Necessary and sufficient condition for oscillations of neutral differential equations. (English) Zbl 0616.34064

This paper deals with the neutral delay differential equation \[ (1)\quad \frac{d}{dt}[y(t)+py(t-\tau)]+\sum^{k}_{i=1}q_ iy(t-\sigma_ i)=0, \] with its characteristic equation \[ (2)\quad \lambda +p\lambda e^{- \lambda \tau}+\sum^{k}_{i=1}q_ ie^{-\lambda \sigma_ i}=0, \] where \(p\in {\mathbb{R}}\), \(\tau\geq 0\), \(q_ i>0\) and \(\sigma_ i\geq 0\), for \(i=1,...,k\). It is known that even though all the characteristic roots of (2) may have negative real parts, it is still possible for (1) to have unbounded solutions. However, the authors of this paper establish the fact that unlike stability, the oscillatory nature of the solutions of (1) is determined by the roots of the characteristic equation (2); that is, every solution of (1) oscillates if and only if the characteristic equation (2) has no real roots.
Reviewer: Ding Tongren


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
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