  In this paper we study the asymptotic behavior of solutions of fractional
differential equations of the form $D^{\alpha}_Cu(t)=Au(t)+f(t)$ on the half
line, where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in
Caputo's sense, $A$ is generally an unbounded closed operator, $f$ is
polynomially bounded. To this end we develop a spectral theory for functions of
polynomial growth on the half line. Our main result claims that if $u$ is mild
solution of the Cauchy problem such that $\lim_{h\downarrow 0} \sup_{t\ge 0} \|
u(t+h)-u(t)\|/(1+t)^n=0$, and $\sup_{t\ge 0} \| u(t)\| /(1+t)^n <\infty$, then,
$\lim_{t\to\infty} u(t)/(1+t)^n =0$ provided that the spectral set $\Sigma
(A,\alpha )\cap i\R$ is countable, where $\Sigma (A,\alpha )$ is defined to be
the set of complex numbers $\xi$ such that $\lambda^{\alpha -1} (\lambda^\alpha
-A)^{-1}$ is analytic in a neighborhood of $\xi$, and $u$ satisfies some
ergodic
