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 \[ \limsup_{t \to \infty}t^{1- m}\lambda \left(\int^ t_{t_ 0}(t-s)^{m-1}Q(s)ds\right)=\infty, \] 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 I. V. Kamenev [Mat. Zametky 23, 249-251 (1978; Zbl 0386.34032)].
Theorem. Let \(m>2\) be an integer. If \[ \limsup_{t \to \infty}t^{1- m}\lambda \left(\int^ t_{t_ 0}(t-s)^{m-1}Q(s)ds\right)=\infty, \] 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 I. V. Kamenev [Mat. Zametky 23, 249-251 (1978; Zbl 0386.34032)].
Reviewer: O.Došlý (Brno)
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34A30 | Linear ordinary differential equations and systems |
