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Compatible and incompatible nonuniqueness conditions for the classical Cauchy problem. (English) Zbl 1222.34010

Summary: Sufficient conditions for nonuniqueness of the classical Cauchy problem
\[ \dot x = f(t, x),\quad x(t_0) = x_0 \]
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.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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[1] 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. · Zbl 1052.33505
[2] J. Kalas, “General nonuniqueness theorem for ordinary differential equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 97-111, 1997. · Zbl 0870.34006
[3] J. Kalas, “Nonuniqueness for the solutions of ordinary differential equations,” Czechoslovak Mathematical Journal, vol. 29, no. 1, pp. 105-112, 1979. · Zbl 0396.34006
[4] J. Kalas, “Nonuniqueness results for ordinary differential equations,” Czechoslovak Mathematical Journal, vol. 48, no. 2, pp. 373-384, 1998. · Zbl 0957.34004
[5] C. Nowak and H. Stettner, “Nonuniqueness results for ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 30, no. 6, pp. 3935-3938, 1997. · Zbl 0887.34003
[6] 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. · Zbl 0465.34013
[7] V. A. \vCe\vcik, “Investigation of systems of ordinary differential equations with a singularity,” Trudy Moskovskogo Matemati\vceskogo Ob\vs\vcestva, vol. 8, pp. 155-198, 1959 (Russian).
[8] 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. · Zbl 1111.34007
[9] J. Diblík and M. Rů\vzi, “Inequalities for solutions of singular initial problems for Carathéodory systems via Wa\Dzewski’s principle,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 12, pp. 4482-4495, 2008. · Zbl 1170.34008
[10] J. Diblík and M. Rů\vzi, “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. · Zbl 1080.34001
[11] J. Kalas, “Nonuniqueness theorem for a singular Cauchy problem,” Georgian Mathematical Journal, vol. 7, no. 2, pp. 317-327, 2000. · Zbl 0970.34005
[12] I. T. Kiguradze, Some Singular Boundary Value Problems for Ordinary Differential Equations, Tbilisi University Press, Tbilisi, Georgia, 1975. · Zbl 0307.34003
[13] 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. · Zbl 0792.34002
[14] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations, Academic Press, New York, NY, USA, 1969. · Zbl 0177.12403
[15] T. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method, Springer, New York, NY, USA, 1977. · Zbl 0364.34022
[16] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, The Mathematical Society of Japan, Tokyo, Japan, 1966. · Zbl 0144.10802
[17] P. Hartman, Ordinary Differential Equations, vol. 38 of Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2002. · Zbl 1009.34001
[18] M. Nekvinda, “On uniqueness of solutions of differential equations,” \vCasopis Pro P Matematiky, vol. 108, no. 1, pp. 1-7, 1983. · Zbl 0524.34004
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