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Sturmian comparison theory for impulsive differential inequalities and equations. (English) Zbl 0856.34033
The authors generalize the Sturmian theory to second-order impulsive differential equations $\left(*\right)$ ${x}^{\text{'}\text{'}}\left(t\right)+p\left(t\right)x\left(t\right)=0$, $t\ne {\tau }_{k}$, ${\Delta }x\left({\tau }_{k}\right)=0$, ${\Delta }{x}^{\text{'}}\left({\tau }_{k}\right)+{p}_{k}x\left({\tau }_{k}\right)=0$. Particularly, a comparison theorem, oscillation and non-oscillation theorems as well as a zero-separation theorem are proved. (Note that all solutions of an impulsive system, given in the special form $\left(*\right)$, are continuous.) Using comparison results and considering various simple (e.g., periodic, with constant coefficients) test systems, the authors present various sufficient conditions for oscillation and non-oscillation in $\left(*\right)$. On the other hand, this theory is also used here in the inverse order, to construct impulsive systems of the form $\left(*\right)$ with previously known oscillatory properties. The last section of the paper contains some applications of the main results to nonlinear impulsive differential equations.
##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A37 Differential equations with impulses 34A40 Differential inequalities (ODE)
##### References:
 [1] D. D.Bainov, V.Lakshmikantham and P. S.Simeonov, Theory of Impulsive Differential Equations. Singapore 1989. [2] D. D.Bainov and P. S.Simeonov, Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Ser. Math. Appl., Chichester 1989. [3] D. D.Bainov and P. S.Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications. Harlow 1993. [4] Yu. I. Domshlak, Sturm-like Comparison Method in the Investigation of Solutions’ Behaviour for Differential-Operator Equations. Baku: ?Ehlm? 1986 (Russian). [5] K. Gopalsamy andB. G. Zhang, On delay differential equations with impulses. J. Math. Anal. Appl.139, 110-122 (1989). · Zbl 0687.34065 · doi:10.1016/0022-247X(89)90232-1