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

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 is an element of [0, +infinity)} 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 is an element of 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).