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Pseudo almost periodic solutions of some differential equations. (English) Zbl 0796.34029
The concept of a pseudo almost periodic function on $$R$$ or $$\Omega\times R$$, $$R\subset C^ n$$, generalizes the concept of an almost periodic (in sense of Bohr) function. A bounded on $$R$$ $$(\Omega\times R)$$ function is said to be pseudo almost periodic if $$f= g+ \varphi$$, where $$g$$ is almost periodic on $$R$$ $$(\Omega\times R)$$ and $$\varphi$$ satisfies the condition $\lim_{t\to\infty} {1\over 2t}\int_{-t}^ t|\varphi(x)| dx=0,$
$\left(\lim_{t\to\infty}{1\over 2t}\int_{-t}^ t |\varphi(z,x)| dx=0\quad\text{uniformly in }z\in\Omega\right).$ The purpose of the paper is to establish existence of pseudo almost periodic solutions of linear and quasi-linear ordinary and parabolic partial differential equations. The following proposition is an emanation of the paper spirit:
Consider the system of the form $${dY\over dx}= AY+ F$$, where $$A$$ is a complex $$n\times n$$ matrix of $$F: R\to R^ n$$ is a vector function, whose components are pseudo almost periodic. If the matrix $$A= (a_{ij})$$ has no eigenvalues with real part zero, then this system admits a unique solution $$Y$$, whose components are pseudo almost periodic.
Reviewer: I.Ginchev (Varna)

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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