On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces

Abstract

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation dψβ,μdtβdψα,μdtα+λx(t)=f(t,x(t)),t∈[a,b],λ∈R, for f∈C[a,b]×R and some critical orders β,α∈(0,1), combined with appropriate initial or boundary conditions, and with general classes of ψ-tempered Hilfer problems with ψ-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied.