Viscosity approximation methods for nonexpansive nonself-mappings

AbstractLet 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)Txt). The explicit iterative sequence {xn} is given byxn+1=P(αnf(xn)+(1−αn)Txn), 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]