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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Necessary and sufficient conditions for oscillations of higher order delay differential equations


Authors: G. Ladas, Y. G. Sficas and I. P. Stavroulakis
Journal: Trans. Amer. Math. Soc. 285 (1984), 81-90
MSC: Primary 34K10; Secondary 34C10, 34K15
DOI: https://doi.org/10.1090/S0002-9947-1984-0748831-8
MathSciNet review: 748831
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Abstract: Consider the $n{\text {th}}$ order delay differential equation (1) \[ {x^{(n)}}(t) + {( - 1)^{n + 1}}\sum \limits _{i = 0}^k {{p_i}x(t - {\tau _i}) = 0, \qquad t \geq {t_0}},\] where the coefficients and the delays are constants such that $0 = {\tau _0} < {\tau _{1}} < \cdots < {\tau _k};{p_0} \geq 0,{p_i} > 0,i = 1,2,\ldots ,k;k \geq 1$ and $n \geq 1$. The characteristic equation of (1) is (2) \[ {\lambda ^n} + {( - 1)^{n + 1}}\;\sum \limits _{i = 0}^k {{p_i}{e^{ - \lambda {\tau _i}}} = 0}. \] We prove the following theorem. Theorem. (i) For $n$ odd every solution of (1) oscillates if and only if (2) has no real roots. (ii) For $n$ even every bounded solution of (1) oscillates if and only if (2) has no real roots in $( - \infty ,0]$. The above results have straightforward extensions for advanced differential equations.


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Keywords: Delay differential equations, advanced arguments, oscillations
Article copyright: © Copyright 1984 American Mathematical Society