Periodic solutions of differential equations in Banach spaces

Let X be a Banach space, DX, f: [0,)xDX continuous and -periodic. In this paper we consider various conditions on D and f sufficient for existence of an -periodic solution of the differential equation u=f(t,u). In the main, we shall assume that D is closed bounded and convex and f satisfies a boundary condition at D such that D is flow invariant for u=f(t,u). The map f is assumed to be either compact or dissipative or a certain perturbation of such maps.