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arxiv: 2510.23952 · v5 · submitted 2025-10-28 · 🧮 math.FA

Fixed Point Theorems for Relaxed Asymptotic Contractions via Two Quasi-Metrics

Jie Shi This is my paper

Pith reviewed 2026-05-18 03:55 UTC · model grok-4.3

classification 🧮 math.FA
keywords fixed point theoremsasymptotic contractionsquasi-metricsBoyd-Wong functioncomplete metric spacesiterative convergenceKirk theorem
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The pith

A new asymptotic contraction using two quasi-metrics defined from the mapping proves unique fixed points and convergent iterates under only continuity and orbit boundedness in complete metric spaces.

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

The paper develops a relaxed form of asymptotic contraction by means of two quasi-metrics built directly from the map under study. These are compared using a sequence of bounding functions that approach a Boyd-Wong function in a locally uniform way. This condition is weaker than those in Kirk's theorem yet includes the earlier result as a particular instance. Under the sole requirements that the map is continuous and possesses at least one bounded orbit inside a complete metric space, the work shows that a unique fixed point exists and that every sequence of iterates converges to it.

Core claim

By introducing two quasi-metrics defined in terms of the mapping and imposing that they satisfy an inequality controlled by bounding functions converging locally uniformly to a Boyd-Wong function, the authors prove that any continuous self-map of a complete metric space that has a bounded orbit possesses a unique fixed point, and that the iterates of the map converge to this fixed point from any initial point.

What carries the argument

A pair of quasi-metrics constructed from the mapping, whose comparison is governed by bounding functions converging locally uniformly to a Boyd-Wong function.

If this is right

  • Existence and uniqueness of a fixed point for this wider class of mappings.
  • Convergence of all iterates to the fixed point.
  • Strict containment of Kirk's asymptotic fixed point theorem as a special case.
  • Fixed point guarantees for maps that do not satisfy stricter classical contraction conditions.

Where Pith is reading between the lines

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

  • The construction of quasi-metrics directly from the mapping may extend to other contraction conditions in generalized metric settings.
  • Local uniform convergence of the bounding functions could support estimates on convergence speed of iterates in applications.
  • The relaxed condition may apply to mappings from nonlinear analysis or optimization that satisfy the inequality but not prior ones.

Load-bearing premise

The bounding functions converge locally uniformly to a Boyd-Wong function and the two quasi-metrics are defined directly in terms of the underlying mapping.

What would settle it

A continuous mapping in a complete metric space with a bounded orbit that satisfies the new contraction condition but has no fixed point or fails to have convergent iterates would falsify the theorem.

read the original abstract

We introduce a new class of asymptotic contractions that employs two quasi-metrics defined directly in terms of the underlying mapping. The contraction condition compares these two quantities via a sequence of bounding functions that converge locally uniformly to a Boyd-Wong function. This framework relaxes the hypotheses of Kirk's asymptotic fixed point theorem and strictly contains it as a special case. Assuming only the continuity of the map and the boundedness of some orbit in a complete metric space, we prove both the existence and uniqueness of a fixed point, along with the convergence of all iterates to that point.

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 manuscript introduces a new class of asymptotic contractions that employs two quasi-metrics defined directly in terms of the underlying mapping. The contraction condition compares these two quantities via a sequence of bounding functions that converge locally uniformly to a Boyd-Wong function. This framework relaxes the hypotheses of Kirk's asymptotic fixed point theorem and strictly contains it as a special case. Assuming only the continuity of the map and the boundedness of some orbit in a complete metric space, the paper proves both the existence and uniqueness of a fixed point, along with the convergence of all iterates to that point.

Significance. If the central claims hold, the result provides a strict generalization of Kirk's theorem under weaker hypotheses on the contraction, which could extend the reach of fixed-point theory to mappings that fail standard asymptotic contraction conditions. The explicit use of map-dependent quasi-metrics and locally uniform convergence to a Boyd-Wong function is a technically interesting relaxation.

major comments (2)
  1. [§3] §3 (main existence theorem): the passage to the limit inside the quasi-metric inequality is load-bearing for both existence and uniqueness. The proof must show that the local-uniform convergence of the bounding functions together with the definition of q1 and q2 (directly from the mapping) still yields the strict Boyd-Wong inequality φ(d(x*,x*)) < d(x*,x*) after extracting a Cauchy sequence in the underlying metric d. The manuscript does not appear to supply an explicit continuity or compatibility relation between q1/q2 and d that would guarantee this step.
  2. [§2] Definition of the quasi-metrics (early in §2): because q1 and q2 are constructed from the iterates of the map, it is not immediate that they inherit the triangle inequality or lower semi-continuity properties needed to preserve completeness of the orbit in the limit. If these properties fail for some sequences, the argument that the orbit is Cauchy in d may not close.
minor comments (2)
  1. [Introduction] The introduction should state the precise new contraction inequality (including the roles of φ_n and the two quasi-metrics) rather than only describing it in words.
  2. [§2] Add a short remark clarifying whether the Boyd-Wong function φ is assumed to satisfy φ(t) < t for t > 0 or only the weaker integral condition; this affects the uniqueness argument.

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 comment below, clarifying the relevant arguments and indicating the revisions we will incorporate to make the proofs fully explicit.

read point-by-point responses

  1. Referee: [§3] §3 (main existence theorem): the passage to the limit inside the quasi-metric inequality is load-bearing for both existence and uniqueness. The proof must show that the local-uniform convergence of the bounding functions together with the definition of q1 and q2 (directly from the mapping) still yields the strict Boyd-Wong inequality φ(d(x*,x*)) < d(x*,x*) after extracting a Cauchy sequence in the underlying metric d. The manuscript does not appear to supply an explicit continuity or compatibility relation between q1/q2 and d that would guarantee this step.

    Authors: We agree that an explicit compatibility argument is needed for rigor. In the revised manuscript we will insert a new lemma immediately preceding the main theorem. The lemma uses the continuity of T together with the bounded-orbit hypothesis to prove that if {x_n} is Cauchy in d and converges to x*, then q1(x_n, x*) → d(x*, x*) and q2(x_n, x*) → d(x*, x*) (or the appropriate one-sided limits required by the inequality). Combined with the given locally uniform convergence of the bounding functions to the Boyd-Wong function φ, this yields the strict inequality φ(d(x*, x*)) < d(x*, x*) after passing to the limit, thereby closing both the existence and uniqueness arguments. revision: yes

  2. Referee: [§2] Definition of the quasi-metrics (early in §2): because q1 and q2 are constructed from the iterates of the map, it is not immediate that they inherit the triangle inequality or lower semi-continuity properties needed to preserve completeness of the orbit in the limit. If these properties fail for some sequences, the argument that the orbit is Cauchy in d may not close.

    Authors: The referee correctly notes that these properties require verification. We will expand the opening of §2 with a short proposition that directly verifies the triangle inequality for q1 and q2 from their explicit construction in terms of the iterates of T and the triangle inequality already satisfied by d. Lower semi-continuity with respect to d follows at once from the continuity of T and the boundedness of the orbit. These added details ensure that the completeness of the orbit in d is preserved and that the subsequent Cauchy-sequence argument remains valid. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation

full rationale

The paper introduces a new contraction condition via two map-dependent quasi-metrics and locally uniformly convergent bounding functions to a Boyd-Wong function, then proves fixed-point existence, uniqueness, and iterate convergence directly from continuity of the map plus bounded orbit in a complete metric space. This is a standard direct proof that relaxes Kirk's theorem as a special case without any step reducing by construction to a fitted input, self-definition, or load-bearing self-citation chain. The abstract and description show an independent argument whose central claims are not tautological with the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard assumptions from metric space theory plus the novel definition of the contraction condition; no free parameters or invented entities are introduced.

axioms (3)
  • domain assumption The underlying space is a complete metric space.
    Invoked as the setting where the fixed point theorem is proved.
  • domain assumption The mapping is continuous.
    Stated explicitly as a hypothesis for existence and convergence.
  • domain assumption Some orbit of the mapping is bounded.
    Required assumption for the theorem as described in the abstract.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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