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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 ${\Omega }×R$, $R\subset {C}^{n}$, generalizes the concept of an almost periodic (in sense of Bohr) function. A bounded on $R$ $\left({\Omega }×R\right)$ function is said to be pseudo almost periodic if $f=g+\varphi$, where $g$ is almost periodic on $R$ $\left({\Omega }×R\right)$ and $\varphi$ satisfies the condition

$\underset{t\to \infty }{lim}\frac{1}{2t}{\int }_{-t}^{t}|\varphi \left(x\right)|dx=0,$
$\left(\underset{t\to \infty }{lim}\frac{1}{2t}{\int }_{-t}^{t}|\varphi \left(z,x\right)|dx=0\phantom{\rule{1.em}{0ex}}\text{uniformly}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}z\in {\Omega }\right)·$

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 $\frac{dY}{dx}=AY+F$, where $A$ is a complex $n×n$ matrix of $F:R\to {R}^{n}$ is a vector function, whose components are pseudo almost periodic. If the matrix $A=\left({a}_{ij}\right)$ has no eigenvalues with real part zero, then this system admits a unique solution $Y$, whose components are pseudo almost periodic.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions of ODE