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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a conjecture for oscillation of second-order ordinary differential systems


Author: Angelo B. Mingarelli
Journal: Proc. Amer. Math. Soc. 82 (1981), 593-598
MSC: Primary 34C10; Secondary 34A30
MathSciNet review: 614884
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Abstract: We present here some results pertaining to the oscillatory behavior at infinity of the vector differential equation

$\displaystyle y'' + Q(t)y = 0,\quad t \in [0,\infty )$

, where $ Q(t)$ is a real continuous $ n \times n$ symmetric matrix function. It has been conjectured (cf., e.g. [6]) that the criterion

$\displaystyle \mathop {\lim }\limits_{t \to \infty } {\lambda _1}\left\{ {\int_0^t {Q(s)\;ds} } \right\} = \infty $

where $ {\lambda _1}( \cdot )$ denotes the maximum eigenvalue of the matrix concerned, implies oscillation. We show that this is so under the tacit assumption

$\displaystyle \mathop {\lim \inf }\limits_{t \to \infty } {t^{ - 1}}{\text{tr}}\left\{ {\int_0^t {Q(s)\;ds} } \right\} > - \infty $

where $ {\text{tr}}( \cdot )$ represents the trace of the matrix under consideration.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0614884-3
PII: S 0002-9939(1981)0614884-3
Keywords: Conjugate points, disconjugacy, oscillation at infinity, differential systems
Article copyright: © Copyright 1981 American Mathematical Society