T Extended Weakly Contractive, Kannan, and Geraghty Mappings Fixed Points, Equivalences,

Fatemeh Fogh; Sara Behnamian

arxiv: 2604.23400 · v1 · submitted 2026-04-25 · 🧮 math.FA

T Extended Weakly Contractive, Kannan, and Geraghty Mappings Fixed Points, Equivalences,

Fatemeh Fogh , Sara Behnamian This is my paper

Pith reviewed 2026-05-08 06:54 UTC · model grok-4.3

classification 🧮 math.FA MSC 47H10

keywords T-extended mappingsweakly contractive mappingsGeraghty mappingsKannan mappingsfixed point theoremsmetric spacesPicard convergence

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The pith

T-extended weakly contractive maps coincide exactly with T-extended Geraghty maps through reduction to an induced map on T(X)

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

The paper develops a unified T-extended framework for weakly contractive, weakly Kannan, and Geraghty self-maps S on a metric space, where distances are taken between T-images and iterations proceed via the composition T and S. It proves that the T-extended weakly contractive class is identical to the T-extended Geraghty class, and that the T-extended weakly Kannan class coincides with the T-extended Kannan-Geraghty class. The mechanism is the transport of the problem to an induced map F on the image space T(X), defined by F(Tx) = T(Sx), under which the extended conditions reduce directly to the classical ones with the same control functions. Fixed point theorems, Picard convergence, and a Delta-type ratio criterion then follow from the classical theory. The results hold under standard assumptions on T and extend to rectangular metric spaces, with examples including Volterra operators.

Core claim

Under the standard assumptions on the auxiliary map T, the T-extended weakly contractive mappings coincide with the T-extended Geraghty mappings, and the T-extended weakly Kannan mappings coincide with the T-extended Kannan-Geraghty mappings, because both sets of conditions reduce exactly to the corresponding classical conditions when the dynamics is rewritten in terms of the induced map F from T(X) to T(X) defined by F(Tx) = T(Sx).

What carries the argument

The induced map F: T(X) to T(X) given by F(Tx) = T(Sx), which carries every T-extended contraction property of S back to a classical contraction property of F with the same control function.

If this is right

Fixed point existence, uniqueness, and Picard convergence hold for each of the four equivalent classes.
A Delta-type ratio criterion on T(X) controls the convergence of the iterates.
Quantitative rates for the Picard sequence are obtained directly from the classical case.
All theorems and the equivalences extend verbatim to rectangular metric spaces.

Where Pith is reading between the lines

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

The same reduction technique could establish equivalences between other families of contractive mappings by transporting them to the image of T.
The Volterra operator examples indicate that the framework may give uniform fixed-point results for certain classes of integral operators.
Relaxing injectivity or continuity of T while retaining the equivalences would enlarge the range of applicable auxiliary maps.

Load-bearing premise

The auxiliary map T must be continuous and injective, with the iterates of the composition admitting convergent subsequences.

What would settle it

A concrete metric space X, continuous injective T, and self-map S such that S satisfies the T-extended weakly contractive inequality but fails the T-extended Geraghty inequality would falsify the claimed equivalence of the two classes.

read the original abstract

We develop a unified T-extended framework for weakly contractive, weakly Kannan, and Geraghty classes of self-maps S on a metric space (X, d), where distances are measured on the auxiliary image via d(Tx, Ty), and the dynamics is governed by the composition of T and S. Under standard assumptions on the auxiliary map T (continuity, injectivity, subsequential convergence), fixed point theorems and Picard convergence are established for each class. The main contribution is twofold. First, it is shown that the T-extended weakly contractive class coincides with the T-extended Geraghty class, and that the T-extended weakly Kannan class coincides with the T-extended Kannan-Geraghty class. Second, the mechanism behind these equivalences is clarified by transporting the problem to an induced map F from T(X) to T(X), defined by F(Tx) = T(Sx), where the extended properties reduce exactly to the classical ones with the same control functions. A Delta-type ratio criterion on T(X) and quantitative Picard convergence rates are also provided. Examples, including Volterra smoothing operators, are presented to highlight the role of the auxiliary map. All results extend naturally to rectangular (Branciari) metric spaces.

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 claimed coincidences between T-extended classes collapse to a false statement about classical weakly contractive and Geraghty maps when T is the identity.

read the letter

The main thing here is that the paper asserts the T-extended weakly contractive class equals the T-extended Geraghty class (and likewise for the Kannan variants), but the reduction they use makes this impossible in general. They introduce an auxiliary map T that is continuous, injective, and has a subsequential convergence property. Distances in the contraction conditions are taken as d(Tx, Ty), and they define an induced self-map F on the image T(X) by F(Tx) = T(Sx). Under this transport, each T-extended condition on S becomes exactly the corresponding classical condition on F with the same control function. Fixed-point theorems and Picard convergence then follow from the classical results, and they add a Delta-type ratio test plus quantitative rates. Examples include Volterra smoothing operators, and the results carry over to rectangular metric spaces. That framework and the concrete examples are the parts that actually work. The transport idea is clean and lets them recycle known theorems without much new machinery. The soft spot is the equivalence claim itself. Setting T to the identity satisfies every listed assumption on T, so T(X) becomes X and F becomes S. The T-extended classes therefore reduce exactly to the ordinary weakly contractive and Geraghty classes. Those two classes are distinct: there exist maps and control functions that satisfy one but not the other. The reduction therefore cannot produce coincidence unless the paper imposes extra, unstated restrictions on the control functions or on T. No such restrictions appear in the mechanism described. The fixed-point theorems for each class separately may still be valid, but the unification via equivalence does not hold. This paper is for people already working inside metric fixed-point theory who want to see how an auxiliary map can move the problem to a subspace. A reader looking for new unification results will be disappointed by the gap. It does not deserve a serious referee because the central claim contradicts a basic fact about the classical setting it reduces to.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a T-extended framework for weakly contractive, weakly Kannan, and Geraghty self-maps S on a metric space (X,d), with distances measured via an auxiliary map T. Under assumptions of continuity, injectivity, and subsequential convergence on T, it establishes fixed-point theorems and Picard convergence for each class. The central claims are that the T-extended weakly contractive class coincides with the T-extended Geraghty class, and the T-extended weakly Kannan class coincides with the T-extended Kannan-Geraghty class; these equivalences are obtained by transporting to an induced map F: T(X) → T(X) given by F(Tx) = T(Sx), under which the extended conditions reduce exactly to the classical ones with the same control functions. Additional results include a Delta-type ratio criterion on T(X), quantitative Picard rates, examples with Volterra operators, and natural extensions to rectangular (Branciari) metric spaces.

Significance. If the equivalences and fixed-point results hold without hidden restrictions, the work offers a unifying transport mechanism that embeds several generalized contraction classes into a common T-extended setting, allowing classical theorems to be applied directly on the image T(X). The explicit reduction via the induced map F, together with quantitative convergence rates and concrete examples such as Volterra smoothing operators, would provide a practical organizing principle for fixed-point theory involving auxiliary maps. The extension to Branciari spaces further broadens applicability.

major comments (1)

[Abstract and main equivalence results] Abstract and main equivalence results: The asserted coincidence of the T-extended weakly contractive class with the T-extended Geraghty class (and the analogous Kannan claim) cannot hold in the stated generality. The described mechanism reduces each T-extended property for S exactly to the corresponding classical property for the induced F on T(X). Specializing to the identity map T = id (which satisfies all listed assumptions on T) yields T(X) = X and F = S, so the T-extended classes reduce exactly to the classical weakly contractive and Geraghty classes. These classical classes are distinct: there exist maps satisfying d(Sx,Sy) ≤ d(x,y) − ϕ(d(x,y)) for suitable ϕ but not the Geraghty form d(Sx,Sy) ≤ α(d(x,y))d(x,y) with lim sup α(t_n) < 1 whenever t_n → 0, and vice versa. The reduction therefore implies the classical classes coincide, which they do not. This logical inconsistency is a

minor comments (1)

The abstract refers to a 'Delta-type ratio criterion' and 'quantitative Picard convergence rates' without stating the precise form of the criterion or the rate estimates; these should be displayed explicitly (e.g., as a displayed inequality or theorem statement) in the introduction or results section for immediate readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and detailed reading of the manuscript. The major comment identifies a logical inconsistency in the claimed equivalences between the T-extended classes, which we address directly below.

read point-by-point responses

Referee: Abstract and main equivalence results: The asserted coincidence of the T-extended weakly contractive class with the T-extended Geraghty class (and the analogous Kannan claim) cannot hold in the stated generality. The described mechanism reduces each T-extended property for S exactly to the corresponding classical property for the induced F on T(X). Specializing to the identity map T = id (which satisfies all listed assumptions on T) yields T(X) = X and F = S, so the T-extended classes reduce exactly to the classical weakly contractive and Geraghty classes. These classical classes are distinct: there exist maps satisfying d(Sx,Sy) ≤ d(x,y) − ϕ(d(x,y)) for suitable ϕ but not the Geraghty form d(Sx,Sy) ≤ α(d(x,y))d(x,y) with lim sup α(t_n) < 1 whenever t_n → 0, and vice versa. The reduction therefore implies the classical classes coincide, which they do not. This logical inconsistency is a

Authors: We agree with the referee's reasoning. The transport via the induced map F: T(X) → T(X) given by F(Tx) = T(Sx) shows that S satisfies a T-extended condition if and only if F satisfies the corresponding classical condition on T(X). Because the classical weakly contractive and Geraghty classes are distinct (as the referee correctly notes), the T-extended versions cannot coincide for arbitrary T satisfying the stated assumptions. The same holds for the weakly Kannan and Kannan-Geraghty cases. This constitutes an error in the manuscript's central claim. We will revise the paper by removing all assertions that the T-extended classes coincide. The revised version will instead focus on the reduction mechanism itself: each T-extended fixed-point theorem follows immediately by applying the corresponding classical theorem to F on the metric subspace T(X). We will update the abstract, introduction, main theorems, and examples to reflect this corrected perspective, while retaining the quantitative rates, Delta-type criterion, Volterra examples, and extension to Branciari spaces. revision: yes

Circularity Check

0 steps flagged

No circularity in the derivation; equivalences via explicit transport to induced map

full rationale

The paper defines the T-extended classes explicitly in terms of distances d(Tx, Ty) and the composition TS, then reduces each extended condition for S to the corresponding classical condition for the induced self-map F on T(X) given by F(Tx) = T(Sx). This reduction is a direct, one-to-one transport of the defining inequalities (with the same control functions) and does not redefine any class in terms of itself, rename a fitted parameter as a prediction, or rely on a load-bearing self-citation whose content is unverified. The argument remains self-contained under the stated assumptions on T; no step collapses the claimed coincidence to a tautology or to the input data by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard metric-space axioms plus domain assumptions on the auxiliary map T; no free parameters or invented entities are introduced.

axioms (2)

standard math Metric space axioms (non-negativity, identity of indiscernibles, symmetry, triangle inequality)
Standard background assumed for any metric-space fixed point result.
domain assumption Continuity, injectivity, and subsequential convergence of the auxiliary map T
Explicitly listed in the abstract as required for the theorems and equivalences.

pith-pipeline@v0.9.0 · 5537 in / 1407 out tokens · 67983 ms · 2026-05-08T06:54:47.794482+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Ariza-Ruiz and A

D. Ariza-Ruiz and A. Jim´ enez-Melado, A continuation me thod for weakly Kannan mappings, Fixed Point Theory Appl. (2010), Article ID 321594

work page 2010

[2] [2]

Beiranvand, S

A. Beiranvand, S. Moradi, M. Omid, and H. Pazandeh, Two ﬁx ed-point theorems for special mappings, arXiv:0903.1504 (2009)

work page arXiv 2009

[3] [3]

Branciari, A ﬁxed point theorem of Banach–Caccioppol i type on a class of generalized metric spaces, Publ

A. Branciari, A ﬁxed point theorem of Banach–Caccioppol i type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31–37

work page 2000

[4] [4]

Caballero, J

J. Caballero, J. Harjani, and K. Sadarangani, A best prox imity point theorem for Geraghty contractions, Abstr. Appl. Anal. (2012), Article ID 35017. 13

work page 2012

[5] [5]

Dugundji and A

J. Dugundji and A. Granas, Weakly contractive mappings a nd elemen- tary domain invariance theorem, Bull. Soc. Math. Gr` ece 19 (1978), 141–151

work page 1978

[6] [6]

F. Fogh, S. Behnamian, and F. Pashaie, On Kannan–Geraght y maps as an extension of Kannan maps, Int. J. Maps Math. 2 (2019), no. 1, 1–13

work page 2019

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Fogh and S

F. Fogh and S. Behnamian, Best proximity points for Gerag hty–type non–self mappings, Int. J. Topol. 3 (2026), no. 2, Article 7

work page 2026

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Gabeleh, Existence and uniqueness results for best pr oximity points, Miskolc Math

M. Gabeleh, Existence and uniqueness results for best pr oximity points, Miskolc Math. Notes 16 (2015), 123–131

work page 2015

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Geraghty, On contractive mappings, Proc

M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604–608

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Kannan, Some results on ﬁxed points, Bull

R. Kannan, Some results on ﬁxed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76

work page 1968

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Khojasteh, A

F. Khojasteh, A. Shukla, and S. Radenovi´ c, A new approa ch to the study of ﬁxed point theorems via simulation functions, Filomat 29 (2015), no. 6, 1189–1194

work page 2015

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Moradi, New extensions of Kannan ﬁxed point theorem o n complete metric and generalized metric spaces, Int

S. Moradi, New extensions of Kannan ﬁxed point theorem o n complete metric and generalized metric spaces, Int. J. Math. Anal. 5 (2011), no. 47, 2313–2320

work page 2011

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Samet, C

B. Samet, C. Vetro, and P. Vetro, Fixed point theorems fo r α –ψ con- tractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165

work page 2012

[14] [14]

Wardowski, Fixed points of a new type of contractive m appings in complete metric spaces, Fixed Point Theory Appl

D. Wardowski, Fixed points of a new type of contractive m appings in complete metric spaces, Fixed Point Theory Appl. 2012 (2012), Article 94

work page 2012

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Behnamian, F

S. Behnamian, F. Fogh, F. Pashaei, and N. Konstantis, So me notes on Boyd–Wong ﬁxed point theorem, Fixed Point Theory Appl. (2020), Article 94. . 14

work page 2020