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Oscillation for a second-order neutral differential equation with impulses. (English) Zbl 1210.34090

The goal of this paper is to study oscillating systems which remain oscillating after the system is perturbed by impulses. The main results give sufficient conditions for the solutions to a class of second-order neutral delay differential systems with impulses to be oscillatory. The results in this work provide extensions to some previous oscillation criteria and are based on some ideas and results included in [L. P. Gimenes and M. Federson, Comput. Math. Appl. 52, 819–828 (2006; Zbl 1134.34040); V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive differential equations. Series in Modern Applied Mathematics 6, Singapore, World Scientific (1989; Zbl 0719.34002) and H.-J. Li, Math. Comput. Modelling 25, No. 3, 69–79 (1997; Zbl 0882.34077)].
To be precise, the following system is studied
\[ [r(t)(x(t)+p(t)x(t-\tau))']'+f(t,x(t),x(t-\delta))=0,\qquad t \geq t_0,\qquad t \neq t_k, \]
\[ x(t_k)=I_k(x(t_k^-)),\qquad x'(t_k)=J_k(x'(t_k^-)),\qquad k=1,2,\dots, \]
\[ x(t)=\phi(t),\qquad t_0-\sigma \leq t \leq t_0, \]
where \(\delta\) and \(\tau\) are positive real numbers, \(\sigma:=\max\{\delta,\tau\}\), \(0\leq t_0<t_1<\dots<t_k<\cdots\) with \(\{t_k\} \rightarrow +\infty\), \(t_{k+1}-t_k>\sigma\) \(\forall k \in \mathbb{N}\), \(p \in PC^1([t_0,+\infty), \mathbb{R}^+)\) and \(\phi,\, \phi':[t_0-\sigma,t_0]\rightarrow \mathbb{R}\) have at most a finite number of discontinuities of the first kind at which they are right continuous.
Some particular oscillatory nonimpulsive neutral delay differential equations of second order are considered to illustrate that the solutions remain oscillatory under the introduction of impulses. An application to the extended Emden-Fowler equation \[ [y(t)+p(t)y(t-\tau)]''+q(t)f(y(t-\delta))=0 \]
is also provided.

MSC:

34K11 Oscillation theory of functional-differential equations
34K45 Functional-differential equations with impulses
34K40 Neutral functional-differential equations
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