Viscosity approximation method for accretive operator in Banach space

Let X$X$ be a uniformly smooth Banach space, C$C$ be a closed convex subset of X$X$, and A$A$ an m-accretive operator with a zero. Consider the iterative method that generates the sequence {_xn}$\left\{x_n\right\}$ by the algorithm

_xn+1=_αnf(_xn)+(1−_αn)_Jrnxn,$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)J_{r_n}{x}_n\text{,}$

where _αn${\alpha }_n$ and _γn${\gamma }_n$ are two sequences satisfying certain conditions, _Jr$J_r$ denotes the resolvent (I+rA)⁻¹${(I+rA)}^{-1}$ for r>0$r>0$, and f:C→C$f:C\to C$ be a fixed contractive mapping. Then as n→∞$n\to \infty$, the sequence {_xn}$\left\{x_n\right\}$ strongly converges to a point in F(A)$F\left(A\right)$. 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].