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Asymptotic behavior of systems of linear ordinary differential equations


Author: E. C. Tomastik
Journal: Proc. Amer. Math. Soc. 90 (1984), 381-390
MSC: Primary 34C11; Secondary 34E05
DOI: https://doi.org/10.1090/S0002-9939-1984-0728353-6
MathSciNet review: 728353
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Abstract: Conditions will be placed on the $ m \times m$ matrices $ G(t)$ and $ {G_i}(t)$ to assure that for any integer $ k = 1, \ldots ,n$, the linear differential system

$\displaystyle {x'_i} = {G_i}(t){x_{i + 1}},\quad i = 1, \ldots ,n - 1,\quad {x'_n} = G(t){x_1},$

where the $ {x_i}$ are $ m \times m$ matrices, has a solution $ ({x_1}, \ldots ,{x_n})$ with the property that $ {x_k}(t) \to I$ (the identity matrix) and if $ k < n$, $ {x_i}(t) \to 0$, $ i = k + 1, \ldots ,n$, as $ t \to \infty $. Furthermore, important bounds on the $ {x_i}(t)$ will be given. Some of these conditions will require that $ \int_a^\infty {\left\vert G \right\vert < \infty } $ while others will not. Corollaries will be given for special cases such as $ (R(t)x'')'' = G(t)x$. No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0728353-6
Keywords: Asymptotic behavior, systems of linear differential equations
Article copyright: © Copyright 1984 American Mathematical Society

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