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