Equivalences of Nonlinear Higher Order Fractional Differential Equations With Integral Equations

Abstract

Equivalences of initial value problems (IVPs) of both nonlinear higher order (Riemann–Liouville type) fractional differential equations (FDEs) and Caputo FDEs with the corresponding integral equations are studied in this paper. It is proved that the nonlinearities in the FDEs can be $L^1$-Carathéodory with suitable conditions. The new results generalize the previous results which assumed that the nonlinearities are continuous. For the Caputo FDEs, it is shown in this paper that the continuity assumptions on the nonlinearities used in the literature before are not sufficient for the obtained equivalences. A counterexample is provided to exhibit this. The previous equivalence results with the continuity assumptions alone in the literature have been widely used to study the existence of solutions and numerical solutions of the Caputo FDEs up to now, so according to the new results obtained in this paper, there are no guarantees that the solutions of the integral equations obtained in the literature are the solutions of the Caputo FDEs. New conditions which are stronger than continuity are provided to ensure the equivalences. Sufficient conditions for solutions of the integral equations to be solutions of the Caputo FDEs are obtained. The new equivalence results and the sufficient conditions will be useful for further studying the existence of solutions and numerical solutions of the nonlinear Caputo FDEs via the corresponding integral equations.