Asymptotic Behavior of Polynomially Bounded Solutions of Linear Fractional Differential Equations

Abstract

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