2016 Filippov-Ważewski theorem for certain second order differential inclusions
Grzegorz Bartuzel, Andrzej Fryszkowski
Topol. Methods Nonlinear Anal. 47(1): 389-403 (2016). DOI: 10.12775/TMNA.2016.013

## Abstract

In the paper we give a generalization of the Filippov-Ważewski Theorem to the second order differential inclusions $$\mathcal{D}y=y^{\prime \prime }-A^{2}y\in F( t,y) , \tag{*}$$ with the initial conditions $$y( 0) =\alpha ,\quad y^{\prime }( 0) =\beta , \tag{**}$$ 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:

Theorem 5.1. Assume that $F\colon[ 0,T] \times \mathbb{R} ^{d}\leadsto c( \mathbb{R}^{d})$ is measurable in $t$, Lipschitz continuous in $x\in \mathbb{R}^{d}$ $($with integrable constant$)$ and integrably bounded. Let $r\in W^{2,1}$ be a solution of the relaxed problem $$\mathcal{D}y=y^{\prime \prime }-A^{2}y\in {\rm cl}\, {\rm co}\,F ( t,y) , \tag{***}$$ with $(**)$. Then, for each $\varepsilon \gt 0$, there exists a solution $y\in W^{2,1}$ of $(*)$ with $(**)$ such that \begin{equation*} \Vert y-r\Vert _{C^{1}[ 0,T] } \lt \varepsilon . \end{equation*}

The proof goes via a version of the Fillipov Lemma (Theorem 4.4) for inclusions ($*$).

## Citation

Grzegorz Bartuzel. Andrzej Fryszkowski. "Filippov-Ważewski theorem for certain second order differential inclusions." Topol. Methods Nonlinear Anal. 47 (1) 389 - 403, 2016. https://doi.org/10.12775/TMNA.2016.013

## Information

Published: 2016
First available in Project Euclid: 23 March 2016

zbMATH: 1362.34030
MathSciNet: MR3469063
Digital Object Identifier: 10.12775/TMNA.2016.013

Rights: Copyright © 2016 Juliusz P. Schauder Centre for Nonlinear Studies

JOURNAL ARTICLE
15 PAGES

Vol.47 • No. 1 • 2016