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**Continuity of solutions to discrete fractional initial value problems.**
*(English)*
Zbl 1197.39002

Summary: We consider a fractional initial value problem (IVP) in the case where the order \(\nu \) of the fractional difference satisfies \(0<\nu \leq 1\). We show that solutions of this IVP satisfy a continuity condition both with respect to the order of the difference, \(\nu \), and with respect to the initial conditions, and we deduce several important corollaries from this theorem. Thus, we address a complication that arises in the fractional case but not in the classical (integer-order) case.

### MSC:

39A10 | Additive difference equations |

26A33 | Fractional derivatives and integrals |

45J05 | Integro-ordinary differential equations |

### Keywords:

discrete fractional calculus; initial value problem; discrete gronwall inequality; continuity with respect to initial conditions
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\textit{C. S. Goodrich}, Comput. Math. Appl. 59, No. 11, 3489--3499 (2010; Zbl 1197.39002)

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### References:

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