Fixed Points of Nonlinear Operators in Banach Space
|School||Harbin University of Science and Technology|
|Keywords||nonexpansive mappings asymptotically nonexpansive mappings fixedpoints viscosity approximative methods|
The problems of fixed point are important branch of modern mathematical, and have close connection with many mathematics subjects. It plays an important role in dealing with the determining solution and the approximate 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 four parts. In the first part, 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.In the second part, the convergence of Ishikawa iterative method for nonexpansive nonself mapping is obtained. In the framework of a Banach space, which is reflexive and has weakly sequential continuous duality mapping, we apply viscosity approximation methods to obtain the convergence of Ishikawa iterative sequence when the sequence of positive numbers satisfies appropriate conditions.In the third part, the convergence of averaging iterative method for asymptotically nonexpansive mapping is obtained. In the framework of a Banach space, which has uniformly Gateaux differential norm, the strong convergence theorems of two types of averaing iterative algorithms for asmptotically nonexpansive mappings in Banach spaces are obtained. Our results extend and improve the corresponding results of Lou, Matsushita and Kuroiwa, Song and Wangkeer.In the last part, a new geometric constant was introduced in this section, it’s properties and the relationship between uniform normal structure, convex modulous and the constant are studied. Our results extend and improve the corresponding results of Marco Baronti and Emanuele Casini.