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On Kneser solutions of higher order nonlinear ordinary differential equations. (English) Zbl 1118.34317
Summary: The equation \(x^{(n)}(t)=(-1)^n|x(t)|^k\), with \(k>1\) is considered. In the case \(n\leq 4\) it is proved that solutions defined in a neighbourhood of infinity coincide with \(C(t-t_0)^{-n/(k-1)}\), where \(C\) is a constant depending only on \(n\) and \(k\). In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times \((t-t_0)^{-n/(k-1)}\). It is shown that they do not necessarily coincide with \(C(t-t_0)^{-n/(k-1)}\). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics.

34C11 Growth and boundedness of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI
[1] Astashova, I. V., Asymptotic behavior of solutions of certain nonlinear differential equations, inReports of the extended sessions of a seminar of the I. N. Vekua Institute of Applied Mathematics 1 (Kiguradze, I. T., ed.), pp. 9–11, Tbilis. Gos. Univ., Tbilisi, 1985 (Russian).
[2] Guckenheimer, J. andHolmes, P.,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci.42, Springer-Verlag, New York, 1983. · Zbl 0515.34001
[3] Kiguradze, I. T., On Kneser solutions of ordinary differential equations,Uspekhi Mat. Nauk 41:4 (1986), 211 (Russian).
[4] Kiguradze, I. T. andChanturia, T. A.,Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer, Dordrecht, 1993.
[5] Kneser, A., Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen bei grossen reelen Werten des Arguments,J. Reine Angew. Math. 116 (1896), 178–212. · JFM 27.0253.03 · doi:10.1515/crll.1896.116.178
[6] Kozlov, V. andMaz’ya, V.,Theory of a Higher-Order Sturm-Liouville Equations, Lecture Notes in Math.1659, Springer-Verlag, Berlin-Heidelberg, 1997.
[7] Kozlov, V. andMaz’ya, V.,Differential equations with Operator Coefficients with Applications to Boundary Value Problems for Partial Differential Equations), Monogr. Math., Springer-Verlag, Berlin-Heidelberg, 1999. · Zbl 0920.35003
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