h-Stability of Nonlinear Hilfer Nabla Fractional Difference Equations

This paper investigates the h-stability of nonlinear Hilfer nabla fractional difference equations, which interpolate between the Riemann–Liouville and Caputo nabla fractional differences through an additional type parameter. To the best of our knowledge, this is the first work devoted to the h-stability analysis of nonlinear Hilfer nabla fractional difference equations. By employing the properties of generalized nabla fractional sums and Taylor monomials together with a comparison-based approach, we establish new sufficient conditions guaranteeing the h-stability of the zero solution under suitable growth assumptions on the nonlinear term. Furthermore, we show that the associated solution map is differentiable with respect to the initial condition, providing a sensitivity framework for Hilfer-type discrete fractional systems. As another contribution, we derive a discrete variation in parameters formula for perturbed Hilfer nabla fractional difference equations, yielding an explicit representation of perturbed solutions. The obtained results extend the existing stability theory for discrete fractional systems and provide a unified framework encompassing both the Riemann–Liouville and Caputo nabla settings as particular cases.