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On oscillations of unbounded solutions


Authors: I. Győri, G. Ladas and L. Pakula
Journal: Proc. Amer. Math. Soc. 106 (1989), 785-792
MSC: Primary 34K15; Secondary 44A10
DOI: https://doi.org/10.1090/S0002-9939-1989-0969316-X
MathSciNet review: 969316
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Abstract: Consider the differential equation with deviating arguments (1)

$\displaystyle \dot y(t) + p[y(t - {\sigma _1}) - y(t - {\sigma _2})] = 0$

where $ p,{\sigma _1}$ and $ {\sigma _2}$ are real numbers. We prove that every unbounded solution of (1) oscillates if and only if the characteristic equation (2)

$\displaystyle \lambda + p({e^{ - \lambda {\sigma _1}}} - {e^{ - \lambda {\sigma _2}}}) = 0$

has no positive roots and 0 is a simple root of (2).

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

  • [1] R. Bellman and K. Cooke, Differential-difference equations, Academic Press, New York, 1963. MR 0147745 (26:5259)
  • [2] C. Corduneanu, Almost-periodic functions, Interscience Publishers, New York, 1968. MR 0481915 (58:2006)
  • [3] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 0210528 (35:1420)
  • [4] D. V. Widder, An introduction to transform theory, Academic Press, New York, 1971.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0969316-X
Article copyright: © Copyright 1989 American Mathematical Society

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