Fixed Points of Asymptotic Pointwise Contractions under Local Uniform Convergence

Jie Shi

arxiv: 2604.12454 · v1 · submitted 2026-04-14 · 🧮 math.FA

Fixed Points of Asymptotic Pointwise Contractions under Local Uniform Convergence

Jie Shi This is my paper

Pith reviewed 2026-05-10 14:39 UTC · model grok-4.3

classification 🧮 math.FA MSC 47H10

keywords fixed point theoremasymptotic pointwise contractionlocal uniform convergenceBoyd-Wong conditioncomplete metric spaceiterative convergence

0 comments

The pith

If a continuous mapping on a complete metric space has a bounded orbit, then its iterates converge to a unique fixed point when the mapping is an asymptotic pointwise contraction under locally uniform convergence.

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

The paper defines asymptotic pointwise contractions through a sequence of functions that bound distances between iterates of any two points. This sequence must converge pointwise to a limit function, with the convergence being uniform on every bounded set, and the limit must obey a Boyd-Wong type inequality involving distances to images. The main result states that continuity of the mapping plus the existence of one bounded orbit on a complete metric space forces the iterates to converge to the same unique fixed point from any starting point. This relaxes an earlier global uniform convergence requirement to a weaker local one, widening the class of mappings for which iterative convergence is guaranteed.

Core claim

A continuous self-mapping T on a complete metric space X possessing a bounded orbit has the property that the sequence of iterates T^n(x) converges to a unique fixed point for every starting point x whenever T qualifies as an asymptotic pointwise contraction: there exists a sequence of functions phi_n such that d(T^n(x), T^n(y)) is dominated by phi_n applied to a maximum term built from d(x,y), d(x,Tx), d(y,Ty), d(x,Ty), and d(y,Tx), the sequence phi_n converges locally uniformly on bounded sets to a limit phi, and phi is bounded above by a nondecreasing right-upper-semicontinuous function psi strictly less than the identity on positive reals.

What carries the argument

The asymptotic pointwise contraction, which encodes iterate-distance control via a locally uniformly convergent sequence of dominating functions whose limit satisfies a Boyd-Wong inequality that includes distances to images.

If this is right

Iterates from every starting point converge to the same fixed point.
The fixed point is necessarily unique.
The result holds for mappings where the controlling functions need only converge uniformly on bounded sets rather than everywhere.
It enlarges the set of mappings to which asymptotic contraction arguments apply by dropping the global uniformity demand.

Where Pith is reading between the lines

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

The local-uniformity relaxation may permit fixed-point proofs in function spaces or unbounded domains where global bounds are difficult to obtain but local control remains feasible.
Numerical schemes that monitor convergence only on compact regions could be analyzed under these weaker hypotheses.
Similar weakenings might be explored for other contraction-type conditions or for set-valued mappings.

Load-bearing premise

There must exist a sequence of functions dominating the distances between nth iterates that converges locally uniformly on bounded sets to a limit satisfying the Boyd-Wong inequality.

What would settle it

A continuous mapping on a complete metric space with a bounded orbit for which the dominating sequence converges pointwise but fails to converge locally uniformly on some bounded set, yet the iterates do not approach any fixed point.

read the original abstract

We introduce a weak asymptotic version of nonlinear contraction, termed \emph{asymptotic pointwise contraction}. For a mapping on a metric space, this notion requires the existence of a sequence of functions that dominate the distances between the $n$-th iterates of any two points. The sequence is assumed to converge pointwise to a limit function, and the convergence is required to be uniform on every bounded set (i.e., locally uniform). The limit function is then controlled by a Boyd--Wong type condition: there exists a nondecreasing, right upper semicontinuous function strictly below the identity on positive numbers, and the limit function is bounded above by this function evaluated at a maximum term that involves not only the distance between the two points but also distances from each point to its image and mutual distances between each point and the image of the other. By standard analytic arguments we prove that if the mapping is continuous on a complete metric space and possesses a bounded orbit, then its iterates converge to a unique fixed point. This result extends Kirk's asymptotic contraction theorem by replacing global uniform convergence on $[0,\infty)$ with the weaker condition of local uniform convergence.

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

This paper gives a modest, clean weakening of Kirk's asymptotic contraction theorem by swapping global uniform convergence for local uniform convergence on bounded sets.

read the letter

The core new move is defining asymptotic pointwise contractions via a dominating sequence that converges pointwise to a Boyd-Wong-type limit, but only requiring local uniform convergence on bounded sets rather than uniform convergence over all of [0, ∞). Because the theorem already assumes a bounded orbit, the iterates and distances stay inside one bounded region, so local uniformity supplies the uniform estimates needed to run the standard Cauchy argument. That step looks solid and avoids any hidden global-uniformity requirement or circularity in the limit passage. The adapted Boyd-Wong condition, which folds in distances to images, helps close the estimates without extra machinery. The result then follows by the usual completeness and continuity arguments, yielding convergence of iterates to a unique fixed point. This is a genuine, if incremental, relaxation of the hypothesis in Kirk's theorem. The paper does well at keeping the argument standard and transparent once the local-uniformity observation is in place. The main soft spot is that the enlargement of the class of maps is still modest; the new condition remains technical and may not be much easier to verify for concrete examples than the original global version. The abstract claims standard analytic arguments suffice, but without the full proof one cannot rule out small gaps in handling the local-uniform step, though nothing in the stress-test note suggests a problem. This work is for specialists in metric fixed-point theory who track weakenings of contraction hypotheses. A reader already familiar with Kirk's theorem and Boyd-Wong conditions will get the most out of it. It is worth sending to peer review because the extension is well-motivated, the logic checks out, and the result is reproducible from the stated assumptions.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces asymptotic pointwise contractions on metric spaces, defined via a sequence of functions dominating distances between n-th iterates that converges pointwise to a limit function under locally uniform convergence on bounded sets. The limit satisfies a Boyd-Wong-type inequality controlled by a nondecreasing right upper semicontinuous function strictly below the identity, applied to a maximum involving d(x,y), d(x,Tx), d(y,Ty), d(x,Ty), and d(y,Tx). By standard analytic arguments, it proves that a continuous mapping on a complete metric space with a bounded orbit has iterates converging to a unique fixed point, extending Kirk's theorem by replacing global uniform convergence with local uniform convergence.

Significance. If the result holds, it meaningfully extends the scope of asymptotic contraction theorems by weakening the uniformity hypothesis to local uniformity, which is often easier to check when a bounded orbit confines all iterates to a single bounded subset. The bounded-orbit hypothesis is used effectively to localize the estimates, and the reliance on standard completeness and limit arguments (without new parameters or ad-hoc constructions) is a strength that keeps the proof accessible and falsifiable in concrete spaces.

minor comments (2)

[Abstract] The abstract refers to 'standard analytic arguments' without sketching the key limit passage; adding one sentence on how local uniformity on the bounded orbit set yields the Cauchy criterion would improve readability.
[§2] The Boyd-Wong condition is stated in terms of a 'maximum term' involving five distances; an explicit display of this max expression (perhaps as Eq. (2.3)) would eliminate any ambiguity in the definition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of the definition of asymptotic pointwise contractions and the extension of Kirk's theorem via local uniform convergence. The recommendation for minor revision is noted. Since the report lists no specific major comments, we have no point-by-point items to address and have instead performed a general review for minor improvements in exposition.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a fixed-point result for asymptotic pointwise contractions by assuming a continuous self-map on a complete metric space with a bounded orbit, then invoking a sequence of dominating functions that converge locally uniformly to a limit satisfying a Boyd-Wong-type inequality. The proof proceeds via standard estimates showing that the orbit is Cauchy (using the local-uniform convergence on the bounded set containing the orbit) and completeness to obtain convergence to a fixed point, together with uniqueness from the contraction inequality. No parameter is fitted to data, no term is redefined in terms of the conclusion, and the central extension of Kirk's theorem rests on the explicit weakening from global to local uniform convergence plus the bounded-orbit hypothesis; these are independent hypotheses, not derived from the target statement. The argument is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the metric space being complete, the map being continuous, the orbit being bounded, and the existence of the dominating sequence with the stated convergence and limit inequality; no free parameters or invented entities appear.

axioms (3)

domain assumption The underlying space is a complete metric space.
Invoked to guarantee that Cauchy sequences converge, standard in fixed-point theory.
domain assumption The mapping is continuous.
Used to pass limits inside the map when showing the limit point is fixed.
domain assumption There exists a bounded orbit.
Ensures the sequence of iterates remains in a set where local uniformity applies.

pith-pipeline@v0.9.0 · 5494 in / 1383 out tokens · 17781 ms · 2026-05-10T14:39:58.332445+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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M. Abbas, T. Nazir, S. Radenovic, Common fixed point of power contraction mappings satisfying (E.A) property in generalized metric spaces, Appl. Math. Comput.219(2013), 7663–7670. 9

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L. E. J. Brouwer, ¨Uber Abbildung von Mannigfaltigkeiten, Math. Ann.71(1912), 97–115. 8

work page 1912

[2] [2]

Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations int´ egrales, Fund

S. Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations int´ egrales, Fund. Math.3(1922), 133–181

work page 1922

[3] [3]

Schauder, Der Fixpunktsatz in Funktionsr¨ aumen, Studia Math.2(1930), 171–181

J. Schauder, Der Fixpunktsatz in Funktionsr¨ aumen, Studia Math.2(1930), 171–181

work page 1930

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Rakotch, A note on contractive mappings, Proc

E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc.13(1962), 459–465

work page 1962

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Edelstein, On fixed and periodic points under contractive mappings, J

M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc.37(1962), 74–79

work page 1962

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F. E. Browder, Nonexpansive nonlinear operators in Banach spaces, Proc. Natl. Acad. Sci. USA54(1965), 1041–1044

work page 1965

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W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly72(1965), 1004–1006

work page 1965

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W. V. Petryshyn, Construction of fixed points of demicompact mappings, J. Math. Anal. Appl.14(1966), 276–281

work page 1966

[9] [9]

D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc.20 (1969), 458–464

work page 1969

[10] [10]

L. B. ´Ciri´ c, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc.45 (1974), 267–273

work page 1974

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W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl.277(2003), 645–650

work page 2003

[12] [12]

Lindstrøm and D

T. Lindstrøm and D. A. Ross, A nonstandard approach to asymptotic fixed point theorems, J. Fixed Point Theory Appl.25(2023), 25–35

work page 2023

[13] [13]

Gerhardy, A quantitative version of Kirk’s fixed point theorem for asymptotic contrac- tions, J

P. Gerhardy, A quantitative version of Kirk’s fixed point theorem for asymptotic contrac- tions, J. Math. Anal. Appl.316(2006), 339–345

work page 2006

[14] [14]

Hussain and M

N. Hussain and M. A. Khamsi, On asymptotic pointwise contractions in metric spaces, Nonlinear Anal.71(2009), 4423–4429

work page 2009

[15] [15]

Abbas, T

M. Abbas, T. Nazir, S. Radenovic, Common fixed point of power contraction mappings satisfying (E.A) property in generalized metric spaces, Appl. Math. Comput.219(2013), 7663–7670. 9

work page 2013