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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 \((*)\) \(x''(t)+ p(t) x(t)= 0\), \(t\neq \tau_k\), \(\Delta x(\tau_k)=0\), \(\Delta x'(\tau_k)+p_k x(\tau_k)= 0\). 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.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A37 Ordinary differential equations with impulses
34A40 Differential inequalities involving functions of a single real variable
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References:

[1] D. D.Bainov, V.Lakshmikantham and P. S.Simeonov, Theory of Impulsive Differential Equations. Singapore 1989. · Zbl 0719.34002
[2] D. D.Bainov and P. S.Simeonov, Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Ser. Math. Appl., Chichester 1989. · Zbl 0676.34035
[3] D. D.Bainov and P. S.Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications. Harlow 1993. · Zbl 0815.34001
[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
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