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Oscillatory properties of linear difference equations with continuous time. (English) Zbl 0868.39006

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)+ k=1 N b k (t)x[r k (t)]=0,tt 0 ·

As a result we obtain:

(a) Sharp conditions for the oscillation of all solutions of the above equation (with N=1 and N=2) in terms of its coefficients;

(b) Upper estimates (and some lower estimates as well) for the length of the sign-preservation intervals for solutions;

(c) Some properties for the DECTs with no regular oscillatory solutions.

39A12Discrete version of topics in analysis
39A10Additive difference equations
26D20Analytical inequalities involving real functions