## Abstract and Applied Analysis

### Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

#### 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 $x\in \Omega$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 949141, 15 pages.

Dates
First available in Project Euclid: 1 April 2013

https://projecteuclid.org/euclid.aaa/1364845183

Digital Object Identifier
doi:10.1155/2012/949141

Mathematical Reviews number (MathSciNet)
MR2872316

Zentralblatt MATH identifier
1235.65072

#### Citation

Liou, Yeong-Cheng; Yao, Yonghong; Tseng, Chun-Wei; Lin, Hui-To; Yang, Pei-Xia. Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 949141, 15 pages. doi:10.1155/2012/949141. https://projecteuclid.org/euclid.aaa/1364845183