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 $D_C^{\alpha }u(t)=Au(t)+f(t)$ on the half line, where $D_C^{\alpha }u(t)$ 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 ${\lim }_{h\downarrow 0}{\sup }_{t\ge 0}\Vert u(t+h)-u(t)\Vert /(1+t{)}^n=0$, and ${\sup }_{t\ge 0}\Vert u(t)\Vert /(1+t{)}^n<\infty$, then, ${\lim }_{t\to \infty }u(t)/(1+t{)}^n=0$ provided that the spectral set $\Sigma (A,\alpha )\cap i\mathbb{R}$ 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

16 pages