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In complete b-metric spaces, mappings that shrink triangle perimeters become graphic contractions on high enough iterates, so they are weakly Picard with at most two fixed points (or exactly one 2-cycle).

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2026-07-12 10:26 UTC pith:HKGEIJL2

load-bearing objection Solid, usable extension of Cvetković’s perimeter-contraction iterates to b-metrics; the sequential-tracking arguments work and the 2-cycle dichotomy is clean.

arxiv 2606.30954 v2 pith:HKGEIJL2 submitted 2026-06-29 math.MG

Perimetric Contractions and Their Iterates in Complete b-Metric Spaces

Mujahid Abbas , Alemayehu G. Negash , Meaza F. Bogale This is my paper
classification math.MG MSC 47H0547J25
keywords b-metric spacemapping contracting perimeters of trianglesgraphic contractionweakly Picard operatorperiodic pointfixed-point setiterate
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies mappings that contract the perimeter of every triangle of three distinct points in a complete b-metric space. These multi-point conditions are independent of ordinary pairwise contractions. When the map has no period-2 orbits, every sufficiently high iterate becomes a continuous graphic contraction (provided the product of the b-coefficient s and the nth power of the contraction constant is less than 1). Consequently the map is weakly Picard and its fixed-point set has cardinality between 1 and 2. When a genuine 2-cycle is present and s q^{2} < 1, the even iterates become graphic contractions and the map has exactly those two periodic points. The proofs carefully avoid simultaneous continuity of the b-metric by using sequential tracking inequalities, and explicit examples show that the class of perimeter contractions properly contains the graphic contractions.

Core claim

Under the sole exclusion of prime period-two orbits, every iterate f^{n} of a mapping that contracts perimeters of triangles becomes a continuous graphic contraction once s q^{n} < 1; the map is therefore weakly Picard and satisfies 1 ≤ |Fix(f)| ≤ 2. When a 2-cycle exists and s q^{2} < 1, the even iterates f^{2n} are continuous graphic contractions and the map possesses exactly two periodic points forming that single 2-cycle.

What carries the argument

The perimeter-contraction inequality P(f(x),f(y),f(z)) ≤ q P(x,y,z) for distinct triples, combined with sequential tracking bounds that replace simultaneous continuity of the b-metric; these force high iterates (or even iterates) to satisfy the graphic-contraction estimate d(f^{n}(x),f^{2n}(x)) ≤ λ d(x,f^{n}(x)) with λ < 1.

Load-bearing premise

The whole existence argument rests on an external theorem that a continuous graphic contraction with closed graph and sλ < 1 already possesses a fixed point in a complete b-metric space.

What would settle it

Exhibit a complete b-metric space and a perimeter-contracting map with no 2-cycle for which some iterate with s q^{n} < 1 fails to be a graphic contraction, or for which the fixed-point set is empty or has more than two points.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies mappings that contract perimeters of triangles (MCPTs) on complete b-metric spaces with coefficient s ≥ 1. Under the exclusion of prime period-two orbits, it proves that iterates f^n become continuous graphic contractions whenever s q^n < 1, so f is weakly Picard and 1 中 |Fix(f)| ≤ 2. When a 2-cycle is present and s q^{2} < 1, the even iterates f^{2n} are continuous graphic contractions and the map has exactly two periodic points forming that unique 2-cycle. Continuity of any MCPT is established by sequential tracking (Lemma 2.2), periods greater than 2 are ruled out, fixed points and 2-cycles are shown to be mutually exclusive, and examples (including a shift map) demonstrate that MCPTs properly contain graphic contractions and that the parameter thresholds are sharp.

Significance. The work cleanly extends Cvetković’s metric-space results on perimetric contractions and their iterates to the b-metric setting, correctly handling the lack of joint continuity of d via sequential inequalities. The continuity proposition, the period dichotomy, the graphic estimates derived from Lemma 2.9, and the cardinality statements are self-contained once the external graphic-contraction theorem of Petruşel–Petruşel is invoked under verified hypotheses. Concrete examples establish that the class of MCPTs is strictly larger than graphic contractions and that the thresholds s q^n < 1 and s q^{2} < 1 are optimal. The contribution is a solid, technically careful generalization that completes the structural picture for this multi-point contraction class in b-metric spaces.

minor comments (6)
  1. Abstract and title use both “CPTM” and “MCPT”; the body consistently uses MCPT. Standardize the acronym throughout.
  2. Theorem 3.2, Step 2: the parenthetical “exactly as in the proof of Theorem 3.2” is a self-reference; rephrase to “as in the estimate leading to (3.4)” or similar.
  3. Proposition 3.1, Step 4: the passage from (3.1) to the limsup inequality is correct via Lemma 2.2, but a one-sentence reminder that the left-hand side is independent of l would improve readability.
  4. Example 3.3: the verification that d_b is complete is clear, yet the claim that the triangle inequality for the original d holds for all triples with k < 4 is asserted without listing the six triples; a short table or explicit check would remove any residual doubt.
  5. Several bibliographic entries (e.g., [7], [12], [24]) appear with future or non-standard volume data; verify final publication details before typesetting.
  6. Notation for the perimeter P(x,y,z) is introduced twice (once in the preliminaries and again before Definition 2.8); a single definition suffices.

Circularity Check

0 steps flagged

No significant circularity: the derivations of graphic-contraction iterates, fixed-point cardinality, and 2-cycle dichotomy are self-contained from the MCPT definition, sequential tracking, and elementary perimeter bounds.

full rationale

The paper is a pure fixed-point theory extension. All load-bearing steps (continuity of any MCPT via sequential tracking and pigeonhole in Prop. 3.1; the graphic estimate d(f^n x, f^{2n}x) ≤ (s q^n)/(1-s q^n) d(x,f^n x) obtained by applying the perimeter inequality n times plus Lemma 2.9 in Thm. 3.2; the period-dichotomy lemmas forbidding periods >2 or coexistence of fixed points with 2-cycles; and the exact-two-periodic-points claim under s q^2 <1) are proved directly from the definitions of b-metric, perimeter P, and MCPT, without fitting parameters, without renaming empirical patterns, and without reducing a claimed prediction to its own input by construction. External theorems (Petruşel–Petruşel graphic-contraction convergence, Rus weakly Picard operators) are invoked only after the paper independently verifies their hypotheses (closed graph via continuity + Hausdorff property of b-metrics; coefficient bound s λ <1 by elementary limit). Self-citations appear only for background lemmas or examples and are not load-bearing for the central claims. The derivation chain is therefore non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside the standard axiomatic framework of b-metric fixed-point theory. No free parameters are fitted; s and q are given data of the space and the map. The only non-standard ingredients are the definition of MCPT (imported from Petrov) and the graphic-contraction convergence theorem (imported from Petruşel–Petruşel), both of which are used with their hypotheses verified.

axioms (4)
  • standard math b-metric axioms (identity of indiscernibles, symmetry, s-relaxed triangle inequality) and completeness of (X,d)
    Definition 2.1 and the standing hypothesis of every main theorem; classical since Bakhtin–Czerwik.
  • domain assumption A mapping is an MCPT if P(fx,fy,fz) ≤ q P(x,y,z) for all mutually distinct triples, q∈[0,1)
    Definition 2.8, taken from Petrov; the entire theory is conditional on this multi-point inequality.
  • domain assumption Graphic contractions with closed graph and sλ<1 are weakly Picard (orbits converge to fixed points)
    Theorem 2.6 (Petruşel–Petruşel); invoked after the paper establishes the graphic estimate and closed graph.
  • ad hoc to paper Exclusion of prime period-two orbits (first main theorem) or presence of such an orbit together with s q^2 <1 (second main theorem)
    Structural hypotheses that split the two cases; without them the graphic-iterate conclusion fails, as shown by the counter-example in §3.

pith-pipeline@v1.1.0-grok45 · 23826 in / 2581 out tokens · 39831 ms · 2026-07-12T10:26:06.567866+00:00 · methodology

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In this paper, the structural and operator-theoretic properties of contracting perimeters of triangles mapping (CPTM) within the generalized topological framework of complete $b$-metric spaces with coefficient $s \geq 1$, is systematically investigated. Extending recent foundational advancements from classical metric spaces, we explore the architectural interplay between multi-point perimetric constraints and path-wise orbital stability under two distinct structural framework. First, assuming the minimal exclusion of periodic orbits of prime period two, we prove that the higher-order iterates $f^{n}$ of an CPTM behave as graphic contractions for all indices satisfying the condition $sq^{n} < 1$. This classifies the operator as a weakly Picard operator and yields a unified existence and cardinality theorem establishing that the fixed-point set satisfies $1 \leq |\mathrm{Fix}(f)| \leq 2$.\\ Second, in the alternative configuration where the operator does possess a periodic orbit of prime period two, we resolve a significant structural gap under the parameter condition $sq^{2} < 1$. We demonstrate that the higher even iterates $f^{2n}$ collapse into continuous graphic contractions, proving that the mapping possesses exactly two periodic points which form a single, isolated 2-cycle. Throughout our proofs, we rigorously navigate the analytical challenges arising from the potential simultaneous non-continuity of the $b$-metric function by relying strictly on sequential tracking inequalities. Finally, we present concrete analytical examples, including a shift map on a discrete metric space, to show that the class of CPTM is strictly larger than the class of graphic contractions, thereby demonstrating the sharpness and optimality of the obtained parameter conditions.

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