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Asymptotic integration of $(1+α)$-order fractional differential equations

\noindent{\bf Abstract} We establish the long-time asymptotic formula of solutions to the $(1+α)$--order fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x+a(t)x=0$, $t>0$, under some simple restrictions on the functional coefficient $a(t)$, where ${}_{0}^{\>i}{\cal O}_{t}^{1+α}$ is one of the fractional differential operators ${}_{0}D_{t}^α(x^{\prime})$, $({}_{0}D_{t}^αx)^{\prime}={}_{0}D_{t}^{1+α}x$ and ${}_{0}D_{t}^α(tx^{\prime}-x)$. Here, ${}_{0}D_{t}^α$ designates the Riemann-Liouville derivative of order $α\in(0,1)$. The asymptotic formula reads as $[a+O(1)]\cdot x_{\scriptstyle small}+b\cdot x_{\scriptstyle large}$ as $t\rightarrow+\infty$ for given $a$, $b\in\mathbb{R}$, where $x_{\scriptstyle small}$ and $x_{\scriptstyle large}$ represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x=0$, $t>0$.