## Oscillating solutions of nonlinear impulsive differential equations with a deviating argument.(English)Zbl 0901.34066

Let $$\{b_i\}$$ be a sequence of real numbers with $$b_i>- 1$$ for every $$i= 1,2,\dots$$. Let $$\{t_i\}$$ be an increasing sequence of positive numbers such that $$t_{i+1}- t_i\geq a> h$$ for every $$i= 1,2,\dots$$ and some constants $$a,h>0$$. Ocillation criteria are proved for semilinear impulse delay problems of type \begin{aligned} -y'(t) &= p(t)f(y(t- h)),\quad t\neq t_i\\ y(t) &= g(t)\quad\text{on }I= [-h, 0]\tag{1}\\ y(t_i+ 0) & = (1+ b_i)y(t_i),\quad i= 1,2,\dots,\end{aligned} where $$f\in C(\mathbb{R},\mathbb{R})$$, $$g\in C(I,\mathbb{R})$$, $$p\in C(\overline{\mathbb{R}}_+, \mathbb{R}_+)$$, $$uf(u)> 0$$ for all $$u\neq 0$$, $$f$$ is increasing, $$| f(u)|\geq C| u|$$ for a constant $$C> 0$$. For example, (1) is oscillatory if $\limsup_{i\to \infty} (1+ b_i)^{-1} \int^{t_i+ h}_{t_i} p(t)dt> C^{-1}.$ Also, an analogous result is obtained for a nonhomogeneous variant of (1). Oscillation theorems for second-order impulse delay problems appear elsewhere [D. D. Bainov and M. Dimitrova, Appl. Anal. 64, No. 1-2, 57-67 (1997; Zbl 0879.34068)].

### MSC:

 34K11 Oscillation theory of functional-differential equations 34A37 Ordinary differential equations with impulses 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Zbl 0879.34068