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

We establish the long-time asymptotic formula of solutions to the (1+α)(1+α)-order fractional differential equation 0iOt1+αx+a(t)x=0, t>0t>0, under some simple restrictions on the functional coefficient a(t)a(t), where 0iOt1+α is one of the fractional differential operators 0Dtα(x′), (0Dtαx)′=0Dt1+αx and 0Dtα(tx′−x). Here, 0Dtα designates the Riemann–Liouville derivative of order α∈(0,1)α∈(0,1). The asymptotic formula reads as [b+O(1)]⋅xsmall+c⋅xlarge[b+O(1)]⋅xsmall+c⋅xlarge as t→+∞t→+∞ for given bb, c∈Rc∈R, where xsmallxsmall and xlargexlarge represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation 0iOt1+αx=0, t>0t>0.