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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quickly oscillating solutions of autonomous ordinary differential equations
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by Stephen R. Bernfeld and A. Lasota PDF
Proc. Amer. Math. Soc. 30 (1971), 519-526 Request permission

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
  • Andrzej Lasota, Convergence to zero of oscillating integrals of an ordinary differential equation of the second order, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat. 6 (1961), 27–33 (Polish, with English and Russian summaries). MR 214855
  • A. Lasota and C. Olech, An optimal solution of Nicoletti’s boundary value problem, Ann. Polon. Math. 18 (1966), 131–139. MR 204742, DOI 10.4064/ap-18-2-131-139
  • A. Lasota and James A. Yorke, Oscillatory solutions of second order ordinary differential equations, Ann. Polon. Math. 25 (1971/72), 175–178. MR 304771, DOI 10.4064/ap-25-2-175-178
  • Marian Łuczyński, On the convergence to zero of oscillating solutions of an ordinary differential equation of order $n$, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat. 7 (1962), 17–20. MR 196187
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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