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On a question in the theory of almost periodic differential equations

Author(s): Zuo Sheng Hu; Angelo B. Mingarelli
Journal: Proc. Amer. Math. Soc. 127 (1999), 2665-2670.
MSC (1991): Primary 34C27
Posted: April 23, 1999
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Abstract: We show that there exists a real homogeneous differential equation of order $n$ with classical almost periodic coefficients such that all solutions are uniformly bounded on the real line yet no non-trivial solution is almost periodic. This now appears to make the search for a Floquet theory of such equations a futile enterprise.

References:

[1]
H. Bohr, Almost periodic functions, Chelsea, New York, 1947. MR 8:512a

[2]
C.C. Conley and R.K. Miller, Asymptotic stability without uniform stability: almost periodic coefficients, J. Differential Equations, 1 (1965), 333-336. MR 34:1619

[3]
M.A. Fink, Almost periodic differential equations, Springer-Verlag, New York-Berlin, 1974.

[4]
A.B. Mingarelli, F.Q. Pu and L. Zheng, A counter-example in the theory of almost periodic differential equations, Rocky Mountain J. of Math. 25 (1995), 437-440. MR 96e:34070

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Additional Information:

Zuo Sheng Hu
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6

Angelo B. Mingarelli
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
Email: angelo_mingarelli@carleton.ca

DOI: 10.1090/S0002-9939-99-04738-3
PII: S 0002-9939(99)04738-3
Keywords: Second-order, almost periodic, boundedness
Received by editor(s): October 7, 1997
Posted: April 23, 1999
Additional Notes: The second author was partially supported by an NSERC research grant.
Communicated by: Hal L. Smith
Copyright of article: Copyright 1999, American Mathematical Society

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