Bounded solutions and periodic solutions to linear differential equations in Banach spaces (Q1411129)

This survey paper is devoted to the study of the existence of bounded and periodic solutions for the nonhomogeneous abstract Cauchy problem \[ x'(t) = A x(t) + f(t), \;\; t \geq 0, \] on a Banach space \(X\), where \(A : D(A) \subseteq X \to X\) is the infinitesimal generator of a strongly continuous semigroup and \(f: [0, \infty) \to X\) is a periodic function. The authors characterize the existence of bounded solutions in terms of some properties of the spectrum of \(A\) and present a Massera-type theorem for the existence of periodic solutions. Finally, a description of the set of bounded and periodic solutions is given.