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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Quickly oscillating solutions of autonomous ordinary differential equations


Authors: Stephen R. Bernfeld and A. Lasota
Journal: Proc. Amer. Math. Soc. 30 (1971), 519-526
MSC: Primary 34.42
DOI: https://doi.org/10.1090/S0002-9939-1971-0285764-4
MathSciNet review: 0285764
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Abstract: We are concerned here with the asymptotic behavior of quickly oscillating solutions of systems of differential equations. It is shown that the limit of the norm of any quickly oscillating solution exists and is either equal to infinity or zero. We then determine asymptotic bounds on the solutions by imposing certain growth conditions on the right-hand side of the equation. Our results, when applied to second order equations, yield asymptotic behavior of both the solutions and its derivatives.


References [Enhancements On Off] (What's this?)

  • [1] Andrzej Lasota, Convergence to zero of oscillating integrals of an ordinary differential equation of the second order, Zeszyty Nauk Uniw. Jagiello. Prace Mat. No. 6 (1961), 27–33 (Polish, with English and Russian summaries). MR 0214855
  • [2] A. Lasota and C. Olech, An optimal solution of Nicoletti’s boundary value problem, Ann. Polon. Math. 18 (1966), 131–139. MR 0204742
  • [3] A. Lasota and James A. Yorke, Oscillatory solutions of second order ordinary differential equations, Ann. Polon. Math. 25 (1971/72), 175–178. MR 0304771
  • [4] Marian Łuczyński, On the convergence to zero of oscillating solutions of an ordinary differential equation of order 𝑛, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. No. 7 (1962), 17–20. MR 0196187

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0285764-4
Keywords: Quickly oscillating solutions, bounds on solutions, asymptotic behavior of solutions and their derivatives, zeros of solutions
Article copyright: © Copyright 1971 American Mathematical Society