The Fixed Point Properties For Asymptotically Mean Nonexpansive Mappings In Ucw-hyperbolic Spaces

The fixed point theory is one of the most important branches of modernmathematics. As is known to all, with the fixed point theory, which is applied tomany fields, for instance, differential equation, cybernetics, optimization, economicbalance theory and game theory and so on, lots of famous mathematical results areobtained. Since Banach proved the fixed point theory in complete distance spaces in1922, the research on fixed point theory for nonexpansive mappings by using thegeometric properties of Banach spaces has developed rapidly. Especially in recentyears, fixed point theory is proved not only in Euclidean space, but also in thegeneral metric space (especially the UCW-hyperbolic space) or linear topologicalspace. Therefore, it is theoretically significant and practically valuable to investigatethe fixed point theory for mean nonexpansive mappings in UCW-hyperbolic spaces.In this dissertation, the existence, demi-closed principle and convergence offixed points for asymptotically mean nonexpansive mappings in UCW-hyperbolicspaces are discussed. The content of this dissertation is divided into three parts.In the first part, the domestic and foreign background and present situation offixed point theory and its development situation in UCW-hyperbolic spaces aresimply introduced.In the second part, this dissertation offers the proof of the existence of fixedpoints for asymptotically mean nonexpansive mappings in UCW-hyperbolic spacesand discusses the demi-closed principle for asymptotically mean nonexpansivemappings.In the last part, some results for asymptotically mean nonexpansive mappingsin UCW-hyperbolic spaces has been extended to CAT(0) spaces which is a specialcase of UCW-hyperbolic spaces. As an application, we obtains a-convergencetheorem in CAT(0) spaces and a strong convergence theorem of Ishikawa iterationhas been discussed.