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

TL;DR

This paper proves that a perturbed iterative algorithm converges strongly to the unique solution of a variational inequality problem in a Hilbert space, under certain conditions on the involved functions and sequences.

Contribution

It establishes strong convergence of a new perturbed algorithm for variational inequalities, extending and unifying previous results in the field.

Findings

Sequence converges strongly to the unique solution.

Conditions on functions and sequences ensure convergence.

The method generalizes existing algorithms.

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.