## 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
Full Text:

### References:

 [1] Bainov D.D., Systems with Impulse Effect: Stability, Theory and Applications (1989) · Zbl 0676.34035 [2] DOI: 10.1142/9789812831804 [3] Erbe L.H., Pure and Appl. Math. 190 (1995) [4] DOI: 10.1016/0022-247X(89)90232-1 · Zbl 0687.34065 [5] Györi I., Oscillation Theory of Delay Differential Equations with Applications (1991) · Zbl 0780.34048 [6] Ladde G.S., Pure and Applied Mathematics 110 (1987) [7] Lakshmikantham V., Theory of Impulsive Differential Equations (1989) · Zbl 0718.34011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.