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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 the dimension of the zero or infinity tending sets for linear differential equations


Author: James S. Muldowney
Journal: Proc. Amer. Math. Soc. 83 (1981), 705-709
MSC: Primary 34A30; Secondary 34D05
MathSciNet review: 630041
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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 }}\vert x(t)\vert = 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 $ \mathop {\lim }\limits_{t \to \omega } \vert x(t)\vert = 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].


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0630041-9
PII: S 0002-9939(1981)0630041-9
Article copyright: © Copyright 1981 American Mathematical Society