Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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].

References [Enhancements On Off] (What's this?)

  • [1] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
  • [2] Philip Hartman, The existence of large or small solutions of linear differential equations, Duke Math. J. 28 (1961), 421–429. MR 0130432
  • [3] Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR 0344555
  • [4] Jack W. Macki and James S. Muldowney, The asymptotic behaviour of solutions to linear systems of ordinary differential equations, Pacific J. Math. 33 (1970), 693–706. MR 0268463
  • [5] Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Inc., Boston, Mass., 1964. MR 0162808
  • [6] H. Milloux, Sur l'équation différentielle $ x'' + A(t)x = 0$, Prace Mat. Fiz. 41 (1934), 39-53.
  • [7] Binyamin Schwarz, Totally positive differential systems, Pacific J. Math. 32 (1970), 203–229. MR 0257466
  • [8] M. Ō. Tnūthail, Algēbar Iolscoile, Oifig an tSolāthair, Baile Ātha Cliath, 1947.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34A30, 34D05

Retrieve articles in all journals with MSC: 34A30, 34D05

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society