Structure of the fixed point set of asymptotically nonexpansive mappings in Banach spaces with weak uniformly normal structure

This paper is concerned with weak uniformly normal structure and the structure of the set of fixed points of Lipschitzian mappings. It is shown that in a Banach space X with weak uniformly normal structure, every asymptotically regular Lipschitzian semigroup of self-mappings defined on a weakly compact convex subset of X satisfies the (!)-fixed point property. We show that if X has a uniformly Gâteaux differentiable norm, then the set of fixed points of every asymptotically nonexpansive mapping is nonempty and sunny nonexpansive retract of C. Our results improve several known fixed point theorems for the class of Lipschitzian mappings in a general Banach space.