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Zbl 0856.34033
Bainov, D.D.; Domshlak, Yu.I.; Simeonov, P.S.
Sturmian comparison theory for impulsive differential inequalities and equations. (English)
[J] Arch. Math. 67, No.1, 35-49 (1996). ISSN 0003-889X; ISSN 1420-8938/e

The authors generalize the Sturmian theory to second-order impulsive differential equations $(*)$ $x''(t)+ p(t) x(t)= 0$, $t\ne \tau_k$, $\Delta x(\tau_k)=0$, $\Delta x'(\tau_k)+p_k x(\tau_k)= 0$. 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 $(*)$, 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 $(*)$. On the other hand, this theory is also used here in the inverse order, to construct impulsive systems of the form $(*)$ with previously known oscillatory properties. The last section of the paper contains some applications of the main results to nonlinear impulsive differential equations.
[S.I.Trofimchuk (Kiev)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34A37 Differential equations with impulses
34A40 Differential inequalities (ODE)

Keywords: Sturmian theory; second-order impulsive differential equations; comparison theorem; non-oscillation theorems

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