Some remarks on "Convergence of Picard's iteration using projection algorithm for noncyclic contractions" [Indag. Math. 30 (2019) 227--239]

Hans-Peter A. K\"unzi; Moosa Gabeleh

arxiv: 1907.01494 · v1 · pith:S7DRR24Gnew · submitted 2019-07-02 · 🧮 math.FA

Some remarks on "Convergence of Picard's iteration using projection algorithm for noncyclic contractions" [Indag. Math. 30 (2019) 227--239]

Moosa Gabeleh , Hans-Peter A. K\"unzi This is my paper

Pith reviewed 2026-05-25 10:38 UTC · model grok-4.3

classification 🧮 math.FA

keywords best proximity pointsrelatively nonexpansive mappingscyclic mappingsnoncyclic mappingsstrictly convex Banach spacesprojection operatorPicard's iteration

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The pith

Existence of best proximity points for cyclic relatively nonexpansive mappings equals existence of best proximity pairs for noncyclic ones in strictly convex Banach spaces.

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

The note proves an equivalence: in strictly convex Banach spaces, a cyclic relatively nonexpansive mapping has a best proximity point if and only if the associated noncyclic mapping has a best proximity pair, with the proof relying on the projection operator. This equivalence immediately yields the main result of an earlier paper on proximal normal structure. The authors further derive convergence of best proximity pairs for noncyclic contractions directly from the known convergence of Picard's iteration sequences applied to the cyclic case, recovering the method of the 2019 Indag. Math. paper exactly.

Core claim

In strictly convex Banach spaces, the existence of best proximity points for cyclic relatively nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic relatively nonexpansive mappings by using the projection operator. As a direct consequence, the main result of the paper 'Proximal normal structure and relatively nonexpansive mappings' follows immediately. Convergence of best proximity pairs for noncyclic contractions is obtained by applying the convergence of iterative sequences for cyclic contractions, and the convergence method of the 2019 paper is recovered exactly from Picard's iteration sequence.

What carries the argument

The projection operator onto a nonempty closed convex subset, which reduces the noncyclic case to the cyclic case and is single-valued in strictly convex Banach spaces.

If this is right

The main theorem on proximal normal structure in the 2005 Studia Math. paper follows immediately from the equivalence.
Convergence results for noncyclic contractions follow from applying known convergence of cyclic contraction iterates.
The 2019 Indag. Math. convergence method for noncyclic contractions is obtained exactly from Picard's iteration on the cyclic case.

Where Pith is reading between the lines

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

The equivalence may allow transferring other fixed-point or proximity results between cyclic and noncyclic settings without separate proofs.
Strict convexity could be relaxed in some extensions if the projection remains single-valued under weaker conditions.
The reduction technique might apply to other classes of mappings beyond relatively nonexpansive ones.

Load-bearing premise

The projection operator onto a nonempty closed convex subset of a strictly convex Banach space is well-defined and single-valued.

What would settle it

A strictly convex Banach space containing a noncyclic relatively nonexpansive mapping with a best proximity pair whose associated cyclic mapping has no best proximity point.

read the original abstract

In this note, at first we prove that the existence of best proximity points for cyclic relatively nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that a main result of the paper "Proximal normal structure and relatively nonexpansive mappings", Studia Math., (171~(2005) 283--293) immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper published in Indag. Math., 30(1) (2019) 227--239 is obtained exactly from Picard's iteration sequence.

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

This note shows that best-proximity existence for cyclic and noncyclic relatively nonexpansive maps are equivalent via projection in strictly convex spaces, and that one convergence claim reduces directly to Picard iteration.

read the letter

The core observation is that existence of best proximity points for cyclic relatively nonexpansive mappings equals existence of best proximity pairs for the noncyclic case in strictly convex Banach spaces once you apply the metric projection. That equivalence immediately yields the 2005 Studia Math result as a corollary. The second part shows that convergence of best proximity pairs for noncyclic contractions is recovered exactly by running Picard iteration on the cyclic version, so the method in the 2019 Indag Math paper is not an independent construction.

Referee Report

0 major / 0 minor

Summary. The manuscript proves that, in strictly convex Banach spaces, the existence of best proximity points for cyclic relatively nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic relatively nonexpansive mappings, using the metric projection. It deduces that a main result from Eldred et al. (Studia Math. 2005) follows immediately. It also shows that convergence of best proximity pairs for noncyclic contractions follows from the convergence of Picard's iteration for cyclic contractions, recovering exactly the method of the 2019 Indag. Math. paper.

Significance. The equivalence provides a reduction technique that links the cyclic and noncyclic settings in best proximity theory. If valid, it allows the 2005 existence result to be obtained as a corollary and unifies the convergence analysis by showing the 2019 approach is a direct application of the cyclic Picard iteration. The arguments rely on the standard property that the metric projection onto a closed convex set is single-valued in strictly convex spaces.

Simulated Author's Rebuttal

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We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The summary accurately captures the equivalence result in strictly convex Banach spaces and its consequences for the 2005 Studia Math. paper as well as the recovery of the 2019 Indag. Math. convergence method via cyclic Picard iteration.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The note proves an equivalence between best-proximity-point existence for cyclic relatively nonexpansive mappings and best-proximity-pair existence for the noncyclic case in strictly convex Banach spaces by invoking the metric projection operator. This projection is single-valued precisely when the space is strictly convex, a standard textbook fact independent of the paper. The 2005 Studia Math. result is then recovered as an immediate corollary of the newly proved equivalence, and the convergence discussion for noncyclic contractions is obtained by direct reduction to the cyclic Picard iteration already analyzed in the literature. No step reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and the cited prior results (2005 and 2019) are external; the central claims rest on the paper's own proof of equivalence plus standard background rather than any self-citation chain or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard properties of strictly convex Banach spaces and the projection operator; no new free parameters or invented entities are introduced.

axioms (1)

standard math In a strictly convex Banach space the metric projection onto a nonempty closed convex set is single-valued.
Invoked when the authors equate cyclic and noncyclic cases via the projection operator (abstract).

pith-pipeline@v0.9.0 · 5693 in / 1199 out tokens · 24143 ms · 2026-05-25T10:38:54.169742+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1: equivalence via projection P on A0∪B0 and strict convexity to obtain fixed points of T and SP
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 1.8 and Theorems 2.3–2.4: use of metric projection and cyclic/non-cyclic reductions for convergence

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
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unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.