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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 $$ L:C^\prime(\bbfR)^n\rightarrow C(\bbfR)^n,\quad y\rightarrow Ly\equiv y^\prime+A(t)y. $$ 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 $f\in PAP_0(\bbfR)^n$, the inhomogeneous equation $Ly=f$ has a unique solution in $C(\bbfR)^n$. Here, $C(\bbfR)^n$ and $C^\prime(\bbfR)^n$ are the $n$th powers of $C(\bbfR)$, respectively, where $C(\bbfR)$ is the space of the bounded continuous functions on the real line $\bbfR$ supplied with the $\sup$ norm and $C^\prime(\bbfR)$ is the space of differentiable functions $\varphi$ with $\varphi^\prime\in C(\bbfR)$ (with the norm $\|\varphi\|_{C^\prime(\bbfR)}= \|\varphi\|_{C(\bbfR)}+\|\varphi^\prime\|_{C(\bbfR)}$). $PAP(\bbfR)$ is the space of pseudo-almost periodic functions and $PAP_0(\bbfR)\subset PAP(\bbfR)$ consists of those $\varphi\in PAP(\bbfR)$ for which $M(\varphi)\equiv\lim_{T\to\infty}{1\over 2T} \int_{-T}^T |\varphi |dt=0$. The main result states that, if the matrix $A(t)$ is such that $a_{ij}=0$ for all $i>j$ and $a_{ii}$, $i=1,\dots n$, are ergodic, then the operator $L$ is regular if and only if $M(\text{Re } a_{ii})\neq 0$, $i=1,\dots,n$. Furthermore, if $A(t)$ and $f$ are in $PAP(\bbfR)^n$ then the unique solution $y$ to $Ly=f$ is again in $PAP(\bbfR)^n$.

MSC:
34C27Almost and pseudo-almost periodic solutions of ODE
34D09Dichotomy, trichotomy
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References:
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