Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
MathSciNet review: 748831
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the $ n{\text{th}}$ order delay differential equation (1)

$\displaystyle {x^{(n)}}(t) + {( - 1)^{n + 1}}\sum\limits_{i = 0}^k {{p_i}x(t - {\tau_i}) = 0, \qquad t \geq {t_0}},$

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)

$\displaystyle {\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.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34K10, 34C10, 34K15

Retrieve articles in all journals with MSC: 34K10, 34C10, 34K15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0748831-8
PII: S 0002-9947(1984)0748831-8
Keywords: Delay differential equations, advanced arguments, oscillations
Article copyright: © Copyright 1984 American Mathematical Society