A discrete fractional Grönwall inequality for dissipative systems via resolvent kernel estimate

We establish a discrete fractional Gronwall inequality for dissipative systems with time-dependent forcing. For completely positive discretizations, we bound individual resolvent entries by integrals of the continuous response kernel, under controlled perturbation to the L1 weights and a stepsize restriction. The proof constructs a supersolution for each resolvent column. The resulting Mittag-Leffler weights retain both the decay of initial data and the historical contributions of forcing. The framework covers L1 and several standard schemes on uniform meshes. For linear subdiffusion, it yields decay-preserving L1 error estimates separating first order and $(2-α)$ order contributions, with stronger long-time decay under additional solution regularity.