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Oscillation criteria for second-order matrix dynamic equations on a time scale. (English) Zbl 1017.34030

The authors investigate oscillatory properties of the matrix Sturm-Liouville dynamic equation on a time scale 𝕋 (which is supposed to be unbounded from above)

[P(t)X Δ ] Δ +Q(t)X σ =0,(*)

where P,Q are symmetric n×n-matrices and P is positive definite. Basic facts of the time scale calculus can be found in the recent monograph of M. Bohner and A. Peterson [Dynamic equations on time scales. An introduction with applications. Basel: Birkhäuser (2001; Zbl 0978.39001)]. Recall that the time scale derivative Δ reduces to the usual derivative d dt if 𝕋= and to the forward difference Δ if 𝕋=. The forward jump operator is defined by σ(t)=inf{s𝕋:s>t} and the graininess by μ(t)=σ(t)-t. One of the main results of the paper reads as follows:

Suppose that for every t 0 𝕋 there exist t 0 a 0 <b 0 such that μ(a 0 )>0, μ(b 0 )>0 and

λ max a 0 b 0 Q(t)Δt1 μ(a 0 )+1 μ(b 0 ),

where λ max stands for the greatest eigenvalue of the matrix indicated. Then (*) with P(t)I is oscillatory.

The general oscillation criteria presented in the paper are illustrated by a number of examples and corollaries.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
39A10Additive difference equations
34B24Sturm-Liouville theory
34B30Special ODE (Mathieu, Hill, Bessel, etc.)