A discrete fractional Gr\"onwall inequality for dissipative systems via resolvent kernel estimate

Lei Li; Yunxiao Liu; Jiwei Zhang

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A discrete fractional Gr\"onwall inequality for dissipative systems via resolvent kernel estimate

Lei Li, Yunxiao Liu, Jiwei Zhang

arXiv:2610.34827v1Oct 26, 2026math.NAcs.NA

TL;DR

Dissipative fractional systems with time-dependent forcing need estimates that preserve both initial-data decay and the history of forcing contributions. The paper bounds discrete resolvent entries using continuous response kernels and derives explicit Mittag–Leffler weights. For linear subdiffusion, the resulting L1 error estimates separate first-order and (2−α)-order contributions, with stronger long-time decay under an additional regularity assumption.

Problem
Existing discrete fractional Gronwall estimates for dissipative systems do not retain the timing and decay of individual time-dependent forcing contributions.
Source: §1 · pp. 1–2
Method
The paper bounds discrete resolvent entries by continuous response-kernel integrals and constructs a supersolution for each resolvent column.
Source: §1 · p. 2, §3.2 · p. 9, §3.3 · p. 10
Results
The L1 temporal error estimate for linear subdiffusion separates first-order and (2−α)-order contributions, with stronger long-time decay under an additional all-time derivative-decay assumption.
Source: §1 · p. 3
Takeaways & Limitations
The resulting Mittag–Leffler weights retain both initial-data decay and the historical contributions of forcing for decay-preserving error analysis.
Source: §1 · p. 2
The results do not resolve the non-dissipative case κ ≥ λ1.
Source: §6.4 · p. 23

Abstract

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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 $(2-\alpha)$ order contributions, with stronger long-time decay under additional solution regularity.

1 Introduction

The paper develops a dissipative discrete fractional Gronwall inequality that preserves initial-data decay and the individual historical contributions of time-dependent forcing. It applies the inequality to L1 temporal error estimates for linear subdiffusion, separating first-order and (2−α)-order contributions under stated conditions.

1 Introduction

The analysis compares individual discrete resolvent entries with integrals of the continuous response kernel to obtain the time-dependent forcing bound.The proof constructs a supersolution for each resolvent column.Source: §1 · p. 2
The inequality uses explicit Mittag–Leffler weights to track each forcing term’s contribution while retaining decay from dissipation.It is proved for completely positive discretizations whose weights satisfy two-sided bounds relative to L1 weights.Source: §1 · p. 2
For the L1 approximation of linear subdiffusion, the temporal error estimate separates first-order and (2−α)-order contributions.The estimate is stated under regularity, mesh, and step-size conditions and is motivated by Conjecture 4.1 of.Source: §1 · pp. 2–3
With an additional all-time derivative-decay assumption for forcing approaching a stationary limit, the second contribution receives stronger long-time decay.The paper distinguishes fixed-time estimates for general smooth forcing from all-time estimates under this added assumption.Source: §1 · p. 3
The estimate concerns temporal error and does not resolve the L2 part of Conjecture 4.1 or coarse-mesh regimes outside the step-size restriction.The introduction also notes that replacing time-dependent forcing by its past maximum loses when individual terms act and how their contributions decay.Source: §1 · pp. 2–3

2 Preliminaries and problem setup

The preliminaries introduce continuous and discrete resolvents, then set up the discrete inequality through pseudo-convolution and comparison principles. The discrete resolvent serves as the scheme’s fundamental solution and motivates the subsequent explicit bounds.

2 Preliminaries and problem setup

The section sets up continuous and discrete resolvent kernels and pseudo-convolution as the framework for the problem.The continuous fractional ODE is equivalent to a Volterra equation with the Abel integral kernel.Source: §2 · p. 3, §2.1 · p. 3

2.1 Continuous equation and continuous resolvents

For the fractional relaxation equation, the solution is represented through resolvents, and rλ is positive and decreasing.The response kernel describes how an earlier perturbation contributes to the current solution.Source: §2.1 · p. 4

2.2 Time discretization and pseudoconvolution

The time-discrete operator is organized as a kernel array, and pseudo-convolution acts as matrix multiplication for that representation.Complete positivity supplies comparison principles for the discretization.Source: §2.2 · pp. 5–6

2.3 The discrete problem and its resolvent representation

The discrete resolvent is the numerical scheme’s fundamental solution, and explicit bounds for it are the route to the discrete Gronwall inequality.The exposition relates discrete resolvent columns to continuous responses generated by piecewise-constant forcing.Source: §2.3 · pp. 6–7

3 Resolvent estimates and the Gr¨onwall inequality

This section derives a dissipative discrete fractional Grönwall inequality by bounding individual discrete resolvent entries through comparison with a continuous response kernel. A supersolution construction under the stated assumptions and stepsize restriction yields the inequality for time-dependent forcing.

3.1 Assumptions and reference gap

The section sets out assumptions on the discretization weights and identifies the L1 operator as the reference scheme for the comparison.The L1 weights define the reference gap used in the assumptions.Source: §3.1 · pp. 7–8

3.2 The equation for the discrete resolvent

The analysis targets entrywise resolvent bounds because cumulative estimates do not recover accurate decay when forcing varies over time.The section notes that individual entries should ideally be bounded by the continuous pulse response.Source: §3 · p. 7, §3.2 · p. 10

3.3 The supersolution and Grönwall inequality

A supersolution is constructed for each fixed resolvent column, enabling comparison with the continuous response and transfer of its decay to discrete entries.The auxiliary kernel is described as having the same algebraic tail as the continuous pulse response.Source: §3 · p. 7, §3.3 · p. 11
Theorem 3.1’s resolvent estimate and Theorem 3.2’s resulting Grönwall inequality apply under Assumption 3.1 and the stated stepsize restriction.The section frames these results as estimates for dissipative systems with varying force terms.Source: §3 · p. 7, §3.1 · p. 8, §3.3 · p. 11

4 Proof of the supersolution property

The proof establishes the supersolution property by controlling the continuous residual, interpolation error, and perturbations from the reference L1 weights. These bounds support the discrete resolvent estimate with constants independent of the final time.

4.1 Continuous residual

The continuous residual Z_m is nonnegative, strictly positive on (0, η_m), and zero for t ≥ η_m.This residual is part of the continuous-side comparison used in the supersolution proof.Source: §4.1 · p. 12

4.2 The reference L1 discretization error estimate

The reference L1 interpolation error e_m vanishes before η_m and at every grid node, and is nonnegative on the tail by convexity.The proof estimates this error over cells beyond η_m.Source: §4.2 · pp. 13–14

4.3 Weight perturbation and error absorption

The weight-perturbation and interpolation errors are absorbed by choosing constants D* and L* under a scale condition.The resulting constants depend only on α and M_ker, not on λ, the time steps, or the final time.Source: §4.3 · pp. 15–16

4.4 Supersolution estimates

The supersolution estimates bound the discrete comparison function by the continuous resolvent kernel, with a final constant independent of λ, the grid, and the final time.The proof uses positivity and monotonicity of the weights and r_λ.Source: §4.4 · p. 16

5 Standard discretization schemes

The paper verifies that several uniform-mesh discretizations satisfy the assumptions needed for its results. The verification includes Grönwald–Letnikov, Li–Liu deconvolution, and θ-method convolution quadrature schemes, with a restriction on θ for complete positivity.

5.1 The Grönwald–Letnikov scheme

The uniform-step Grönwald–Letnikov scheme satisfies Assumption 3.1 through bounds comparing its weights with the reference L1 weights.The argument uses weight monotonicity and verifies the relative deviation condition, including the k = 0 case.Source: §5.1 · p. 17

5.2 A deconvolution scheme

The Li–Liu deconvolution scheme satisfies Assumption 3.1 with the stated constants, so Theorem 3.1 applies under the stepsize condition.Source: §5.2 · p. 18

5.3 The θ-method convolution quadrature

For θ ≥ 1, the θ-method convolution quadrature weights are positive and decreasing, verifying complete positivity and the assumptions for applying Theorem 3.1.The constants in the assumptions need not remain uniform as θ increases without bound.Source: §5.3 · pp. 18–19
Values θ < (1 + α)/2 violate complete positivity, excluding the trapezoidal choice θ = 1/2 for every 0 < α < 1.Source: §5.3 · p. 19

6 Decay-preserving estimate for linear subdiffusion models

For dissipative linear subdiffusion, the L1 bounds separate first-order and (2−α)-order error contributions. With additional solution regularity and controlled forcing decay, the long-time estimates are uniform in the final time and capture stronger decay of recent errors.

6.1 Model problem and parameter-dependent convergence

For A=−∆−κI with dissipative gap λ=λ1−κ>0, parameter-dependent L1 convergence rates motivate bounds separating first- and (2−α)-order contributions.Reported rates in approach first order under further refinement, while practical meshes can show rates nearer 2−α.Source: §6.1 · p. 19

6.2 Finite time estimates for general smooth forcing

Under λ>0, a bounded mesh ratio, and λτ^α≤1, finite-time L1 estimates for smooth forcing separate first-order and (2−α)-order contributions.The estimates apply on fixed time intervals and require the stated regularity and stepsize conditions.Source: §6.1 · p. 19, §6.2 · p. 21

6.3 Long time estimates for forcing with a stationary limit

With stationary-limit forcing, (6.11), λ>0, and λτ^α≤1, the L1 error estimates hold uniformly in final time and recent errors gain additional decay.The forcing derivatives must have controlled temporal decay; convergence of f alone does not ensure the derivative bound.Source: §6.3 · pp. 22–23

6.4 Discussion

The two-term bounds distinguish accumulated initial singularity from recent interpolation errors; at fixed T and λ refinement favors first order, while long times suppress recent errors.The (2−α)-order scale may be observed at small times or when time-varying forcing dominates, but an upper-bound comparison does not establish an exact observed order.Source: §6.4 · p. 23

A Proof of Lemma 4.1

The proof derives a spectral representation for the response kernel, establishes its positivity and shape, and obtains integral and cumulative estimates.

A Proof of Lemma 4.1

The Laplace transform identity and a keyhole-contour argument yield an integrable density representation for the response kernel, with uniqueness of the Laplace transform identifying the kernel.The proof uses (z^α + κz + λ)bϕ(z) = λ.Source: A Proof of Lemma 4.1 · p. 24
The response kernel ϕ is positive, decreasing, and convex.These properties follow by differentiating under the integral sign and applying monotone convergence.Source: A Proof of Lemma 4.1 · p. 24
For p = 0, 1, 2, the integral I_p(σ, θ) is comparable to (1 + θ)^−2.For θ ≥ 1, the proof splits the integral at θ/2 and bounds the exponential tail.Source: A Proof of Lemma 4.1 · p. 24
The cumulative estimate uses the spectral representation to express bϕ(1/t) as a ratio comparable to θ/(1 + θ).The proof invokes 1 − e^−x ≍ x/(1 + x) for x > 0.Source: A Proof of Lemma 4.1 · p. 25