Strong convergence of approximation fixed points for nonexpansive nonself-mapping

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(t f (xt) + (1− t)Txt), the explicit iterative sequence {xn} is given by xn+1 = P(αn f (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).