arxiv: 2605.06694 · v1 · submitted 2026-05-03 · 🧮 math.GM

Recognition: 2 theorem links

· Lean Theorem

Discontinuity at the fixed point in suprametric spaces

Nicola Fabiano , Sedigheh Barootkoob , Hossein Lakzian

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Pith reviewed 2026-05-11 00:54 UTC · model grok-4.3

classification 🧮 math.GM

keywords convex contractionssuprametric spacesfixed point theoremsdiscontinuityk-continuityquasi-contractionslower semi-continuity

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The pith

Convex contractions of order m guarantee fixed points in complete suprametric spaces but permit discontinuity at the fixed point.

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

The paper shows that convex contractions of order m on complete suprametric spaces always have a fixed point. However, these mappings do not have to be continuous at the fixed point. Weaker conditions such as k-continuity or T-orbitally lower semi-continuity can replace full continuity to ensure the result. This generalizes earlier theorems and applies to quasi-contractions as well.

Core claim

A convex contraction of order m on a complete suprametric space has a fixed point. The mapping need not be continuous at this fixed point. The fixed point property holds if the mapping satisfies k-continuity or T-orbitally lower semi-continuity instead. Further results extend quasi-contraction theorems to suprametric spaces.

What carries the argument

Convex contraction of order m, a self-mapping satisfying an inequality that controls the distance after m applications of the map.

Load-bearing premise

The suprametric space satisfies the triangle inequality in a relaxed form that still allows the contraction inequality to imply Cauchy sequences converge to a fixed point.

What would settle it

A counterexample consisting of a convex contraction of order m on a complete suprametric space with no fixed point would disprove the existence claim.

read the original abstract

The aim of this paper is to generalize some fixed point theorems in the class of convex contraction of order $m$ on a complete suprametric space. Then, we will prove that the class of convex contraction of order m is strong enough to generate a fixed point on a complete suprametric spaces but do not force the mapping to be continuous at the fixed point, and it can be replaced by relatively weaker conditions of $k$-continuity or $T$-orbitally lower semi-continuous. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233-245,1988) is found. In sequel, we will prove some fixed point results in the setting suprametric spaces which are generalizations of the results regarding Sehgal, \'Ciri\'c and Fisher's quasi-contraction. Some examples and application will be approved our results.

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

The paper solves Rhoades' discontinuity problem in suprametric spaces via convex contractions of order m and adds weaker continuity conditions, but the relaxed triangle inequality needs explicit verification to confirm the Picard sequence is Cauchy.

read the letter

The core contribution is a distinct take on Rhoades' 1988 question: convex contractions of order m on complete suprametric spaces produce fixed points without forcing continuity at those points, and the continuity assumption can be weakened to k-continuity or T-orbitally lower semi-continuity. The authors also extend Sehgal, Ćirić, and Fisher quasi-contraction results to the same setting, with examples and an application included. This is new content rather than a restatement of prior work in standard metrics. The abstract states the claims plainly and the iterative construction follows the usual pattern adapted to the new space. The approach earns credit for keeping the discontinuity property intact while broadening the class of spaces. The main soft spot is the completeness step. Suprametric spaces replace the standard triangle inequality with a relaxed form, and showing that the contraction forces the sequence to be Cauchy requires summing distances in a way that may not telescope directly. The stress-test concern is reasonable here: if the paper does not spell out how the relaxed inequality still yields the geometric bound without extra uniform continuity, the existence and discontinuity claims rest on an unverified step. No circularity appears in the abstract, and the results are derived from the contraction and completeness rather than fitted parameters. This work is aimed at researchers in metric fixed point theory who track generalizations of classical theorems and open problems like Rhoades'. A reader already familiar with quasi-contractions would get value from the new setting and the weaker continuity replacements. It deserves a serious referee because the claims are specific, the setting is distinct, and the open-problem angle makes it worth checking the details rather than desk-rejecting outright. I would send it for review.

Referee Report

3 major / 3 minor

Summary. The paper generalizes fixed point theorems for convex contractions of order m on complete suprametric spaces. It claims these mappings admit fixed points without necessarily being continuous at the fixed point, and that weaker conditions (k-continuity or T-orbitally lower semi-continuity) suffice in place of continuity. This is presented as a new solution to Rhoades' 1988 open problem. The paper also extends results of Sehgal, Ćirić, and Fisher on quasi-contractions to the suprametric setting, with examples and an application.

Significance. If the proofs are correct, the work identifies a class of mappings in generalized metric spaces that possess fixed points while remaining discontinuous at those points, extending classical fixed-point theory and providing a distinct approach to Rhoades' problem. The generalizations of quasi-contraction theorems and the identification of weaker continuity conditions could broaden the scope of fixed-point results in non-standard metric settings.

major comments (3)

[Proof of the main existence theorem (convex contraction of order m)] In the proof that the Picard iterates {x_n} form a Cauchy sequence (main existence theorem, following the convex contraction inequality of order m): the repeated application of the suprametric triangle inequality must be shown to produce a telescoping bound of the form d(x_n, x_m) ≤ C r^n /(1-r) with r < 1. The relaxed form of the triangle inequality (whatever constant or supremum it involves) may inflate the effective ratio above 1, so the standard geometric-series argument does not automatically carry over; this step is load-bearing for both existence and the subsequent discontinuity claim.
[Example demonstrating discontinuity] The example constructed to show discontinuity at the fixed point (the concrete mapping satisfying the order-m convex contraction but failing continuity at the fixed point): it must be verified that the mapping obeys the exact contraction inequality while the underlying space satisfies the suprametric axioms and completeness. If the example inadvertently forces continuity or violates the relaxed triangle inequality, it does not resolve Rhoades' problem in the claimed way.
[Quasi-contraction generalizations] The generalizations of Sehgal, Ćirić, and Fisher quasi-contractions (later theorems): each proof must explicitly track how the suprametric inequality replaces the ordinary triangle inequality when deriving the Cauchy property and the fixed-point conclusion; any hidden uniform-continuity assumption on the space would undermine the claim that the results are direct generalizations.

minor comments (3)

[Abstract] Abstract: 'approved our results' should read 'support our results'.
[Preliminaries] The definition of k-continuity and T-orbitally lower semi-continuity should appear explicitly in the preliminaries before being invoked in the main theorems.
[Definition of convex contraction] Notation for the convex contraction of order m (the precise inequality involving the m-fold composition or convex combination) should be stated once in a numbered display equation and then referenced consistently.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and for pinpointing these key points that merit clarification. We address each major comment below and will make the indicated revisions to improve the rigor and transparency of the proofs and example.

read point-by-point responses

Referee: In the proof that the Picard iterates {x_n} form a Cauchy sequence (main existence theorem, following the convex contraction inequality of order m): the repeated application of the suprametric triangle inequality must be shown to produce a telescoping bound of the form d(x_n, x_m) ≤ C r^n /(1-r) with r < 1. The relaxed form of the triangle inequality (whatever constant or supremum it involves) may inflate the effective ratio above 1, so the standard geometric-series argument does not automatically carry over; this step is load-bearing for both existence and the subsequent discontinuity claim.

Authors: We appreciate the referee drawing attention to this load-bearing step. The convex contraction of order m employs a convex combination whose weights sum to a factor strictly less than 1; when combined with the fixed multiplicative constant arising from the suprametric inequality, the composite ratio remains strictly less than 1. We will revise the proof to display the explicit inductive bound, incorporating the suprametric constant at each iteration and verifying that the resulting geometric ratio is less than 1. This explicit calculation will also support the subsequent claim that continuity at the fixed point is not required. revision: yes
Referee: The example constructed to show discontinuity at the fixed point (the concrete mapping satisfying the order-m convex contraction but failing continuity at the fixed point): it must be verified that the mapping obeys the exact contraction inequality while the underlying space satisfies the suprametric axioms and completeness. If the example inadvertently forces continuity or violates the relaxed triangle inequality, it does not resolve Rhoades' problem in the claimed way.

Authors: The example was constructed so that the mapping satisfies the precise order-m convex contraction inequality on a complete suprametric space whose relaxed triangle inequality holds with a constant strictly greater than 1. Direct verification confirms both the contraction condition and discontinuity at the fixed point. To remove any doubt, we will expand the example section in the revision with explicit numerical checks of the inequality for representative points, confirmation of the suprametric axiom, and a proof of completeness. revision: yes
Referee: The generalizations of Sehgal, Ćirić, and Fisher quasi-contractions (later theorems): each proof must explicitly track how the suprametric inequality replaces the ordinary triangle inequality when deriving the Cauchy property and the fixed-point conclusion; any hidden uniform-continuity assumption on the space would undermine the claim that the results are direct generalizations.

Authors: We agree that explicit tracking is necessary. In each of the quasi-contraction proofs the suprametric inequality is substituted directly for the classical triangle inequality at the steps where distance estimates are summed; the resulting bounds remain valid without invoking any form of uniform continuity of the space. We will revise these proofs to insert brief remarks or annotations at each substitution, making the replacement transparent and confirming the absence of hidden continuity assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained from contraction and completeness

full rationale

The paper establishes fixed-point existence for convex contractions of order m on complete suprametric spaces by constructing the Picard iterates, applying the contraction inequality to bound distances, and invoking the relaxed triangle inequality plus completeness to conclude that the sequence is Cauchy and converges to a fixed point. This is a direct proof from the stated axioms and does not reduce any step to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation. The subsequent claims that continuity at the fixed point is not forced (and can be weakened to k-continuity or T-orbital lower semi-continuity) are shown via explicit counter-examples rather than by construction. Generalizations of Sehgal, Ćirić and Fisher results follow the same pattern. No quoted equation or definition collapses the claimed result into its own input.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of suprametric space (relaxed triangle inequality), completeness, and the specific form of convex contraction of order m. No free parameters or invented entities are mentioned.

axioms (2)

domain assumption Suprametric space satisfies a relaxed triangle inequality sufficient for Cauchy sequences to converge.
Invoked to ensure the iterative sequence converges to the fixed point.
domain assumption The mapping satisfies the convex contraction inequality of order m.
This is the load-bearing contraction condition used to prove existence.

pith-pipeline@v0.9.0 · 5454 in / 1420 out tokens · 29366 ms · 2026-05-11T00:54:05.562866+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references

[1]

Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations in- teg´ erales, Fundamenta Mathematicae, 3 (1922), 132–181

S. Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations in- teg´ erales, Fundamenta Mathematicae, 3 (1922), 132–181

1922
[2]

Berzig, First Results in Suprametric Spaces with Applications, Mediterr

M. Berzig, First Results in Suprametric Spaces with Applications, Mediterr. J. Math. (2022) 19:226

2022
[3]

R. K. Bisht, N. Ozgur, Discontinuous convex contraction and their applications in neural networks, Computational and applied mathematics, 1–11 (2021)

2021
[4]

R. K. Bisht, V. Rakoˇ cevi´ c, Fixed points of convex and generalized convex contractions, Rendiconti del Circolo Matematico di Palermo Series 2, 69 (2018), 21–28

2018
[5]

R. K. Bisht, N. K. Singh, V. Rakoˇ cevi´ c, B. Fisher, On discontinuity at fixed point via power quasi- contraction, Publ. de L’Institut Math., 108(122), (2020) 5–11

2020
[6]

Lj. B. ´Ciri´ c, On Sehgal’s map with a contractive iterate at a point, Publ. Inst. Math.33(1983), 59–62

1983
[7]

Fisher, Quasi-contractions on metric spaces, Proc

B. Fisher, Quasi-contractions on metric spaces, Proc. Amer. Math. Soc. , 75 (2), (1979) 321–325

1979
[8]

L. F. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 26, (1970) 615–618

1970
[9]

V. I. Istrˇ atescu, Some fixed point theorems for convex contraction mappings and nonexpansive mapping . I, Libertas Math. , 1, (1981), 151–163

1981
[10]

V. I. Istrˇ atescu, Fixed Point Theory, An Introduction. Mathematics and Its Applications, vol. 7, D. Reidel Publishing Company, Dordrecht,1981. 16

1981
[11]

B. E. Rhoades, Contractive definitions and continuity. Contemp. Math. 72, (1988), 233–245

1988
[12]

Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc

V.M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Amer. Math. Soc. 23(1969), 631–634

1969
[13]

Suzuki, W

T. Suzuki, W. Takahashi, Fixed point theorems and characterizations of metric completeness, Top. Methods in Nonlinear Anal., J. of the Juliusz Schauder Center, (1996), 371-382

1996
[14]

Miandaragh et al., Generalized convex contractions, Math

M.A. Miandaragh et al., Generalized convex contractions, Math. Comput. Modelling 57 (2013) 2547–2552. 17

2013