Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

We provide global continuation principle of periodic solutions for the equation $\dot{u}=-Au+F(t,u)$, where $A:D(A)\to X$ is a sectorial operator on a Banach space $X$ and $F:[0,+\mathrm{\infty })\times X^{\alpha }\to X$ is a nonlinear map defined on fractional space $X^{\alpha }$. The approach that we use in this paper is based upon the theory of topological invariants that applies in the situation when Poincar'e operator associated with the equation is endowed with some form of compactness.