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Zbl 0868.39006
Domshlak, Yuri
Oscillatory properties of linear difference equations with continuous time. (English)
[J] Differ. Equ. Dyn. Syst. 1, No.4, 311-324 (1993). ISSN 0971-3514; ISSN 0974-6870/e

Summary: We establish the analogues of classical Sturmian theorems (comparison theorem, oscillation theorem, zeros-separation theorem) for linear difference equations (and inequalities) with continuous time (DECT) $$x[p(t)]- a(t)x(t)+ \sum^N_{k=1} b_k(t) x[r_k(t)]=0, \qquad t\geq t_0.$$ As a result we obtain: \par (a) Sharp conditions for the oscillation of all solutions of the above equation (with $N=1$ and $N=2$) in terms of its coefficients; \par (b) Upper estimates (and some lower estimates as well) for the length of the sign-preservation intervals for solutions; \par (c) Some properties for the DECTs with no regular oscillatory solutions.

MSC 2000:: *39A12 Discrete version of topics in analysis
39A10 Difference equations
26D20 Analytical inequalities involving real functions

Keywords: difference inequalities; comparison theorem; linear difference equations with continuous time; Sturmian theorems; oscillation theorem; zeros-separation theorem; oscillatory solutions

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