Riccati techniques and variational principles in oscillation theory for linear systems

Butler, G. J.; Erbe, L. H.; Mingarelli, A. B.

doi:10.1090/S0002-9947-1987-0896022-5

Riccati techniques and variational principles in oscillation theory for linear systems
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by G. J. Butler, L. H. Erbe and A. B. Mingarelli PDF

Trans. Amer. Math. Soc. 303 (1987), 263-282 Request permission

Abstract:

We consider the seond order differential system $(1) Y'' + Q(t)Y = 0$, where $Q$, $Y$ are $n \times n$ matrices with $Q = Q(t)$ a continuous symmetric matrix-valued function, $t \in [a, + \infty ]$. We obtain a number of sufficient conditions in order that all prepared solutions $Y(t)$ of $(1)$ are oscillatory. Two approaches are considered, one based on Riccati techniques and the other on variational techniques, and involve assumptions on the behavior of the eigenvalues of $Q(t)$ (or of its integral). These results extend some well-known averaging techniques for scalar equations to system $(1)$.

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