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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Conjugate points of vector-matrix differential equations


Author: Roger T. Lewis
Journal: Trans. Amer. Math. Soc. 231 (1977), 167-178
MSC: Primary 34C10
MathSciNet review: 0442364
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Abstract: The system of equations

$\displaystyle \sum\limits_{k = 0}^n {{{( - 1)}^{n - k}}{{\left( {{P_k}(x){y^{(n - k)}}(x)} \right)}^{(n - k)}}} = 0\quad (0 \leqslant x < \infty )$

is considered where the coefficients are real, continuous, symmetric matrices, y is a vector, and $ {P_0}(x)$ is positive definite.

It is shown that the well-known quadratic functional criterion for existence of conjugate points for this system can be further utilized to extend results of the associated scalar equation to the vector-matrix case, and in some cases the scalar results are also improved. The existence and nonexistence criteria for conjugate points of this system are stated in terms of integral conditions on the eigenvalues or norms of the coefficient matrices.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0442364-0
PII: S 0002-9947(1977)0442364-0
Article copyright: © Copyright 1977 American Mathematical Society