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**Necessary and sufficient condition for oscillation of a neutral differential system with several delays.**
*(English)*
Zbl 0691.34054

The reason for the growing interest in the study of delay differential equations whose solutions exhibit oscillatory behavior is that there are applications, such as modelling fluctuating populations, for which it has been shown that delay equations give a better description than the ordinary ones.

In the oscillation theory of linear delay differential equations, one of the most important objectives is to give a necessary and sufficient condition for oscillation via the characteristic equation. Such a result for scalar delay differential equations was proved using various methods (see, e.g., [the authors and A. Jawhari [J. Differ. Equations 53, 115-123 (1984; Zbl 0547.34060)]) and it was extended to delay systems too [J. Ferreira and the second author, J. Math. Anal. Appl. 128, 332- 346 (1987; Zbl 0653.34047)]. In the neutral case, many special scalar equations have been investigated using different techniques, but no general result was found.

In this paper, the authors prove that in the general scalar as well as system cases, a neutral delay differential equation with several delays has a nonoscillatory solution if and only if its characteristic equation has a real root. The proof is based on the method of the Laplace transform, using the fact that the solutions of a neutral equation are not growing faster than exponentially.

In the oscillation theory of linear delay differential equations, one of the most important objectives is to give a necessary and sufficient condition for oscillation via the characteristic equation. Such a result for scalar delay differential equations was proved using various methods (see, e.g., [the authors and A. Jawhari [J. Differ. Equations 53, 115-123 (1984; Zbl 0547.34060)]) and it was extended to delay systems too [J. Ferreira and the second author, J. Math. Anal. Appl. 128, 332- 346 (1987; Zbl 0653.34047)]. In the neutral case, many special scalar equations have been investigated using different techniques, but no general result was found.

In this paper, the authors prove that in the general scalar as well as system cases, a neutral delay differential equation with several delays has a nonoscillatory solution if and only if its characteristic equation has a real root. The proof is based on the method of the Laplace transform, using the fact that the solutions of a neutral equation are not growing faster than exponentially.

Reviewer: D.Savin

### MSC:

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

### Keywords:

delay differential equations; modelling fluctuating populations; neutral delay differential equation
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\textit{O. Arino} and \textit{I. Györi}, J. Differ. Equations 81, No. 1, 98--105 (1989; Zbl 0691.34054)

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### References:

[1] | Arino, O.; Győri, I.; Jawhari, J., Oscillation criteria in delay equations, J. Differential Equations, 53, 115-123 (1984) · Zbl 0547.34060 |

[2] | Ferreira, J.; Győri, I., Oscillatory behavior in linear retarded functional differential equations, J. Math. Anal. Appl., 128, 332-342 (1987) · Zbl 0653.34047 |

[3] | Győri, I., Oscillation and comparison results in neutral differential equations with application to the delay logistic equation, (Proceedings of the Conference on Mathematical Problems in Population Dynamics. Proceedings of the Conference on Mathematical Problems in Population Dynamics, Oxford, Mississippi (1986)), in press · Zbl 0697.92023 |

[4] | Hale, J., Theory of Functional Differential Equations, (Applied Mathematical Sciences, Vol. 3 (1977), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0222.34003 |

[5] | Kulenovic, M. R.S; Ladas, G., Linearized Oscillations in Population Dynamics, Bull. Math. Biol., 49, 615-617 (1987) · Zbl 0634.92013 |

[6] | Kulenovic, M. R.S; Ladas, G.; Meimaridon, A., Necessary and sufficient condition for oscillations of neutral differential equations, J. Austral. Math. Soc. Ser. B, 28, 362-375 (1987) · Zbl 0616.34064 |

[7] | Ladas, G.; Stavroulakis, J., Oscillations caused by several retarded and advanced arguments, J. Differential Equations, 44, 134-152 (1982) · Zbl 0452.34058 |

[8] | Nisbet, P. M.; Gurney, W. S.C, Modelling Fluctuating Populations (1982), Wiley: Wiley New York · Zbl 0593.92013 |

[9] | Tramov, M. I., Conditions for oscillatory solutions of first order differential equations with a delayed argument, Izv. Vyssh. Uchebn. Zaved. Mat., 19, No. 3, 92-96 (1975) · Zbl 0319.34070 |

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