Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

Abstract

We establish here that under some simple restrictions on the functional coefficient $a\left(t\right)$ the fractional differential equation ${}_{0}D{}_t^{\alpha }[t{x}^{\prime }-x+x\left(0\right)]+a\left(t\right)x=0$, $t>0$, has a solution expressible as $ct+d+o\left(1\right)$ for $t\to +\infty$, where ${}_{0}D{}_t^{\alpha }$ designates the Riemann-Liouville derivative of order $\alpha \in (0,1)$ and $c,d\in ℝ$.