Abstract and Applied Analysis
Volume 2011 (2011), Article ID 743815, 15 pages
http://dx.doi.org/10.1155/2011/743815
Research Article

Compatible and Incompatible Nonuniqueness Conditions for the Classical Cauchy Problem

Josef Diblík1,2 and Christine Nowak3

1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, University of Technology, 602 00 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic
3Institute for Mathematics, University of Klagenfurt, 9020 Klagenfurt, Austria

Received 21 September 2010; Revised 23 November 2010; Accepted 10 March 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 Josef Diblík and Christine Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the first part of this paper sufficient conditions for nonuniqueness of the classical Cauchy problem ̇ 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.