Bounded and periodic solutions of differential equations in Banach space

Abstract We study the equation u′(t)=Au(t)+f(t), t 0 , (1) 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 u t (t, x)= Σ |α|≤2m c α (x)D α u(t, x)+f(t, x). (2)