{"id":"544fdef4-76e0-4dc4-a16d-9db051e4f529","arxiv_id":"1908.07225","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove existence and stability of coupled best proximity points for p-cyclic contractions, but the uniqueness proof contains a logical contradiction.","lead":"This paper defines p-cyclic contraction and p-cyclic nonexpansive maps on pairs of points and proves existence, uniqueness, and Ulam-Hyers stability for coupled best proximity points in uniformly convex Banach spaces. The main contraction result is a plausible extension of earlier best proximity theory, but the uniqueness proof as written is internally inconsistent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unique coupled best proximity point is not proven in Theorem 2.6: (2.7) forces the distance D to equal dist(A,B), while the strict-contraction step in the uniqueness proof asserts D>dist(A,B). The central uniqueness and stability claims are therefore unsupported as written.","rationale":"The paper attempts a genuinely new existence-uniqueness-stability theorem for p-cyclic contractions. I checked the existence machinery: given Lemma 2.4 and Lemma 2.5, the Cauchy argument for {(x_{2n},y_{2n})} is coherent, and those lemmas, though unproved, are very likely salvageable by coordinatewise application of the Eldred-Veeramani technique. The collapse is in uniqueness. For any coupled best proximity point, the defining equalities and (2.7) give D = dist(A,B). The proof later needs D > dist(A,B) to make a strict inequality; asserting it is a direct contradiction, not a typo. The Ulam-Hyers part then rests on the unproven uniqueness. The reader's REJECT verdict is appropriate; I would not accept the paper as a citable theorem in this form. The only difference from the reader is that I rank the uniqueness contradiction above the unproved Lemma 2.4 as the load-bearing defect, so my agreement with the reader's weakest-assumption identification is only partial.","tokens_in":12560,"tokens_out":30488,"duration_ms":298317,"concrete_test":"Analytical check: distinguish the two candidate points as (x,y) and (x_bar, y_bar) throughout the uniqueness segment. Compute D1 = ||(x,y)-(T(x,y),T(y,x))|| and D2 = ||(x_bar,y_bar)-(T(x_bar,y_bar),T(y_bar,x_bar))|| from the defining equalities; by (2.7) both equal dist(A,B). Substitute these values into the displayed chain 'D = max{...} <= lambda D + (1-lambda)dist(A,B) < lambda D + (1-lambda)D = D'. The middle strict inequality fails because it requires D > dist(A,B); the argument yields only D <= D. This isolates the exact false step; a repaired proof must supply a different contradiction or show non-uniqueness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After establishing (2.7), the uniqueness proof in Theorem 2.6 considers a second coupled best proximity point and later asserts that ||(x,y)-(T(x,y),T(y,x))|| > dist(A,B). But (2.7) says exactly the opposite for every coupled best proximity point: ||(x,y)-(T(x,y),T(y,x))|| = dist(A,B), because both coordinates already have distance dist(A,B). The displayed contraction chain D = max{...} <= lambda D + (1-lambda)dist(A,B) < lambda D + (1-lambda)D = D would yield the desired contradiction only if D > dist(A,B). Since D = dist(A,B), the inequality gives only D <= D, so no contradiction follows. This is not a missing detail or a repairable estimate; it is a false assertion in the central proof. The proof of Ulam-Hyers stability in the same theorem is built on the uniqueness result, so it inherits the gap. Separately, Lemma 2.4 is stated without proof, but that lemma appears to be repairable by applying Eldred-Veeramani componentwise in the max-norm product; the uniqueness step is the hard blocker.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12825,"tokens_out":11792,"duration_ms":103612,"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":[{"comment":"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.","section":"Section 2, Lemma 2.4"},{"comment":"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.","section":"Theorem 2.6, uniqueness proof"},{"comment":"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.","section":"Theorem 2.11 and its proof"}],"minor_comments":[{"comment":"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.","section":"Example 2.10"},{"comment":"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)‖\".","section":"Proposition 2.2, proof"},{"comment":"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.","section":"Abstract and Theorem 2.6"},{"comment":"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.","section":"Definition 2.1"},{"comment":"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.","section":"Theorem 2.6, stability proof"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a genuinely new idea and the central claims may be salvageable, but the current manuscript is not correct: the uniqueness proof in Theorem 2.6 contains a false assertion, and Lemma 2.4 is unproved. If the authors can supply a correct uniqueness argument and a proof of Lemma 2.4, resubmission would be worth considering. Otherwise, I would not recommend publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nWorse 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.\n\nOther 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.\n\nThe 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.","headline":"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.","tokens_in":13305,"tokens_out":3887,"would_cite":false,"duration_ms":37352,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H10","47H09","41A65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["coupled best proximity point","p-cyclic contraction mapping","p-cyclic nonexpansive mapping","uniformly convex Banach space","Ulam-Hyers stability","best proximity point","fixed point theory"],"falsifier":"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.","tokens_in":12291,"feed_emoji":"🎯","tokens_out":16401,"duration_ms":161387,"temperature":0.7,"pith_summary":"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.","feed_headline":"Unique optimal pairs exist for p-cyclic contractions","feed_subtitle":"In uniformly convex spaces, one contractive condition yields existence, uniqueness, and Ulam-Hyers stability.","key_machinery":"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 $S$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the scalar 'approximants coalesce' lemma and the cyclic contraction theorem that the paper's product-space arguments extend.","marker":"[1]"},{"why":"Establishes that $A_0$ and $B_0$ are nonempty for closed convex bounded $A$, used in the nonexpansive existence proof.","marker":"[6]"},{"why":"Introduces cyclic nonexpansive mappings and their best proximity point theorem, which the p-cyclic nonexpansive results generalize.","marker":"[10]"},{"why":"Defines coupled fixed points, the notion that reduces to the coupled best proximity point concept when $A\\cap B\\neq\\emptyset$.","marker":"[4]"},{"why":"Supplies the Ulam stability framework for operatorial equations that Definition 1.4 adapts to the coupled best proximity problem.","marker":"[11]"}],"fun_headline_variants":["Unique coupled best proximity points for p-cyclic contractions","p-cyclic contractions ensure coupled best proximity and Ulam-Hyers stability","Coupled best proximity points: existence, uniqueness, stability","Ulam-Hyers stable best proximity points for p-cyclic contractions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Unique coupled best proximity points for p-cyclic contractions","p-cyclic contractions ensure coupled best proximity and Ulam-Hyers stability","Coupled best proximity points: existence, uniqueness, stability","Ulam-Hyers stable best proximity points for p-cyclic contractions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000629,"raw_usage":{"total_tokens":2903,"prompt_tokens":939,"completion_tokens":1964,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":1893}},"tokens_in":555,"tokens_out":1964,"duration_ms":15498,"temperature":1.0,"reasoning_tokens":1893,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:24:08.730697+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the scalar 'approximants coalesce' lemma and the cyclic contraction theorem that the paper's product-space arguments extend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that $A_0$ and $B_0$ are nonempty for closed convex bounded $A$, used in the nonexpansive existence proof."},{"cited_title":"Sankar Raj and P","cited_arxiv_id":null,"evidence_quote":"Introduces cyclic nonexpansive mappings and their best proximity point theorem, which the p-cyclic nonexpansive results generalize."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines coupled fixed points, the notion that reduces to the coupled best proximity point concept when $A\\cap B\\neq\\emptyset$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Ulam stability framework for operatorial equations that Definition 1.4 adapts to the coupled best proximity problem."}],"review_version":1}