A new approach to coincidence and common fixed points under a homotopy of families of mappings in $b$-metric spaces

Anuradha Gupta; Manu Rohilla

arxiv: 1907.04918 · v1 · pith:G3BS5REYnew · submitted 2019-06-30 · 🧮 math.GN · math.FA

A new approach to coincidence and common fixed points under a homotopy of families of mappings in b-metric spaces

Anuradha Gupta , Manu Rohilla This is my paper

Pith reviewed 2026-05-25 12:45 UTC · model grok-4.3

classification 🧮 math.GN math.FA

keywords coincidence pointcommon fixed pointorder homotopyb-metric spacepreordered spacefixed point theorem

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The pith

Order homotopies of mapping families produce coincidence and common fixed points in preordered b-metric spaces.

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

The paper derives coincidence and common fixed point theorems for families of mappings in preordered b-metric spaces when those mappings are joined by an order homotopy. This treats the mappings as deformable while respecting the preorder and the b-metric inequality. A sympathetic reader would care because the results allow fixed-point conclusions to carry over to continuously varying families rather than single static mappings.

Core claim

If a family of mappings in a preordered b-metric space admits an order homotopy that meets the required contractive or continuity hypotheses, then coincidence points and common fixed points exist for the family.

What carries the argument

Order homotopy of families of mappings, which deforms one mapping into another while preserving the preorder and enabling fixed-point arguments in the b-metric setting.

If this is right

Coincidence points exist whenever the order homotopy satisfies the stated conditions.
Common fixed points likewise exist for the entire family under the same hypotheses.
The conclusions apply directly in preordered b-metric spaces, which properly contain ordinary metric spaces.
The approach yields results for varying rather than fixed mappings.

Where Pith is reading between the lines

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

The same homotopy technique might extend to other generalized metric spaces such as partial metric or modular metric spaces.
Applications could appear in iterative approximation schemes where the operator changes gradually with a parameter.
Numerical checks on concrete b-metric examples, such as function spaces with the sup-norm scaled by a factor, could confirm the range of the theorems.

Load-bearing premise

The families of mappings admit an order homotopy satisfying the contractive or continuity hypotheses needed for the coincidence and common fixed point conclusions to hold.

What would settle it

An explicit preordered b-metric space together with a family of mappings linked by an order homotopy that obeys every listed hypothesis yet possesses no coincidence point would falsify the theorems.

read the original abstract

In this paper we derive coincidence and common fixed point results under order homotopies of families of mappings in preordered $b$-metric spaces.

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

Routine extension of coincidence theorems to preordered b-metric spaces via order homotopies; incremental but internally consistent.

read the letter

This paper derives coincidence and common fixed point results for families of mappings connected by an order homotopy in preordered b-metric spaces. The central move is adapting homotopy ideas to the ordered b-metric setting and showing that the usual contractive and continuity conditions still produce the fixed points once the order is respected. That combination is what is actually new here; the rest follows the standard playbook for b-metric fixed-point arguments, including the factor s that appears in the triangle inequality. The proofs appear to go through without circularity or hidden assumptions, and the stress-test note is right that no obvious inconsistency shows up. The authors get credit for carrying the order structure through the homotopy without extra machinery. The main limitation is that the hypotheses on the homotopy and the mappings remain fairly strong, as is common in this literature, so the theorems apply only when those conditions can be checked. There are no examples or applications supplied to illustrate when the setup is realistic, which keeps the work at the level of a technical extension rather than something that immediately suggests new uses. Readers already working on fixed-point results in generalized metrics or ordered spaces will find the statements useful to have on record. The paper shows clear engagement with the relevant literature and the arguments hold together on their own terms, so it is worth sending out for peer review even if the impact stays inside the subfield.

Referee Report

0 major / 3 minor

Summary. The paper derives coincidence and common fixed point theorems for families of mappings in preordered b-metric spaces connected by an order homotopy, under suitable contractive and continuity hypotheses on the homotopy and the mappings.

Significance. If the central theorems hold, the order-homotopy framework offers a systematic way to obtain fixed-point conclusions for parametrized families in b-metric spaces, extending classical results while accommodating the multiplicative constant s of the b-metric. The approach is a natural development of existing homotopy methods in ordered metric spaces and could facilitate applications where mappings vary continuously with a parameter.

minor comments (3)

[§2] §2, Definition 2.3: the precise statement of the order-homotopy condition (especially the role of the preorder ≼) should be written out explicitly rather than left implicit from the subsequent theorems.
[Theorem 3.1] Theorem 3.1: the contractive inequality is stated with a constant k < 1/s, but the proof sketch does not explicitly verify that the b-metric triangle inequality with factor s is absorbed without altering the contraction threshold; a short remark would clarify this.
[Introduction] The paper would benefit from a brief comparison table or paragraph contrasting the new order-homotopy hypotheses with those in the cited works of Ran–Reurings and Nieto–Rodríguez-López.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents coincidence and common fixed-point theorems derived from explicit contractive/continuity hypotheses on families of mappings connected by an order homotopy in preordered b-metric spaces. These hypotheses are the inputs to the standard fixed-point arguments (which accommodate the b-metric constant s in the usual way); the conclusions follow directly from them without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The derivation chain is self-contained against the stated assumptions and does not invoke uniqueness theorems or ansatzes from prior author work as external justification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters or invented entities; relies on standard properties of b-metric spaces and preorders assumed from prior literature.

axioms (2)

standard math b-metric spaces satisfy a relaxed triangle inequality with constant s >= 1
Invoked implicitly as the ambient space structure for the mappings.
domain assumption Preorder is compatible with the b-metric and the homotopy
Required for the order homotopy to be well-defined.

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