This paper establishes a Lyapunov-type inequality for a class of fractional boundary value problems (BVPs) involving two Hadamard fractional derivatives of different orders with Dirichlet boundary conditions. The method is based on the construction of the corresponding Green's function and establishing its maximum value through rigorous analytical techniques. The obtained inequality provides the necessary conditions for the existence of nontrivial solutions to the proposed problem. Finally, we illustrate the applicability of our results by establishing nonexistence criteria for nontrivial solutions to certain problems, providing examples.
In \cite{s=0}, the new formula for $L^p(\mathbb{R}^N)$ norms was given by Gu and Yung which is characterized by the standard weak-$L^p(\mathbb{R}^{2N})$ norms. At first glance, it seems to be able to measure the degree of fractional smoothness of functions. In this paper, our proof goes in parallel as in \cite{s=0}, but gives another different representation which is obviously independent of the smoothness of function.
