A linear, almost periodic equation with an almost automorphic solution

Johnson, Russell A.

doi:10.1090/S0002-9939-1981-0609651-0

A linear, almost periodic equation with an almost automorphic solution
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Proc. Amer. Math. Soc. 82 (1981), 199-205 Request permission

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.

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