Viscosity approximation method for accretive operator in Banach space

Abstract

Let X be a uniformly smooth Banach space, C be a closed convex subset of X , and A an m-accretive operator with a zero. Consider the iterative method that generates the sequence { x n } by the algorithm x n + 1 = α n f ( x n ) + ( 1 − α n ) J r n x n , where α n and γ n are two sequences satisfying certain conditions, J r denotes the resolvent ( I + r A ) − 1 for r > 0 , and f : C → C be a fixed contractive mapping. Then as n → ∞ , the sequence { x n } strongly converges to a point in F ( A ) . The results presented extends and improves the corresponding results of Hong-Kun Xu [Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643].