Existence and approximation of fixed points of nonlinear mappings in spaces with weak uniform normal structure

Abstract

It is shown that in a Banach space X with weak uniform normal structure, every demicontinuous asymptotically regular nearly Lipschitzian self-mapping T with lim supn→∞η(Tn)<WCS(X) defined on a weakly compact convex subset C 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 pseudocontractive nearly nonexpansive mapping T is nonempty and a sunny nonexpansive retract of C. Further, we study the approximation of fixed points of T by Halpern type iteration process: xn+1=αnu+(1−αn)Tnxnfor alln∈N, where u∈C and {αn} is a sequence in (0,1) satisfying appropriate conditions. Our results improve several known existence and convergence fixed point theorems in general Banach spaces for a wider class of nonlinear mappings which are not necessarily Lipschitzian.