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arxiv: 2604.02333 · v1 · submitted 2026-01-06 · 🧮 math.MG

Fixed point theorems on perturbed metric space with an application

Dipti Barman , T. Bag This is my paper

Pith reviewed 2026-05-16 17:12 UTC · model grok-4.3

classification 🧮 math.MG
keywords fixed point theoremsperturbed metric spaceF-perturbed mappingsboundary value problemsexistence results
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The pith

Fixed point theorems hold for F-perturbed mappings in complete perturbed metric spaces and apply to boundary value problems.

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

The paper takes the definition of a perturbed metric space and proves fixed point theorems for mappings that satisfy F-perturbed conditions within complete such spaces. It supplies a counterexample to show why those conditions matter. The theorems are then used to prove existence of a solution to a second-order boundary value problem.

Core claim

In a complete perturbed metric space, every F-perturbed mapping possesses at least one fixed point; the result is justified by a counterexample showing the conditions are necessary and is applied to establish existence of solutions for a second-order boundary value problem.

What carries the argument

The F-perturbed mapping, defined by a contraction-type inequality involving a function F applied to the perturbed distance between points and their images.

If this is right

  • Existence of fixed points follows directly once the F-perturbed condition holds in a complete perturbed metric space.
  • The same theorems guarantee solutions exist for the second-order boundary value problem when the operator is shown to be F-perturbed.
  • The counterexample demonstrates that dropping completeness or the F-condition can destroy the fixed-point guarantee.

Where Pith is reading between the lines

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

  • The approach could be tested on other classes of mappings or on perturbed versions of standard metric spaces arising in optimization.
  • If perturbed metrics model small perturbations of distances, the theorems may apply to stability questions in geometric fixed-point problems.

Load-bearing premise

The space must be complete under the perturbed metric and the mapping must obey the F-perturbed condition as stated in the paper.

What would settle it

A concrete counterexample in a complete perturbed metric space where the mapping violates the F-perturbed condition and has no fixed point.

Figures

Figures reproduced from arXiv: 2604.02333 by Dipti Barman, T. Bag.

Figure 1
Figure 1. Figure 1: Curve type picture indicates the value of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The upper picture indicates F(D(x, y)) where, as the lower picture indicates τ + F(D(T x, T y)) where G(t, s) is Green’s function given by G(t, s) = ( s(1 − t), 0 < s ≤ t ≤ 1 t(1 − s), 0 < t < s ≤ 1. The Green’s function satisfies the boundary conditions u(0) = u(1) = 0. Let X = C[0, 1], the class of all real-valued continuous functions defined on [0, 1] and define a function D(u1(t), u2(t)) = sup t∈[0,1] … view at source ↗
Figure 3
Figure 3. Figure 3: (a) Iteration un(t) vs t (b) ||un+1(t) − un(t)||∞ vs n For numerical iteration, we choose initial guess u0(t) = t. Figures 3 (a) show that the sequence un+1(t) = T un(t) = R 1 0 G(t, s)· (s+0.5) 2 ·sin u(s) ds converges to 0. Here, ||un+1(t) − un(t)||∞ = max |un+1(t) − un|(t)| = sup-norm error of the iteration. 6 Another generalized fixed-point theorem In this section, we prove another theorem on the fixed… view at source ↗
read the original abstract

Following the definition of perturbed metric space, in this paper, some fixed point theorems are established for $ F $-perturbed mappings in complete perturbed metric spaces and justify the result by counter example. Finally, an application of this theorem for the existence of a solution for the second-order boundary value problem is given.

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 / 1 minor

Summary. The manuscript defines perturbed metric spaces and proves fixed point theorems for F-perturbed mappings on complete perturbed metric spaces. It includes a counterexample and applies the main result to prove existence of a solution to a second-order boundary value problem.

Significance. If the central theorems are correct, the work extends Banach-type fixed point results to a perturbed setting and supplies a concrete application to BVPs. The counterexample and BVP example are positive features that help anchor the abstract claims.

major comments (2)
  1. [§3] §3 (proof of the main fixed-point theorem): the argument that the Picard iterates form a Cauchy sequence in the perturbed metric d_F and that the limit is a fixed point does not establish that F(x_n, x) → 0 or that the perturbation preserves the necessary limit passage; without an explicit continuity or vanishing condition on F the standard completeness argument fails to close.
  2. [Application section] Application section (BVP): the reduction of the boundary-value problem to an F-perturbed contraction on the integral operator is stated but the verification that the operator satisfies the precise F-perturbed inequality with a uniform constant <1 is omitted; this step is load-bearing for the claimed existence result.
minor comments (1)
  1. [Abstract] The abstract claims the results are 'justified by counter example' but the manuscript never states what property the counterexample is meant to demonstrate (necessity of completeness, sharpness of the contraction constant, etc.).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript to strengthen the presentation where the concerns are valid.

read point-by-point responses

  1. Referee: [§3] §3 (proof of the main fixed-point theorem): the argument that the Picard iterates form a Cauchy sequence in the perturbed metric d_F and that the limit is a fixed point does not establish that F(x_n, x) → 0 or that the perturbation preserves the necessary limit passage; without an explicit continuity or vanishing condition on F the standard completeness argument fails to close.

    Authors: We agree that the limit passage for F(x_n, x) requires explicit justification. The current proof shows the sequence is Cauchy in d_F via the contraction inequality but does not detail why F(x_n, x) → 0 when passing to the limit. We will revise Section 3 by adding a short lemma or remark establishing this limit under the standing assumptions on F (or by introducing a mild continuity condition on F if needed to close the argument). This will make the completeness step fully rigorous. revision: yes

  2. Referee: [Application section] Application section (BVP): the reduction of the boundary-value problem to an F-perturbed contraction on the integral operator is stated but the verification that the operator satisfies the precise F-perturbed inequality with a uniform constant <1 is omitted; this step is load-bearing for the claimed existence result.

    Authors: We acknowledge that the explicit verification for the integral operator was omitted. In the revised version we will insert the missing calculation: starting from the Green's function representation, we bound the difference of the operator values in the perturbed metric d_F and show that the contraction constant is strictly less than 1 under the stated hypotheses on the nonlinearity. This step is indeed essential and will be written out in full. revision: yes

Circularity Check

0 steps flagged

No circularity: standard fixed-point derivation from definitions and completeness

full rationale

The paper defines perturbed metric spaces and F-perturbed mappings, then establishes fixed-point results in complete spaces via direct arguments (likely Banach-type iteration) and applies them to a BVP. No quoted step reduces a prediction to a fitted input, renames a known result, or relies on a self-citation chain whose justification is internal to the paper. The derivation remains self-contained against the stated axioms and completeness assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, providing no details on specific free parameters, axioms, or invented entities used in the proofs or definitions.

pith-pipeline@v0.9.0 · 5327 in / 880 out tokens · 28018 ms · 2026-05-16T17:12:56.111595+00:00 · methodology

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Reference graph

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