J Austral Math Soc Ser A 52 pp261--284, 1992.

Oscillations of Higher Order Neutral Differential Equations

S. J. Bilchev, M. K. Grammatikopoulos and I. P. Stavroulakis

Abstract

Consider the nth order neutral differential equation

dn dtn [x(t) + å J  pix(t - ti)] + d å K  qkx(t - sk) = 0

(E)

where n ³ 1, d = ±1, J, K are initial segments of natural numbers, p_i, t_i, s_k Î R and q_k ³ 0 for i Î J and k Î K. Then a necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation

ln + ln å J  pie-lti + d å K  qke-lsk = 0

has no real roots. The method of proof has the advantage that it results is easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutions of Equation (E).

1991 AMS Subject Classification: primary 34K40; secondary 34K15, 34C10

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Authors

S. J. Bilchev

Technical University, 7017 Rousse, Bulgaria.

M. K. Grammatikopoulos

University of Ioannina, 451 10 Ioannina, Greece.

I. P. Stavroulakis

University of Ioannina, 451 10 Ioannina, Greece.