Boundedness and almost periodicity of solutions of linear differential systems
In this paper there is investigated the system \[ x^{\prime}= A(t)x+b(t),\quad t\in \mathbb{R}, \] where \(b:\mathbb{R}\to \mathbb{R}^n\) is an almost periodic function and \(A\) is an \(n\times n\)-matrix that is almost periodic. The authors prove that if all solutions of the corresponding homogeneous system are almost periodic, then there is an almost periodic function \(b\) for which the considered nonhomogeneous system has no bounded solutions. Moreover, in the paper it is proved that if for any almost periodic function \(b\) there is a bounded solution of the considered nonhomogeneous function, then there is at least one solution of the corresponding homogeneous system that is not almost periodic.
scientific article; zbMATH DE number 7604037
scientific article