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Ergodicity and its applications in regularity and solutions of pseudo-almost periodic equations. (English) Zbl 1004.34033

A relation between ergodicity and regularity in pseudo-almost periodic equations is established. For this purpose the following operator is considered

$L:{C}^{\text{'}}{\left(ℝ\right)}^{n}\to C{\left(ℝ\right)}^{n},\phantom{\rule{1.em}{0ex}}y\to Ly\equiv {y}^{\text{'}}+A\left(t\right)y·$

It is shown that the following three statements are equivalent:

(1) The operator $L$ is regular.

(2) An solution to the homogeneous equation $Ly=0$ exhibits an exponential dichotomy.

(3) For every $f\in PA{P}_{0}{\left(ℝ\right)}^{n}$, the inhomogeneous equation $Ly=f$ has a unique solution in $C{\left(ℝ\right)}^{n}$.

Here, $C{\left(ℝ\right)}^{n}$ and ${C}^{\text{'}}{\left(ℝ\right)}^{n}$ are the $n$th powers of $C\left(ℝ\right)$, respectively, where $C\left(ℝ\right)$ is the space of the bounded continuous functions on the real line $ℝ$ supplied with the $sup$ norm and ${C}^{\text{'}}\left(ℝ\right)$ is the space of differentiable functions $\varphi$ with ${\varphi }^{\text{'}}\in C\left(ℝ\right)$ (with the norm ${\parallel \varphi \parallel }_{{C}^{\text{'}}\left(ℝ\right)}={\parallel \varphi \parallel }_{C\left(ℝ\right)}+{\parallel {\varphi }^{\text{'}}\parallel }_{C\left(ℝ\right)}$). $PAP\left(ℝ\right)$ is the space of pseudo-almost periodic functions and $PA{P}_{0}\left(ℝ\right)\subset PAP\left(ℝ\right)$ consists of those $\varphi \in PAP\left(ℝ\right)$ for which $M\left(\varphi \right)\equiv {lim}_{T\to \infty }\frac{1}{2T}{\int }_{-T}^{T}|\varphi |dt=0$.

The main result states that, if the matrix $A\left(t\right)$ is such that ${a}_{ij}=0$ for all $i>j$ and ${a}_{ii}$, $i=1,\cdots n$, are ergodic, then the operator $L$ is regular if and only if $M\left(\text{Re}\phantom{\rule{4.pt}{0ex}}{a}_{ii}\right)\ne 0$, $i=1,\cdots ,n$. Furthermore, if $A\left(t\right)$ and $f$ are in $PAP{\left(ℝ\right)}^{n}$ then the unique solution $y$ to $Ly=f$ is again in $PAP{\left(ℝ\right)}^{n}$.

MSC:
 34C27 Almost and pseudo-almost periodic solutions of ODE 34D09 Dichotomy, trichotomy