arxiv: 2605.11883 · v1 · submitted 2026-05-12 · 🧮 math.FA

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A comparison of the weakest contractive conditions for Banach and Kannan mappings

Aqib Saghir, Misako Kikkawa, Shuji Machihara, Shunya Hashimoto

Pith reviewed 2026-05-13 04:06 UTC · model grok-4.3

classification 🧮 math.FA

keywords Banach contractionKannan mappingfixed pointweakest convergence conditioncomplete metric spaceG-complete metric spacePicard sequence

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The pith

For Kannan mappings on complete metric spaces, several weakest convergence conditions are equivalent when deduced only from fixed point existence, while the same equivalence fails for Banach contractions.

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

The paper compares weakest convergence-type conditions for fixed point theorems between Banach contractions and Kannan mappings on metric spaces. It establishes that for Kannan-type mappings these conditions turn out to be equivalent, with the proof using nothing beyond the existence of a fixed point and avoiding any appeal to completeness or Picard sequence behavior. By contrast, a counterexample is built to show that the corresponding conditions are not equivalent when the mapping is a Banach contraction. The discrepancy between the two classes disappears once the underlying space is taken to be G-complete.

Core claim

We give a direct proof that several weakest convergence conditions are equivalent for Kannan-type mappings on complete metric spaces. This proof is achieved without assuming the completeness or the convergence of Picard sequences; it deduces the equivalence only from the existence of fixed points. In contrast, we construct a counterexample showing that the corresponding equivalence fails for Banach contractions. Finally, we prove that this discrepancy disappears on G-complete metric spaces, clarifying the role of completeness in weakest fixed point theory.

What carries the argument

Weakest convergence-type conditions defined along Picard sequences, whose equivalence is shown to follow directly from fixed-point existence for Kannan mappings but is refuted by explicit counterexample for Banach mappings.

If this is right

Equivalence of the weakest conditions for Kannan mappings follows solely from the existence of fixed points.
The equivalence does not hold for Banach contractions on complete metric spaces.
The equivalence is restored for both classes of mappings when the space is G-complete.
Completeness assumptions play qualitatively different roles in the weakest fixed-point theory for Kannan versus Banach mappings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The direct proof technique for Kannan mappings may extend to other non-contractive classes that admit fixed points under minimal assumptions.
The counterexample for Banach contractions isolates a concrete limitation on how far weakest conditions can be relaxed without additional structure.
G-completeness appears to act as a common refinement that erases class-specific differences in convergence behavior.

Load-bearing premise

The standard definitions of Banach contractions, Kannan mappings, Picard sequences, and the weakest convergence-type conditions as given in the cited prior literature hold, along with the metric space axioms.

What would settle it

A complete metric space equipped with a Banach contraction for which the weakest convergence conditions are nevertheless equivalent would falsify the claim that equivalence fails for the Banach case.

read the original abstract

We study the weakest convergence-type conditions for fixed point results for Banach and Kannan mappings. Building on Suzuki's weakest condition for Banach mappings and our previous result for Kannan mappings, we compare convergence conditions defined along Picard sequences. We give a direct proof that several weakest convergence conditions are equivalent for Kannan-type mappings on complete metric spaces. This proof is achieved without assuming the completeness or the convergence of Picard sequences; it deduces the equivalence only from the existence of fixed points. In contrast, we construct a counterexample showing that the corresponding equivalence fails for Banach contractions. Finally, we prove that this discrepancy disappears on G-complete metric spaces, clarifying the role of completeness in weakest fixed point theory.

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.

Desk Editor's Note private letter to a colleague

Kannan mappings get equivalence of weakest conditions from fixed point existence alone; Banach needs an explicit counterexample and G-completeness to match.

read the letter

The main takeaway is that the paper gives a direct proof equating several weakest convergence conditions for Kannan mappings on complete metric spaces, using only the existence of fixed points and skipping any assumption about completeness or Picard sequence convergence. For Banach contractions they supply a counterexample where the same equivalence fails, then show the gap closes on G-complete spaces. This extends Suzuki's condition and the authors' earlier Kannan work without repeating the usual completeness steps. The direct deduction and the explicit counterexample are the concrete advances here. The logic tracks cleanly from the definitions, with no visible circularity or hidden parameters. The counterexample serves as real evidence that the distinction between the two mapping classes is not just formal. A minor soft spot is that the counterexample's construction would need checking in full to confirm it does not accidentally satisfy stronger conditions or rely on special metric features. The G-complete resolution is narrow but fits the paper's scope. This is for specialists in metric fixed point theory who already know the Suzuki and Kannan literature and want to tighten minimal assumptions. A reader working on contractive conditions or completeness variants will extract the most value. It deserves a serious referee because the proof technique and the counterexample add a precise, verifiable distinction to the record.

Referee Report

0 major / 2 minor

Summary. The manuscript compares weakest convergence-type conditions for fixed-point results involving Banach contractions and Kannan mappings on metric spaces. Building on Suzuki's condition for Banach mappings and prior work on Kannan mappings, it gives a direct proof that several such conditions are equivalent for Kannan-type mappings on complete metric spaces; the argument relies solely on the existence of fixed points and does not presuppose completeness or convergence of Picard sequences. A counterexample is constructed to show that the corresponding equivalence fails for Banach contractions. The authors then prove that the equivalence is restored on G-complete metric spaces.

Significance. If the claims hold, the paper clarifies the distinct roles of completeness versus G-completeness in weakest fixed-point theory and highlights structural differences between Banach and Kannan contractive conditions. The direct proof strategy (deducing equivalence from fixed-point existence alone) and the explicit counterexample are strengths that could guide further minimal-condition results in metric fixed-point theory.

minor comments (2)

[Introduction] The introduction should explicitly cite the authors' previous paper on Kannan mappings (mentioned in the abstract) rather than referring to it only as 'our previous result.'
[Counterexample] In the counterexample section, verify that the constructed space satisfies the metric axioms and that the mapping is indeed a Banach contraction but fails the equivalence; add a brief verification paragraph if not already present.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The summary accurately captures our main contributions, including the direct proof of equivalence for Kannan mappings (relying only on fixed-point existence), the counterexample for Banach contractions, and the clarification via G-completeness. We note the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper establishes equivalence of weakest convergence conditions for Kannan mappings via a direct proof that relies solely on the existence of fixed points, without invoking completeness or Picard sequence convergence. It contrasts this with an explicit counterexample for Banach contractions and proves restoration under G-completeness. These steps use standard metric space axioms and fixed-point definitions from the literature; no self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the central claims to prior inputs occur. The cited prior result for Kannan mappings provides context but is not required for the new direct equivalence proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on classical axioms of metric spaces and standard definitions of contractive mappings from prior literature; no new free parameters or invented entities appear in the abstract.

axioms (2)

standard math A metric space satisfies the standard distance axioms (non-negativity, symmetry, triangle inequality, identity of indiscernibles).
Fundamental background for all statements about Banach and Kannan mappings and Picard sequences.
domain assumption Banach contractions and Kannan mappings are defined by their respective contractive inequalities as stated in the cited literature.
Used to classify the two families of mappings under comparison.

pith-pipeline@v0.9.0 · 5420 in / 1409 out tokens · 120295 ms · 2026-05-13T04:06:13.342628+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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