Oscillation of Impulsive Neutral Delay Differential Equations

• Published in: Journal of mathematical analysis and applications 2002-04, Vol.268 (1), p.310-333
• Main Authors: Graef, J.R., Shen, J.H., Stavroulakis, I.P.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; impulse; Mathematical analysis; Mathematics; neutral differential equation; nonoscillation; Ordinary differential equations; oscillation; Sciences and techniques of general use
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1006/jmaa.2001.7836

In this paper, effective sufficient conditions for the oscillation of all solutions of impulsive neutral delay differential equations of the form [x(t) − P(t)x(t − τ)]′ + Q(t)|x(t − σ)| λsgn x(t − σ) = 0, t ≥ t 0, ((*)) x t + k = b kx(t k), k = 1, 2, … ((**)) are established. Our results reveal the fact that the oscillatory properties of all solutions of Eqs. (*) and (**) may be caused by the impulsive perturbations (**) though the corresponding neutral delay differential equation without impulses, i.e., Eq. (*), admits a nonoscillatory solution. It is also seen that the oscillatory behavior of all solutions of Eq. (*) can be inherited by Eqs. (*) and (**) under certain impulsive perturbations (**). Some examples are also given to illustrate the applicability of the results obtained.