Fillipov-Ważewski Thorem for certain second order differential inclusions

Abstract

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 ($*$).