# zbMATH — the first resource for mathematics

On a question in the theory of almost periodic differential equations. (English) Zbl 0924.34039
It 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.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems
Full Text: