  We consider the existence and uniqueness of bounded solutions of periodic
evolution equations of the form $u'=A(t)u+\epsilon H(t,u)+f(t)$, where $A(t)$
is, in general, an unbounded operator depending 1-periodically on $t$, $H$ is
1-periodic in $t$, $\epsilon$ is small, and $f$ is a bounded and continuous
function that is not necessarily uniformly continuous. We propose a new
approach to the spectral theory of functions via the concept of "circular
spectrum" and then apply it to study the linear equations $u'=A(t)u+f(t)$ with
general conditions on $f$. For small $\epsilon$ we show that the perturbed
equation inherits some properties of the linear unperturbed one. The main
results extend recent results in the direction, saying that if the unitary
spectrum of the monodromy operator does not intersect the circular spectrum of
$f$, then the evolution equation has a unique mild solution with its circular
spectrum contained in the circular spectrum of $f$.
