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

Abstract EN

Let X be a real reflexive Banach space with a weakly continuous duality mapping Jφ.

Let C be a nonempty weakly closed star-shaped (with respect to u) subset of X.

Let ℱ = {T(t):t∈[0,+∞)} 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: yn=αnu+(1−αn)T(tn)yn, for all n∈ℕ, converges strongly to a common fixed point of the semigroup ℱ for some suitably chosen parameters {αn} and {tn}.

Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).