  Let $Q$ be a nonempty closed and convex subset of a real Hilbert space $%
\mathcal{H}$, $T:Q\rightarrow Q$ a nonexpansive mapping which has a least one
fixed point $f:Q\rightarrow \mathcal{H}$ a Lipschitzian function, and $%
F:Q\rightarrow \mathcal{H}$ 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 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.
