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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)

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