On the solution set for a class of sequential fractional differential equations. (English) Zbl 1216.34004

The authors consider the linear homogeneous Riemann-Liouville fractional differential equation with non-constant coefficients
\[ D^{\alpha+1} x(t) + a(t) x(t) = 0,\quad 0 < \alpha <1, \]
and determine sufficient conditions on the coefficient function \(a\) such that the differential equation has a solution whose Riemann-Liouville derivative of order \(\alpha\) converges to a finite limit as \(t\to\infty\), and has a solution with a prescribed asymptotic behaviour of the form
\[ x(t) = (X_0 + O(1)) t^{\alpha-1} + (X_1 + o(1)) t^\alpha\text{ as }t\to\infty, \]
where \(X_0\) and \(X_1\) are arbitrarily prescribed real numbers.


34A08 Fractional ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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