Lyapunov-type inequality for fractional BVPs involving two Hadamard fractional derivatives of different orders

TL;DR

This paper derives a Lyapunov-type inequality for fractional boundary value problems involving two Hadamard derivatives of different orders, aiding in understanding solution existence.

Contribution

It introduces a new inequality for fractional BVPs with Hadamard derivatives, using Green's function and analytical methods.

Findings

Provides necessary conditions for nontrivial solutions.

Establishes nonexistence criteria for certain fractional BVPs.

Includes illustrative examples demonstrating applicability.

Abstract

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 and providing examples.