Electron. J. Differential Equations, Vol. 2016 (2016), No. 297, pp. 1-13.

Convolutions and Green's functions for two families of boundary value problems for fractional differential equations

Paul W. Eloe, Jeffrey T. Neugebauer

We consider families of two-point boundary value problems for fractional differential equations where the fractional derivative is assumed to be the Riemann-Liouville fractional derivative. The problems considered are such that appropriate differential operators commute and the problems can be constructed as nested boundary value problems for lower order fractional differential equations. Green's functions are then constructed as convolutions of lower order Green's functions. Comparison theorems are known for the Green's functions for the lower order problems and so, we obtain analogous comparison theorems for the two families of higher order equations considered here. We also pose a related open question for a family of Green's functions that do not apparently have convolution representations.

Submitted August 31, 2016 Published November 22, 2016.
Math Subject Classifications: 26A33, 34A08, 34A40, 26D20.
Key Words: Fractional boundary value problem; fractional differential inequalities.

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Paul W. Eloe
Department of Mathematics
University of Dayton
Dayton, Ohio 45469, USA
email: peloe1@udayton.edu
Jeffrey T. Neugebauer
Department of Mathematics and Statistics
Eastern Kentucky University
Richmond, Kentucky 40475, USA
email: Jeffrey.Neugebauer@eku.edu

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