Strong convergence of an iterative method for nonexpansive and accretive operators

Abstract

Let X be a Banach space and A an m-accretive operator with a zero. Consider the iterative method that generates the sequence $\{x_n\}$ by the algorithm $x_{n+1}={\alpha }_{n}u+(1-{\alpha }_n)J_{r_n}{x}_n$, where $\{{\alpha }_n\}$ and $\{r_n\}$ are two sequences satisfying certain conditions, and $J_r$ denotes the resolvent ${(I+rA)}^{\mathrm{-1}}$ for $r>0$. Strong convergence of the algorithm $\{x_n\}$ is proved assuming X either has a weakly continuous duality map or is uniformly smooth.