  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 $F:[0,+\infty)\times X^\alpha\to X$ 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.
