# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations. (English) Zbl 0855.34055

Linear autonomous ordinary differential equations of the form (1) $dx/dt=Ax\left(t\right)+f\left(t\right)+\mu G\left(x\left(t\right),t\right)$, where $A$ is a continuous or an almost periodic square matrix, $f\left(t\right)$ is an almost periodic function, $G\left(t\right)$ is a Green function and $\mu$ is a small parameter are considered.

The theory of almost periodic functions (a.p.f.) and almost periodic solutions of the ordinary differential equations (1) is well known. In the article a generalization of a.p.f., first, to a so-called pseudo almost periodic function (p.a.p.f.) and, second, to generalized pseudo almost periodic functions (g.p.a.p.f.) is given.

Definition 1. A function $f$ which can be written as a sum $f=g+h$, where $g$ is a.p.f. and $h$ is a continuous bounded function with $M\left(|h|\right)=0$, $\left(M\left(·\right)$ is the asymptotic mean value, defined by $M\left(w\right)={lim}_{\tau \to \infty }\left(\frac{1}{2\tau }{\int }_{-\tau }^{r}w\left(s\right)ds\right)\right)$ is called p.a.p.f.

Definition 2. A function $f$ is called g.a.p.f. if in contrary to Definition 1 we assume that the function $f$ is neither continuous nor bounded.

Under the assumption that the linear system of differential equations has an exponential dichotomy a theorem of existence of the pseudo almost periodic solutions of (1) is proved.

##### MSC:
 34D05 Asymptotic stability of ODE 34C27 Almost and pseudo-almost periodic solutions of ODE