2012 Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems
Yeong-Cheng Liou, Yonghong Yao, Chun-Wei Tseng, Hui-To Lin, Pei-Xia Yang
Abstr. Appl. Anal. 2012(SI12): 1-15 (2012). DOI: 10.1155/2012/949141

## Abstract

We consider a general variational inequality and fixed point problem, which is to find a point ${x}^{*}$ with the property that (GVF): ${x}^{*}\in \text{GVI}(C,A)$ and $g({x}^{*})\in \text{Fix}(S)$ where $\text{GVI}(C,A)$ is the solution set of some variational inequality $\text{Fix}(S)$ is the fixed points set of nonexpansive mapping $S$, and $g$ is a nonlinear operator. Assume the solution set $\Omega$ of (GVF) is nonempty. For solving (GVF), we suggest the following method $g({x}_{n+1})=\beta g({x}_{n})+(1-\beta )S{P}_{C}[{\alpha }_{n}F({x}_{n})+(1-{\alpha }_{n})(g({x}_{n})-\lambda A{x}_{n})]$, $n\ge 0$. It is shown that the sequence $\{{x}_{n}\}$ converges strongly to ${x}^{*}\in \Omega$ which is the unique solution of the variational inequality $〈F({x}^{*})-g({x}^{*}),g(x)-g({x}^{*})〉\le 0$, for all $x\in \Omega$.

## Citation

Yeong-Cheng Liou. Yonghong Yao. Chun-Wei Tseng. Hui-To Lin. Pei-Xia Yang. "Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems." Abstr. Appl. Anal. 2012 (SI12) 1 - 15, 2012. https://doi.org/10.1155/2012/949141

## Information

Published: 2012
First available in Project Euclid: 1 April 2013

zbMATH: 1235.65072
MathSciNet: MR2872316
Digital Object Identifier: 10.1155/2012/949141