Oscillation of Impulsive Neutral Delay Differential Equations

By John R. Graef, Jianhua Shen, Ioannis P. Stavroulakis. 2002

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Abstract 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)| λ sgn x(t − σ) = 0, t ≥ t 0 , ((*)) x t + k = b k x(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.

Subfields: Applied Mathematics, Numerical Analysis, Public Health, Environmental and Occupational Health

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