Fixed Point Properties For Asymptotically Nonexpansive Mappings In W-hyperbolic Spaces

Geometric theory of Banach space is an important branch of modern functional analysis. A reflexive Banach space with normal structure which is proved by Kirk has Fixed Point Property in1965. Since then, the research on the class of asymptotically nonexpansive mappings by using the geometric theory of Banach space developed rapidly. Recently, the fixed point theory of metric spaces, especially CAT(O) spaces, has been widely studied. As well known, fixed point theory has wide applications in differential equation, control theory, algebra, economy balance theory, game theory and so on. Therefore, studying the applications of geometric constants of Banach spaces for metric fixed point theory play an important role both in theoretical significance and application values.K-uniformly convex W-hyperbolic spaces with monotone modulus of K-uniform Convexity are natural generalization of uniformly convex W-hyperbolic spaces which contain both of uniformly convex normed spaces and CAT(O) spaces. For asymptotically nonexpansive mappings, we discuss some properties of K-uniformly convex W-hyperbolic spaces, and we have get that the fixed points exist. These results extend the corresponding recent results in uniformly convex W-hyperbolic spaces and CAT(O) spaces.The main results are as follows:For asymptotically nonexpansive mappings, we prove that it has a fixed point in K-UCW-hyperbolic spaces with a monotone modulus of uniform convexity which contain CAT(O) spaces and the demiclosedness principle with respect to A-convergence holds. As applications, we show the A-convergence theorem of iterative sequences for continuous mappings of asymptotically nonexpansive in CAT(O) spaces.