Abstract and Applied Analysis
Volume 2012 (2012), Article ID 453452, 14 pages
http://dx.doi.org/10.1155/2012/453452
Research Article

Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities

Xionghua Wu,1 Yeong-Cheng Liou,2 Zhitao Wu,1 and Pei-Xia Yang1

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 27 June 2012; Accepted 21 August 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Xionghua Wu 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 { 𝑡 𝑛 } ⊂ ( 0 , 1 ) be such that 𝑡 𝑛 → 1 as 𝑛 → ∞ , let 𝛼 and 𝛽 be two positive numbers such that 𝛼 + 𝛽 = 1 , and let 𝑓 be a contraction. If 𝑇 be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence { 𝑡 𝑛 } , we show the existence of a sequence { 𝑥 𝑛 } 𝑛 satisfying the relation 𝑥 𝑛 = ( 1 − 𝑡 𝑛 / 𝑘 𝑛 ) 𝑓 ( 𝑥 𝑛 ) + ( 𝑡 𝑛 / 𝑘 𝑛 ) 𝑇 𝑛 𝑥 𝑛 and prove that { 𝑥 𝑛 } converges strongly to the fixed point of 𝑇 , which solves some variational inequality provided 𝑇 is uniformly asymptotically regular. As an application, if 𝑇 be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by 𝑧 0 ∈ 𝐾 , 𝑧 𝑛 + 1 = ( 1 − 𝑡 𝑛 / 𝑘 𝑛 ) 𝑓 ( 𝑧 𝑛 ) + ( 𝛼 𝑡 𝑛 / 𝑘 𝑛 ) 𝑇 𝑛 𝑧 𝑛 + ( 𝛽 𝑡 𝑛 / 𝑘 𝑛 ) 𝑧 𝑛 converges strongly to the fixed point of 𝑇 .