Filippov Lemma for matrix fourth order differential inclusions
Tom 101 / 2014
Streszczenie
In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions \begin{equation} \mathcal{D}y=y^{\prime\prime\prime\prime}-(A^{2}+B^{2})y^{\prime\prime} +A^{2}B^{2}y\in F( t,y) , \tag{$\hbox{*}$} \end{equation} with the initial conditions \begin{equation} y( 0) = y'( 0) = y^{\prime\prime}(0) = y^{\prime\prime\prime}( 0) =0, \tag{**} \end{equation} where the matrices $A,B\in \mathbb{R}^{d\times d}$ are commutative and the multifunction $F:[ 0,1] \times \mathbb{R}^{d}\leadsto \mathop{\rm cl}( \mathbb{R}^{d}) $ is Lipschitz continuous in $y$ with a $t$-independent constant $l<\Vert A\Vert ^{2}\Vert B\Vert ^{2}$.
Main theorem. Assume that $F:[ 0,1] \times \mathbb{R}^{d}\leadsto \mathop{\rm cl}( \mathbb{R}^{d}) $ is measurable in $t$ and integrably bounded. Let $y_{0}\in W^{4,1}$ be an arbitrary function satisfying (**) and such that \[ d_{H}( \mathcal{D}y_{0}( t) ,F( t,y_{0}( t) ) ) \leq p_{0}( t) \text{ a.e. in }[ 0,1], \] where $p_{0}\in L^{1}[ 0,1] $. Then there exists a solution $y\in W^{4,1}$ of (*) with (**) such that \begin{align*} \vert \mathcal{D}y( t) -\mathcal{D}y_{0}( t)\vert &\leq p_{0}( t) +l ( Y_{4}( \,\cdot\, ,\alpha,\beta ) \ast p_{0}) ( t) \\ \vert y( t) -y_{0}( t) \vert &\leq (Y_{4}( \,\cdot\, ,\alpha ,\beta ) \ast p_{0}) ( t) \text{ a.e. in } [ 0,1] , \end{align*} where \[ Y_{4}( x,\alpha ,\beta ) =\frac{\alpha^{-1}\sinh ( \alpha x) -\beta^{-1}\sinh ( \beta x) }{\alpha^{2}-\beta ^{2}} \] and $\alpha ,\beta $ depend on $\Vert A\Vert $, $\Vert B\Vert $ and $l$.
