Viscosity approximation methods for nonexpansive nonself-mappings

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*, and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and T: K -> E be nonexpansive mappings satisfying the weakly inward condition and F(T) not equal 0, and f : K -> K be a fixed contractive mapping. The implicit iterative sequence {x(t)} is defined by for t is an element of (0, 1) x(t) = P(tf (x(t)) + (1 - t) Tx(t)). The explicit iterative sequence {x(n)} is given by x(n+1) =P(alpha(n)f(x(n))+(1 - alpha(n))Tx(n)), where alpha(n) is an element of (0, 1) and P is sunny nonexpansive retraction of E onto K. We prove that {x(t)} strongly converges to a fixed point of T as t -> 0, and {x(n)} strongly converges to a fixed point of T as alpha(n) satisfying appropriate conditions. The results presented extend and improve the corresponding results of [H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291]. (c) 2005 Elsevier Inc. All rights reserved.