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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 Ω×R, RC n , generalizes the concept of an almost periodic (in sense of Bohr) function. A bounded on R (Ω×R) function is said to be pseudo almost periodic if f=g+φ, where g is almost periodic on R (Ω×R) and φ satisfies the condition

lim t 1 2t -t t |φ(x)|dx=0,
lim t 1 2t -t t |φ(z,x)|dx=0uniformlyinzΩ·

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 dx=AY+F, where A is a complex n×n matrix of F:RR 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.

34C27Almost and pseudo-almost periodic solutions of ODE