Journal of Applied Mathematics
Volume 2012 (2012), Article ID 641479, 19 pages
http://dx.doi.org/10.1155/2012/641479
Research Article

Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

Haiqing Wang,1 Yongfu Su,1 and Hong Zhang2

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Mathematics, Tianjin No. 8 Middle School, Tianjin 300252, China

Received 11 November 2011; Accepted 17 December 2011

Academic Editor: Rudong Chen

Copyright © 2012 Haiqing Wang 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.

Abstract

Let 𝑋 be a uniformly convex Banach space and 𝒮 = { 𝑇 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } be a nonexpansive semigroup such that ⋂ 𝐹 ( 𝒮 ) = 𝑠 > 0 𝐹 ( 𝑇 ( 𝑠 ) ) ≠ ∅ . Consider the iterative method that generates the sequence { 𝑥 𝑛 } by the algorithm 𝑥 𝑛 + 1 = 𝛼 𝑛 𝑓 ( 𝑥 𝑛 ) + 𝛽 𝑛 𝑥 𝑛 + ( 1 − 𝛼 𝑛 − 𝛽 𝑛 ) ( 1 / 𝑠 𝑛 ) ∫ 𝑠 𝑛 0 𝑇 ( 𝑠 ) 𝑥 𝑛 𝑑 𝑠 , 𝑛 ≥ 0 , where { 𝛼 𝑛 } , { 𝛽 𝑛 } , and { 𝑠 𝑛 } are three sequences satisfying certain conditions, 𝑓 ∶ 𝐶 → 𝐶 is a contraction mapping. Strong convergence of the algorithm { 𝑥 𝑛 } is proved assuming 𝑋 either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.