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.