Abstract and Applied Analysis
Volume 2012 (2012), Article ID 949141, 15 pages
http://dx.doi.org/10.1155/2012/949141
Research Article

Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

Yeong-Cheng Liou,1 Yonghong Yao,2 Chun-Wei Tseng,1 Hui-To Lin,1 and Pei-Xia Yang2

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

Received 21 September 2011; Revised 16 November 2011; Accepted 17 November 2011

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Yeong-Cheng Liou 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

We consider a general variational inequality and fixed point problem, which is to find a point 𝑥 ∗ with the property that (GVF): 𝑥 ∗ ∈ G V I ( 𝐶 , 𝐴 ) and 𝑔 ( 𝑥 ∗ ) ∈ F i x ( 𝑆 ) where G V I ( 𝐶 , 𝐴 ) is the solution set of some variational inequality F i x ( 𝑆 ) is the fixed points set of nonexpansive mapping 𝑆 , and 𝑔 is a nonlinear operator. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following method 𝑔 ( 𝑥 𝑛 + 1 ) = 𝛽 𝑔 ( 𝑥 𝑛 ) + ( 1 − 𝛽 ) 𝑆 𝑃 𝐶 [ 𝛼 𝑛 𝐹 ( 𝑥 𝑛 ) + ( 1 − 𝛼 𝑛 ) ( 𝑔 ( 𝑥 𝑛 ) − 𝜆 𝐴 𝑥 𝑛 ) ] , 𝑛 ≥ 0 . It is shown that the sequence { 𝑥 𝑛 } converges strongly to 𝑥 ∗ ∈ Ω which is the unique solution of the variational inequality ⟨ 𝐹 ( 𝑥 ∗ ) − 𝑔 ( 𝑥 ∗ ) , 𝑔 ( 𝑥 ) − 𝑔 ( 𝑥 ∗ ) ⟩ ≤ 0 , for all 𝑥 ∈ Ω .