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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 $n\times n$ 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 $$\limsup\sb{t \to \infty}t\sp{1- m}\lambda \left(\int\sp t\sb{t\sb 0}(t-s)\sp{m-1}Q(s)ds\right)=\infty,$$ where $\lambda(\cdot)$ 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 {\it I. V. Kamenev} [Mat. Zametky 23, 249-251 (1978; Zbl 0386.34032)].
Reviewer: O.Došlý (Brno)

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