Circular Spectrum and Bounded Solutions of Periodic Evolution Equations

In this paper we consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u = A(t)u + ǫH(t, u) + f(t), where A(t) is an in general unbounded operator depending 1-periodically on t, H is periodic in t with the same period as A, ǫ 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 this spectral theory to study a similar problem for difference equation of the form u(t) = B(t)u(t − 1) + f(t), where B is an operator in a Banach space X that is 1-periodic, strongly continuous in t, f is an X-valued bounded function. The solution of this problem turns out to yield a solution to the above-mentioned problem for the unperturbed evolution equations with general conditions on f . For small ǫ we show that the perturbed equation inherits some properties of the linear unperturbed equation on the existence and uniqueness of the bounded solutions. The main result extends 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 .