Viscosity approximation method for accretive operator in Banach space

Let XX be a uniformly smooth Banach space, CC be a closed convex subset of XX, and AA an m-accretive operator with a zero. Consider the iterative method that generates the sequence {xn}{xn} by the algorithm xn+1=αnf(xn)+(1−αn)Jrnxn,xn+1=αnf(xn)+(1−αn)Jrnxn, where αnαn and γnγn are two sequences satisfying certain conditions, JrJr denotes the resolvent (I+rA)−1(I+rA)−1 for r>0r>0, and f:C→Cf:C→C be a fixed contractive mapping. Then as n→∞n→∞, the sequence {xn}{xn} strongly converges to a point in F(A)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].