{"id":"1530247a-682d-4875-bfe6-7646f1cd6c2b","arxiv_id":"1908.06734","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit rates of convergence are derived for implicit and Ishikawa-type iterative algorithms finding zeros of set-valued accretive operators, unifying several known strong convergence proofs.","lead":"This paper derives explicit rates of convergence for iterative algorithms that locate zeros of set-valued accretive operators in Banach spaces. It shows that several previously separate convergence proofs share one common abstract pattern, making the rates modular and applicable across different iteration schemes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof-mining rates are correct conditional on the stated quantitative inputs.","rationale":"The reader found no red flags and accepted the paper with high confidence. I agree after a careful pass over the abstract lemma, the four applications, and their proofs. Potential pitfalls all check out: the choice of j from Lemma 2.6 is legitimate because the preceding inequalities hold for every j in the duality map; in Theorem 7.4 uniform smoothness makes J single-valued, so the use of j_n is justified; the joint rate phi for alpha_n and beta_n is used only via the stated bounds; and the conditionality on supplied moduli and bounds is fully transparent. The only respect in which I might adjust emphasis is the reader's weakest assumption: the dependence on quantitative data is real, but it is a standard feature of proof mining and is exactly what the theorems assert. Thus the verdict should remain ACCEPT, with no change needed.","tokens_in":18313,"tokens_out":27815,"duration_ms":266510,"concrete_test":"Recompute the conversion from the squared-norm rate to the norm rate in Theorem 6.6: since Lemma 3.4 is applied to theta_n = ||x_n - q||^2, a target epsilon for the norm must enter as epsilon^2 in the lemma, so the Theta argument in the final formula should be Theta_K(epsilon), not Theta_K(sqrt(epsilon)). If this substitution is made incorrectly, the rate formula would be off by a square root. This one-line check is the most useful verification of the displayed formulas.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked Lemma 3.4 and its four instantiations (Theorems 4.2, 5.6, 6.6, 7.4) step by step. The telescoping argument in Lemma 3.4 is valid, and each application supplies the required quantitative witnesses (N, phi, K^2) correctly. The only soft spot is the one the reader noted: every rate is conditional on user-supplied data, namely a modulus Theta of uniform accretivity at zero, rates phi and r for the scalar sequences, and a priori bounds K, K', K0, K1, K2. If any of these is unavailable, the displayed Phi is not a numerical rate. But this is exactly the proof-mining framework the paper advertises, and the theorems state the conditionality explicitly. I find no internal inconsistency or hidden assumption that would make the central claim false. The caveat is a practical limitation, not a load-bearing error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18482,"tokens_out":21465,"duration_ms":191357,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§5.1, Definition 5.1"},{"comment":"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.","section":"§5.3, Theorem 5.7"},{"comment":"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.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The referee's assessment is close to the reader's: the central technical content is sound, all key derivations are present, and the paper does not require a major revision. The only issues are local (a Hausdorff-distance typo and a strict-inequality slack in the proof of Theorem 5.7). I would recommend acceptance after minor revision. The paper is well within the scope of the journal and should be of interest to researchers in proof mining and quantitative operator theory."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":18997,"tokens_out":1039,"would_cite":true,"duration_ms":12950,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","47J25","03F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["accretive operators","uniform accretivity at zero","rates of convergence","Ishikawa iterations","implicit iteration schemes","uniformly smooth Banach spaces","modulus of accretivity"],"falsifier":"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.","tokens_in":18137,"feed_emoji":"🧮","tokens_out":6486,"duration_ms":61430,"temperature":0.7,"pith_summary":"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.","feed_headline":"Explicit convergence rates for accretive zero-finding algorithms","feed_subtitle":"One abstract lemma turns four known convergence proofs into ready-to-use error bounds.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the implicit iteration schemes and the qualitative convergence theorems whose proofs are turned into explicit rates in Sections 4 and 5.","marker":"[1]"},{"why":"Introduces uniform accretivity at zero and its modulus $\\Theta$, the central assumption used in every rate in the paper.","marker":"[16]"},{"why":"Source of the uniformly continuous Ishikawa-type scheme that is analyzed quantitatively in Theorem 6.6.","marker":"[24]"},{"why":"Provides the qualitative convergence result for Ishikawa-type iterations in uniformly smooth spaces that underlies Section 7.","marker":"[8]"},{"why":"Generalizes [8] to two operators; its Theorem 2.1 is the qualitative statement made quantitative in Theorem 7.4.","marker":"[23]"},{"why":"Gives the quantitative modulus for norm-to-norm continuity of the duality mapping in uniformly smooth spaces, used in the proof of Theorem 7.4.","marker":"[17]"},{"why":"Earlier quantitative version of the abstract convergence lemma for Lipschitzian pseudocontractions; Lemma 3.4 generalizes this result.","marker":"[19]"}],"fun_headline_variants":["Proof mining yields explicit convergence rates for accretive operators","One lemma transforms four known proofs into explicit error bounds","Weaker accretivity condition still yields quantitative convergence","Explicit rates from a modulus of uniform accretivity at zero","Proof-mining framework unifies convergence proofs with explicit rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Proof mining yields explicit convergence rates for accretive operators","One lemma transforms four known proofs into explicit error bounds","Weaker accretivity condition still yields quantitative convergence","Explicit rates from a modulus of uniform accretivity at zero","Proof-mining framework unifies convergence proofs with explicit rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000597,"raw_usage":{"total_tokens":2762,"prompt_tokens":882,"completion_tokens":1880,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":1802}},"tokens_in":498,"tokens_out":1880,"duration_ms":12552,"temperature":1.0,"reasoning_tokens":1802,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:36:21.236370+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Alber, S","cited_arxiv_id":null,"evidence_quote":"Supplies the implicit iteration schemes and the qualitative convergence theorems whose proofs are turned into explicit rates in Sections 4 and 5."},{"cited_title":"Kohlenbach and A","cited_arxiv_id":null,"evidence_quote":"Introduces uniform accretivity at zero and its modulus $\\Theta$, the central assumption used in every rate in the paper."},{"cited_title":"Moore and B","cited_arxiv_id":null,"evidence_quote":"Source of the uniformly continuous Ishikawa-type scheme that is analyzed quantitatively in Theorem 6.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the qualitative convergence result for Ishikawa-type iterations in uniformly smooth spaces that underlies Section 7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Generalizes [8] to two operators; its Theorem 2.1 is the qualitative statement made quantitative in Theorem 7.4."},{"cited_title":"Kohlenbach and L","cited_arxiv_id":null,"evidence_quote":"Gives the quantitative modulus for norm-to-norm continuity of the duality mapping in uniformly smooth spaces, used in the proof of Theorem 7.4."},{"cited_title":"K¨ ornlein and U","cited_arxiv_id":null,"evidence_quote":"Earlier quantitative version of the abstract convergence lemma for Lipschitzian pseudocontractions; Lemma 3.4 generalizes this result."}],"review_version":1}