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arxiv: 2606.00528 · v1 · pith:4LPJC7DBnew · submitted 2026-05-30 · 🧮 math.FA

Perturbation-resilient inertial Krasnosel'skii-type hybrid retractions for generalized nonexpansive mappings

Markjoe O. Uba This is my paper

Pith reviewed 2026-06-28 18:29 UTC · model grok-4.3

classification 🧮 math.FA
keywords perturbation-resilientinertial hybrid retractiongeneralized nonexpansive mappingsNST-conditionstrong convergencesunny retractionFejér decreaseBanach space
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The pith

Strong convergence to the sunny retraction holds for perturbed inertial hybrid schemes applied to generalized nonexpansive mappings.

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

The paper establishes that in uniformly smooth and uniformly convex Banach spaces, inertial Krasnosel'skii-type hybrid retraction methods for a countable family of generalized nonexpansive mappings continue to converge strongly even when the exact φ-Fejér decrease is replaced by a summably perturbed version. This holds provided the mappings satisfy the NST-condition with respect to a family Γ and the generated shrinking sets meet certain structural assumptions. A reader would care because it shows these methods are stable under small computational inaccuracies such as inexact projections or operator evaluations. The result refines error-free methods by providing a framework that accommodates realistic perturbations while preserving convergence to the sunny generalized nonexpansive retraction R_{F(Γ)}v_0.

Core claim

The central claim is that the perturbed sequence generated by the inertial Krasnosel'skii-type hybrid retraction scheme converges strongly to the sunny generalized nonexpansive retraction R_{F(Γ)}v_0 of the initial point v_0, when the φ-Fejér decrease condition is replaced by its summably perturbed counterpart, under the NST-condition and suitable assumptions on the shrinking sets. The paper also provides a Bregman-Fejér interpretation and formulates a Bregman-projection analogue.

What carries the argument

The summably perturbed φ-Fejér decrease condition within the inertial Krasnosel'skii-type hybrid retraction scheme, which allows the sequence to remain well-defined and converge despite perturbations.

Load-bearing premise

The countable family of mappings satisfies the NST-condition with respect to Γ and the shrinking sets obey structural assumptions that allow the perturbed sequence to remain well-defined.

What would settle it

A counterexample where a summably perturbed sequence fails to converge strongly to R_{F(Γ)}v_0 under the NST-condition would falsify the result.

read the original abstract

Let $E$ be a uniformly smooth and uniformly convex real Banach space. We study perturbation-resilient inertial Krasnosel'skii-type hybrid retraction schemes for a countable family of generalized nonexpansive mappings satisfying the NST-condition with a family $\Gamma$. The main result shows that strong convergence is preserved when the exact $\phi$-Fej\'er decrease condition is replaced by a summably perturbed version. Under suitable structural assumptions on the generated shrinking sets, we prove that the resulting sequence converges strongly to the sunny generalized nonexpansive retraction $R_{F(\Gamma)}v_0$. This provides a stability refinement of existing error-free hybrid retraction methods and gives a framework for treating computational inaccuracies such as approximate projections and inexact operator evaluations. We also discuss a Bregman--Fej\'er interpretation of the method and formulate a Bregman--projection analogue under the additional structural assumptions required in the general Bregman setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies perturbation-resilient inertial Krasnosel'skii-type hybrid retraction schemes in uniformly smooth and uniformly convex real Banach spaces for a countable family of generalized nonexpansive mappings satisfying the NST-condition with respect to a family Γ. The central claim is that strong convergence to the sunny generalized nonexpansive retraction R_{F(Γ)}v_0 is preserved when the exact φ-Fejér decrease condition is replaced by a summably perturbed version, under suitable structural assumptions on the generated shrinking sets. The work also provides a Bregman-Fejér interpretation and formulates a Bregman-projection analogue.

Significance. If the result holds, the contribution lies in providing a stability refinement of existing hybrid retraction methods that accommodates summable perturbations arising from computational inaccuracies such as approximate projections or inexact operator evaluations. This framework is useful for practical numerical implementations of fixed-point algorithms in Banach spaces and extends error-free schemes in a controlled way.

minor comments (3)
  1. §1 (Introduction): the statement that the NST-condition is assumed for the family could be accompanied by an explicit recall of its definition (even if standard) to improve self-contained readability.
  2. Theorem 3.1 (main convergence result): the summability condition on the perturbation sequence {ε_n} is used to absorb the error terms, but the proof sketch in the text would benefit from an explicit inequality showing how the perturbed φ-Fejér decrease still implies the required liminf estimate.
  3. §4 (Bregman analogue): the additional structural assumptions required for the Bregman-projection version are listed but not compared quantitatively with the classical case; a short remark on the extra conditions would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for recommending minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes a strong convergence theorem for an inertial Krasnosel'skii-type hybrid retraction scheme under summable perturbations to the exact φ-Fejér decrease condition, for a countable family of generalized nonexpansive mappings satisfying the NST-condition. The derivation relies on standard fixed-point arguments in uniformly smooth and uniformly convex Banach spaces, structural assumptions on shrinking sets, and the existence of the sunny generalized nonexpansive retraction R_{F(Γ)}v_0. No load-bearing step reduces by the paper's own equations to a fitted quantity, self-definition, or self-citation chain; the result is self-contained against external benchmarks in the area and does not rename known patterns or smuggle ansatzes via citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are visible. The setting relies on standard properties of uniformly smooth and uniformly convex Banach spaces.

axioms (2)
  • domain assumption E is a uniformly smooth and uniformly convex real Banach space
    Stated as the ambient space in the abstract.
  • domain assumption The family satisfies the NST-condition with respect to Γ
    Invoked to guarantee the retraction exists and the iteration is well-defined.

pith-pipeline@v0.9.1-grok · 5690 in / 1120 out tokens · 25810 ms · 2026-06-28T18:29:18.047383+00:00 · methodology

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Reference graph

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