Rates of convergence for iterative solutions of equations involving set-valued accretive operators

Thomas Powell; Ulrich Kohlenbach

Rates of convergence for iterative solutions of equations involving set-valued accretive operators

T0 review · 0 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Set-valued accretive operators with a modulus of uniform accretivity at zero admit explicit rates of convergence for implicit, approximating, and Ishikawa-type iterations, all derived from one abstract recurrence lemma.

desk verdict A rigorous proof-mining paper that delivers the first explicit rates for Ishikawa-type schemes and a genuinely unifying quantitative lemma; the rates are conditional on supplied moduli and bounds, but that is the advertised framework, not a flaw. read the letter →

arxiv 1908.06734 v2 pith:X7I54LMM submitted 2019-08-19 math.OC math.FAmath.LO

Ulrich Kohlenbach , Thomas Powell This is my paper · ORCID

classification math.OCmath.FAmath.LO MSC 47H0547J2503F10

keywords accretiveoperatorsuniformaccretivityatzeroratesofconvergenceIshikawaiterationsimplicititerationschemesuniformlysmoothBanachspacesmodulus

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that, for set-valued accretive operators (multivalued maps satisfying a Banach-space monotonicity condition) that are uniformly accretive at zero, explicit rates of convergence can be written down for several standard iterative schemes. The rates are expressed in terms of a modulus of uniform accretivity at zero, rates for the coefficient sequences, and a priori bounds on the iterates or on the ranges of the relevant operators. The authors show that four existing strong-convergence proofs, which rest on apparently different assumptions such as range conditions, uniform continuity, or uniform smoothness, all instantiate a single abstract recurrence lemma. If the required quantitative data are supplied, the formulas give a concrete step count after which the iterates are guaranteed to be within ε of the unique zero.

What carries the argument

The central object is Lemma 3.4, an abstract recurrence lemma: if a nonnegative sequence $\theta_n$ is bounded by $K$, the series $\sum \alpha_i$ diverges with rate $r$, and for every $\varepsilon$ there exist $N(\varepsilon)$ and $\varphi(\varepsilon)$ such that whenever $\theta_{n+1} > \varepsilon$ one has $\theta_{n+1} \le \theta_n - \alpha_n \varphi(\varepsilon)$, then $\theta_n \to 0$ with rate $r(N(\varepsilon), K/\varphi(\varepsilon)) + 1$. The companion Lemma 2.6 converts uniform accretivity at zero into an estimate for the pseudocontractive operator $I - A$: whenever $\varepsilon \le \|x - q\| \le K$, some duality selection $j$ satisfies $\langle u - q, j \rangle \le \|x - q\|^2 - \Theta_K(\varepsilon)$. Each application of Lemma 3.4 chooses $\theta_n$, $\alpha_n$, $N$, and $\varphi$ so that this estimate supplies the required one-step decrease. Lemma 7.2 additionally provides a quantitative modulus for norm-to-norm continuity of the duality mapping in uniformly smooth spaces, which enters the rate in that case.

What would settle it

Take the implicit scheme on $X = \mathbb{R}$ with $A x = c x$ for $c > 0$, $q = 0$, $\alpha_n = 1/(n+1)$, and $x_0 = 1$, so that $\Theta_K(\varepsilon) = c \varepsilon$. Simulate the exact map $x_{n+1} = x_n/(1 + \alpha_n c)$ and compare the first $n$ with $x_n < \varepsilon$ against the bound $r(0, 1/(c\varepsilon)) + 1$ from Theorem 4.2; an $\varepsilon$ where the bound is smaller than the actual hitting time would refute the stated rate, while equality would confirm the expected sharp behavior.

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Extended reading notes

Core claim

Under the stated quantitative assumptions, the paper establishes strong convergence with explicit rates for an implicit scheme $x_{n+1} = x_n - \alpha_n u_n$ with $u_n \in A x_{n+1}$, for an implicit scheme using approximating operators, for an Ishikawa-type scheme with uniformly continuous $A$, and for an Ishikawa-type scheme in uniformly smooth spaces. The model result is Theorem 4.2: if $A$ has modulus $\Theta$ of uniform accretivity at zero, the coefficient sequence $\alpha_n$ has rate of divergence $r$, and $\|x_0 - q\| < K$, then $\|x_n - q\| \to 0$ with rate $r(0, K^2/\Theta_K(\varepsilon)) + 1$. The same pattern, applied to $\theta_n = \|x_n - q\|^2$, yields the Ishikawa-type rates, with $\Theta_K(\sqrt{\varepsilon})$ in place of $\Theta_K(\varepsilon)$. Along the way, the paper replaces strong accretivity and quasi-accretivity with the weaker uniform accretivity at zero, so the results strengthen earlier qualitative theorems not only by adding rates but by enlarging the class of operators covered.

Load-bearing premise

The rates are conditional on explicit quantitative inputs: a modulus $\Theta$ of uniform accretivity at zero, rates of divergence and convergence for the scalar sequences, and a priori bounds on the iterates or on the ranges of $I - A$ (or $I - A_i$), and if those bounds are not known the formulas do not produce a concrete number of steps.

Editorial extensions

If this is right

For any application where a modulus $\Theta$, coefficient rates, and a bound $K$ are known, each of the analyzed algorithms comes with a guaranteed $\varepsilon$-complexity certificate: a finite number of iterations that suffices to reach accuracy $\varepsilon$.
The Krasnoselskii–Mann iteration, viewed as the Ishikawa-type scheme with $\beta_n = 0$, is covered by the same quantitative treatment whenever the operator is uniformly accretive at zero.
The results unify the implicit schemes of [1] and the Ishikawa schemes of [24], [8], and [23]: the differing hypotheses appear only in how the abstract lemma's $N(\varepsilon)$ and $\varphi(\varepsilon)$ are supplied.
Several qualitative convergence theorems that assumed $\psi$-strong accretivity or uniform $\varphi$-accretivity go through under the weaker hypothesis of uniform accretivity at zero, with explicit rates of convergence in hand.
When $\alpha_n$ is bounded below by a positive constant and $\psi$ grows linearly, the refined analysis in Remark 3.6 gives linear convergence, improving the general polynomial-scale bound from Lemma 3.4.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's claims, the same abstract recurrence lemma should apply to other scalar-driven iterations, such as Halpern-type or proximal algorithms in Hilbert spaces, whenever their convergence proofs can be arranged to produce the one-step decrease $\theta_{n+1} \le \theta_n - \alpha_n \varphi$ on the event $\theta_{n+1} > \varepsilon$.
A testable extension is to replace the a priori boundedness assumptions by computable bounds derived from the operator itself, for example using the range conditions already present in the original qualitative theorems, which would make the rates fully self-contained.
The linear-convergence refinement suggests that the general formula systematically undercounts speed when the decrease is proportional to $\theta_{n+1}$; deriving logarithmic rates from Lemma 3.7 in the linearly perturbed cases would give a sharper practical bound.
In monotone-operator settings in Hilbert spaces, uniform accretivity at zero is a weak form of regularity at the solution, and the rates should transfer to forward-backward or Douglas–Rachford splittings whenever a modulus of that form is available.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, it really does deliver the first explicit worst-case iteration counts for the Ishikawa-type schemes in Sections 6 and 7, and it does so through a single abstract lemma (Lemma 3.4) that unifies several existing convergence proofs. Second, after checking the derivations, I find them correct. The telescoping argument in Lemma 3.4 is sound, and each of the four applications (Theorems 4.2, 5.6, 6.6, 7.4) supplies the required quantitative witnesses correctly.

What is genuinely new: the explicit rates in Sections 6 and 7 for Ishikawa-type iterations in the uniformly continuous and uniformly smooth settings. These are not in the original papers by Chang, Moore–Nnoli, or Lin. The paper also weakens the qualitative assumptions by replacing various strong accretivity notions with uniform accretivity at zero, which is a real conceptual simplification. Section 4 recovers known rates from Alber–Reich–Shoikhet as a special case, which is honest and useful.

The paper is also transparent. Each theorem states exactly what quantitative data are needed: a modulus of uniform accretivity at zero, rates for the scalar sequences, and a priori bounds on the iterates or ranges. If those are missing, you get no concrete rate. That is a practical limitation, but it is the proof-mining framework the authors advertise, and they do not hide it.

The only soft spot I see is the dependence on a priori bounds like K, K', K0, K1, K2 in every theorem. In applications, boundedness of the range is often the kind of qualitative input that is hard to verify numerically. But the paper is explicit about this, and the original convergence theorems had the same assumptions, so it is not a regression.

The citation pattern looks fine. Lemma 7.2 is correctly cited to Kohlenbach–Leuștean, and the authors are careful to distinguish their contribution from earlier work. I found no circularity: the convergence statements are not used as premises.

Who is this for? Proof miners, obviously, and anyone working on quantitative fixed point theory or accretive operator iterations. It is a niche audience, but within that audience this is a solid, publishable contribution. It deserves a serious referee, and I would recommend acceptance. I would cite the modular lemma in my own work if I were doing related quantitative analysis.

Referee Report

0 major / 3 minor

Summary. This paper develops a proof-theoretic, quantitative analysis of convergence proofs for iterative algorithms that solve equations involving set-valued accretive operators. The authors introduce a unifying abstract lemma (Lemma 3.4) which converts a pointwise decrease inequality with a quantitative witness into an explicit rate of convergence, provided the step-size series has a rate of divergence and the iterates admit a priori bounds. They instantiate this lemma in four settings: an implicit scheme for uniformly accretive operators (Theorem 4.2), an implicit scheme using approximating operators (Theorems 5.6 and 5.7), an Ishikawa-type scheme for uniformly continuous operators (Theorem 6.6), and an Ishikawa-type scheme in uniformly smooth spaces (Theorem 7.4). In each case the extracted rate is an explicit expression in the modulus of uniform accretivity at zero, rates for the scalar sequences, and the relevant quantitative bounds. The paper also claims that several apparently unrelated convergence results are all instances of the same abstract pattern.

Significance. The paper is a solid contribution to quantitative nonlinear analysis in the proof-mining tradition. Its central results are new explicit rates for algorithms for which, in general, no computable rate can be expected without strong quantitative assumptions. The main technical lemma is simple and carefully proved, and each application supplies all quantitative witnesses needed to instantiate it; the calculations in Sections 4–7 are detailed and checkable. The dependence of the rates on user-supplied moduli and bounds is stated transparently and is inherent to the proof-mining approach, rather than a hidden weakness, although it means that the formulas are not numerical rates unless such data are available. The paper also strengthens existing non-effective convergence theorems (e.g., results of Alber–Reich–Shoikhet, Moore–Nnoli, and Lin) by making them quantitative and by replacing restrictive strong-accretivity assumptions with uniform accretivity at zero.

minor comments (3)

[§5.1, Definition 5.1] The displayed definition of the Hausdorff distance quantifies over x∈X and y∈Y; it should quantify over x∈P and y∈Q for the two sets being compared, since X and Y are not the variables in the formula.
[§5.3, Theorem 5.7] In the proof, the assertion H(A_n q, A q) < h_n ξ*(K_1) is only justified in general with ≤; to obtain w_n with the strict bound used in inequality (7), one should either choose ξ* with a strict slack or argue using the strictness of the a priori bounds K_1 and K_2. This is a local fix and does not affect the validity of the stated rate.
[Throughout] There are several typographical slips (e.g., 'i terative', 'acc retive', and 'zer_A' in the introduction) that should be corrected in a final revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the extracted rates are conditional theorems whose premises (moduli, divergence rates, bounds) are never derived from the target convergence.

full rationale

The derivation chain is self-contained in the proof-mining sense. Lemma 3.4 proves convergence of (theta_n) from an explicit quantitative recurrence (∗), using only a rate of divergence r for the partial sums and a uniform bound K; these are stated premises, not consequences of the conclusion theta_n -> 0. Each subsequent theorem instantiates Lemma 3.4 by proving the recurrence from the assumed modulus of uniform accretivity at zero, the stated scalar-sequence rates, the approximation/continuity/smoothness moduli, and the a priori bounds on the iterates or ranges. For example, Theorem 4.2 obtains theta_n = ||x_n - q||, proves ||x_{n+1}-q|| <= ||x_n-q|| - alpha_n Theta_K(epsilon)/K under epsilon < ||x_{n+1}-q||, and then invokes Lemma 3.4 with N(epsilon)=0 and phi(epsilon)=Theta_K(epsilon)/K. The convergence statement is never assumed; the rates are functions of the quantitative inputs. The caveat that these rates are not numerical unless the user supplies bounds/moduli is stated explicitly in the theorems and is a practical limitation, not circularity. The cited prior work introduces the concept of uniform accretivity at zero and supplies auxiliary quantitative facts, but the present claims do not reduce to those citations: the core inequalities are reproved within the paper. No fitted parameter is relabeled as a prediction, and no target result is imported from the authors' earlier papers as a premise.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted; all moduli and bounds are inputs from the problem. The only new technical objects are reformulations (H*, modulus concepts) of existing quantitative conditions, none postulating new physical or mathematical entities.

assumptions (7)

standard math Basic norm inequalities and properties of the normalized duality mapping J (Lemma 2.1)
Every application uses Lemma 2.1 to convert norm-squared estimates into inner product inequalities; standard Banach space background.
domain assumption A is uniformly accretive at zero with modulus Theta (Definition 2.4)
The central regularity hypothesis; all extracted rates are expressed in terms of Theta.
domain assumption A zero q with 0 in Aq exists and the iterates are well-defined
Stated explicitly in Theorems 4.2, 5.6, 6.6 and 7.4; the paper drops the range conditions that would guarantee existence in the original theorems.
domain assumption Scalar parameter sequences alpha_n, beta_n satisfy alpha_n, beta_n -> 0 with joint rate phi and Sum alpha_i diverges with rate r
Necessary input data for the rates; taken as assumptions from the problem setting.
domain assumption A priori boundedness of the iterates and/or ranges: ||x_n-q|| < K, ||q|| < K', R(I-A_i) bounded by K0
Appears as hypotheses in the quantitative theorems; Theorem 5.7 derives boundedness from Sum alpha_i h_i < infinity, but the other theorems assume it.
standard math Uniform continuity modulus for A or uniform smoothness modulus tau for X, and Lemma 7.2 from [17] for the duality map modulus
Sections 6 and 7 invoke these moduli; Lemma 7.2 is cited from the literature and not reproved here.
domain assumption The approximating operators A_n uniformly approximate A with rate mu (Section 5)
Definition 5.4 is the quantitative replacement of the Hausdorff approximation condition from [1].

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Cite this review

Pith. "Pith review of Rates of convergence for iterative solutions of equations involving set-valued accretive operators." pith.science (2026). https://pith.science/paper/X7I54LMM

@misc{pith2026190806734,
  author       = {Pith},
  title        = {Pith review of: Rates of convergence for iterative solutions of equations involving set-valued accretive operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X7I54LMM}},
  note         = {Machine review of arXiv:1908.06734}
}

read the original abstract

This paper studies proofs of strong convergence of various iterative algorithms for computing the unique zeros of set-valued accretive operators that also satisfy some weak form of uniform accretivity at zero. More precisely, we extract explicit rates of convergence from these proofs which depend on a modulus of uniform accretivity at zero, a concept first introduced by A. Koutsoukou-Argyraki and the first author in 2015. Our highly modular approach, which is inspired by the logic-based proof mining paradigm, also establishes that a number of seemingly unrelated convergence proofs in the literature are actually instances of a common pattern.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alber, S

Y. Alber, S. Reich, and D. Shoikhet. Iterative approximations of null points of uniformly accretive operators with estimates of the convergen ce rate. Communications in Applied Analysis , 6(1):89–104, 2002

work page 2002

[2] [2]

V. Barbu. Nonlinear semigroups and diﬀerential equations in Bana ch spaces, Noordhoﬀ International Publishing, Leyden, The Netherlands, 19 76. 22

work page

[3] [3]

V. Barbu. Nonlinear diﬀerential equations of monotone types in B anach spaces. Springer Monographs in Mathematics, x+272pp., Springer -Verlag, 2010

work page 2010

[4] [4]

Bauschke, P.L

H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone O pera- tor Theory in Hilbert Spaces, Springer, New York-Dordrecht-Heid elberg- London, 2010

work page 2010

[5] [5]

H. H. Bauschke, E. Matouˇ skov´ a, S. Reich. Projection and pr oximal point methods: convergence results and counterexamples. Nonlinear Analysis , 50:715–738, 2004

work page 2004

[6] [6]

F. E. Browder, Nonlinear accretive operators in Banach spaces . Bulletin of the American Mathematical Society , 73:470–476, 1967

work page 1967

[7] [7]

R. E. Bruck, S. Reich. Nonexpansive projections and resolvent s of accretive operators in Banach spaces. Houston Journal of Mathematics , 3:459–470, 1977

work page 1977

[8] [8]

S.S. Chang. On Chidume’s open questions and approximate solution of multi-valued strongly accretive mapping equations in Banach spaces . Jour- nal of Mathematical Analysis and Applications , 216:94–111, 1994

work page 1994