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A linear, almost periodic equation with an almost automorphic solution


Author: Russell A. Johnson
Journal: Proc. Amer. Math. Soc. 82 (1981), 199-205
MSC: Primary 34C28; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9939-1981-0609651-0
MathSciNet review: 609651
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Abstract: We construct a scalar, first order, almost periodic $ {\text{ODE}}( * )x + A(t)x = B(t)$ which admits bounded solutions, but no almost periodic solutions. Using this equation, we give an example of a two-dimensional, almost periodic system whose projective flow admits two minimal subsets, one of which is almost automorphic but not almost periodic. Finally, we show that some equation in the hull of $ ( * )$ admits an almost automorphic, nonalmost periodic solution.


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

  • [1] C. C. Conley and R. A. Miller, Asymptotic stability without uniform stability: almost periodic coefficients, J. Differential Equations 1 (1965), 333-336. MR 0201737 (34:1619)
  • [2] A. M. Fink, Almost periodic differential equations, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin and New York, 1974. MR 0460799 (57:792)
  • [3] R. Johnson, Bounded solutions of scalar, almost periodic differential equations, Illinois J. Math. (to appear). MR 630840 (82k:34052)
  • [4] -, On a Floquet theory for almost periodic, two-dimensional linear systems, J. Differential Equations 37 (1980), 184-205. MR 587221 (81j:58069)
  • [5] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320-358. MR 0501182 (58:18604)
  • [6] W. A. Veech, Point-distal flows, Amer. J. Math. 92 (1970), 205-242. MR 0267560 (42:2462)

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DOI: https://doi.org/10.1090/S0002-9939-1981-0609651-0
Article copyright: © Copyright 1981 American Mathematical Society

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