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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(t)+f(t)+μG(x(t),t), where A is a continuous or an almost periodic square matrix, f(t) is an almost periodic function, G(t) is a Green function and μ 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(|h|)=0, (M(·) is the asymptotic mean value, defined by M(w)=lim τ (1 2τ -τ r w(s)ds)) 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.

34D05Asymptotic stability of ODE
34C27Almost and pseudo-almost periodic solutions of ODE