A generalized Picard-Lindelöf theorem

We generalize the Picard-Lindelöf theorem on the unique solvability of initial value problems $\dot x=f(t,x)$, $x(t_0)=x_0$, by replacing the sufficient classical Lipschitz condition of $f$ with respect to $x$ with a more general Lipschitz condition along hyperspaces of the $(t,x)$-space. A comparis...

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Main Authors:	Stefan Siegmund, Christine Nowak, Josef Diblik
Format:	Article
Language:	English
Published:	University of Szeged 2016-05-01
Series:	Electronic Journal of Qualitative Theory of Differential Equations
Subjects:	picard-lindelöf theorem initial value problem generalized lipschitz condition unique solvability
Online Access:	http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=4576

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record_format	Article
spelling	doaj-931fd7bb54ef44a0894068e28efa3d592021-07-14T07:21:28ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38751417-38752016-05-012016281810.14232/ejqtde.2016.1.284576A generalized Picard-Lindelöf theoremStefan Siegmund0Christine Nowak1Josef Diblik2Institute for Analysis, Technische Universität Dresden, 01062 Dresden, GermanyInstitute for Mathematics, University of Klagenfurt, 9020 Klagenfurt, AustriaBrno University of Technology, Brno, Czech RepublicWe generalize the Picard-Lindelöf theorem on the unique solvability of initial value problems $\dot x=f(t,x)$, $x(t_0)=x_0$, by replacing the sufficient classical Lipschitz condition of $f$ with respect to $x$ with a more general Lipschitz condition along hyperspaces of the $(t,x)$-space. A comparison with known results is provided and the generality of the new criterion is shown by an example.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=4576picard-lindelöf theoreminitial value problemgeneralized lipschitz conditionunique solvability
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Stefan Siegmund Christine Nowak Josef Diblik
spellingShingle	Stefan Siegmund Christine Nowak Josef Diblik A generalized Picard-Lindelöf theorem Electronic Journal of Qualitative Theory of Differential Equations picard-lindelöf theorem initial value problem generalized lipschitz condition unique solvability
author_facet	Stefan Siegmund Christine Nowak Josef Diblik
author_sort	Stefan Siegmund
title	A generalized Picard-Lindelöf theorem
title_short	A generalized Picard-Lindelöf theorem
title_full	A generalized Picard-Lindelöf theorem
title_fullStr	A generalized Picard-Lindelöf theorem
title_full_unstemmed	A generalized Picard-Lindelöf theorem
title_sort	generalized picard-lindelöf theorem
publisher	University of Szeged
series	Electronic Journal of Qualitative Theory of Differential Equations
issn	1417-3875 1417-3875
publishDate	2016-05-01
description	We generalize the Picard-Lindelöf theorem on the unique solvability of initial value problems $\dot x=f(t,x)$, $x(t_0)=x_0$, by replacing the sufficient classical Lipschitz condition of $f$ with respect to $x$ with a more general Lipschitz condition along hyperspaces of the $(t,x)$-space. A comparison with known results is provided and the generality of the new criterion is shown by an example.
topic	picard-lindelöf theorem initial value problem generalized lipschitz condition unique solvability
url	http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=4576
work_keys_str_mv	AT stefansiegmund ageneralizedpicardlindeloftheorem AT christinenowak ageneralizedpicardlindeloftheorem AT josefdiblik ageneralizedpicardlindeloftheorem AT stefansiegmund generalizedpicardlindeloftheorem AT christinenowak generalizedpicardlindeloftheorem AT josefdiblik generalizedpicardlindeloftheorem
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A generalized Picard-Lindelöf theorem

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