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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) $dx/dt = Ax(t) + f(t) + \mu G (x(t),t)$, 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 $\mu$ 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 $M (|h |) = 0$, $(M (\cdot)$ is the asymptotic mean value, defined by $M(w) = \lim_{\tau \to \infty} ({1 \over 2 \tau} \int^r_{- \tau} w(s) ds))$ 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
Full Text: DOI
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