vector space cone b-metric space over Banach algebra and then study the existence of fixed points of some contractions under the c-metric by the iterative method.In the fourth chapter we study several topological degree calculation problems and obtain several new results of s a kind of semi-closed 1-set-contractive operators in M-PN-spaces.