Bounded and periodic solutions of differential equations in Banach space (Q1344021)

This article deals with generalizations of classical results about boundedness and ultimate boundedness of solutions of finite systems of linear differential equations to equations of the type \(u'(t) = Au(t) + f(t)\) \((0 < t < \infty)\) (1) with \(A\) being a generator of an analytic (or a strongly continuous) semigroup \(S(t)\). The main results are a theorem about boundedness and ultimate boundedness of solutions to (1) in terms of Lyapunov-like functions and a theorem about existence of a \(T\)- periodic solution to (1) provided that the solutions to (1) are bounded and ultimately bounded, \(S(t)\) is compact for sufficiently large \(t\) and \(f(t)\) is \(T\)-periodic. Applications to parabolic differential equations are presented.