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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 743815, 15 pages
Compatible and Incompatible Nonuniqueness Conditions for the Classical Cauchy Problem
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.
- R. P. Agarwal and V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, vol. 6 of Series in Real Analysis, World Scientific Publishing, River Edge, NJ, USA, 1993.
- J. Kalas, “General nonuniqueness theorem for ordinary differential equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 97–111, 1997.
- J. Kalas, “Nonuniqueness for the solutions of ordinary differential equations,” Czechoslovak Mathematical Journal, vol. 29, no. 1, pp. 105–112, 1979.
- J. Kalas, “Nonuniqueness results for ordinary differential equations,” Czechoslovak Mathematical Journal, vol. 48, no. 2, pp. 373–384, 1998.
- C. Nowak and H. Stettner, “Nonuniqueness results for ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 30, no. 6, pp. 3935–3938, 1997.
- K. Balla, “Solution of singular boundary value problems for non-linear systems of ordinary differential equations,” U.S.S.R. Computational Mathematics and Mathematical Physics, vol. 20, no. 4, pp. 100–115, 1980.
- V. A. Čečik, “Investigation of systems of ordinary differential equations with a singularity,” Trudy Moskovskogo Matematičeskogo Obščestva, vol. 8, pp. 155–198, 1959 (Russian).
- J. Diblík and C. Nowak, “A nonuniqueness criterion for a singular system of two ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 4, pp. 637–656, 2006.
- J. Diblík and M. Růžičková, “Inequalities for solutions of singular initial problems for Carathéodory systems via Ważewski's principle,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 12, pp. 4482–4495, 2008.
- J. Diblík and M. Růžičková, “Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 3, pp. 923–944, 2004.
- J. Kalas, “Nonuniqueness theorem for a singular Cauchy problem,” Georgian Mathematical Journal, vol. 7, no. 2, pp. 317–327, 2000.
- I. T. Kiguradze, Some Singular Boundary Value Problems for Ordinary Differential Equations, Tbilisi University Press, Tbilisi, Georgia, 1975.
- C. Nowak, “Some remarks on a paper by M. Samimi on: “Nonuniqueness criteria for ordinary differential equations“,” Applicable Analysis, vol. 47, no. 1, pp. 39–44, 1992.
- V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations, Academic Press, New York, NY, USA, 1969.
- T. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov's Direct Method, Springer, New York, NY, USA, 1977.
- T. Yoshizawa, Stability Theory by Liapunov's Second Method, The Mathematical Society of Japan, Tokyo, Japan, 1966.
- P. Hartman, Ordinary Differential Equations, vol. 38 of Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2002.
- M. Nekvinda, “On uniqueness of solutions of differential equations,” Časopis Pro Pěstování Matematiky, vol. 108, no. 1, pp. 1–7, 1983.