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Journal of the London Mathematical Society 1990 s2-41(2):244-260; doi:10.1112/jlms/s2-41.2.244
© 1990 by London Mathematical Society
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© Oxford University Press

Necessary and Sufficient Conditions for Oscillation of Neutral Equations with Deviating Arguments

M. K. Grammatikopoulos and I. P. Stavroulakis

Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece

Consider the first-order neutral differential equation

Formula
, (1)where p, r, q, g, {tau}, p isin R and {sigma}, µ isin[0, {infty}). A necessary and sufficient condition for the oscillation of all solutions of (1) is that its characteristic equation {lambda}+{lambda}pe–{lambda}{tau}+{lambda}re–{lambda}p+qe–{lambda}{sigma}+ge–{lambda}µ=0 has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutions of (1).


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