non-resonant nabla fractional boundary value problems

We consider two simple non-resonant boundary value problems for a nabla fractional difference equation. first, we construct associated green's functions and obtain some of their properties. under suitable constraints on the nonlinear part of the nabla fractional difference equation, we deduce sufficient conditions for the existence of solutions to the considered problems through an appropriate fixed point theorem. we also provide two examples to demonstrate the applicability of the established results.