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Necessary and sufficient condition for oscillations of neutral differential equations

Published online by Cambridge University Press:  17 February 2009

M. R. S. Kulenović
Affiliation:
Department of Mathematics, University of Rhode Island, Kingston, R.I. 02881, USA. Department of Mathematics, University of Sarajevo, Sarajevo 71000, Yugoslavia.
G. Ladas
Affiliation:
Department of Mathematics, University of Rhode Island, Kingston, R.I. 02881, USA.
A. Meimaridou
Affiliation:
Department of Mathematics, University of Rhode Island, Kingston, R.I. 02881, USA. Department of Electrical Engineering, Democritus University of Thrace, Xanthi 67100, Greece.
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Abstract

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Consider the neutral delay differential equation

where pR, τ ≥ 0, q1 > 0, σ1 ≥ 0, for i = 1, 2, …, k. We prove the following result.

Theorem. A necessary and sufficient condition for the oscillation of all solutions of Eq. (1) is that the characteristic equation

has no real roots.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Brayton, R. K. and Willoughby, R. A., “On the numerical integration of a symmetric system of difference-differential equations of neutral type”, J. Math. Anal. Appl. 18 (1967), 182189.CrossRefGoogle Scholar
[2]Grammatikopoulos, M. K., Grove, E. A. and Ladas, G., “Oscillation of first order neutral delay differential equations”, J. Math. Anal. Appl. (to appear).Google Scholar
[3]Grove, E. A., Ladas, G. and Meimaridou, A., “A necessary and sufficient condition for the oscillation of neutral equations”, J. Math. Anal. Appl. (to appear).Google Scholar
[4]Hale, J., Theory of functional differential equations, (Springer-Verlag, New York, 1977).CrossRefGoogle Scholar
[5]Ladas, G. and Sficas, Y. G., “Oscillations of neutral delay differential equations”, Canad. Math. Bull. (to appear).Google Scholar
[6]Ladas, G., Sficas, Y. G. and Stavroulakis, I. P., “Necessary and sufficient conditions for oscillations”, Amer. Math. Monthly 90 (1983), 637640.CrossRefGoogle Scholar
[7]Sficas, Y. G. and Stavroulakis, I. P., “Necessary and sufficient conditions for oscillations of neutral differential equations”, Proc. Int. Conf. on Theory and Applic. of Differential Equations, Pan American University, Edinburg, Texas 78539, USA, May 20–23, 1985.Google Scholar
[8]Slemrod, M. and Infante, E. F., “Asymptotic stability criteria for linear systems of differential equations of neutral type and their discrete analogues”, J. Math. Anal. Appl. 38 (1972), 399415.CrossRefGoogle Scholar
[9]Snow, W., “Existence, uniqueness, and stability for nonlinear differential-difference equations in the neutral case”, N.Y.U. Courant Inst. Math. Sci. Rep. IMM-NYU 328, (February 1965).Google Scholar
[10]Tramov, M. I., “Conditions for oscillatory solutions of first order differential equations with a delayed argument”, Izv. Vyssh. Uchebn. Zaved. Mat. 19 (1975), 9296.Google Scholar