On a question in the theory of almost periodic differential equations.

*(English)*Zbl 0924.34039It is well-known that every bounded solution to a linear homogeneous equation
\[
x^{(n)}+p_1(t)x^{(n-1)}+\cdots + p_n(t)x=0 \tag{1}
\]
with periodic coefficients \(p_i(t)\) is almost-periodic on account of Floquet theory. The analogous result for almost-periodic coefficients \(p_i(t)\) is known to be false.

For \(n=1\), C. C. Conley and R. K. Miller [J. Differ. Equations 1, 333-336 (1965; Zbl 0145.11401)] gave an example of equation (1) where a bounded solution is not almost-periodic; however, this result did not extend to \(n>1\).

A. B. Mingarelli, F. Q. Pu and L. Zheng [Rocky Mt. J. Math. 25, No. 1, 437-440 (1995; Zbl 0833.34041)] constructed an example, for each \(n>1\), of an equation (1) with almost-periodic coefficients in which there exists a bounded solution which is not almost-periodic; however, for each such case \(n>1\) there is always another unbounded solution. This raises the following question: If \(p_i(t)\) are almost-periodic and all solutions to (1) are bounded, does it necessarily follow that all solutions are almost-periodic?

In the paper the answer to this question is shown to be negative annihilating any hope of a Floquet-type theory for linear almost-periodic differential equations. Namely, for each \(n\geq 2\), there exists an equation of form (1) with almost-periodic coefficients for which every solution is bounded on \(\mathbb{R}\) but yet no solution (except the trivial solution) is almost-periodic.

For \(n=1\), C. C. Conley and R. K. Miller [J. Differ. Equations 1, 333-336 (1965; Zbl 0145.11401)] gave an example of equation (1) where a bounded solution is not almost-periodic; however, this result did not extend to \(n>1\).

A. B. Mingarelli, F. Q. Pu and L. Zheng [Rocky Mt. J. Math. 25, No. 1, 437-440 (1995; Zbl 0833.34041)] constructed an example, for each \(n>1\), of an equation (1) with almost-periodic coefficients in which there exists a bounded solution which is not almost-periodic; however, for each such case \(n>1\) there is always another unbounded solution. This raises the following question: If \(p_i(t)\) are almost-periodic and all solutions to (1) are bounded, does it necessarily follow that all solutions are almost-periodic?

In the paper the answer to this question is shown to be negative annihilating any hope of a Floquet-type theory for linear almost-periodic differential equations. Namely, for each \(n\geq 2\), there exists an equation of form (1) with almost-periodic coefficients for which every solution is bounded on \(\mathbb{R}\) but yet no solution (except the trivial solution) is almost-periodic.

Reviewer: E.Ershov (St.Peterburg)

##### MSC:

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |