The reviewed record of science sign in
Pith

arxiv: 2606.30954 · v1 · 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

Reviewed by Pith2026-07-01 01:21 UTCgrok-4.3pith:HKGEIJL2open to challenge →

classification math.MG
keywords perimetric contractionsMCPTb-metric spacesgraphic contractionsfixed point theoryperiodic orbitsweakly Picard operatorsiterates
0
0 comments X

The pith

Mappings contracting perimeters of triangles in complete b-metric spaces have iterates that are graphic contractions under a parameter condition, limiting fixed points to one or two.

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

The paper systematically investigates mappings contracting perimeters of triangles, known as MCPTs, in complete b-metric spaces. It proves that without prime period-two periodic orbits, the iterates f^n of such a mapping behave as graphic contractions whenever s q^n is less than 1, classifying the mapping as a weakly Picard operator with the number of fixed points between one and two inclusive. When a prime period-two orbit exists and s q^2 is less than 1, the even iterates are continuous graphic contractions, and there are exactly two periodic points forming an isolated 2-cycle. The work uses sequential tracking to handle possible discontinuity of the b-metric and provides examples showing MCPTs are a strictly larger class than graphic contractions.

Core claim

Assuming the minimal exclusion of periodic orbits of prime period two, we prove that the higher-order iterates f^n of an MCPT behave as graphic contractions for all indices satisfying the condition s q^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 ≤ |Fix(f)| ≤ 2. In the case with a prime period-two orbit, under s q^2 <1 the even iterates f^{2n} are continuous graphic contractions and the mapping has exactly two periodic points forming a single isolated 2-cycle.

What carries the argument

Mappings contracting perimeters of triangles (MCPTs) defined by a multi-point perimetric constraint in b-metric spaces.

If this is right

  • The higher-order iterates behave as graphic contractions under the condition s q^n <1.
  • The mapping is a weakly Picard operator.
  • The fixed-point set has cardinality satisfying 1 ≤ |Fix(f)| ≤ 2.
  • With a prime period-two orbit and s q^2 <1, even iterates are continuous graphic contractions with exactly one isolated 2-cycle.

Where Pith is reading between the lines

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

  • The perimetric approach may allow fixed-point results in spaces where standard contraction mappings do not apply directly.
  • Sequential tracking methods could be useful for other non-continuous metric-like structures.

Load-bearing premise

The mapping satisfies the perimetric contraction inequality that defines an MCPT.

What would settle it

An example of an MCPT in a complete b-metric space with no prime period-two orbit, s q^n <1 for some n, but with three or more fixed points would falsify the cardinality result.

read the original abstract

In this paper, we systematically investigate the structural and operator-theoretic properties of mappings contracting perimeters of triangles (MCPTs) within the generalized topological framework of complete $b$-metric spaces with coefficient $s \geq 1$. 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 scenarios. First, assuming the minimal exclusion of periodic orbits of prime period two, we prove that the higher-order iterates $f^{n}$ of an MCPT 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 MCPTs is strictly larger than the class of graphic contractions, thereby demonstrating the sharpness and optimality of the obtained parameter conditions.

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

2 major / 2 minor

Summary. The paper investigates mappings contracting perimeters of triangles (MCPTs) in complete b-metric spaces with coefficient s ≥ 1. Assuming no prime period-2 orbits, it proves that iterates f^n behave as graphic contractions whenever sq^n < 1, establishing that f is a weakly Picard operator with fixed-point set satisfying 1 ≤ |Fix(f)| ≤ 2. In the case of a prime period-2 orbit and sq^2 < 1, even iterates f^{2n} are shown to be continuous graphic contractions, yielding exactly two periodic points forming an isolated 2-cycle. Proofs rely on sequential tracking inequalities to handle possible discontinuity of the b-metric; examples (including a shift map) demonstrate that the MCPT class properly contains graphic contractions.

Significance. If the derivations hold, the work extends fixed-point theory by introducing a new structural class (MCPTs) strictly larger than graphic contractions, with unified existence/cardinality results and a clean case distinction for period-2 orbits. The parameter conditions that absorb the b-coefficient s and the avoidance of continuity assumptions via sequential inequalities are technically useful contributions.

major comments (2)
  1. [Main results / Theorem on iterates] The central claim that 1 ≤ |Fix(f)| ≤ 2 follows from the graphic-contraction property of the iterates under sq^n < 1; the manuscript must explicitly verify (in the relevant theorem, presumably around the main results section) that the perimetric inequality plus completeness directly yields the graphic contraction constant without additional hidden assumptions on the b-metric.
  2. [Section on period-2 orbits] In the period-2 case, the claim that f^{2n} are continuous graphic contractions under sq^2 < 1 is load-bearing for the isolated 2-cycle conclusion; the sequential tracking argument needs to be checked to confirm that continuity of the even iterates follows without invoking continuity of f itself.
minor comments (2)
  1. [Abstract / Introduction] The abstract invokes the MCPT definition but does not restate the precise perimetric inequality; including the exact form (with the constant q) in the introduction would improve readability.
  2. [Introduction] Notation for the b-coefficient s and the contraction parameter q should be introduced consistently before the first use of sq^n < 1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Main results / Theorem on iterates] The central claim that 1 ≤ |Fix(f)| ≤ 2 follows from the graphic-contraction property of the iterates under sq^n < 1; the manuscript must explicitly verify (in the relevant theorem, presumably around the main results section) that the perimetric inequality plus completeness directly yields the graphic contraction constant without additional hidden assumptions on the b-metric.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised version, we will expand the proof of the main theorem on iterates to include a direct derivation showing how the perimetric contraction inequality, together with the completeness of the b-metric space, yields the graphic contraction constant for f^n when sq^n < 1. This derivation relies only on the stated properties of the b-metric and sequential inequalities, with no hidden assumptions. revision: yes

  2. Referee: [Section on period-2 orbits] In the period-2 case, the claim that f^{2n} are continuous graphic contractions under sq^2 < 1 is load-bearing for the isolated 2-cycle conclusion; the sequential tracking argument needs to be checked to confirm that continuity of the even iterates follows without invoking continuity of f itself.

    Authors: The sequential tracking argument in the period-2 section is designed precisely to establish the continuity of the even iterates f^{2n} as a consequence of the graphic contraction property under sq^2 < 1, without any appeal to the continuity of f. To address the referee's point, we will add a clarifying paragraph in the revised manuscript that explicitly outlines the steps of the sequential tracking to confirm this independence. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from MCPT definition and b-metric completeness

full rationale

The paper starts from the perimetric contraction inequality that defines an MCPT and derives iterate behavior (graphic contractions when sq^n < 1), weak Picard property, and fixed-point cardinality bounds (1 ≤ |Fix(f)| ≤ 2) directly via sequential tracking in complete b-metric spaces. No fitted parameters are renamed as predictions, no self-citation chain supplies the central premise, and the case split on prime period-2 orbits follows from the stated assumptions without reducing any result to its own input by construction. The abstract and described proofs treat the MCPT inequality as an independent structural hypothesis whose consequences are tracked analytically.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on the standard axioms of b-metric spaces and the newly introduced MCPT property; no numerical parameters are fitted and no new physical entities are postulated.

axioms (2)
  • domain assumption The ambient space is a complete b-metric space with coefficient s ≥ 1
    Explicitly stated as the generalized topological framework in the opening sentence of the abstract.
  • domain assumption The mapping satisfies the perimetric contraction inequality that defines an MCPT
    This is the central hypothesis from which all iterate and fixed-point conclusions are derived.
invented entities (1)
  • MCPT (mapping contracting perimeters of triangles) no independent evidence
    purpose: New class of operators whose perimeter-contraction property is used to obtain the iterate and periodicity results
    Introduced in the paper as an extension beyond graphic contractions; no independent falsifiable evidence outside the definition is supplied.

pith-pipeline@v0.9.1-grok · 5839 in / 1720 out tokens · 53823 ms · 2026-07-01T01:21:49.241229+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Aghajani, M

    A. Aghajani, M. Abbas, and J. R. Roshan,Common fixed point of generalized weak con- tractive mappings in partially ordered b-metric spaces, Mathematica Slovaca, 64 (2014), 941–960

    work page 2014

  2. [2]

    H. Aydi, M. Bota, E. Karapinar, and S. Mitrovic,A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory and Applications, 2012 (2012), Article 88

    work page 2012

  3. [3]

    I. A. Bakhtin,The contraction mapping principle in quasimetric spaces, Functional Anal- ysis (Ulianowsk Gos. Ped. Inst.), 30 (1989), 26–37. 21

    work page 1989

  4. [4]

    Berinde,Iterative Approximation of Fixed Points, Second Edition, Berlin, Heidelberg: Springer Berlin Heidelberg, 2007

    V . Berinde,Iterative Approximation of Fixed Points, Second Edition, Berlin, Heidelberg: Springer Berlin Heidelberg, 2007

    work page 2007

  5. [5]

    Berinde and M

    V . Berinde and M. P ˘acurar,The early developments in fixed point theory on b-metric spaces, Carpathian Journal of Mathematics 38.3 (2022): 523–538

    work page 2022

  6. [6]

    L. B. ´Ciri´c,A generalization of Banach’s contraction principle, Proceedings of the Amer- ican Mathematical Society 4592(2) (1974), 267–273

    work page 1974

  7. [7]

    Cvetkovi ´c,Iterates of mappings contracting perimeters of triangles, Bulletin of the Iranian Mathematical Society, 52 (4) (2026), Article 36

    M. Cvetkovi ´c,Iterates of mappings contracting perimeters of triangles, Bulletin of the Iranian Mathematical Society, 52 (4) (2026), Article 36

    work page 2026

  8. [8]

    Czerwik,Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1(1) (1993), 5–11

    S. Czerwik,Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1(1) (1993), 5–11

    work page 1993

  9. [9]

    Kunzi and H

    H. Kunzi and H. Pajoohesh,b-metrics, Topology and its Applications, 309 (2022), 107905

    work page 2022

  10. [10]

    Malkowsky and V

    E. Malkowsky and V . Rakoˇcevi´c,Advanced Functional Analysis, CRC Press, 2019

    work page 2019

  11. [11]

    Miculescu and A

    R. Miculescu and A. Mihail,New fixed point theorems for set-valued contractions in b- metric spaces, Journal of Fixed Point Theory and Applications, 19(3) (2017), 2153–2163

    work page 2017

  12. [12]

    Negash, A

    A. Negash, A. Ayalew, and M. Bogale,A Common Fixed Point Theorem For General- ized Contraction Pair Of Selfmaps In B-Metric Spaces, Parana Journal of Science and Education, 11(4) (2025), 25–32

    work page 2025

  13. [13]

    Petrov,Fixed point theorem for mappings contracting perimeters of triangles, Journal of Fixed Point Theory and Applications, 25 (2023), Article 74

    E. Petrov,Fixed point theorem for mappings contracting perimeters of triangles, Journal of Fixed Point Theory and Applications, 25 (2023), Article 74

    work page 2023

  14. [14]

    Petrov and R

    E. Petrov and R. Salimov,Fixed point theorem for mappings contracting perimeters of tri- angles and its generalizations, Algebraic and Geometric Methods of Analysis, 25 (2023), 84–95

    work page 2023

  15. [15]

    Petrus ¸el and I

    A. Petrus ¸el and I. A. Rus,Graphic contraction principle and applications, in: T. Rassias and P. Pardalos (eds.), Mathematical Analysis and Applications, Springer International Publishing, Springer, Cham, 2019, pp. 411–432

    work page 2019

  16. [16]

    Petrus ¸el, I

    A. Petrus ¸el, I. A. Rus, and M.-A. S ¸erban,Nonexpansive operators as graphic contrac- tions, Journal of Nonlinear and Convex Analysis, 17(7) (2016), 1409–1415

    work page 2016

  17. [17]

    Petrus ¸el, G

    A. Petrus ¸el, G. Petrus ¸el, and J.-C. Yao,Fixed points and coupled fixed points for graphic contractions in complete b-metric spaces, Journal of Nonlinear and Convex Analysis, 19(11) (2018), 1869–1882

    work page 2018

  18. [18]

    Petrus ¸el and G

    A. Petrus ¸el and G. Petrus ¸el,Graphical contractions and common fixed points in b-metric spaces, Arabian Journal of Mathematics, 12 (2) (2023), 423–430

    work page 2023

  19. [19]

    I. A. Rus,Weakly Picard mappings, Commentationes Mathematicae Universitatis Caroli- nae, 34(4) (1993), 769–773

    work page 1993

  20. [20]

    I. A. Rus,Weakly Picard operators and applications, Seminar on Fixed Point Theory (Cluj-Napoca), 2 (2001), 41–58. 22

    work page 2001

  21. [21]

    I. A. Rus,Picard operators and applications, Universitatea Babes-Bolyai. Faculty of Mathematics and Computer Science, 1996

    work page 1996

  22. [22]

    I. A. Rus,Generalized Contractions and Applications, Cluj University Press, Cluj- Napoca, 2001

    work page 2001

  23. [23]

    Suzuki,Basic inequality on a b-metric space and its applications, Journal of Inequali- ties and Applications, 2017 (2017), Article 256

    T. Suzuki,Basic inequality on a b-metric space and its applications, Journal of Inequali- ties and Applications, 2017 (2017), Article 256

    work page 2017

  24. [24]

    Woldegiorgis, M

    M. Woldegiorgis, M. F. Bogale, A. G. Negash, and S. Hussein,A point of coincidence and common fixed point theorem for expansive type mappings in b-metric spaces, Fixed Point Methods and Optimization, 2(3) (2025), 220–228. 23

    work page 2025