Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On oscillation of nonlinear delay differential equations


Authors: M. R. S. Kulenović, G. Ladas and A. Meimaridou
Journal: Quart. Appl. Math. 45 (1987), 155-164
MSC: Primary 34K15
DOI: https://doi.org/10.1090/qam/885177
MathSciNet review: 885177
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Abstract: Necessary, sufficient, and necessary and sufficient conditions are obtained for all solutions of the nonlinear differential equation

$\displaystyle \frac{{dy}}{{dt}} + \sum\limits_{j = 1}^n {{q_j}f\left( {y\left( {t - {\tau _j}} \right)} \right) = 0,\qquad t \ge 0,} \qquad \left( * \right)$

to be oscillatory. These conditions are expressed in terms of the characteristic equation of the corresponding linear ``variational'' equation

$\displaystyle \frac{{dy}}{{dt}} + \sum\limits_{j = 1}^n {{q_j}y\left( {t - {\tau _j}} \right) = 0, \qquad t \ge 0. \qquad \left( { * * } \right)} $

Our results show that for a certain class of nonlinear functions $ f,\left( * \right)$ oscillates if and only if $ \left( { * * } \right)$ oscillates. As an application of our results, we obtain simple sufficient and necessary and sufficient conditions for the oscillation of several nonlinear delay differential equations which appear in applications.

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DOI: https://doi.org/10.1090/qam/885177
Article copyright: © Copyright 1987 American Mathematical Society


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