Linear first order Riemann-Liouville fractional differential and perturbed Abel's integral equations

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's inte-gral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical prob-lems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is dif-ferent from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives. (c) 2021 Elsevier Inc. All rights reserved.