The Fixed Point Theorems For Mean Nonexpansive Mappings In CAT(O) Spaces

Fixed point theory is one of the most important research subjects of Functional Analysis. It has close connection with many branches of modern mathematics and many famous mathematical results are acquired by means of it. Since KIRK W A proved that a reflexive Banach space with normal structure has Fixed Point Property in1965, the research on fixed point theory for nonexpansive mappings by using the geometric properties of Banach spaces has developed rapidly. Especially in recent years, fixed point theory is not only limited in Euclidean space, but more expanded to the general metric space (especially the CAT(O) space) or linear topological space. As is known to all, fixed point theory has a very wide range of applications in many fields, for instance, differential equation, cybernetics, optimization, economic balance theory and game theory and so on. Therefore, it has important theoretical significance and practical value to investigate the fixed point theory for mean nonexpansive mappings in CAT(O) spaces.In this paper, the existence and convergence of fixed points for mean nonexpansive mappings in CAT(O) spaces is discussed. The content of this paper is divided into three parts.In the first part, this dissertation introduces the background and simple history of Fixed point Theory and its development situation in CAT(O) spaces, which provide the right direction for the study.In the second part, this dissertation offers the proof of the existence of fixed points for mean nonexpansive mappings in CAT(O) spaces and discusses the demi-closed principle for mean nonexpansive mappings so as to generalize some results from nonexpansive mappings to mean nonexpansive mappings.In the last part, firstly we extend some lemmas and propositions existed for nonexpansive mappings to mean nonexpansive mappings, which provide convenience for the following proof. Secondly, this dissertation obtains a△-convergence theorem in CAT(O) spaces. Finally, a strong convergence theorem of Ishikawa iteration for mean nonexpansive mappings in CAT(O) spaces will be discussed by use of the above conclusions.