# Fixed point iterative schemes for variational inequality problems

Risultato della ricerca: Other

### Abstract

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:$$\label{PDS}\frac{dx(t)}{dt}= \Pi_{\mathbb{K}}\left(x(t),-F(x(t))\right), \quad x(0)=x_0 \in \mathbb{K}, \, t \in [0,+\infty[,\tag{P}$$ 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.
Lingua originale English 36-37 2 Published - 2015

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Variational Inequality Problem
Projected Dynamical Systems
Iterative Scheme
Pi
Projection Operator
Fixed point
Initial Value Problem
Polyhedral Sets
Directional derivative
Convergence Theorem
Convex Sets
Critical point
Hilbert space
Equivalence
Perturbation
Computing
Operator
Model

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2015. 36-37.

Risultato della ricerca: Other

title = "Fixed point iterative schemes for variational inequality problems",
abstract = "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:$$\label{PDS}\frac{dx(t)}{dt}= \Pi_{\mathbb{K}}\left(x(t),-F(x(t))\right), \quad x(0)=x_0 \in \mathbb{K}, \, t \in [0,+\infty[,\tag{P}$$ 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.",
author = "Calogero Vetro and Elena Toscano",
year = "2015",
language = "English",
pages = "36--37",

}

TY - CONF

T1 - Fixed point iterative schemes for variational inequality problems

AU - Vetro, Calogero

AU - Toscano, Elena

PY - 2015

Y1 - 2015

N2 - 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:$$\label{PDS}\frac{dx(t)}{dt}= \Pi_{\mathbb{K}}\left(x(t),-F(x(t))\right), \quad x(0)=x_0 \in \mathbb{K}, \, t \in [0,+\infty[,\tag{P}$$ 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.

AB - 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:$$\label{PDS}\frac{dx(t)}{dt}= \Pi_{\mathbb{K}}\left(x(t),-F(x(t))\right), \quad x(0)=x_0 \in \mathbb{K}, \, t \in [0,+\infty[,\tag{P}$$ 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.

UR - http://hdl.handle.net/10447/145106