Oscillation in odd-order neutral delay differential equations
type: Item
instance of: scholarly article
title: Oscillation in odd-order neutral delay differential equations
zbMATH Open document ID: 0855.34082
DOI: 10.1007/BF02880368
author: Pitambar Das
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1997-02-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 903354
zbMATH Keywords: odd-order neutral delay differential equation
zbMATH Keywords: oscillation criterion
reviewed by: V. Petrov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 903354