STRONG CONVERGENCE OF A HYBRID VISCOSITY APPROXIMATION METHOD WITH PERTURBED MAPPINGS FOR NONEXPANSIVE AND ACCRETIVE OPERATORS

Abstract

Recently, H. K. Xu [J. Math. Anal. Appl. 314 (2006) 631-643] considered the iterative method for approximation to zeros of an $m$-accretive operator $A$ in a Banach space $X$. In this paper, we propose a hybrid viscosity approximation method with perturbed mapping that generates the sequence $\{x_n\}$ by the algorithm $x_{n+1} = \alpha_n(u+f(x_n)) + (1-\alpha_n) [J_{r_n} x_n - \lambda_n F(J_{r_n} x_n)]$, where $\{\alpha_n\}$, $\{r_n\}$ and $\{\lambda_n\}$ are three sequences satisfying certain conditions, $f$ is a contraction on $X$, $J_r$ denotes the resolvent $(I+rA)^{-1}$ for $r \gt 0$, and $F$ is a perturbed mapping which is both $\delta$-strongly accretive and $\lambda$-strictly pseudocontractive with $\delta + \lambda \geq 1$. Under the assumption that $X$ either has a weakly continuous duality map or is uniformly smooth, we establish some strong convergence theorems for this hybrid viscosity approximation method with perturbed mapping.