Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

• Main Authors: Yali Li, Jianjun Liu, Lei Deng
• Format: Article
• Language: English
• Published: Hindawi Limited 2008-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/2008/687815

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.