*(English)*Zbl 0855.34055

Linear autonomous ordinary differential equations of the form (1) $dx/dt=Ax\left(t\right)+f\left(t\right)+\mu G\left(x\right(t),t)$, 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 $M\left(\right|h\left|\right)=0$, $\left(M\right(\xb7)$ is the asymptotic mean value, defined by $M\left(w\right)={lim}_{\tau \to \infty}\left(\frac{1}{2\tau}{\int}_{-\tau}^{r}w\left(s\right)ds\right))$ 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.