Circular spectrum and bounded solutions of periodic evolution equations

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 $f$.