Viscosity approximation methods for pseudocontractive mappings in Banach spaces

Let K be a closed convex subset of a Banach space E and let T: K -> E be a continuous weakly inward pseudocontractive mapping. Then for t epsilon (0, 1), there exists a sequence {y(t)} subset of K satisfying y(t) = (1 - t)f(y(t)) + tT(y(t)), where f epsilon Pi(K):= {f: K -> K, a contraction with a suitable contractive constant}. Suppose further that F(T) not equal theta and E is reflexive and strictly convex which has uniformly Gateaux differentiable norm, Then it is proved that {y(t)} converges strongly to a fixed point of T which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of T and hence to a solution of certain variational inequality is constructed provided that T is Lipschitzian. (c) 2006 Elsevier Inc. All rights reserved.