Iteration Algorithms For Approximating To Fixed Points Of Non-linear Operators

The problems of fixed point have close connection with many branches of modern mathematics. Especially, it plays an important role in the existence and uniqueness of the solution to various equations. Among many directions of the fixed point researches, it becomes main problem that the convergence problem about making approximating fixed point sequences and its application in control, optimization, non-linear operator and derivative equation etc.In this thesis, the problem on approximating to the fixed points of non-linear operators is discussed. Contents of this thesis are divided into the following parts. Firstly, the background and simple history of non-linear operators and iterative methods are recalled, which offer the right direction to the study of this thesis.Secondly, the existence and uniqueness of fixed point for mean nonexpansive mapping is studied. And two examples are given to show that nonexpansive mapping is a special case of mean nonexpansive mapping. At the same time, several properties similar to nonexpansive mapping including the existence of the approximate fixed point sequence, demi-closed principle and Opial's condition are discussed.The strong convergence theorems of the averaging iterative algorithms for nonexpansive self-mappings and noself-mappings in Banach spaces are obtained by using viscosity approximation methods. Our results extend and improve the corresponding results of Xu, Matsushita and Kuroiwa, Song and Wangkeer.In the end, a new approach to study the convergence of Noor iterative algorithm based on the previous results is used, and it not only weakened the condition of the space, but also the rules of judgment and the description of conclusions are given. It provides the basis for looking for more general conditions to the approximate problem of Noor iterative sequence.