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 ) = ∅, and f :K →K be a fixed contractive mapping. The implicit iterative sequence {xt } is defined by for t ∈ (0, 1) xt = P tf (xt )+ (1− t)T xt . The explicit iterative sequence {xn} is given by xn+1 = P αnf (xn) +(1−αn)T xn , where αn ∈ (0, 1) and P is sunny nonexpansive retraction of E onto K. We prove that {xt } strongly converges to a fixed point of T as t →0, and {xn} strongly converges to a fixed point of T as αn 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]. © 2005 Elsevier Inc. All rights reserved