Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 01, pp. 1-20.

Title: Linear higher-order fractional differential and integral equations
       
Author: Kunquan Lan (Toronto Metropolitan Univ., Toronto, Ontario, Canada)
         
Abstract:
We study the equivalences and the implications between linear (or homogeneous) nth order
fractional differential equations (FDEs) and integral equations in
the spaces  L<sup>1</sup>(a,b) and  C[a,b] when n&ge; 2.
We establish the equivalence in C[a,b] of the IVP  of the  nth order FDE subject
to the initial condition u<sup>(i)</sup>(a)=u<sub>I</sub> for 
i in {0,1,...,n-1} when
n&ge;2. The difficulty is that the known conditions for such equivalence
for the first order  FDEs are not sufficient for equivalence in
the nth order FDEs with n&ge;2.
In this article we provide additional conditions to ensure the equivalence for the
nth order FDEs with n&ge;2.
In particular, we obtain conditions under which solutions of the integral equations are
solutions of the linear nth order FDEs.
These results are keys for further studying the existence 
of solutions and nonnegative solutions to initial and
boundary value problems of nonlinear nth order FDEs.
This is done via the corresponding integral equations by topological methods
such as the Banach contraction principle, fixed point index theory, and degree theory.

Submitted November 1, 2022. Published January 04, 2023.
Math Subject Classifications: 34A08, 26A33, 34A12, 45D05.
Key Words: Fractional differential equation; integral equation;  equivalence; initial value problem.