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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Riccati techniques and variational principles in oscillation theory for linear systems


Authors: G. J. Butler, L. H. Erbe and A. B. Mingarelli
Journal: Trans. Amer. Math. Soc. 303 (1987), 263-282
MSC: Primary 34C10; Secondary 34A30, 34C29
MathSciNet review: 896022
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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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DOI: https://doi.org/10.1090/S0002-9947-1987-0896022-5
Keywords: Oscillation, Riccati equation, variational techniques
Article copyright: © Copyright 1987 American Mathematical Society