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**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.

\[ D^{\alpha+1} x(t) + a(t) x(t) = 0,\quad 0 < \alpha <1, \]

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.

Reviewer: Kai Diethelm (Braunschweig)

### MSC:

34A08 | Fractional ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

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\textit{D. Băleanu} et al., J. Phys. A, Math. Theor. 43, No. 38, Article ID 385209, 7 p. (2010; Zbl 1216.34004)

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