doi:10.1155/2008/687815 Research Article Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, F {Th : h ≥ 0} a generalized asymptotically nonexpansive self-mapping semigroup of K, and f: K → K a fixed contractive mapping with contractive coefficient β ∈ 0, 1. We prove that the following implicit and modified implicit viscosity iterative schemes {xn} defined by xn αnfxn 1 − αnTtnxn and xn αnyn 1 − αnTtnxn, yn βnfxn−1 1−βnxn−1 strongly converge to p ∈ F as n → ∞ and p is the unique solution to the following variational inequality: 〈fp − p, jy − p 〉 ≤ 0 for all y ∈ F. Copyright q 2008 Yali Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1