Structure of the fixed point set of asymptotically nonexpansive mappings in Banach spaces with weak uniformly normal structure
Daya Ram Sahu, Ravi P. Agarwal, Donal O’Regan · Journal of Applied Analysis · 2011
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 ( ω )-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.