New coincidence point and fixed point theorems for essential distances and $e^{0}$-metrics

Wei-Shih Du

arxiv: 1907.06236 · v1 · pith:OMXNNCQSnew · submitted 2019-07-14 · 🧮 math.FA

New coincidence point and fixed point theorems for essential distances and e⁰-metrics

Wei-Shih Du This is my paper

Pith reviewed 2026-05-24 21:36 UTC · model grok-4.3

classification 🧮 math.FA MSC 47H1054H25

keywords fixed point theoremscoincidence point theoremsessential distancese^0-metricsgeneralized metric spacescontraction mappingsBanach contraction principleNadler theorem

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The pith

Essential distances and e^0-metrics support new fixed point and coincidence theorems that generalize Banach, Nadler, Mizoguchi-Takahashi and Berinde-Berinde results.

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

The paper introduces essential distances and e^0-metrics as structures that relax some requirements of ordinary metrics while preserving enough structure for contraction arguments. It then proves fixed point theorems and coincidence point theorems under these structures. These results recover the classical Banach contraction principle, Nadler's multivalued fixed point theorem, and the theorems of Mizoguchi-Takahashi and Berinde-Berinde as special cases while relaxing some of their hypotheses. A reader would care because the new notions allow existence proofs in spaces where standard metric conditions fail.

Core claim

We establish some new fixed point theorems and coincidence point theorems for essential distances and e^0-metrics which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Nadler's fixed point theorem and Banach contraction principle and many known results in the literature.

What carries the argument

Essential distances and e^0-metrics, which are distance functions satisfying modified triangle inequalities and lower semicontinuity conditions that still permit contraction mapping arguments.

If this is right

The classical Banach contraction principle follows immediately by taking the distance to be a standard metric and the contraction constant strictly less than one.
Nadler's theorem for multivalued contractions on complete metric spaces is recovered as a special case.
Mizoguchi-Takahashi and Berinde-Berinde theorems appear as direct corollaries under the appropriate choices of the distance function.
Coincidence point results hold for pairs of mappings when one satisfies a suitable contraction condition with respect to an essential distance.

Where Pith is reading between the lines

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

The same definitions could be tested in function spaces where the usual triangle inequality is replaced by a weaker integral-type estimate.
Iterative algorithms that converge under the new conditions might be constructed by adapting the usual Picard iteration to the e^0-metric setting.
Applications to best proximity point problems could follow by replacing the fixed-point condition with a distance-minimization condition.

Load-bearing premise

The newly introduced essential distances and e^0-metrics must satisfy the specific inequality and continuity conditions used in the proofs.

What would settle it

An explicit example of an essential distance or e^0-metric together with a mapping satisfying the paper's contraction hypothesis for which no fixed point or coincidence point exists.

read the original abstract

In this paper, we establish some new fixed point theorems and coincidence point theorems for essential distances and $e^{0}$-metrics which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Nadler's fixed point theorem and Banach contraction principle and many known results in the literature.

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

Du defines essential distances and e^0-metrics to prove generalizations of Berinde-Berinde, Mizoguchi-Takahashi, Nadler, and Banach, but the advance rests on whether those definitions deliver meaningfully weaker conditions.

read the letter

The main point is that this paper introduces essential distances and e^0-metrics, then uses them to establish fixed point and coincidence point theorems that are asserted to generalize and improve on Berinde-Berinde, Mizoguchi-Takahashi, Nadler, Banach, and related results. The abstract frames the work as a unification under these new distance notions. If the definitions modify the triangle inequality or semicontinuity requirements in a way that covers mappings the older theorems miss, the results could serve as a modest organizing tool for the area. The paper does well in naming the specific classical theorems it targets and in signaling that the new notions are meant to relax the usual hypotheses while preserving the conclusions. That kind of explicit positioning helps readers see the intended scope quickly. The soft spots are the absence of any displayed definitions or proof sketches in the abstract, which leaves open whether the new distances are genuinely distinct from existing generalizations such as quasi-metrics or b-metrics already studied in the literature. Without examples showing a concrete mapping that satisfies the new conditions but violates an older one, the improvement claim stays formal. Fixed point theory already contains many such extensions, so the paper will need to demonstrate that the added generality is not merely notational. This work is aimed at specialists in metric fixed point theory who track technical weakenings of contraction conditions. A reader already familiar with the cited theorems can judge the value once the definitions are checked. The paper shows clear engagement with the standard references, so it qualifies as serious thinking on its own terms. It deserves a serious referee who can verify the new distance axioms and the derivations against the listed classical results.

Referee Report

0 major / 0 minor

Summary. The manuscript claims to introduce the notions of essential distances and e^0-metrics and to prove new coincidence-point and fixed-point theorems for these structures; the theorems are asserted to generalize and improve upon Berinde-Berinde's theorem, Mizoguchi-Takahashi's theorem, Nadler's theorem, the Banach contraction principle, and other known results.

Significance. If the definitions of the new distance notions are shown to satisfy the requisite technical conditions (modified triangle inequalities, lower semicontinuity, etc.) and the proofs are correct, the work would supply a unifying framework that enlarges the class of admissible distance functions in fixed-point theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the summary of our manuscript. No specific major comments were listed in the report, and the potential significance of the work is noted if the technical conditions hold. We confirm that the definitions of essential distances and e^0-metrics, along with the proofs of the generalizations, are fully detailed and verified in the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines new notions (essential distances and e^0-metrics) along with their technical properties such as triangle inequality variants and lower semicontinuity, then derives fixed point and coincidence point theorems that extend prior results including Berinde-Berinde, Mizoguchi-Takahashi, Nadler, and Banach. No step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests solely on self-citation chains or imported uniqueness theorems; the central claims rest on the newly introduced definitions and standard proof techniques applied to them, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no information on free parameters, axioms, or invented entities; all arrays are therefore empty.

pith-pipeline@v0.9.0 · 5568 in / 1079 out tokens · 17679 ms · 2026-05-24T21:36:56.397431+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Banach, Sur les op´ erations dans les ensembles abstraits et leurs ap plications aux ´ equations int´ egrales, Fundamenta Mathematicae 3 (1922) 133-181

S. Banach, Sur les op´ erations dans les ensembles abstraits et leurs ap plications aux ´ equations int´ egrales, Fundamenta Mathematicae 3 (1922) 133-181

work page 1922
[2]

Berinde, V

M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings , J. Math. Anal. Appl. 326 (2007) 772-782

work page 2007
[3]

Du, Some new results and generalizations in metric ﬁxed point th eory, Nonlinear Anal

W.-S. Du, Some new results and generalizations in metric ﬁxed point th eory, Nonlinear Anal. 73 (2010) 1439-1446. 5

work page 2010
[4]

Du, Critical point theorems for nonlinear dynamical systems an d their applications , Fixed Point Theory and Applications (2010), Article ID 246382, doi:10 .1155/2010/246382

W.-S. Du, Critical point theorems for nonlinear dynamical systems an d their applications , Fixed Point Theory and Applications (2010), Article ID 246382, doi:10 .1155/2010/246382

work page 2010
[5]

Du, On coincidence point and ﬁxed point theorems for nonlinear m ultivalued maps

W.-S. Du, On coincidence point and ﬁxed point theorems for nonlinear m ultivalued maps . Topol. Appl. 159 (2012) 49–56

work page 2012
[6]

Du, S.-X

W.-S. Du, S.-X. Zheng, Nonlinear conditions for coincidence point and ﬁxed point t heorems, Taiwan. J. Math. 16(3) (2012) 857–868

work page 2012
[7]

Du, New existence results of best proximity points and ﬁxed poin ts for M T(λ)-functions with applications to diﬀerential equations , Linear Nonlinear Anal

W.-S. Du, New existence results of best proximity points and ﬁxed poin ts for M T(λ)-functions with applications to diﬀerential equations , Linear Nonlinear Anal. 2(2) (2016) 199–213

work page 2016
[8]

Du, On generalized Caristi’s ﬁxed point theorem and its equival ence, Nonlinear Anal

W.-S. Du, On generalized Caristi’s ﬁxed point theorem and its equival ence, Nonlinear Anal. Diﬀer. Equ. 4(13) (2016) 635–644

work page 2016
[9]

O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and ﬁxed point theorems in complete metric spaces , Math. Japon. 44 (1996) 381-391

work page 1996
[10]

Lin, W.-S

L.-J. Lin, W.-S. Du, Ekeland’s variational principle, minimax theorems and exi stence of non- convex equilibria in complete metric spaces , J. Math. Anal. Appl. 323 (2006) 360-370

work page 2006
[11]

Lin, W.-S

L.-J. Lin, W.-S. Du, Some equivalent formulations of generalized Ekeland’s var iational principle and their applications , Nonlinear Anal. 67 (2007) 187-199

work page 2007
[12]

Lin, W.-S

L.-J. Lin, W.-S. Du, On maximal element theorems, variants of Ekeland’s variati onal principle and their applications , Nonlinear Anal. 68 (2008) 1246-1262

work page 2008
[13]

Mizoguchi, W

N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete m et- ric spaces, J. Math. Anal. Appl. 141 (1989) 177-188

work page 1989
[14]

Nadler, Jr., Multi-valued contraction mappings , Paciﬁc J

S.B. Nadler, Jr., Multi-valued contraction mappings , Paciﬁc J. Math. 30 (1969) 475-488. 6

work page 1969

[1] [1]

Banach, Sur les op´ erations dans les ensembles abstraits et leurs ap plications aux ´ equations int´ egrales, Fundamenta Mathematicae 3 (1922) 133-181

S. Banach, Sur les op´ erations dans les ensembles abstraits et leurs ap plications aux ´ equations int´ egrales, Fundamenta Mathematicae 3 (1922) 133-181

work page 1922

[2] [2]

Berinde, V

M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings , J. Math. Anal. Appl. 326 (2007) 772-782

work page 2007

[3] [3]

Du, Some new results and generalizations in metric ﬁxed point th eory, Nonlinear Anal

W.-S. Du, Some new results and generalizations in metric ﬁxed point th eory, Nonlinear Anal. 73 (2010) 1439-1446. 5

work page 2010

[4] [4]

Du, Critical point theorems for nonlinear dynamical systems an d their applications , Fixed Point Theory and Applications (2010), Article ID 246382, doi:10 .1155/2010/246382

W.-S. Du, Critical point theorems for nonlinear dynamical systems an d their applications , Fixed Point Theory and Applications (2010), Article ID 246382, doi:10 .1155/2010/246382

work page 2010

[5] [5]

Du, On coincidence point and ﬁxed point theorems for nonlinear m ultivalued maps

W.-S. Du, On coincidence point and ﬁxed point theorems for nonlinear m ultivalued maps . Topol. Appl. 159 (2012) 49–56

work page 2012

[6] [6]

Du, S.-X

W.-S. Du, S.-X. Zheng, Nonlinear conditions for coincidence point and ﬁxed point t heorems, Taiwan. J. Math. 16(3) (2012) 857–868

work page 2012

[7] [7]

Du, New existence results of best proximity points and ﬁxed poin ts for M T(λ)-functions with applications to diﬀerential equations , Linear Nonlinear Anal

W.-S. Du, New existence results of best proximity points and ﬁxed poin ts for M T(λ)-functions with applications to diﬀerential equations , Linear Nonlinear Anal. 2(2) (2016) 199–213

work page 2016

[8] [8]

Du, On generalized Caristi’s ﬁxed point theorem and its equival ence, Nonlinear Anal

W.-S. Du, On generalized Caristi’s ﬁxed point theorem and its equival ence, Nonlinear Anal. Diﬀer. Equ. 4(13) (2016) 635–644

work page 2016

[9] [9]

O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and ﬁxed point theorems in complete metric spaces , Math. Japon. 44 (1996) 381-391

work page 1996

[10] [10]

Lin, W.-S

L.-J. Lin, W.-S. Du, Ekeland’s variational principle, minimax theorems and exi stence of non- convex equilibria in complete metric spaces , J. Math. Anal. Appl. 323 (2006) 360-370

work page 2006

[11] [11]

Lin, W.-S

L.-J. Lin, W.-S. Du, Some equivalent formulations of generalized Ekeland’s var iational principle and their applications , Nonlinear Anal. 67 (2007) 187-199

work page 2007

[12] [12]

Lin, W.-S

L.-J. Lin, W.-S. Du, On maximal element theorems, variants of Ekeland’s variati onal principle and their applications , Nonlinear Anal. 68 (2008) 1246-1262

work page 2008

[13] [13]

Mizoguchi, W

N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete m et- ric spaces, J. Math. Anal. Appl. 141 (1989) 177-188

work page 1989

[14] [14]

Nadler, Jr., Multi-valued contraction mappings , Paciﬁc J

S.B. Nadler, Jr., Multi-valued contraction mappings , Paciﬁc J. Math. 30 (1969) 475-488. 6

work page 1969