On the strong convergence of a perturbed algorithm to the unique solution of a variational inequality problem

Abstract

Let $Q$ be a nonempty closed and convex subset of a real Hilbert space $% \mathcal{H}$. $T:Q\rightarrow Q$ is a nonexpansive mapping which has a least one fixed point. $f:Q\rightarrow \mathcal{H}$ is a Lipschitzian function, and $% F:Q\rightarrow \mathcal{H}$ is a Lipschitzian and strongly monotone mapping. We prove, under appropriate conditions on the functions $f$ and $F$, the control real sequences $\{\alpha _{n}\}$ and $\{\beta _{n}\},$ and the error term $\{e_{n}\},$ that for any starting point $x_{0}$ in $Q,$ the sequence $% \{x_{n}\}$ generated by the perturbed iterative process \[ x_{n+1}=\beta _{n}x_{n}+(1-\beta _{n})P_{Q}\left( \alpha _{n}f(x_{n})+(I-\alpha _{n}F)Tx_{n}+e_{n}\right) \] converges strongly to the unique solution of the variational inequality problem \[ \text{Find }q\in C\text{ such that }\langle F(q)-f(q),x-q\rangle \geq 0\text{ for all }x\in C \] where $C=F_{ix}(T)$ is the set of fixed points of $T.$ Our main result unifies and extends many well-known previous results.