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Kamenev type theorems for second order matrix differential systems. (English) Zbl 0777.34024

The authors deal with oscillation criteria for self-adjoint differential systems (*) (P(t)y ' ) ' +Q(t)y=0, where P,Q are n×n symmetric matrices of real-valued functions and y is an n-dimensional vector. As a consequence of the general oscillation criterion for (*) the following result is proved.

Theorem. Let m>2 be an integer. If

lim sup t t 1-m λ t 0 t (t-s) m-1 Q(s)ds=,

where λ(·) stands for the largest eigenvalue, then the system y '' +Q(t)y=0 is oscillatory.

If Q and y are scalar quantities then this statement reduces to the oscillation criterion of I. V. Kamenev [Mat. Zametky 23, 249-251 (1978; Zbl 0386.34032)].

Reviewer: O.Došlý (Brno)

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general