On the dimension of the zero or infinity tending sets for linear differential equations
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- by James S. Muldowney
- Proc. Amer. Math. Soc. 83 (1981), 705-709
- DOI: https://doi.org/10.1090/S0002-9939-1981-0630041-9
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Abstract:
There are well-known conditions which guarantee that all solutions to a system of $n$ differential equations $x’ = A(t)x$, $t \in [0,\omega )$, satisfy ${\lim _{t \to \omega }}|x(t)| = 0$. Under certain stability assumptions on the system, Hartman [2], Coppel [1] and Macki and Muldowney [4] give necessary and sufficient [sufficient] conditions that the system has at least one nontrivial solution satisfying $\lim \limits _{t \to \omega } |x(t)| = 0[\infty ]$. These results are extended by studying a sequence of matrices ${A^{[k]}}(t)$, $k = 1, \ldots ,n$, related to $A(t)$ such that, under the same stability assumptions as before, the given system has an $(n - k + 1)$-dimensional zero [infinity] tending solution set if and only if [if] all nontrivial solutions of the system $y’ = {A^{[k]}}(t)y$ tend to zero [infinity].References
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Bibliographic Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 83 (1981), 705-709
- MSC: Primary 34A30; Secondary 34D05
- DOI: https://doi.org/10.1090/S0002-9939-1981-0630041-9
- MathSciNet review: 630041