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 $g\left(x^{*}\right)\in \text{Fix}\left(S\right)$ where $\text{GVI}(C,A)$ is the solution set of some variational inequality $\text{Fix}\left(S\right)$ 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 $g\left(x_{n+1}\right)=\beta g\left(x_n\right)+(1-\beta )S{P}_C\left[{\alpha }_{n}F\right(x_n)+(1-{\alpha }_n\left)\right(g\left(x_n\right)-\lambda A{x}_n\left)\right]$, $n\ge 0$. It is shown that the sequence $\left\{x_n\right\}$ converges strongly to $x^{*}\in \Omega$ which is the unique solution of the variational inequality $\langle F\left(x^{*}\right)-g\left(x^{*}\right),g\left(x\right)-g\left(x^{*}\right)\rangle \le 0$, for all $x\in \Omega$.