On coupled best proximity points and Ulam-Hyers stability

Anuradha Gupta; Manu Rohilla

REVIEW 3 major objections 5 minor 13 references

On coupled best proximity points and Ulam-Hyers stability

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that p-cyclic contractions between closed convex sets in a uniformly convex Banach space always have a unique coupled best proximity point, and that the problem is Ulam-Hyers stable.

desk verdict A plausible extension to coupled best proximity points that is not yet a theorem: the uniqueness proof contradicts the paper's own equation (2.7), and the product-space lemma it relies on is unproved. read the letter →

arxiv 1908.07225 v1 pith:EZKTOZDL submitted 2019-08-20 math.FA

Anuradha Gupta , Manu Rohilla This is my paper · ORCID

classification math.FA MSC 47H1047H0941A65

keywords coupledbestproximitypointp-cycliccontractionmappingnonexpansiveuniformlyconvexBanachspaceUlam-Hyersstabilityfixedtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces two new classes of mappings, p-cyclic contractions and p-cyclic nonexpansive mappings, on the union $(A\times B)\cup(B\times A)$, and asks whether an iteration of such a mapping yields a pair that is as close as possible to being fixed: a coupled best proximity point. The central result is that in a uniformly convex Banach space, every p-cyclic contraction between two nonempty closed convex sets has exactly one such pair, and the defining equations are stable under small perturbations. For p-cyclic nonexpansive mappings, existence and Ulam-Hyers stability are obtained under a compactness condition on the best-proximity sets $A_0\times B_0$. These statements matter because they extend classical best proximity point theorems from single-valued cyclic maps to maps of pairs while adding a quantitative stability guarantee.

What carries the argument

The load-bearing object is the p-cyclic contraction condition itself, applied to the product domain with the norm $\|(x,y)\|=\max\{\|x\|,\|y\|\}$; the iteration is $(x_n,y_n)=(T(x_{n-1},y_{n-1}),T(y_{n-1},x_{n-1}))$, which alternates between $A\times B$ and its mirror. Proposition 2.2 shows this iteration drives the proximity error down to $\operatorname{dist}(A,B)$, and the key step is a product-space analogue of the standard scalar lemma asserting that two sequences that both become best approximations to the same target sequence must coalesce. That lemma (Lemma 2.4 in the paper) turns the approximate best-proximity property into a Cauchy property for the even subsequence, after which Proposition 2.3 extracts the coupled best proximity point. For the nonexpansive case, the machinery is a perturbation $T_n=\frac1n S(x_0,y_0)+(1-\frac1n)S$, which is a p-cyclic contraction and whose coupled best proximity points converge weakly to a solution for $S$.

What would settle it

Exhibit a p-cyclic contraction $T$ between two disjoint closed convex subsets $A,B$ of a uniformly convex Banach space for which the even iterates $(x_{2n},y_{2n})$ have two distinct accumulation points; Theorem 2.6 says none exists. A concrete check in $\mathbb{R}^2$ with $A$ and $B$ as parallel unit segments, testing all affine p-cyclic contractions, would either produce such an example or confirm the uniqueness theorem in that setting.

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Extended reading notes

Core claim

The paper's main claim is Theorem 2.6: if $A$ and $B$ are nonempty, closed, and convex subsets of a uniformly convex Banach space $X$, and $T:(A\times B)\cup(B\times A)\to A\cup B$ satisfies $T(A,B)\subset B$, $T(B,A)\subset A$, and the p-cyclic contraction inequality $\|T(x_1,y_1)-T(x_2,y_2)\|\le \lambda\|(x_1,y_1)-(x_2,y_2)\|+(1-\lambda)\operatorname{dist}(A,B)$ for some $\lambda\in(0,1)$, then $T$ has a unique coupled best proximity point $(x^*,y^*)$, meaning $\|x^*-T(x^*,y^*)\|=\|y^*-T(y^*,x^*)\|=\operatorname{dist}(A,B)$. The same theorem asserts that the coupled best proximity point problem is Ulam-Hyers stable: any $(u,v)$ satisfying the two proximity inequalities up to an additive $\epsilon$ lies within $\alpha\epsilon+\beta\operatorname{dist}(A,B)$ of $(x^*,y^*)$, with explicit constants. For p-cyclic nonexpansive mappings the paper proves existence and Ulam-Hyers stability when $A_0\times B_0$ is compact, and notes via an example that the solution need not be unique in that case.

Load-bearing premise

The proof of existence rests on Lemma 2.4, a product-space coalescence lemma that the paper states without proof; if that lemma is false, the even iteration sequence cannot be shown to be Cauchy and the existence theorem collapses.

Editorial extensions

If this is right

Any p-cyclic contraction between closed convex sets in a uniformly convex Banach space has a unique coupled best proximity point, and the natural iteration $(x_n,y_n)=(T(x_{n-1},y_{n-1}),T(y_{n-1},x_{n-1}))$ converges to it.
Approximate solutions, where the two defining equations hold only up to an additive $\epsilon$, are guaranteed to stay within $\alpha\epsilon+\beta\operatorname{dist}(A,B)$ of the exact solution, with $\alpha=1/(1-\lambda)$ and $\beta=(3-\lambda)/(1-\lambda)$.
For p-cyclic nonexpansive mappings, existence and Ulam-Hyers stability hold whenever $A_0\times B_0$ is compact, but uniqueness can fail.
When $A\cap B\neq\varnothing$, the coupled best proximity point problem reduces to the classical coupled fixed point problem, so the new theorems recover that older setting as a special case.
The paper's closing remark points toward extending the same treatment to finitely many sets and multidimensional best proximity points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves the Ulam-Hyers constants at $\alpha=1/(1-\lambda)$, $\beta=(3-\lambda)/(1-\lambda)$; a natural next step is to test whether these are sharp, especially as $\lambda\to1$, where the bound degrades.
Because the product norm decouples componentwise, the two coalescence lemmas may be provable by applying the scalar best-proximity lemma to each coordinate; if that works, Theorem 2.6 would extend to any Banach space where the scalar lemma holds, not only uniformly convex spaces.
The sine-map example in the nonexpansive case has a continuum of coupled best proximity points; describing this solution set for general p-cyclic nonexpansive mappings is a question the paper does not address.
Theorem 2.11 already suggests that strict convexity plus compactness of $A_0\times B_0$ can replace uniform convexity; testing whether the same replacement works in the contraction case is a concrete open direction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely new: it defines p-cyclic contraction and p-cyclic nonexpansive mappings on A×B and proves that iterative sequences approach dist(A,B). The existence strategy is a recognizable adaptation of Eldred–Veeramani, and Proposition 2.2 is correct. The stability definition for coupled best proximity points is also a reasonable extension.

The soft spots are serious, though. Lemma 2.4 is stated without proof, saying it is parallel to Eldred–Veeramani's Lemma 3.7, but the product space with the max norm is not uniformly convex, so the transfer is anything but automatic. That alone leaves the Cauchy argument without a foundation.

Worse is the uniqueness proof in Theorem 2.6. Equation (2.7) correctly derives that every coupled best proximity point (x,y) satisfies ||(x,y)-(T(x,y),T(y,x))|| = dist(A,B). But the uniqueness proof later asserts that the same quantity is strictly greater than dist(A,B), using that to derive a contradiction. That is the opposite of what the paper already proved. The contradiction is not a missing detail; it is a false statement inside the central argument. Consequently, the uniqueness claim is unsupported, and the theorem's headline conclusion fails as written. The stability proof in Theorem 2.6 does not actually use uniqueness, so it may be salvageable even without it, but the theorem is false in its present form.

Other issues are minor by comparison: Theorem 2.9 divides by dist(A,B) without assuming dist(A,B)>0, and Theorem 2.11's added assumption (the best proximity point is a best approximation against every point) is ad hoc and not derived from the mapping condition.

The paper is for a narrow audience of metric fixed point specialists. The definitions and the existence route could be repairable, but the current manuscript is not a citable theorem. I would not send it to peer review as is; it needs a corrected proof and a proper treatment of Lemma 2.4. The underlying idea is plausible, so a serious rewrite could yield a publishable paper, but the current version is broken.

Referee Report

3 major / 5 minor

Summary. The paper introduces the notion of p-cyclic contraction and p-cyclic nonexpansive mappings on the product space (A×B)∪(B×A) for closed convex subsets A,B of a uniformly convex Banach space, and studies coupled best proximity points. The main results are Theorem 2.6, asserting existence, uniqueness, and Ulam-Hyers stability for p-cyclic contractions, and Theorems 2.8, 2.9, and 2.11 for p-cyclic nonexpansive mappings. The proofs rely on a product-space analogue of a lemma of Eldred and Veeramani (Lemma 2.4) and on constructing a Cauchy sequence of even iterates. The paper also provides an example for the nonexpansive case.

Significance. If correct, the paper would extend classic best proximity point theory to a coupled setting with a stability statement, which is a natural and potentially useful contribution to the fixed-point literature. The definitions of p-cyclic contractions and Ulam-Hyers stability for coupled best proximity points are new, and the existence argument for contractions is a recognizable adaptation of cyclic contraction techniques. However, the correctness of the central claims is currently undermined by a false assertion in the uniqueness proof and by an unproved essential lemma, so the contribution is not yet established.

major comments (3)

[Section 2, Lemma 2.4] Lemma 2.4 is stated without proof and is essential for the Cauchy-sequence argument in Theorem 2.6. The lemma purports to transfer Eldred and Veeramani's Lemma 3.7 to the product space X×X equipped with the max norm, but (X×X, ‖·‖∞) is not uniformly convex even when X is, so the cited lemma does not automatically apply. A proof or a precise reference to a genuinely applicable generalization must be supplied; without Lemma 2.4 the existence proof in Theorem 2.6 collapses.
[Theorem 2.6, uniqueness proof] The line "Observe that ‖(x,y) − (T(x,y),T(y,x))‖ > dist(A,B)" is false for a coupled best proximity point (x,y), since by definition ‖x−T(x,y)‖=dist(A,B) and ‖y−T(y,x)‖=dist(A,B), which makes the max exactly dist(A,B). Consequently the subsequent strict inequality "< λ‖(x,y)−(T(x,y),T(y,x))‖ + (1−λ)‖(x,y)−(T(x,y),T(y,x))‖" is unjustified, and the contradiction proving uniqueness does not follow. The equality case can potentially be used with Lemma 2.5 to conclude (x,y)=(x,y), but that argument is not what is written.
[Theorem 2.11 and its proof] The additional assumption in Theorem 2.11 uses the symbol T instead of S, writing ‖x − T(u,v)‖ ≤ ‖u − T(u,v)‖ where S is the mapping under study; the same notational confusion appears in the proof when citing Proposition 2.3 for T rather than for the averaged mappings T_n. These are not mere typos because the stability proof depends on the intended contraction inequality for S.

minor comments (5)

[Example 2.10] The example claims to illustrate non-uniqueness of coupled best proximity points for nonexpansive mappings, but the equation sin x = x on [0,1] has only the solution x=0, so the second family described reduces to the same point already exhibited.
[Proposition 2.2, proof] The norm notation "‖(x_{n−1},y_{n−1}), (x_n,y_n)‖" is missing a minus sign; it should read "‖(x_{n−1},y_{n−1}) − (x_n,y_n)‖".
[Abstract and Theorem 2.6] The abstract assumes A and B are bounded, but Theorem 2.6 only assumes closed and convex; the boundedness is not used in the proof of Theorem 2.6. The hypotheses should be aligned.
[Definition 2.1] The term "p-cyclic" is used without a parameter p; the definition does not involve any integer p. Consider naming the notion simply "cyclic contraction on the product" or explicitly defining p.
[Theorem 2.6, stability proof] The stability proof concludes with bounds of the form ‖x−u‖ ≤ ε/(1−λ) + ((3−λ)/(1−λ))dist(A,B), but the constants α and β from Definition 1.4 are not explicitly identified; stating them would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the existence/stability theorem is not defined into its conclusion; the visible defects are proof gaps, not circular reductions.

full rationale

The paper's central claim is an existence, uniqueness and Ulam-Hyers stability theorem for coupled best proximity points of p-cyclic contractions. The object class is fixed independently of the conclusion: Definition 2.1 defines p-cyclic contraction by a contraction inequality involving dist(A,B), and a coupled best proximity point is the standard pair of optimal approximate solutions; neither definition presupposes existence or uniqueness. The proof draws on external prior results (Eldred–Veeramani [1], Sankar Raj–Veeramani [10], Kirk–Reich–Veeramani [6]) and on the paper's own Lemma 2.4, stated as a 'parallel result' to [1, Lemma 3.7] without proof. These are independent citations, not self-citations carrying the argument. The only self-citation, [5], appears in a list of recent Ulam-Hyers stability studies and is not load-bearing. No parameter is fitted to data, and no quantity called a prediction is recovered from a fitted value. Two mathematical defects are visible: Lemma 2.4/2.5 are asserted without proof, so the transfer of the uniform-convexity argument to the max-norm product space is not automatic; and in Theorem 2.6 the uniqueness proof asserts that for another coupled best proximity point one has ||(x,y)-(T(x,y),T(y,x))|| > dist(A,B), although the defining equations of a coupled best proximity point, together with (2.7), force this norm to equal dist(A,B). These are correctness and proof-gap issues, not circular reductions in which an output equals an input by construction. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are involved. The central claim relies on three assumptions: the standard uniform convexity of X, the unproved Lemma 2.4, and the ad hoc stability condition in Theorem 2.11. The latter two are introduced for this paper and are load-bearing.

assumptions (3)

standard math Uniform convexity of X and its consequences, including reflexivity and strict convexity.
The paper assumes X is uniformly convex throughout Theorems 2.6, 2.8, and 2.9, and uses reflexivity and a product-space version of Eldred and Veeramani's approximation lemma without proof.
ad hoc to paper Lemma 2.4: product-max-norm analogue of Eldred and Veeramani Lemma 3.7.
Stated as 'A parallel result ... can be obtained' with no proof. It is not a standard theorem in the cited references and is load-bearing for the Cauchy argument in Theorem 2.6.
ad hoc to paper Theorem 2.11 stability assumption: if (x,y) is a coupled best proximity point of S, then for every (u,v), ||x-S(u,v)|| <= ||u-S(u,v)|| and ||y-S(v,u)|| <= ||v-S(v,u)||.
This assumption is introduced in the statement of Theorem 2.11 without motivation and is needed for its Ulam-Hyers stability conclusion. It is not proved and may not hold for a general p-cyclic nonexpansive map.

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Pith. "Pith review of On coupled best proximity points and Ulam-Hyers stability." pith.science (2026). https://pith.science/paper/EZKTOZDL

@misc{pith2026190807225,
  author       = {Pith},
  title        = {Pith review of: On coupled best proximity points and Ulam-Hyers stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZKTOZDL}},
  note         = {Machine review of arXiv:1908.07225}
}

abstract

For two nonempty, closed, bounded and convex subsets $A$ and $B$ of a uniformly convex Banach space $X$ consider a mapping $T:(A \times B) \cup (B \times A) \rightarrow A \cup B$ satisfying $T(A,B) \subset B$ and $T(B, A) \subset A$. In this paper the existence of a coupled best proximity point is established when $T$ is considered to be a p-cyclic contraction mapping and a p-cyclic nonexpansive mapping. The Ulam-Hyers stability of the best proximity point problem is also studied.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

A. A. Eldred and P. Veeramani, Existence and convergence of be st proximity points, J. Math. Anal. Appl. 323 (2006), no. 2, 1001–1006

work page 2006
[2]

Esp ´ ınola and A

R. Esp ´ ınola and A. Fern´ andez-Le´ on, On best proximity pointsin metric and Banach spaces, Canad. J. Math. 63 (2011), no. 3, 533–550

work page 2011
[3]

Gabeleh and C

M. Gabeleh and C. Vetro, A note on best proximity point theory us ing proximal contractions, J. Fixed Point Theory Appl. 20 (2018), no. 4, Art. 149, 11 pp

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[5]

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A. Gupta, Ulam-Hyers stability theorem by tripled ﬁxed point theo rem, Fasc. Math. No. 56 (2016), 77–97

work page 2016
[6]

W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and be st proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 851–862

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[7]

J. T. Markin, A best approximation theorem for nonexpansive se t-valued mappings in hyper- convex metric spaces, Rocky Mountain J. Math. 35 (2005), no. 6, 2059–2063

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V. Sankar Raj and P. Veeramani, Best proximity pair theorems f or relatively nonexpansive mappings, Appl. Gen. Topol. 10 (2009), no. 1, 21–28. 12

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

A. A. Eldred and P. Veeramani, Existence and convergence of be st proximity points, J. Math. Anal. Appl. 323 (2006), no. 2, 1001–1006

work page 2006

[2] [2]

Esp ´ ınola and A

R. Esp ´ ınola and A. Fern´ andez-Le´ on, On best proximity pointsin metric and Banach spaces, Canad. J. Math. 63 (2011), no. 3, 533–550

work page 2011

[3] [3]

Gabeleh and C

M. Gabeleh and C. Vetro, A note on best proximity point theory us ing proximal contractions, J. Fixed Point Theory Appl. 20 (2018), no. 4, Art. 149, 11 pp

work page 2018

[4] [4]

D. J. Guo and V. Lakshmikantham, Coupled ﬁxed points of nonlinea r operators with applica- tions, Nonlinear Anal. 11 (1987), no. 5, 623–632

work page 1987

[5] [5]

Gupta, Ulam-Hyers stability theorem by tripled ﬁxed point theo rem, Fasc

A. Gupta, Ulam-Hyers stability theorem by tripled ﬁxed point theo rem, Fasc. Math. No. 56 (2016), 77–97

work page 2016

[6] [6]

W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and be st proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 851–862

work page 2003

[7] [7]

J. T. Markin, A best approximation theorem for nonexpansive se t-valued mappings in hyper- convex metric spaces, Rocky Mountain J. Math. 35 (2005), no. 6, 2059–2063

work page 2005

[8] [8]

Mle¸ snit ¸e, Existence and Ulam-Hyers stability results for coincidence problems, J

O. Mle¸ snit ¸e, Existence and Ulam-Hyers stability results for coincidence problems, J. Nonlinear Sci. Appl. 6 (2013), no. 2, 108–116

work page 2013