Iterative Algorithm For Fixed Points Of Nonexpansive Mappings And Asymptotically Expansive Semigroups

Fixed point theory, which is closely related to many branches of modern mathematics,is a very important part of the fast developing nonlinear functional analysis theory. It contributes a great deal to forming unique solutions to varies equations, including all kinds of linear and nonlinear, con?rmed and uncon?rmed di?erential equation, integral equation and varies operation equation.The idea of ?xed point was ?rst used to prove Poincar′e Final Theorem by the French mathematician Poincar′e between 1895 and 1900, who pointed out that the existence of the periodic solution to restrictive three-body problem was just a matter of plane continuous transformation when certain conditions are met. In 1910 L.E.J.Brouwer proved that there was at least one ?xed point in continuous mapping of polyhedron in the ?nite dimension,thus setting about the study of the ?xed point theory. The perfect result and the successful solutions to implicit function theorem, the existence of di?erential equation initial-value problem and etc. made the ?xed point theory a miracle in the mathematics ?eld owing to the Poland mathematician Banach, who proved the contracting mapping principle by using the method of Picard. Especially in recent years, the fast development of the computers has made it possible for people to use various iterative methods to get closer to the ?xed point of the nonlinear mapping and to apply them to some practical problems.Thanks to the theory of the ?xed point, great achievements have been achieved in many?elds such as mathematics and physics and the theory ?nally has become an important part of the nonlinear functional analysis theory.This paper is divided into two chapters:In the ?rst chapter, by introducing a new Ishikawa iterative algorithm, we get strong convergence theorems of the following iterative process about nonexpansive mappings in a real Banach space with a uniformly G?ateaux di?erentiable norm or a weakly sequentially continuous duality mapping.In the second chapter, under appropriate conditions, we obtain the following general iterative process to strongly converge to a ?xed point of an asymptotically nonexpansive semigroup in the framework of a re?exive and strictly convex real Banach space with a uniformly G?ateaux di?erentiable norm.