Strong convergence of approximation fixed points for nonexpansive nonself-mapping

Abstract

Let C be a closed convex subset of a uniformly smooth Banach space E, and T:C→E a nonexpansive nonself-mapping satisfying the weakly inwardness condition such that F(T)≠∅, and f:C→C a fixed contractive mapping. For t∈(0,1), the implicit iterative sequence {xt} is defined by xt=P(tf(xt)+(1−t)Txt), the explicit iterative sequence {xn} is given by xn+1=P(αnf(xn)+(1−αn)Txn), where αn∈(0,1) and P is a sunny nonexpansive retraction of E onto C. 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 Hong-Kun Xu (2004) and Yisheng Song and Rudong Chen (2006).