This is the abstract of the paper
  "An exponential stability criterion for q-periodic evolution"
by C. Buse.

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Abstract: We consider a mild solution $u_{\mu x}$ of a well-posed
inhomogeneous, Cauchy
problem
 $$\dot u(t)=A(t)u(t)+e^{i\mu t}x\quad (t\ge 0), \mu\in{\bf R}, u(0)=0$$
on a separable Hilbert space $H$, where $A(\cdot)$ is periodic. We prove that
if
 $$\sup\limits_{\mu\in{\bf R}}\sup\limits_{t>0}||u_{\mu x}(t)||<\infty,\quad\forall x\in H$$
then the solution of the Cauchy problem
 $$\dot u(t)=A(t)u(t)\quad (t\ge 0),\quad u(0)=x\in H$$
is uniformly exponentially stable for every $x\in H.$ The basic idea for the
proof is to apply the spectral mapping theorem for an evolution semigroup
associated with a $q$-periodic evolution family ${\cal U}$ on a
$H$-valued almost periodic functions space.

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