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Oscillation of sub- and superlinear impulsive differential equations with constant delay. (English) Zbl 0879.34068

This paper contains oscillation criteria for semilinear impulse delay problems of type \[ \left\{ \begin{matrix} & -\bigl(r(t)y'(t)\bigr)' =f\bigl(t,y(t), y(t-c)\bigr),\;t\neq t_i \\ & y(t) =h(t) \text{ on } I=[-c,0],\;y'(0) =a \\ & J_i(y) =0,\;J_i(ry') =g_i\bigl(y(t_i), y(t_i-c) \bigr) \\ & i=1,2, \dots, c>0,\end{matrix} \right. \tag{1} \] where \(a\), \(c\) are constants, \(0<t_1 <t_2 <\cdots\), \(t_i \to\infty\) as \(i\to \infty\), \(J_i(u)\) denotes \(u(t_i+) -u(t_i)\) for left-continuous functions \(u\) at each \(t_i\), \(f\in C(\overline \mathbb{R}_+ \times \mathbb{R}^2,\mathbb{R})\), \(g_i\in C(\mathbb{R}^2, \mathbb{R})\), \(h\in C (I,\mathbb{R})\), and \(r:\overline\mathbb{R}_+ \to\mathbb{R}_+\) is piecewise continuous with jump discontinuities at each \(t_i\) (only). For both sublinear and superlinear problems, the theorems give necessary and sufficient conditions for (1) to be oscillatory at \(\infty\).

MSC:

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

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