Description

• Title:
  Fillipov-Ważewski Thorem for certain second order differential inclusions
• Creator:
  Bartuzel, Grzegorz;
  Fryszkowski, Andrzej
• Description:
  In the paper we give a generalization of the Filippov-Ważewski Theorem to the second order differential inclusions \begin{equation} \mathcal{D}y=y^{\prime \prime }-A^{2}y\in F( t,y) , \tag{$*$} \end{equation} with the initial conditions \begin{equation} y( 0) =\alpha ,\quad y^{\prime }( 0) =\beta , \tag{$**$} \end{equation} where $A\in \mathbb{R}^{d\times d}$ and $F\colon [ 0,T] \times \mathbb{R}^{d}\leadsto c( \mathbb{R}^{d}) $ is a multifunction satisfying for each $t\in [ 0,T] $ the Lipschitz condition in $y$ \begin{equation*} d_{H}( F( t,y_{1}) ,F( t,y_{2}) ) \leq l( t) \vert y_{1}-y_{2}\vert , \end{equation*} where $l(\,\cdot\,) $ is integrable. The main result is the following: {\sc Theorem \ref{th2}}. {\it Assume that $F\colon[ 0,T] \times \mathbb{R}^{d}\leadsto c( \mathbb{R}^{d}) $ is measurable in $t$, Lipschitz continuous in $x\in \mathbb{R}^{d}$ \rom{(}with integrable constant\rom{)} and integrably bounded. Let $r\in W^{2,1}$ be a solution of the relaxed problem \begin{equation} \mathcal{D}y=y^{\prime \prime }-A^{2}y\in \rom{cl}\,\rom{co}\,F ( t,y) , \tag{$**$$*$} \end{equation} with $(**)$. Then, for each $\varepsilon \ge 0$, there exists a solution $y\in W^{2,1}$ of $(*)$ with $(**)$ such that \begin{equation*} \Vert y-r\Vert _{C^{1}[ 0,T] }\le \varepsilon . \end{equation*}} The proof goes via a version of the Fillipov Lemma (Theorem~\ref{th1}) for inclusions ($*$).
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2016.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 47, No 1 (March 2016); 389 - 403;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2016.013/7792 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2016.013/7793
• Subject:
  Differential inclusion;
  differential operator;
  Lipschitz multifunction;
  Filippov Lemma;
  Filippov-Ważewski Theorem;
  Gronwall inequality;
  26A24;
  28A15;
  46G05;
  39A05;
  28A05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2016.013
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other