The Research On Fixed Point Theorems

Fixed point problem is one of the main direction of research in functional analysis and it widely used in the algebraic equation, differential equation and implicit function theories. There are two ways to study the fixed point theorems in this paper. One way is that we can study the fixed point theorems by researching the properties of operators.The other way is that we can get the fixed points of the operators by constructing appropriate iterative process and researching the convergence.Firstly, it is an important way to study the fixed point theorems by constructing the appropriate iterative process.In chapter 1, we introduce the conception of the convex metric structure and the function satisfying some special conditions.And then,we will discuss the common fixed point theorem of weakly compatible mappings.In chapter 2, we introduce the conception of sunny mapping and sunny nonexpansive retraction. And then we introduce two new viscosity approximation methods: ’the first type viscosity approximation methods) (the second type viscosity approximation methods) Where X is a Banach space, C and D be nonempty subsets of X,D (?)C. A mapping P:Câ†'D is a sunny nonexpansive retraction.In this chapter,we will study the convergence problems of the iterative sequences,which are constructed by above viscosity approximation methods.Secondly, we can also study the fixed point theorems by researching the properties of operators.In chapter 3, we introduce the conception of the D metric space and the fundamental properties of it. And then we introduce a new type expansion mapping:D-â…¢ expansion mapping. In this paper, we will study the fixed point theorem and common fixed point theorem in D metric space.