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 differential equations , , satisfy . 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 . These results are extended by studying a sequence of matrices , , related to such that, under the same stability assumptions as before, the given system has an -dimensional zero [infinity] tending solution set if and only if [if] all nontrivial solutions of the system tend to zero [infinity].

**[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*, 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.

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0630041-9

Article copyright:
© Copyright 1981
American Mathematical Society