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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

Citations:

Zbl 0879.34068
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