Oscillations of second order neutral delay differential equations. (English) Zbl 0581.34051
Consider the neutral delay differential equation
\[
(1)\quad \frac{d^ 2}{dt^ 2}[y(t)+P(t)y(t-\tau)]+Q(t)y(t-\sigma)=0,\quad t\geq t_ 0,
\]
where \(P,Q\in C([t_ 0,\infty),{\mathbb{R}})\) and \(\tau\) and \(\sigma\) are nonnegative real numbers. We prove the following theorems.
Theorem 1. Assume that \(0\leq P(t)\leq 1\), Q(t)\(\geq 0\) and that \(\int^{\infty}_{t_ 0}Q(s)[1-P(s-\sigma)]ds=\infty.\) Then every solution of Equation (1) oscillates.
Theorem 2. Assume that P(t)\(\equiv p\geq 0\) and that \(\int^{\infty}_{t_ 0}Q(s)ds=\infty\). Then the derivative of every differentiable solution of (1) oscillates.
Theorem 1. Assume that \(0\leq P(t)\leq 1\), Q(t)\(\geq 0\) and that \(\int^{\infty}_{t_ 0}Q(s)[1-P(s-\sigma)]ds=\infty.\) Then every solution of Equation (1) oscillates.
Theorem 2. Assume that P(t)\(\equiv p\geq 0\) and that \(\int^{\infty}_{t_ 0}Q(s)ds=\infty\). Then the derivative of every differentiable solution of (1) 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 |
