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 { x t } is defined by x t = P ( t f ( x t ) + ( 1 − t ) T x t ) , the explicit iterative sequence { x n } is given by x n + 1 = P ( α n f ( x n ) + ( 1 − α n ) T x n ) , where α n ∈ ( 0 , 1 ) and P is a sunny nonexpansive retraction of E onto C . We prove that { x t } strongly converges to a fixed point of T as t → 0 , and { x n } strongly converges to a fixed point of T as α n satisfying appropriate conditions. The results presented extend and improve the corresponding results of Hong-Kun Xu (2004) and Yisheng Song and Rudong Chen (2006).