Gronwall's inequality and stability analysis of nonlinear fractional difference equations

This paper investigates the existence and uniqueness of solutions for nonlinear fractional difference equations of the Hilfer type using Brouwer's and Banach's fixed-point theorems. The study builds on the fundamental properties of linear fractional difference equations, the discrete comparison principle, and key concepts in fractional calculus. Hilfer-type nabla fractional differences, which generalize the Riemann-Liouville and Caputo nabla differences, are analyzed. Solutions for linear Hilfer-like fractional difference equations are derived using the successive approximation method. Gronwall's inequality and its generalized form are presented and applied to examine asymptotic stability. The theoretical results are validated through numerical examples, simulations, and the Newton's iteration method, demonstrating the practical relevance of the findings.