The Fixed Point Property And Convergence Of Mean Nonexpansive Mapping

Functional Analysis is an important basic concept in mathematics,it is the foundation of modern mathematics,and an important tool for the study and application of mathematics and other fields.As an important part of the branch of mathematics,theory of functional analysis is a complete,rich and independent discipline.Among them,the theory of fixed point is one of the core contents of functional analysis,the research results of the fixed point problem are not only involved a lot in the mathematics discipline itself,but also many problems in other fields can be transformed into the fixed point problem of nonexpansion mappings,and thus have been widely used in other fields.In many researches on fixed point problems,the iterative format is more important because it is a widely used research tool.Many scholars at home and abroad have conducted extensive and in-depth research on this,making the theory increasingly rich and perfect.This article mainly explores the fixed point properties of the mean nonexpansive mapping and divides it into three chapters.The main contents of the study are as follows:In this paper: the development of fixed point theory in geometric space is reviewed,and the related achievements made by scholars at home and abroad are summarized.In the second chapter:Firstly,in a uniformly convex Banach space,given two iterative schemes with monotonic coefficients,it is proved that there are fixed points on the mapping,and {x _n}_n-1^? convergence to this fixed point.After that,the most famous result by W.kirk is that the nonexpansive mapping has the fixed point property in a Banach space with normal structure reflexive is extended to a more general form of mapping,a reflexive Banach space X with normal structure has the fixed point property for the mapping of iterative schemes with monotonic coefficients.Finally,the structure of a fixed point set on a strictly convex Banachspace with respect to the mean nonexpansive mapping fixed point set is discussed.In the third chapter,the iterative methods of fixed point of strong pseudo-contraction mappings and accretive operators are studied in Banach spaces,we give a new three step Ishikawa iteration,it is proved that the iterative scheme sequence convergent to the fixed point of strong pseudo-contraction mapping or accretive operators.Meanwhile,the result mentioned above is extended to n steps of Ishikawa iteration for strong pseudo-contraction mappings and accretive operators.