Bounded and periodic solutions of differential equations in Banach space

摘要

We study the equation in a Banach space X with A the generator of an analytic (or a strongly continuous) semigroup S(·) and prove that if solutions of (1) are bounded and ultimate bounded with f T-periodic and S(T) compact, then (1) has a T-periodic solution. We also show that the existence of a proper Liapunov function implies the boundedness and ultimate boundedness of solutions of (1). These results extend earlier results in finite dimensional spaces. We then apply the results to a parabolic partial differential equation