Linear autonomous ordinary differential equations of the form (1) , where is a continuous or an almost periodic square matrix, is an almost periodic function, is a Green function and 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 which can be written as a sum , where is a.p.f. and is a continuous bounded function with , is the asymptotic mean value, defined by is called p.a.p.f.
Definition 2. A function is called g.a.p.f. if in contrary to Definition 1 we assume that the function 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.