A general Lipschitz uniqueness criterion for scalar ordinary differential equations
The classical Lipschitz-type criteria guarantee unique solvability of the scalar initial value problem $\dot x=f(t,x)$, $x(t_0)=x_0,$ by putting restrictions on $|f(t,x)-f(t,y)|$ in dependence of $|x-y|$. Geometrically it means that the field differences are estimated in the direction of the $x$-axi...
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doaj-80339858eedd4c78856b2b4e6e10b7462021-07-14T07:21:26ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38751417-38752014-07-012014341610.14232/ejqtde.2014.1.343021A general Lipschitz uniqueness criterion for scalar ordinary differential equationsJosef Diblik0Christine Nowak1Stefan Siegmund2Brno University of Technology, Brno, Czech RepublicInstitute for Mathematics, University of Klagenfurt, 9020 Klagenfurt, AustriaInstitute for Analysis, Technische Universit\"{a}t Dresden, 01062 Dresden, GermanyThe classical Lipschitz-type criteria guarantee unique solvability of the scalar initial value problem $\dot x=f(t,x)$, $x(t_0)=x_0,$ by putting restrictions on $|f(t,x)-f(t,y)|$ in dependence of $|x-y|$. Geometrically it means that the field differences are estimated in the direction of the $x$-axis. In 1989, Stettner and the second author could establish a generalized Lipschitz condition in both arguments by showing that the field differences can be measured in a suitably chosen direction $v=(d_{t},d_{x})$, provided that it does not coincide with the directional vector $(1,f(t_{0},x_{0}))$. Considering the vector $v$ depending on $t$, a new general uniqueness result is derived and a short proof based on the implicit function theorem is developed. The advantage of the new criterion is shown by an example. A comparison with known results is given as well.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3021fundamental theory of ordinary differential equationsinitial value problemsuniquenesslipschitz type conditions |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Josef Diblik Christine Nowak Stefan Siegmund |
spellingShingle |
Josef Diblik Christine Nowak Stefan Siegmund A general Lipschitz uniqueness criterion for scalar ordinary differential equations Electronic Journal of Qualitative Theory of Differential Equations fundamental theory of ordinary differential equations initial value problems uniqueness lipschitz type conditions |
author_facet |
Josef Diblik Christine Nowak Stefan Siegmund |
author_sort |
Josef Diblik |
title |
A general Lipschitz uniqueness criterion for scalar ordinary differential equations |
title_short |
A general Lipschitz uniqueness criterion for scalar ordinary differential equations |
title_full |
A general Lipschitz uniqueness criterion for scalar ordinary differential equations |
title_fullStr |
A general Lipschitz uniqueness criterion for scalar ordinary differential equations |
title_full_unstemmed |
A general Lipschitz uniqueness criterion for scalar ordinary differential equations |
title_sort |
general lipschitz uniqueness criterion for scalar ordinary differential equations |
publisher |
University of Szeged |
series |
Electronic Journal of Qualitative Theory of Differential Equations |
issn |
1417-3875 1417-3875 |
publishDate |
2014-07-01 |
description |
The classical Lipschitz-type criteria guarantee unique solvability of the scalar initial value problem $\dot x=f(t,x)$, $x(t_0)=x_0,$ by putting restrictions on $|f(t,x)-f(t,y)|$ in dependence of $|x-y|$. Geometrically it means that the field differences are estimated in the direction of the $x$-axis. In 1989, Stettner and the second author could establish a generalized Lipschitz condition in both arguments by showing that the field differences can be measured in a suitably chosen direction $v=(d_{t},d_{x})$, provided that it does not coincide with the directional vector $(1,f(t_{0},x_{0}))$.
Considering the vector $v$ depending on $t$, a new general uniqueness result is derived and a short proof based on the implicit function theorem is developed. The advantage of the new criterion is shown by an example. A comparison with known results is given as well. |
topic |
fundamental theory of ordinary differential equations initial value problems uniqueness lipschitz type conditions |
url |
http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3021 |
work_keys_str_mv |
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1721303611232747520 |