Fixed point iterative schemes for variational inequality problems

In a wide class of evolutionary processes, the problem of computing the solutions of an initial value problem is encountered. Here, we consider projected dynamical systems in the sense of \cite{Daniele} and references therein. Precisely, a projected dynamical system is an operator which solves the initial value problem: \begin{equation}\label{PDS}\frac{dx(t)}{dt}= \Pi_{\mathbb{K}}\left(x(t),-F(x(t))\right), \quad x(0)=x_0 \in \mathbb{K}, \, t \in [0,+\infty[,\tag{P}\end{equation} where $\mathbb{K}$ is a convex polyhedral set in $\mathbb{R}^n$, $F: \mathbb{K} \to \mathbb{R}^n$ and $\Pi_{\mathbb{K}}: \mathbb{R} \times \mathbb{K} \to \mathbb{R}^n$ is given as follows $\Pi_{\mathbb{K}}(x,-F(x))= \lim_{t \to 0^+} \frac{P_{\mathbb{K}}(x-t F(x))-x}{t},$ by using the directional derivative in the sense of G\^{a}teaux of the projection operator $P_{\mathbb{K}}: \mathbb{R}^n \to \mathbb{K}$. The set of critical points of \eqref{PDS} coincides with the set of solutions of a variational inequality problem (VIP) in the sense of \cite{Stampacchia}. In view of this equivalence, established in \cite{Du-Na}, we study the extension and application of fixed point iterative schemes (see \cite{Berinde2,Rus}) to a (VIP), via admissible perturbations of projection operators in real Hilbert spaces. As a sample model, we prove convergence theorems for Krasnoselskij-type fixed point iterative schemes.