Existence, Uniqueness, and Continuous Dependence on Initial/Final Values for Liouville–Caputo Fractional Difference Equations

This paper develops a unified qualitative framework for four classes of Liouville–Caputo fractional difference equations arising from different combinations of fractional sums and integer-order differences. Based on the equivalence between initial/final value problems and Volterra-type summation equations, sufficient conditions for the existence and uniqueness of solutions are established by applying the Banach contraction mapping principle together with refined combinatorial estimates. Furthermore, the continuous dependence of solutions on prescribed initial or final data is investigated. By deriving explicit error estimates through a discrete fractional Gronwall-type inequality, we prove that Lipschitz solutions depend continuously on perturbations of boundary data. Numerical experiments for a representative case are presented to verify the theoretical results, including the influence of the fractional order and the sensitivity with respect to boundary data, while additional examples illustrate the applicability of the framework. The obtained results extend the unified discrete fractional calculus framework by providing a rigorous well-posedness analysis and offering a theoretical foundation for further applications of discrete fractional models with memory effects and diverse boundary conditions.