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

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 $\frac{d_{\psi }^{\beta ,\mu }}{d{t}^{\beta }}\left(\frac{d_{\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}+\lambda \right)x\left(t\right)=f(t,x\left(t\right)),t\in [a,b],\lambda \in \mathbb{R}$, for $f\in C\left([a,b]\times \mathbb{R}\right)$ and some critical orders $\beta ,\alpha \in (0,1)$, combined with appropriate initial or boundary conditions, and with general classes of $\psi$-tempered Hilfer problems with $\psi$-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied.