A Decomposition Theorem for Bounded Solutions and the Existence of Periodic Solutions of Periodic Differential Equations

We prove a decomposition theorem for bounded uniformly continuous mild solutions to τ-periodic evolution equations of the form dx/dt=A(t) x+f(t) (*) with (in general, unbounded) τ-periodic A(•), τ-periodic f (•), and compact monodromy operator. By this theorem, every bounded uniformly continuous mild solution to (*) is a sum of a τ-periodic solution to (*) and a quasi periodic solution to its homogeneous equation. An analog of this for bounded solutions has been proved for abstract functional differential equations dx/dt=Ax+F(t) xt+f(t) with finite delay, where A generates a compact semigroup. As an immediate consequence, the existence of such a solution implies the existence of a τ-periodic solution to the inhomogeneous equation as well as a formula for its Fourier coefficients. This, even for the classical case of equations, improves considerably the previous results on the subject.