A decomposition theorem for bounded solutions and the existence of periodic solutions of periodic differential equations

We prove a decomposition theorem fbr bounded uniformly continuous mild solu tions to tau-periodic evolution equations of the form dx/dt = A(t) x + f(t) (*) with (in general, unbounded) tau-periodic A(.), tau-periodic f(.), and compact monodromy operator. By this theorem, every bounded uniformly conti nuous mild solution to (*) is a sum of a tau-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 equ ations dx/dt = Ax + F(t) x(t) + f(t) with finite delay, where A generates a compact semigroup, As an immediate consequence, the existence of such a so lution implies the existence of a tau-periodic solution to the inhomogeneou s equation as well as a formula for its Fourier coefficients. This, even fo r the classical case of equations, improves considerably the previous resul ts on the subject. (C) 2000 Academic Press.