Abstract and Applied Analysis
Volume 2012 (2012), Article ID 605389, 18 pages
http://dx.doi.org/10.1155/2012/605389
Review Article

Approximate Iteration Algorithm with Error Estimate for Fixed Point of Nonexpansive Mappings

Yongfu Su

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 12 July 2012; Accepted 7 August 2012

Academic Editor: Xiaolong Qin

Copyright © 2012 Yongfu Su. 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

The purpose of this article is to present a general viscosity iteration process { 𝑥 𝑛 } which defined by 𝑥 𝑛 + 1 = ( 𝐼 − 𝛼 𝑛 𝐴 ) 𝑇 𝑥 𝑛 + 𝛽 𝑛 𝛾 𝑓 ( 𝑥 𝑛 ) + ( 𝛼 𝑛 − 𝛽 𝑛 ) 𝑥 𝑛 and to study the convergence of { 𝑥 𝑛 } , where T is a nonexpansive mapping and A is a strongly positive linear operator, if { 𝛼 𝑛 } , { 𝛽 𝑛 } satisfy appropriate conditions, then iteration sequence { 𝑥 𝑛 } converges strongly to the unique solution 𝑥 ∗ ∈ 𝑓 ( 𝑇 ) of variational inequality ⟨ ( 𝐴 − 𝛾 𝑓 ) 𝑥 ∗ , 𝑥 − 𝑥 ∗ ⟩ ≥ 0 , for all 𝑥 ∈ 𝑓 ( 𝑇 ) . Meanwhile, a approximate iteration algorithm is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality, the error estimate is also given. The results presented in this paper extend, generalize, and improve the results of Xu, G. Marino and Xu and some others.