Publication detail

Compatible and incompatible nonuniqueness conditions for the classical Cauchy problem

DIBLÍK, J. NOWAK, C.

Original Title

Compatible and incompatible nonuniqueness conditions for the classical Cauchy problem

English Title

Compatible and incompatible nonuniqueness conditions for the classical Cauchy problem

Type

journal article - other

Language

en

Original Abstract

In the first part of this paper sufficient conditions for nonuniqueness of the classical Cauchy problem $\dot{x}=f(t,x)$, $x(t_0)=x_0$ are given. As the essential tool serves a method which estimates the ``distance'' between two solutions with an appropriate Lyapunov function and permits to show that under certain conditions the ``distance'' between two different solutions vanishes at the initial point. In the second part attention is paid to conditions that are obtained by a formal inversion of uniqueness theorems of Kamke-type but cannot guarantee nonuniqueness because they are incompatible.

English abstract

In the first part of this paper sufficient conditions for nonuniqueness of the classical Cauchy problem $\dot{x}=f(t,x)$, $x(t_0)=x_0$ are given. As the essential tool serves a method which estimates the ``distance'' between two solutions with an appropriate Lyapunov function and permits to show that under certain conditions the ``distance'' between two different solutions vanishes at the initial point. In the second part attention is paid to conditions that are obtained by a formal inversion of uniqueness theorems of Kamke-type but cannot guarantee nonuniqueness because they are incompatible.

Keywords

Fundamental theory of ordinary differential equations, nonuniqueness of solutions, incompatible set of conditions

RIV year

2011

Released

02.08.2011

ISBN

1085-3375

Periodical

Abstract and Applied Analysis

Year of study

2011

Number

1

State

US

Pages from

1

Pages to

15

Pages count

15

Documents

BibTex


@article{BUT72872,
  author="Josef {Diblík} and Christine {Nowak}",
  title="Compatible and incompatible nonuniqueness conditions for the classical Cauchy problem",
  annote="In the first part of this paper sufficient conditions for nonuniqueness of the classical Cauchy problem
$\dot{x}=f(t,x)$, $x(t_0)=x_0$
are given. As the essential tool serves a method which estimates
the ``distance'' between two solutions with an appropriate Lyapunov function
and permits to show that under certain conditions the ``distance'' between two different solutions
vanishes at the initial point. In the second part attention is paid to
conditions that are obtained by a formal inversion of uniqueness theorems of
Kamke-type but cannot guarantee nonuniqueness because they are
incompatible.",
  chapter="72872",
  number="1",
  volume="2011",
  year="2011",
  month="august",
  pages="1--15",
  type="journal article - other"
}