## Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations

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## Abstract:

A linear $n$th order (Riemann-Liouville) fractional differential equation with $m+1$ initial values, together with a suitable assumption, is proved to be equivalent to a Volterra integral equation of the second kind involving an $n$th order (Riemann-Liouville) fractional integral operator. Two special cases of the result are given: one shows that a well-known result on the solution of an $n$th order fractional differential equation needs an additional condition to hold, and another strengthens a previous result on an $n$th order fractional integral operator composed with an $n$th order fractional differential operator.## References

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## Additional Information

**Kunquan Lan**- Affiliation: Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3 Canada
- MR Author ID: 256493
- Email: klan@ryerson.ca
- Received by editor(s): March 3, 2020
- Received by editor(s) in revised form: April 16, 2020
- Published electronically: September 18, 2020
- Additional Notes: The author was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant no. 135752-2018.
- Communicated by: Wenxian Shen
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5225-5234 - MSC (2010): Primary 34A08; Secondary 26A33, 34A12, 45D05
- DOI: https://doi.org/10.1090/proc/15169
- MathSciNet review: 4163834