arxiv: 2604.24728 · v2 · submitted 2026-04-27 · 🧮 math.FA

Recognition: unknown

Partial extended b-metric and some fixed point theorem

Ivan Hadinata, Muhamad Abdillah Ahen, Raudhatul Mufizah

Pith reviewed 2026-05-07 17:22 UTC · model grok-4.3

classification 🧮 math.FA

keywords partial extended b-metric spacefixed point theoremcontractive mapping0-completenessgeneralized metric spacediscrete dynamical systems

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The pith

The paper introduces partial extended b-metric spaces that unify extended b-metrics and partial b-metrics and establishes fixed point theorems for contractive mappings.

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

This paper defines partial extended b-metric spaces as a broader structure that merges extended b-metric spaces with partial b-metric spaces. The new space uses a point-dependent control function and allows a point to have positive distance to itself. The authors develop basic properties such as convergence and 0-completeness for sequences in these spaces. They then prove that certain contractive mappings possess unique fixed points, generalizing earlier results, and apply the framework to the stability of discrete dynamical systems.

Core claim

We introduce the concept of partial extended b-metric spaces (PEBMS) as a unification and generalization of extended b-metric spaces and partial b-metric spaces. This new structure incorporates a point-dependent control function together with the possibility of non-zero self-distance. We establish several fundamental properties of PEBMS, including convergence, Cauchy sequences, and 0-completeness. We prove fixed point theorems for contractive mappings and show the existence and uniqueness of fixed points under suitable conditions. Furthermore, we demonstrate that every extended b-metric space can be viewed as a special case of a PEBMS.

What carries the argument

The partial extended b-metric, a distance function that satisfies relaxed triangle inequalities with a point-dependent control function and permits non-zero self-distances, which enables the extension of fixed point results from prior metric generalizations.

If this is right

Fixed point theorems for contractive mappings hold in PEBMS under suitable conditions, ensuring existence and uniqueness.
Every extended b-metric space is a special case of a PEBMS.
The framework applies to studying the stability of discrete dynamical systems.
Results generalize and enrich existing theories in metric-type spaces.

Where Pith is reading between the lines

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

The 0-Cauchy notion could support fixed point arguments in incomplete spaces where only 0-convergence matters.
Similar point-dependent controls might be added to other metric generalizations to recover more fixed point theorems.
Stability results for dynamical systems suggest the space could model systems with asymmetric or self-referential distances.

Load-bearing premise

The point-dependent control function and non-zero self-distance allowance still allow the space to maintain sufficient structure for defining convergence, 0-completeness, and contractive conditions that guarantee fixed points.

What would settle it

Construct a specific PEBMS and a contractive mapping on it that has no fixed point, or show a sequence that is 0-Cauchy but does not converge in the space.

read the original abstract

In this paper, we introduce the concept of partial extended b-metric spaces (PEBMS) as a unification and generalization of extended b-metric spaces and partial b-metric spaces. This new structure incorporates a point-dependent control function together with the possibility of non-zero self-distance, providing a more flexible framework for the study of generalized metric spaces. We establish several fundamental properties of PEBMS, including convergence, Cauchy sequences, and 0-completeness. By introducing the notion of 0-Cauchy sequences, we extend various results from extended b-metric spaces to the PEBMS setting. In particular, we prove fixed point theorems for contractive mappings and show the existence and uniqueness of fixed points under suitable conditions. Furthermore, we demonstrate that every extended b-metric space can be viewed as a special case of a PEBMS. As an application, we study the stability of discrete dynamical systems within this framework. The results presented here generalize and enrich existing theories in metric-type spaces and open new directions for further research.

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

A modest unification of partial and extended b-metrics whose fixed-point results largely recycle standard arguments and may not need the full new structure.

read the letter

The paper introduces partial extended b-metric spaces by blending the non-zero self-distance from partial metrics with the point-dependent control function from extended b-metrics. It then establishes basic properties like convergence and 0-completeness before proving fixed point theorems for contractive mappings and applying them to the stability of discrete dynamical systems. What the work does well is to explicitly show that every extended b-metric space is a special case of this new structure. That embedding is straightforward and helps clarify how the ideas fit together. The fixed point results follow the usual pattern of assuming a contraction condition and deriving uniqueness, which is fine as far as it goes. The soft spots come from the incremental nature of the contribution. The field of generalized metric spaces already has many overlapping extensions, and combining two of them with a variable θ does not appear to produce results that could not be obtained by tweaking existing theorems. The key question is whether the interaction between the partial metric axiom and the θ function preserves the necessary completeness properties without additional restrictions. If the proofs require θ to satisfy extra bounds or if positive self-distances prevent proper limits, then the claimed generality might not hold up. Since the abstract only summarizes the claims, the actual derivations need close checking to confirm there are no gaps. This paper is aimed at specialists in fixed point theory on metric-type spaces. A reader already deep in that literature might appreciate having the combined axioms in one place, but it is unlikely to interest anyone outside that niche or to shift broader understanding of the subject. I would accept it for peer review in a journal focused on functional analysis or fixed points, because the formal setup is there and the claims are testable, even if the impact is limited. The authors should be asked to provide concrete examples distinguishing the new space from prior ones and to verify the 0-completeness step explicitly.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces the concept of partial extended b-metric spaces (PEBMS) as a unification of extended b-metric spaces and partial b-metric spaces. The new structure incorporates a point-dependent control function θ together with the possibility of non-zero self-distances d(x,x). The authors establish convergence, Cauchy sequences, and 0-completeness in this setting, prove fixed-point theorems for contractive mappings (including existence and uniqueness), show that every extended b-metric space is a special case of a PEBMS, and apply the framework to the stability of discrete dynamical systems.

Significance. If the central claims hold, the work supplies a strictly more general setting for fixed-point theory that combines two previously studied generalizations. The explicit demonstration that extended b-metric spaces embed as special cases and the application to dynamical-system stability are positive features. The significance is tempered by the fact that the added generality (point-dependent θ and positive self-distances) must be shown not to obstruct 0-completeness or to render the fixed-point results already covered by earlier theorems.

major comments (1)

[Definition of PEBMS and 0-completeness section] Definition of PEBMS and subsequent 0-completeness results: the interaction between the partial-metric axiom (d(x,x) possibly positive) and the extended-b inequality involving the point-dependent control function θ(x,y) is not shown to preserve 0-completeness for sequences that are 0-Cauchy. This verification is load-bearing for the fixed-point theorems; without it the claimed extension may fail or reduce to prior results.

minor comments (1)

[Abstract and Introduction] The abstract states that 'several fundamental properties' are established; these should be listed explicitly with theorem numbers in the introduction for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below and will incorporate the necessary clarification in the revised version.

read point-by-point responses

Referee: [Definition of PEBMS and 0-completeness section] Definition of PEBMS and subsequent 0-completeness results: the interaction between the partial-metric axiom (d(x,x) possibly positive) and the extended-b inequality involving the point-dependent control function θ(x,y) is not shown to preserve 0-completeness for sequences that are 0-Cauchy. This verification is load-bearing for the fixed-point theorems; without it the claimed extension may fail or reduce to prior results.

Authors: We appreciate this observation. The manuscript establishes 0-completeness for PEBMS in the relevant theorem by combining the axioms, but the explicit step-by-step verification of how a 0-Cauchy sequence behaves under the simultaneous presence of positive self-distances d(x,x) and the point-dependent θ(x,y) in the triangle inequality is not isolated as a separate lemma. We will add a short auxiliary result (or expanded remark) in the revised manuscript that directly confirms: if {x_n} is 0-Cauchy, then lim d(x_n,x_m) = 0 as n,m→∞ implies the limit point satisfies the partial-metric and extended-b conditions without contradiction, thereby preserving 0-completeness. This will make the load-bearing step fully transparent and strengthen the foundation for the fixed-point theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: new space definition and standard fixed-point proofs are self-contained

full rationale

The paper defines PEBMS by combining the axioms of extended b-metric spaces (with point-dependent θ) and partial b-metric spaces (allowing d(x,x) ≥ 0), then directly verifies that prior spaces embed as special cases by setting θ ≡ 1 and d(x,x) = 0. It establishes convergence, 0-Cauchy, and 0-completeness from these axioms, and proves fixed-point results for contractive mappings via the usual iterative estimates. No step reduces a derived quantity to a fitted parameter, renames a known result, or relies on a self-citation chain; the theorems follow from the stated axioms without tautological redefinition. The central claim therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework rests on a new definition of distance that modifies the triangle inequality via an arbitrary control function; standard metric axioms are relaxed but not derived from more primitive principles.

free parameters (1)

point-dependent control function
Introduced as part of the PEBMS definition; its specific form is chosen per application rather than derived.

axioms (1)

domain assumption The distance function satisfies a controlled triangle inequality together with non-negativity, symmetry, and the possibility of positive self-distance.
Invoked in the opening definition of PEBMS and used to establish convergence and completeness properties.

invented entities (1)

partial extended b-metric space no independent evidence
purpose: Unify extended b-metric and partial b-metric spaces while allowing point-dependent control and non-zero self-distance.
New mathematical object defined in the paper; no independent empirical or formal verification supplied beyond the definition itself.

pith-pipeline@v0.9.0 · 5487 in / 1285 out tokens · 58207 ms · 2026-05-07T17:22:06.698491+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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