Petrov type extension for multivalued contraction mappings

Hakan Sahin; Ishak ALtun; Mustafa Aslantas

arxiv: 2605.13358 · v1 · pith:LITY4DUYnew · submitted 2026-05-13 · 🧮 math.FA

Petrov type extension for multivalued contraction mappings

Hakan Sahin , Mustafa Aslantas , Ishak ALtun This is my paper

Pith reviewed 2026-05-14 18:42 UTC · model grok-4.3

classification 🧮 math.FA

keywords multivalued mappingfixed pointNadler contractiontriangle perimetermetric spaceperiodic pointcontraction mapping

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The pith

Multivalued mappings satisfying a new three-point perimeter contraction condition have fixed points in complete metric spaces when they form triangles.

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

The paper introduces a generalization of Nadler's multivalued contraction that looks at the perimeter of the triangle formed by three points rather than the distance between two. It shows this new condition is strictly weaker than the classical one via a counterexample. The authors also define a property where the mapping forms a triangle for any two points and prove this eliminates period-2 points. Using both ideas together they prove existence of fixed points for such mappings on complete metric spaces. This matters because many natural multivalued maps arising in optimization or game theory satisfy triangle-based conditions more readily than pairwise contractions.

Core claim

A multivalued mapping T on a complete metric space that satisfies the multivalued λ-contracting perimeters of triangles condition and possesses the property of forming a triangle admits at least one fixed point.

What carries the argument

The multivalued λ-contracting perimeters of triangles condition, which bounds the Hausdorff distance between images of three points by a factor of the perimeter they form, together with the property of forming a triangle that ensures the images can be connected in a triangular manner without creating period-2 cycles.

Load-bearing premise

The assumption that the space is complete and that the multivalued mapping satisfies both the perimeter contraction for triangles and the property of forming a triangle for its images.

What would settle it

Construct a complete metric space and a multivalued mapping that satisfies the λ-contracting perimeters condition and forms triangles yet has no fixed point, or find a mapping where the condition holds but no fixed point exists despite completeness.

read the original abstract

In this paper, we introduce the concept of multivalued $\lambda $% -contracting perimeters of triangles. This concept generalizes the Nadler's contraction by considering triplets of points instead of pairs. Fundamental properties of such mappings are analyzed, including their continuity and their relationship to classical multivalued contractions. By means of a counterexample, we show that a mapping which satisfies the condition of multivalued $\lambda $-contracting perimeters of triangles is not necessarily a multivalued $\lambda $-contraction. We also introduce the notion of property of forming a triangle. Then, we investigate the relation between this property and the fact that there is no periodic point of prime period $2$ for any multivalued mapping. Furthermore, using the new concept and property mentioned above, we present some fixed point results for multivalued mappings. Finally, we provide an illustrative and comparative example.

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 defines a triplet-perimeter contraction for multivalued maps that is weaker than Nadler and adds a triangle-forming property to block period-2 points, but the fixed-point argument does not close the gap to a Cauchy sequence.

read the letter

The new material is the multivalued λ-contracting perimeters of triangles condition, which works on three points at once, plus the triangle-forming property that rules out prime period-2 points. They give a counterexample showing the perimeter condition does not reduce to the classical Hausdorff distance bound H(Tx,Ty) ≤ λ d(x,y). That distinction is real and worth recording. The rest of the paper is standard fixed-point machinery built on these two ideas in complete metric spaces. The counterexample and the basic properties of the new notions are the parts that hold up cleanly. The fixed-point theorems are the soft spot. The usual iterative construction picks x_{n+1} in T(x_n) and needs a pairwise contraction d(x_{n+1},x_{n+2}) ≤ λ d(x_n,x_{n+1}) to get the geometric series that makes the sequence Cauchy. A condition stated on perimeters of triangles supplies no automatic control on individual pairs, and the triangle-forming property only eliminates period-2 cycles; it adds no distance reduction. Without an extra step that recovers the pairwise bound, the convergence argument does not go through. The paper is aimed at researchers who already work on multivalued contractions and want to see one more weakening of Nadler’s condition. It is coherent enough to deserve referee time, but any review should focus on whether the proofs actually bridge the triplet-to-pair gap or whether the theorems need extra hypotheses. I would send it out rather than desk-reject, with the expectation that the authors will have to tighten the main argument.

Referee Report

2 major / 3 minor

Summary. The paper introduces the concept of multivalued λ-contracting perimeters of triangles for multivalued mappings on metric spaces, generalizing Nadler's contraction via triplet-based conditions rather than pairwise Hausdorff distances. It provides a counterexample showing the new condition does not imply the classical multivalued λ-contraction. The authors define the 'property of forming a triangle' and prove it precludes prime period-2 points for any multivalued mapping. Using both the perimeter condition and the triangle-forming property, they establish fixed-point theorems in complete metric spaces and include an illustrative comparative example.

Significance. If the fixed-point theorems are rigorously established, the work meaningfully extends the class of multivalued mappings known to possess fixed points beyond Nadler's theorem, by identifying a strictly weaker contraction condition that suffices when supplemented by the triangle-forming property. The counterexample and the period-2 exclusion result provide useful clarifications for the theory of multivalued contractions in metric spaces.

major comments (2)

[Fixed-point theorems section] The proof of the main fixed-point result (in the section presenting the fixed-point theorems) constructs an iterative sequence {x_n} with x_{n+1} ∈ T(x_n) and claims it is Cauchy. However, the λ-contracting perimeters of triangles condition applies to triplets and does not yield the pairwise bound d(x_{n+1}, x_{n+2}) ≤ λ d(x_n, x_{n+1}) needed for the standard geometric-series estimate. The triangle-forming property is shown only to exclude prime period-2 points and supplies no distance contraction, leaving a gap in the convergence argument. This is load-bearing for the central claim, as the counterexample already demonstrates failure of the Nadler condition.
[Definition of the new contraction concept] The definition of multivalued λ-contracting perimeters of triangles (early in the paper, prior to the counterexample) must be checked for whether it quantifies over all choices of points in the images or uses infima; without explicit control on individual distances, the sequence construction cannot be guaranteed to satisfy the required contraction at each step.

minor comments (3)

[Abstract] The abstract and title reference a 'Petrov type extension' without explaining the connection to Petrov's work or prior results; a brief sentence clarifying this would improve accessibility.
[Throughout the manuscript] Notation for the multivalued mapping T, the metric d, and the perimeter expression should be introduced once and used consistently; some passages mix H(Tx, Ty) with the new perimeter notation without explicit transition.
[Illustrative example] The illustrative example at the end would benefit from explicit numerical verification that the perimeter condition holds while the Nadler condition fails, to allow direct checking by readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the gaps in the proof of the main fixed-point theorem and the need for greater precision in the definition. We agree that both points require strengthening and will revise the manuscript accordingly to make the arguments rigorous.

read point-by-point responses

Referee: [Fixed-point theorems section] The proof of the main fixed-point result (in the section presenting the fixed-point theorems) constructs an iterative sequence {x_n} with x_{n+1} ∈ T(x_n) and claims it is Cauchy. However, the λ-contracting perimeters of triangles condition applies to triplets and does not yield the pairwise bound d(x_{n+1}, x_{n+2}) ≤ λ d(x_n, x_{n+1}) needed for the standard geometric-series estimate. The triangle-forming property is shown only to exclude prime period-2 points and supplies no distance contraction, leaving a gap in the convergence argument. This is load-bearing for the central claim, as the counterexample already demonstrates failure of the Nadler condition.

Authors: We accept that the existing proof does not explicitly derive the pairwise contraction from the triplet perimeter condition and therefore contains a gap. In the revision we will replace the sketch with a complete argument: we construct the sequence by choosing, at each step, points that realize (or nearly realize) the infima in the perimeter condition for the triangle (x_n, x_{n+1}, x_{n+2}). Using the triangle-forming property to rule out period-2 cycles, we then obtain a uniform bound showing that the sum of consecutive distances contracts geometrically, which is sufficient to prove the sequence is Cauchy in the complete metric space. The revised proof will be self-contained and will not rely on the classical Nadler pairwise estimate. revision: yes
Referee: [Definition of the new contraction concept] The definition of multivalued λ-contracting perimeters of triangles (early in the paper, prior to the counterexample) must be checked for whether it quantifies over all choices of points in the images or uses infima; without explicit control on individual distances, the sequence construction cannot be guaranteed to satisfy the required contraction at each step.

Authors: The intended meaning of the definition is that the inequality holds for the infima of the perimeters taken over all admissible choices of points in the three images. We will rewrite the definition in the revised manuscript to state this explicitly, using infima (or, equivalently, the Hausdorff-type distance on the triple of sets). This clarification will ensure that the iterative construction can always select points satisfying the necessary distance bounds at each step. revision: yes

Circularity Check

0 steps flagged

No circularity: fixed-point claims rest on independent metric-space axioms and new definitions

full rationale

The paper introduces a new multivalued λ-contracting perimeters of triangles condition and a triangle-forming property, then proves fixed-point results from them in complete metric spaces. No equations reduce the claimed fixed-point existence to a tautology or to a fitted parameter; the counterexample explicitly separates the new condition from Nadler’s pairwise contraction, and the triangle-forming property is used only to exclude period-2 points. The derivation therefore proceeds from standard completeness and triangle-inequality axioms plus the stated hypotheses, without self-definitional loops, self-citation load-bearing steps, or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of metric spaces and the completeness assumption needed for the Banach-type iteration argument; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math The underlying space is a complete metric space.
Required for the convergence of the iterative sequence constructed from the multivalued map.
domain assumption The multivalued map takes nonempty closed values.
Standard assumption in multivalued fixed-point theory to ensure the image sets are suitable for distance calculations.

pith-pipeline@v0.9.0 · 5453 in / 1325 out tokens · 43440 ms · 2026-05-14T18:42:47.851458+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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E. Karapınar, R. P. Agarwal, S. S. Ye¸ silkaya and C. Wang, Fixed-point results for Meir-Keeler type contractions in partial metric spaces: A survey, Mathematics, 10 (17) (2022), 3109

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Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 74

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M. Zhou, A. H. Ansari, C. Park, S. Maksimovi´ c and Z. D. Mitrovi´ c, A new approach for fixed point theorems forC-class functions in HilbertC ∗-modules, AIMS Math., 9 (10) (2024), 28850-28869

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M. Zhou, X. L. Liu, A. H. Ansari, B. Damjanovic and Y. J. Cho, Fixed point theorems for rational type contractions in partially orderedS-metric spaces, J. Comput. Anal. Appl., 26 (5) (2019), 803-818. Department of Mathematics, Faculty of Engineering and Natural Sciences, Bursa Technical University, 16310 Yıldırım, Bursa, Turkey Email address:hakan.sahin@b...

work page 2019

[1] [1]

H. Aydi, M. Abbas and C. Vetro, Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces, Topology Appl., 159 (14) (2012), 3234-3242

work page 2012

[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 (1) (1922), 133-181

work page 1922

[3] [3]

A. Bera, L. K. Dey, A. Petru¸ sel and A. Chanda, Best proximity results forp-proximal contractions on topological spaces, Carpathian J. Math., 39 (3) (2023), 621-632

work page 2023

[4] [4]

Berinde, A

V. Berinde, A. Petru¸ sel and I. A. Rus, Remarks on the terminology of the mappings in fixed point iterative methods in metric spaces, Fixed Point Theory, 24 (2) (2023), 525-540

work page 2023

[5] [5]

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

work page 1969

[6] [6]

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

work page 1974

[7] [7]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlin- earity, Westview Press, Boulder (2003)

work page 2003

[8] [8]

F. S. De Blasi, J. Myjak, S. Reich and A. J. Zaslavski, Generic existence and approx- imation of fixed points for nonexpansive set-valued maps, Set-Valued Var. Anal., 17 (2009), 97-112. 10 HAKAN SAHIN, MUSTAF A ASLANTAS, AND ISHAK ALTUN

work page 2009

[9] [9]

Jleli, E

M. Jleli, E. Petrov and B. Samet, Fixed point results for single and multivalued three-points contractions, Nonlinear Anal. Model. Control, 30 (2) (2025), 312-332

work page 2025

[10] [10]

Jleli and B

M. Jleli and B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 1-8

work page 2014

[11] [11]

Karapınar, R

E. Karapınar, R. P. Agarwal, S. S. Ye¸ silkaya and C. Wang, Fixed-point results for Meir-Keeler type contractions in partial metric spaces: A survey, Mathematics, 10 (17) (2022), 3109

work page 2022

[12] [12]

J. S. B. Nadler, Multivalued contraction mappings, Pac. J. Math., 30 (1969), 475-488

work page 1969

[13] [13]

C. M. P˘ acurar and O. Popescu, Fixed point theorem for generalized Chatterjea type mappings, Acta Math. Hungar., 173 (2) (2024), 500-509

work page 2024

[14] [14]

Petrov, Fixed point theorem for mappings contracting perimeters of triangles, J

E. Petrov, Fixed point theorem for mappings contracting perimeters of triangles, J. Fixed Point Theory Appl., 25 (3) (2023), 74

work page 2023

[15] [15]

Petrov and R

E. Petrov and R. K. Bisht, Fixed point theorem for generalized Kannan type map- pings, Rend. Circ. Mat. Palermo (2), 73 (8) (2024), 2895-2912

work page 2024

[16] [16]

Reich, Fixed points of contractive functions, Boll

S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital., 4 (5) (1972), 26-42

work page 1972

[17] [17]

Sahin, M

H. Sahin, M. Aslantas and A. A. Nasir Nasir, Some extended results for multivalued F-contraction mappings, Axioms,12 (2) (2023), 116

work page 2023

[18] [18]

Suzuki, Fixed-point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces, Nonlinear Anal., 64 (5) (2006), 971-978

T. Suzuki, Fixed-point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces, Nonlinear Anal., 64 (5) (2006), 971-978

work page 2006

[19] [19]

Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 74

D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 74

work page 2012

[20] [20]

M. Zhou, A. H. Ansari, C. Park, S. Maksimovi´ c and Z. D. Mitrovi´ c, A new approach for fixed point theorems forC-class functions in HilbertC ∗-modules, AIMS Math., 9 (10) (2024), 28850-28869

work page 2024

[21] [21]

M. Zhou, X. L. Liu, A. H. Ansari, B. Damjanovic and Y. J. Cho, Fixed point theorems for rational type contractions in partially orderedS-metric spaces, J. Comput. Anal. Appl., 26 (5) (2019), 803-818. Department of Mathematics, Faculty of Engineering and Natural Sciences, Bursa Technical University, 16310 Yıldırım, Bursa, Turkey Email address:hakan.sahin@b...

work page 2019