Almost periodic solutions of differential equations in Banach spaces: Some new results and methods.
The article under review is a survey on recent results on almost-periodic solutions to differential and functional-differential equations of the form \(u'(t)=A(t)u(t)+f(t)\) and \(u'(t)=Au(t)+F(t)u_{t}+f(t)\) in Banach spaces. At the beginning, autonomous equations are investigated. The authors' approach, based on the spectral theory in the context of an evolution semigroup associated to the equation, allows the extension to the higher-order case. The main results are various criteria for the admissibility of function spaces for the equations under investigation. In the following sections, with similar techniques, the authors consider almost-periodic solutions to periodic equations (here, the conditions are stated in terms of spectral properties of the monodromy operators), and to functional-differential equations (in autonomous and periodic case). Several examples are provided.
scientific article; zbMATH DE number 1885681