Existence and Uniqueness of Mild Solutions of Nonlinear Fractional Sum-Difference Equation with Nonlocal Condition

Abstract

In this study, we explore the solvability of nonlinear difference equations subject to nonlocal conditions. Our focus lies on identifying conditions under which mild solutions exist and are unique. To achieve this, we employ a combination of the Leray–Schauder alternative and Bihari’s integral inequalities, which together provide a robust framework for addressing nonlinearities and nonlocal constraints in discrete settings.
Beyond existence and uniqueness, the work also examines qualitative features of the solutions, including boundedness and sensitivity to changes in the initial data. These results are not only of theoretical interest but also relevant for discrete models in applied mathematics, where stability and parameter dependence play a critical role. Illustrative examples are presented to confirm the applicability and effectiveness of the theoretical developments.