Proceedings of the American Mathematical Society

Abstract:

Denote by $S_{F}^{\prime }\left ( \zeta \right )$ the set of derivatives of all absolutely continuous solutions of a Lipschitz differential inclusion\begin{equation*} \left \{ \begin {array}{cc} x^{\prime }\in F\left ( t,x\right ) , & \ t\in \left [ 0,1\right ] =I ,\\ x\left ( 0\right ) =\zeta . & \end{array}\right . \end{equation*} It is known that the set $S_{F}^{\prime }\left ( \zeta \right )$ is an absolute retract. We show the following:

Theorem. For every $\varepsilon >0$ there exists a continuous mapping $r:X\times L^{1}\rightarrow L^{1}$ such that for every $\zeta \in X$ the map r$\left ( \zeta ,\cdot \right )$ is a retraction of $L^{1}$ onto $S_{F}^{^{\prime }}\left ( \zeta \right )$ and for all $\left ( \zeta ,u\right ) \in X\times L^{1}$ and almost all $t\in I$ we have a Filippov type pointwise estimate \begin{gather*} \left \vert r\left ( \zeta ,u\right ) \left ( t\right ) -u\left ( t\right ) \right \vert \\ \leq \varepsilon \left ( 1+l\left ( t\right ) \right ) \left \Vert p\left ( \zeta ,u\right ) \right \Vert +l\left ( t\right ) \int \limits _{0}^{t}e^{m\left ( t\right ) -m\left ( s\right ) }p\left ( \zeta ,u\right ) \left ( s\right ) ds+p\left ( \zeta ,u\right ) \left ( t\right ) , \end{gather*} where \begin{equation*} p\left ( \zeta ,u\right ) \left ( t\right ) =\operatorname {dist}\left ( u\left ( t\right ) ,F\left ( t,\zeta +\int \limits _{0}^{t}u\left ( \tau \right ) d\tau \right ) \right ) \ \ a.e.\ in\ I \end{equation*} and the functions $l$ and $m$ are related with the Lipschitz condition.