Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

Let $X$ be a real reflexive Banach space with a weakly continuous duality mapping $J_{\phi }$. Let $C$ be a nonempty weakly closed star-shaped (with respect to $u$) subset of $X$. Let $ℱ$ = $\{T(t):t\in [0,+\infty )\}$ be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of $C$, which is uniformly continuous at zero. We will show that the implicit iteration scheme: $y_n={\alpha }_{n}u+(1-{\alpha }_n)T(t_n)y_n$, for all $n\in \mathbb{N}$, converges strongly to a common fixed point of the semigroup $ℱ$ for some suitably chosen parameters $\{{\alpha }_n\}$ and $\{t_n\}$. Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).