Necessary and sufficient condition for oscillation of a neutral differential system with several delays
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 \textit{A. Jawhari} [J. Differ. Equations 53, 115-123 (1984; Zbl 0547.34060)]) and it was extended to delay systems too [\textit{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.
scientific article; zbMATH DE number 4130822