Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces
Yali Li, Jianjun Liu, Lei Deng · International Journal of Mathematics and Mathematical Sciences · 2008
Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, ℱ = {T(h) : 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 = αnf(xn) + (1 − αn)T(tn)xn and xn = αnyn + (1 − αn)T(tn)xn,yn = βnf(xn−1) + (1 − βn)xn−1 strongly converge to p ∈ F as n → ∞ and p is the unique solution to the following variational inequality: 〈f(p) − p, j(y − p)〉 ≤ 0 for all y ∈ F.