Oscillations of higher-order neutral equations. (English) Zbl 0566.34055

Ladas, G.; Sficas, Y. G.

J. Aust. Math. Soc., Ser. B 27, 502-511 (1986).

Consider the neutral delay differential equation of order n \((*)\quad (d^ n/dt^ n)[y(t)+py(t-\tau)]+qy(t-\sigma)=0,\) \(t\geq t_ 0\) where q is a positive constant, the delays \(\tau\) and \(\sigma\) are nonnegative constants and the coefficient p is a real number. Theorem 1. (a) Assume that n is odd and that \(p<-1\). Then every nonoscillatory solution of (*) tends to \(+\infty\) or -\(\infty\) as \(t\to \infty\). (b) Assume that n is odd or even and that \(p>-1\). Then every nonoscillatory solution of (*) tends to zero as \(t\to \infty\). Theorem 2. Assume that n is odd. Then each of the following four conditions implies that every solution of (*) oscillates: (i) \(p<-1\) and \((-q/(1+p))^{1/n}(\tau -\sigma)/n>1/e\); (ii) \(p=-1\); (iii) \(p>-1\) and \((q/(1+p))^{1/n}(\sigma -\tau)/n>1/e\); (iv) \(- 1<p<0\) and \(q^{1/n}\sigma /n>1/e.\)
Theorem 3. Assume that n is even. Then each of the following two conditions implies that all solutions of (*) oscillate: (i) \(p\geq 0\); (ii) \(-1\leq p<0\) and \((q/p)^{1/n}(\sigma -\tau)/n>1/e\). Theorem 4. Assume that n is even, \(p<-1\), and \((q/p)^{1/n}(\sigma -\tau)/n>1/e\). Then every bounded solution of (*) oscillates.

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:

neutral delay differential equation