Green\u27s Functions and Lyapunov Inequalities for Nabla Caputo Boundary Value Problems

Lyapunov inequalities have many applications for studying solutions to boundary value problems. In particular, they can be used to give existence-uniqueness results for certain nonhomogeneous boundary value problems, study the zeros of solutions, and obtain bounds on eigenvalues in certain eigenvalue problems. In this work, we will establish uniqueness of solutions to various boundary value problems involving the nabla Caputo fractional difference under a general form of two-point boundary conditions and give an explicit expression for the Green\u27s functions for these problems. We will then investigate properties of the Green\u27s functions for specific cases of these boundary value problems. Using these properties, we will develop Lyapunov inequalities for certain nabla Caputo BVPs. Further applications and extensions will be explored, including applications of the Contraction Mapping Theorem to nonlinear versions of the BVPs and a development of Green\u27s functions for a more general linear nabla Caputo fractional operator