Pointwise estimates in the Filippov lemma and Filippov-Ważewski theorem for fourth order differential inclusions

Abstract

In this work we give a generalization of the Filippov-Ważewski Theorem to the fourth order differential inclusions in a separable complex Banach space $\mathbb{X}$ \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) , \end{equation*} with the initial conditions in $c\in [ 0,T]$ \begin{equation} y( c) =\alpha ,\qquad y^{\prime }( c) =\beta ,\qquad y^{\prime \prime }( c) =\gamma ,\qquad y^{\prime \prime \prime }( c) =\delta ,\tag{*} \end{equation} We assume that the multifunction $F\colon[ 0,T] \times \mathbb{X} \leadsto c( \mathbb{X}) $ is Lipschitz continuous in $y$ with the integrable Lipschitz constant $l( .) $, while $A^{2},B^{2}\in B( \mathbb{X}) $ are the infinitesimal generators of two cosine families of operators. The main result is the following version of Filippov Lemma:

Theorem: Let $y_{0}\in W^{4,1}=W^{4,1}([ 0,T] ,\mathbb{X}) $ be such function with (*) that \begin{equation*} \mathrm{dist}( \mathcal{D}y_{0}( t) ,F( t,y_{0}( t) ) ) \leq p_{0}( t) \quad \text{a.e. in } [ c,d] \subset [ 0,T] , \end{equation*} where $p_{0}\in L^{1}[ 0,T] $. Then there are $\mathcal{\sigma }_{0}$ $($depending on $p_{0})$ and $\varphi $ such that for each $\varepsilon >0$ there exists a solution $y\in W^{4,1}$ of the above problem such that almost everywhere in $t\in [c,d]$ we have $\vert \mathcal{D}y( t) -\mathcal{D}y_{0}( t) \vert \leq \mathcal{\sigma }_{0}( t) $, \begin{alignat*}2 \vert y( t) -y_{0}( t) \vert &\leq(\varphi \ast _{c}\sigma _{0})( t) , &\qquad \vert y^{\prime }( t) -y_{0}^{\prime }( t) \vert \leq ( \varphi ^{\prime }\ast _{c}\sigma _{0}t) ( t ) , \\ \vert y^{\prime \prime }( t) -y_{0}^{\prime \prime }( t) \vert &\leq ( \varphi ^{\prime \prime }\ast _{c}\sigma _{0}) ( t) &\qquad \vert y^{\prime \prime \prime }( t) -y_{0}^{\prime \prime \prime }( t) \vert \leq( \varphi ^{\prime \prime \prime }\ast _{c}\sigma _{0})( t) , \end{alignat*} where $\ast _{c}$ stands for the convolution started at $c$.

Our estimates are constructive and more precise then those in the known versions of Filippov Lemma.