Stability and almost periodicity of trajectories of periodic processes
Based on the De Leeuw-Glicksberg decomposition theorem, the author proves that if the monodromy operator \(V\) is power-bounded and its spectrum \(\sigma (V)\) has countable intersection with the unit circle \(\Gamma\), and the point spectrum \(P \sigma (V^*)\) of \(V^*\) satisfies a certain condition, then all solutions on \(\mathbb{R}_ +\) of (1) \(u'(t)=A(t)u(t)\), where \(A(t)\) are closed linear operators in a Banach space \(E\), which depend on \(t\) in a periodic manner, are asymptotically almost periodic. In particular, if \(P \sigma (V^*) \cap \Gamma = \emptyset\), then all solutions on \(\mathbb{R}_ +\) of (1) converge to 0 as \(t \to \infty\). It is also proved that if \(\sigma (V) \cap \Gamma\) is countable and \(f\) is almost periodic, then every bounded uniformly continuous mild solution on \(\mathbb{R}\) of (2) \(u'(t) = A(t)u(t) + f(t)\) is almost periodic, provided an additional condition. Some examples of applications and conditions under which nontrivial bounded solution on \(\mathbb{R}\) of (1) are given.
De Leeuw-Glicksberg decomposition theorem
scientific article; zbMATH DE number 716262