Oscillation in odd-order neutral delay differential equations
The authors consider the odd-order neutral delay differential equation (1) \((x(t) - \alpha x(t - \tau))^{(n)} + p(t) f(x(t - \sigma)) = 0\), where \(0 \leq \alpha < 1\), \(\tau, \sigma \in (0, \infty)\), \(p \in C ([0, \infty), (0, \infty))\), \(f \in C^1 (\mathbb{R}, \mathbb{R})\) such that \(f\) is increasing, \(xf(x) > 0\) for \(x \neq 0\). The main result of the paper is the following oscillation criterion: Let the following conditions hold \[ \liminf_{x \to 0} \left |\left( {df \over dx} \right) \right |= 1; \] \[ \bigl |f(x) \bigr |\to \infty \text{ as } |x |\to \infty; \] \[ \liminf_{t \to \infty} \int^t_{t - \sigma/n} \sigma^{n - 1} p(s) ds > {1 \over e} (1 - \alpha) (n - 1)! \left( {n \over n - 1} \right)^{n - 1}. \] Then if \(n > 1\) is an odd integer each solution of (1) is oscillatory. The authors show that their criterion generalizes the known results on oscillation of all solutions of (1). They give an example such that the known oscillation criteria are not applicable while their result guarantees that all solutions of (1) oscillate.
odd-order neutral delay differential equation
scientific article; zbMATH DE number 903354