Asymptotic behavior of systems of linear ordinary differential equations
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- by E. C. Tomastik PDF
- Proc. Amer. Math. Soc. 90 (1984), 381-390 Request permission
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 \[ {x’_i} = {G_i}(t){x_{i + 1}},\quad i = 1, \ldots ,n - 1,\quad {x’_n} = G(t){x_1},\] 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 $\int _a^\infty {\left | G \right | < \infty }$ 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.References
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Additional Information
- © Copyright 1984 American Mathematical Society
- 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