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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pointwise estimates for retractions on the solution set to Lipschitz differential inclusions


Authors: Andrzej Fryszkowski and Tadeusz Rzeżuchowski
Journal: Proc. Amer. Math. Soc. 139 (2011), 597-608
MSC (2010): Primary 34A60, 54C60; Secondary 34A12, 49J53
Published electronically: July 26, 2010
MathSciNet review: 2736341
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Abstract: Denote by $ S_{F}^{\prime }\left( \zeta \right) $ the set of derivatives of all absolutely continuous solutions of a Lipschitz differential inclusion

\begin{displaymath} \left\{ \begin{array}{cc} x^{\prime }\in F\left( t,x\right)... ...ight] =I ,\\ x\left( 0\right) =\zeta . & \end{array}\right. \end{displaymath}

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}\emph{ }$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

$\displaystyle \left\vert r\left( \zeta ,u\right) \left( t\right) -u\left( t\right) \right\vert$    
$\displaystyle \par \leq \varepsilon \left( 1+l\left( t\right) \right) \left\Ver... ...t( \zeta ,u\right) \left( s\right) ds+p\left( \zeta ,u\right) \left( t\right) ,$    

where

$\displaystyle p\left( \zeta ,u\right) \left( t\right) =\dist\left( u\left( t\ri... ...eta +\int\limits_{0}^{t}u\left( \tau \right) d\tau \right) \right) \ a.e. in I $

and the functions $ l$ and $ m$ are related with the Lipschitz condition.


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Additional Information

Andrzej Fryszkowski
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland
Email: fryszko@alpha.mini.pw.edu.pl

Tadeusz Rzeżuchowski
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland
Email: tarz@alpha.mini.pw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10502-6
PII: S 0002-9939(2010)10502-6
Keywords: Differential inclusion, solution set, Filippov Lemma, retraction of the solution set.
Received by editor(s): May 8, 2009
Received by editor(s) in revised form: March 14, 2010
Published electronically: July 26, 2010
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.