Some Studies On The Existence And Uniqueness Of Solutions Of Nonlinear Operator Equations In Metric Spaces

Nonlinear functional analysis is an important branch of mathematics,which comes from theoretical studies and practical applications of physics?biology?economics and other disciplines.The fixed point theory of nonlinear operators has become one of the most active research fields and is of important theoretical significance and application value.In the present thesis,we mainly study the existence and uniqueness of positive solutions of fractional differential equations with infinite-point boundary value conditions and the fixed point theorems of a kind of nonlinear operators satisfying some contractive conditions on generalized cone metric spaces over Banach algebra and applications.The thesis is organized as follows:In chapter 1,we briefly introduce the developments of the fixed point theory of nonlinear operators and fractional differential equations,and simply state the source and backgrounds of the relative research problems and some basic conclusions.In chapter 2,we firstly introduce the Lipschitz constant related to the first eigenvalue corresponding to the relevant operator and the notion of u₀-bounded positive operators,then prove the existence and uniqueness of positive solutions of the fractional differential equation?2.1.3?.In chapter 3,we introduce a monotone iterative technique and establish the existence and uniqueness of positive solutions of singular high-order fractional differential equation?3.1.3?.In chapter 4,we introduce the concept of JS-GM space over Banach algebra which generalizes the conception of JS-GM space,then present several fixedpoint results of some nonlinear mappings with some contractive condition in JS-GM space over Banach algebra.In chapter 5,we briefly talked about future research directions.