Proximal quasi-normal structure and existence of best proximity points

Ali Abkar; Farhad Fouladi

arxiv: 1907.10656 · v1 · pith:T52A2WTOnew · submitted 2019-07-24 · 🧮 math.FA

Proximal quasi-normal structure and existence of best proximity points

Farhad Fouladi , Ali Abkar This is my paper

Pith reviewed 2026-05-24 16:27 UTC · model grok-4.3

classification 🧮 math.FA

keywords best proximity pointsproximal quasi-normal structurecyclic mappingscyclic contractionsKannan nonexpansive mappingsorbitally nonexpansive mappingsmetric spaces

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The pith

Proximal quasi-normal structure ensures existence of best proximity points for cyclic mappings and related nonexpansive mappings.

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

The paper applies the proximal quasi-normal structure property of pairs of subsets in metric spaces to prove existence of best proximity points. It covers cyclic mappings, cyclic contractions, relatively Kannan nonexpansive mappings, and orbitally nonexpansive mappings. These conclusions generalize several earlier existence results for such points. A reader would care because best proximity points extend fixed-point ideas to pairs of sets that may not intersect, giving a minimal-distance guarantee under the mapping.

Core claim

If a pair of nonempty subsets in a metric space satisfies proximal quasi-normal structure, then every cyclic mapping, cyclic contraction, relatively Kannan nonexpansive mapping, and orbitally nonexpansive mapping on that pair admits at least one best proximity point.

What carries the argument

Proximal quasi-normal structure (P.Q-N.S.), a geometric property of pairs of subsets that is used to guarantee convergence of iterative sequences to a point realizing the minimal distance between the sets.

If this is right

Cyclic mappings on pairs satisfying proximal quasi-normal structure have best proximity points.
Cyclic contractions on such pairs have best proximity points.
Relatively Kannan nonexpansive mappings on such pairs have best proximity points.
Orbitally nonexpansive mappings on such pairs have best proximity points.
Several earlier existence theorems for best proximity points become special cases of the new results.

Where Pith is reading between the lines

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

The same property might support existence proofs for additional mapping classes beyond those treated here.
Checking proximal quasi-normal structure on concrete pairs in Euclidean or Hilbert spaces could produce new concrete examples.
The property may connect to geometric conditions already used for ordinary fixed points, allowing transfer of techniques between the two settings.

Load-bearing premise

The pair of subsets in the metric space satisfies the proximal quasi-normal structure property.

What would settle it

An explicit pair of closed subsets with proximal quasi-normal structure together with a cyclic contraction that has no best proximity point.

read the original abstract

In this paper, we use the concept of proximal quasi-normal structure (P. Q-N. S) to study the existence of best proximity points for cyclic mappings, cyclic contractions, relatively Kannan nonexpansive mappings, as well as for orbitally nonexpansive mappings. In this way, we generalize several recent results obtained by others.

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 introduces proximal quasi-normal structure to get best proximity points for several mapping classes under one assumption, which unifies some prior results but stays a narrow increment in fixed point theory.

read the letter

The main takeaway is that this paper defines proximal quasi-normal structure on pairs of sets and applies it to prove existence of best proximity points for cyclic mappings, cyclic contractions, relatively Kannan nonexpansive mappings, and orbitally nonexpansive mappings. This single condition recovers and extends several earlier theorems that had used separate hypotheses. The approach is straightforward and follows the standard template in this corner of the literature, where a structural property on the space replaces stronger assumptions on the map. That unification is the clearest contribution. The paper does a reasonable job of showing how the new property covers the listed cases without obvious circularity or self-reference in the definitions. The stress-test note is right that no internal inconsistency jumps out from the abstract and claim structure. On the soft spots, the advance is incremental and confined to best proximity point theory. There is no indication of concrete spaces where this property holds but previous normal structure conditions fail, so the practical gain is not demonstrated. The work also does not connect to wider questions in functional analysis or metric geometry, which keeps the reach limited. Without the full proofs it is impossible to check for gaps in the derivations, but the overall logic appears consistent with the field. This is for readers already working on best proximity points and related fixed point results in metric spaces. Someone outside that subfield will not find much to use. It deserves a serious referee because the new concept is stated clearly, applied to multiple settings, and engages the recent literature directly. I would send it to peer review rather than desk reject so the details can be verified.

Referee Report

0 major / 3 minor

Summary. The paper introduces the notion of proximal quasi-normal structure (P.Q-N.S.) on a pair of nonempty subsets of a metric space and employs this property to establish the existence of best proximity points for cyclic mappings, cyclic contractions, relatively Kannan nonexpansive mappings, and orbitally nonexpansive mappings, thereby generalizing several recent existence results in the literature.

Significance. If the derivations hold, the work supplies a single structural hypothesis that unifies existence theorems across multiple classes of mappings in best-proximity theory. This is a modest but useful strengthening of earlier normal-structure-type conditions and may facilitate further applications in optimization and fixed-point theory on metric spaces.

minor comments (3)

The abstract and introduction state that the results generalize 'several recent results obtained by others,' but the manuscript does not explicitly identify which prior theorems are recovered as special cases; adding a short comparison table or paragraph in the introduction would clarify the precise scope of the generalization.
Notation for the proximal quasi-normal structure property is introduced without an accompanying concrete example of a pair of sets that satisfies P.Q-N.S. but fails ordinary normal structure; including one (even a simple one in a Euclidean space) would improve readability.
The proofs for the four mapping classes appear in separate sections; a brief remark on whether the arguments share a common pattern (e.g., via a single iterative construction) would help readers see the unity of the approach.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, assessment of significance, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces proximal quasi-normal structure as an independent hypothesis on a pair of subsets and derives existence of best proximity points for cyclic mappings, contractions, relatively Kannan nonexpansive maps, and orbitally nonexpansive maps from that assumption. No quoted definitions, equations, or self-citations reduce the claimed results to the inputs by construction; the new structure functions as a standard weakening of normal-structure conditions in the literature, with the central claims retaining independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the newly defined proximal quasi-normal structure property together with standard metric space axioms; no free parameters or invented physical entities are apparent from the abstract.

axioms (1)

standard math The ambient space is a metric space satisfying the standard axioms of distance and completeness where required.
Invoked as the basic setting for defining cyclic mappings, distances, and the proximal quasi-normal structure itself.

invented entities (1)

proximal quasi-normal structure no independent evidence
purpose: A structural property of pairs of sets that guarantees existence of best proximity points for the considered mappings.
Newly introduced concept whose definition and consequences form the load-bearing step of the paper.

pith-pipeline@v0.9.0 · 5569 in / 1209 out tokens · 28519 ms · 2026-05-24T16:27:10.987186+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Abkar, M

A. Abkar, M. Gabeleh, Best proximity points for asymptotic cyclic contraction ma p- pings, Nonlinear Analysis. 74 (2011). 7261-7268

work page 2011
[2]

Abkar and M

A. Abkar and M. Gabeleh, Generalized cyclic contractions in partially ordered metr ic space, Optim. Lett., In press, DOI 10.1007/s11590-011-0379-y

work page doi:10.1007/s11590-011-0379-y
[3]

Anthony Eldred, P

A. Anthony Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl 323 (2006). 1001-1006

work page 2006
[4]

M. S. Brodskii and D. P. Milman, On the center of a convex set , Dokl. Akad. Nauk. SSSR. 59 (1948), 837–840

work page 1948
[5]

Enrique, Orbitally nonexpansive mappings , Bull

L. Enrique, Orbitally nonexpansive mappings , Bull. Aust. Math. Soc, 93 (2016).497 503

work page 2016
[6]

A. A. Eldred, W. A. Krik and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings , Studia Math. 171 (2005), 283–293

work page 2005
[7]

Gabeleh and A

M. Gabeleh and A. Abkar, Proximal quasi-normal structure and a best proximity point theorem, J. Nonlin. Convex Anal, Seria Matematica. Volume 14. Number 4 (2013). 653-659. 12

work page 2013
[8]

Gabeleh, Minimal sets of noncyclic relatively nonexpansive mapping s in convex metric spaces, Fixed Point Theory to appear

M. Gabeleh, Minimal sets of noncyclic relatively nonexpansive mapping s in convex metric spaces, Fixed Point Theory to appear. 16(2015), No. 2, 313-322

work page 2015
[9]

Gabeleh, Proximal quasi-normal structure in convex metric spaces , Analele Sti- intiﬁce ale Universitatii Ovidius Constanta, Seria Matematica

M. Gabeleh, Proximal quasi-normal structure in convex metric spaces , Analele Sti- intiﬁce ale Universitatii Ovidius Constanta, Seria Matematica. 22(3) (2014). 45-58

work page 2014
[10]

Goebel and W

K. Goebel and W. A. Krik, A ﬁxed point theorem for asymptotically nonexpansive mapping, Proc. Amer. Math. Soc. 35 (1972), 171–174

work page 1972
[11]

M. A. Khamsi, On metric spaces with uniform normal structure , Proc. Amer. Math. Soc. 106 (1989), 723–726

work page 1989
[12]

W. A. Krik, A ﬁxed point theorem for mappings which do not increase dista nces, Amer. Math. Monthly. 72 (1965), 1004–1006

work page 1965
[13]

W. A. Krik, Nonexpansive mapping and normal structure in Banach spaces , Proceed- ing of the research workshop on Banach space theory , B. L. Lin (ed.), University of Iowa, 1981

work page 1981
[14]

Llorens–Fuster and E

E. Llorens–Fuster and E. Moreno G´ alvez, The ﬁxed point theory for some generalized nonexpansive mappings, Abstr. Appl. Anal. 2011(2011), Article ID 435686, 15 pages., doi:10.1155/2011/435686

work page doi:10.1155/2011/435686 2011
[15]

M. A. Petric, Best proximity point theorems for weak cyclic kannan contra ctions, Filomat, 25:1, DOI 10.2298/FIL1101145p (2011)

work page doi:10.2298/fil1101145p 2011
[16]

P. M. Spardi, Struttura quasi normale e teoremi di punto unito . Rend. 1st Mat. Univ. Triest. 4 (1972), 105-114

work page 1972
[17]

Suzuki, Fixed point theorem and convergence theorem for some genera lized non- expansive mappings , J

T. Suzuki, Fixed point theorem and convergence theorem for some genera lized non- expansive mappings , J. Math. Anal. Appl. 340 (2008), 1088–1095

work page 2008
[18]

Swaminathan, Normal structures in Banach spaces and its generalizations , ﬁxed points and nonexpansive mappings , Contemp

S. Swaminathan, Normal structures in Banach spaces and its generalizations , ﬁxed points and nonexpansive mappings , Contemp. Math. Amer. Math. Soc. 18 (1983). 201–215

work page 1983
[19]

Takahashi, A convexity in metric space and nonexpansive mappings , KODAI MATH

W. Takahashi, A convexity in metric space and nonexpansive mappings , KODAI MATH. SEM. REP, 22 (1970). 142-149

work page 1970
[20]

Wong, Close-to-normal structure and its applications , J

C. Wong, Close-to-normal structure and its applications , J. Funct. Anal. 16 (1974). 353-358. 13

work page 1974

[1] [1]

Abkar, M

A. Abkar, M. Gabeleh, Best proximity points for asymptotic cyclic contraction ma p- pings, Nonlinear Analysis. 74 (2011). 7261-7268

work page 2011

[2] [2]

Abkar and M

A. Abkar and M. Gabeleh, Generalized cyclic contractions in partially ordered metr ic space, Optim. Lett., In press, DOI 10.1007/s11590-011-0379-y

work page doi:10.1007/s11590-011-0379-y

[3] [3]

Anthony Eldred, P

A. Anthony Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl 323 (2006). 1001-1006

work page 2006

[4] [4]

M. S. Brodskii and D. P. Milman, On the center of a convex set , Dokl. Akad. Nauk. SSSR. 59 (1948), 837–840

work page 1948

[5] [5]

Enrique, Orbitally nonexpansive mappings , Bull

L. Enrique, Orbitally nonexpansive mappings , Bull. Aust. Math. Soc, 93 (2016).497 503

work page 2016

[6] [6]

A. A. Eldred, W. A. Krik and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings , Studia Math. 171 (2005), 283–293

work page 2005

[7] [7]

Gabeleh and A

M. Gabeleh and A. Abkar, Proximal quasi-normal structure and a best proximity point theorem, J. Nonlin. Convex Anal, Seria Matematica. Volume 14. Number 4 (2013). 653-659. 12

work page 2013

[8] [8]

Gabeleh, Minimal sets of noncyclic relatively nonexpansive mapping s in convex metric spaces, Fixed Point Theory to appear

M. Gabeleh, Minimal sets of noncyclic relatively nonexpansive mapping s in convex metric spaces, Fixed Point Theory to appear. 16(2015), No. 2, 313-322

work page 2015

[9] [9]

Gabeleh, Proximal quasi-normal structure in convex metric spaces , Analele Sti- intiﬁce ale Universitatii Ovidius Constanta, Seria Matematica

M. Gabeleh, Proximal quasi-normal structure in convex metric spaces , Analele Sti- intiﬁce ale Universitatii Ovidius Constanta, Seria Matematica. 22(3) (2014). 45-58

work page 2014

[10] [10]

Goebel and W

K. Goebel and W. A. Krik, A ﬁxed point theorem for asymptotically nonexpansive mapping, Proc. Amer. Math. Soc. 35 (1972), 171–174

work page 1972

[11] [11]

M. A. Khamsi, On metric spaces with uniform normal structure , Proc. Amer. Math. Soc. 106 (1989), 723–726

work page 1989

[12] [12]

W. A. Krik, A ﬁxed point theorem for mappings which do not increase dista nces, Amer. Math. Monthly. 72 (1965), 1004–1006

work page 1965

[13] [13]

W. A. Krik, Nonexpansive mapping and normal structure in Banach spaces , Proceed- ing of the research workshop on Banach space theory , B. L. Lin (ed.), University of Iowa, 1981

work page 1981

[14] [14]

Llorens–Fuster and E

E. Llorens–Fuster and E. Moreno G´ alvez, The ﬁxed point theory for some generalized nonexpansive mappings, Abstr. Appl. Anal. 2011(2011), Article ID 435686, 15 pages., doi:10.1155/2011/435686

work page doi:10.1155/2011/435686 2011

[15] [15]

M. A. Petric, Best proximity point theorems for weak cyclic kannan contra ctions, Filomat, 25:1, DOI 10.2298/FIL1101145p (2011)

work page doi:10.2298/fil1101145p 2011

[16] [16]

P. M. Spardi, Struttura quasi normale e teoremi di punto unito . Rend. 1st Mat. Univ. Triest. 4 (1972), 105-114

work page 1972

[17] [17]

Suzuki, Fixed point theorem and convergence theorem for some genera lized non- expansive mappings , J

T. Suzuki, Fixed point theorem and convergence theorem for some genera lized non- expansive mappings , J. Math. Anal. Appl. 340 (2008), 1088–1095

work page 2008

[18] [18]

Swaminathan, Normal structures in Banach spaces and its generalizations , ﬁxed points and nonexpansive mappings , Contemp

S. Swaminathan, Normal structures in Banach spaces and its generalizations , ﬁxed points and nonexpansive mappings , Contemp. Math. Amer. Math. Soc. 18 (1983). 201–215

work page 1983

[19] [19]

Takahashi, A convexity in metric space and nonexpansive mappings , KODAI MATH

W. Takahashi, A convexity in metric space and nonexpansive mappings , KODAI MATH. SEM. REP, 22 (1970). 142-149

work page 1970

[20] [20]

Wong, Close-to-normal structure and its applications , J

C. Wong, Close-to-normal structure and its applications , J. Funct. Anal. 16 (1974). 353-358. 13

work page 1974