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Fixed Points of Meir-Keeler and Leader Contractions with bounded orbits in b-Metric Spaces

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Pith's one-line read In a complete b-metric space, every non-expansive Leader contraction with bounded orbits has a fixed point.

desk verdict The main theorem is unproven due to an invalid sup-inf step, but the paper asks a meaningful question and deserves a revision attempt. read the letter →

arxiv 2506.09074 v1 pith:UEFQSGUE submitted 2025-06-09 math.MG

Hassan Khandani This is my paper
classification math.MG MSC 47H0947H10
keywords b-metricspacesfixedpointMeir-KeelercontractionsLeadernon-expansivemappingsboundedorbitscontractionhierarchyMatkowski
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 claims that a subtle failure in fixed-point theory for b-metric spaces can be repaired by strengthening the contraction condition and adding a mild boundedness hypothesis. It introduces non-expansive Leader contractions, a class sitting properly between Meir-Keeler/Matkowski contractions and general Leader contractions, and proves that every such map on a complete b-metric space with bounded orbits has a fixed point. This matters because a known counterexample shows that classical Meir-Keeler contractions can lack fixed points in exactly this setting; the new theorem identifies boundedness of orbits as the missing hypothesis. If correct, the result restores a coherent contraction hierarchy in b-metric spaces without relying on the triangle inequality.

What carries the argument

The machinery is the triple of orbit quantities $\sigma(m,n,p) = \inf_{k \geq 0} \Delta(T^{m(k)+p}x, T^{n(k)+p}x)$, the infimum over all admissible index pairs $\sigma(p) = \inf_{(m,n) \in \Sigma} \sigma(m,n,p)$, and the supremum $\theta(p) = \sup_{(m,n) \in \Sigma} \sigma(m,n,p)$. A Leader contraction is a map for which, for every $\epsilon > 0$, there exist $\delta > 0$ and $r \in \mathbb{N}$ such that any pair closer than $\epsilon + \delta$ has its $r$-th iterates closer than $\epsilon$. Non-expansiveness makes $\sigma(p)$ nonincreasing; the Leader condition is invoked to show the limit of $\sigma(p)$ is 0. The step that drives $\theta(p) \to 0$ is the transfer from pointwise decrease $\sigma(m,n,p+1) \le \sigma(m,n,p)$ to the global inequality $\theta(p+1) \le \sigma(p)$, after which a Cauchy orbit follows and convergence gives the fixed point.

What would settle it

In a two-point space $\{a,b\}$ with $\Delta(a,b)=1$, let $T$ swap the points. Then $T$ is non-expansive and all orbits are bounded, and direct computation gives $\sigma(p)=0$ and $\theta(p)=1$ for every $p$, so the inequality $\theta(p+1) \le \sigma(p)$ is false. This isolates the step that would require a new argument; to refute Theorem 2.2 itself, one would need a non-expansive Leader contraction on a complete b-metric space with all orbits bounded and no fixed point, and the failure of this inequality in the swap example is a concrete place to look for such a map.

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Extended reading notes

Core claim

The central claim is Theorem 2.2: given a complete b-metric space $(X,\Delta)$ with coefficient $s \ge 1$ and a non-expansive Leader contraction $T \colon X \to X$ whose orbits are all bounded, $T$ admits a fixed point. The proof tracks two quantities attached to an orbit $\{T^n x\}$: the infimum $\sigma(p)$ over all pairs of tail indices of the shifted distances $\Delta(T^{m(k)+p}x, T^{n(k)+p}x)$, and the corresponding supremum $\theta(p)$. Non-expansiveness makes $\sigma(p)$ nonincreasing; the Leader condition is used to force its limit to be 0. The proof then argues that the pointwise decrease of each pair transfers to the global inequality $\theta(p+1) \le \sigma(p)$, which makes $\theta(p) \to 0$; a bounded orbit whose worst-case shifted distances shrink to 0 is Cauchy, and completeness plus continuity of $T$ produce the fixed point.

Load-bearing premise

The proof depends on turning the pointwise decrease of every shifted pair distance into the global inequality that the worst-case distance after one more iterate is no larger than the best-case distance before it; that step is not justified by non-expansiveness alone.

Editorial extensions

If this is right

  • Every Meir-Keeler or Matkowski contraction with bounded orbits in a complete b-metric space has a fixed point, since both classes are contained in the non-expansive Leader class.
  • The fixed-point theorem does not use the triangle inequality; Remark 2.4 states it holds in any space with unique limits and preserved contraction properties.
  • For a non-expansive map, having one bounded orbit is equivalent to all orbits being bounded and also to having a fixed point, as stated in Remark 2.3.
  • The paper's hierarchy places non-expansive Leader contractions strictly between Meir-Keeler/Matkowski contractions and Leader contractions, so the standard contraction diagram survives in b-metric spaces with the bounded-orbit hypothesis added.

Reading between the lines

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

  • If the theorem is correct, boundedness of orbits could become the standard hypothesis for restoring Meir-Keeler-type fixed points in b-metric spaces, replacing stronger or more artificial conditions.
  • The proof's transfer from pointwise decrease to the global sup-vs-inf inequality is not a consequence of non-expansiveness alone; a reader who wants to rely on the theorem will want to see this step justified from the Leader condition and boundedness together.
  • The known counterexample to Meir-Keeler fixed points in b-metric spaces has unbounded orbits, so the bounded-orbit hypothesis is exactly what excludes that pathology; varying that example to keep orbits bounded while retaining the Leader condition would probe the sharpness of the theorem.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a fixed-point theorem for non-expansive Leader contractions in complete b-metric spaces under the assumption of bounded orbits, together with a hierarchy of contraction classes in which Matkowski and Meir-Keeler contractions are proper subclasses of non-expansive Leader contractions, which in turn form a proper subclass of Leader contractions. The main theorem is proven by introducing auxiliary quantities σ(m,n,p), σ(p), and θ(p), showing that σ(p) tends to 0, then claiming that θ(p) also tends to 0, and finally using this to prove that Picard sequences are Cauchy. The paper also discusses Lu et al.'s counterexample and provides two examples and a diagram intended to support the hierarchy.

Significance. If Theorem 2.2 were correct, it would be a meaningful addition to fixed-point theory in b-metric spaces, since it would show that boundedness of orbits compensates for the failure of ordinary Meir-Keeler arguments in that setting. The paper contains a self-contained proof attempt and correctly notes that Lu et al.'s counterexample has unbounded orbits and therefore does not contradict the new hypothesis. However, the proof of the central theorem rests on a logically invalid inference, and the example that underpins the claimed proper inclusion N.Le⊊Le is mathematically false. As a result, neither the main fixed-point claim nor the hierarchy claim is currently established. The paper does not rely on parameter fitting or external computations, and its declared scope is clear, but the substantive results are unsupported.

major comments (3)
  1. [§2, proof of Theorem 2.2, between (2.4) and (2.9)] The step "Since T is non-expansive... This implies that sup_{(m,n)∈Σ} σ(m,n,p+1) ≤ inf_{(m,n)∈Σ} σ(m,n,p)" is invalid. Pointwise monotonicity σ(m,n,p+1) ≤ σ(m,n,p) for each pair does not imply that the supremum of the left-hand side is bounded by the infimum of the right-hand side. For example, on the two-point b-metric space X={0,1} with Δ(0,1)=1 and T the transposition, the family consisting of pairs with even index difference and pairs with odd index difference satisfies σ(m,n,p+1)=σ(m,n,p) for every pair, while sup_Σ σ(m,n,p)=1 and inf_Σ σ(m,n,p)=0 for every p. Thus (2.9), the conclusion lim_{p→∞} θ(p)=0, is unsupported. Since the proof of the Cauchy property uses ϵ≤θ(p) for arbitrary p and then passes to the limit, the fixed point does not follow.
  2. [§2, definition of Σ in (2.1)] The set Σ allows m(k)=n(k) for all k, which makes σ(m,n,p)=0 for such a pair and hence σ(p)=0 identically. This trivializes the earlier "claim σ=0" argument and shows that the Leader condition is not actually needed to obtain σ(p)→0. The substantive possibility of proving θ(p)→0 is not addressed: the passage from an infimum to a supremum is not justified, and the Leader condition is not invoked at that point. Even if σ(p)=0 were nontrivial, the proof would still need an argument controlling sup_Σ σ(m,n,p) in terms of inf_Σ σ(m,n,p), and no such argument is given.
  3. [§3, Example 2] The verification that T is a Leader contraction contains a false inequality. For x∈[0,1/2] and y∈(1/2,3/4], exact iteration gives |T^n x - T^n y| = |x-y|/3^n + (3/8)(1-3^{-n}), which exceeds the claimed bound |x-y|/3^n + 1/(4·3^{n-1}) already for n=2. In fact, for ε=1/4 and x=0, y=3/4, one has |T^r x - T^r y| > 1/4 for every r (the distance tends to 3/8), so T is not a Leader contraction. Proposition 2.1 and Figure 1 rely on this example for the strict inclusion N.Le⊊Le, so the hierarchy claim is unsubstantiated.
minor comments (4)
  1. [Throughout] The b-metric is denoted Δ in Section 1 and as d from Theorem 2.2 onward; please use a single symbol consistently.
  2. [§2, equations (2.2)-(2.4)] The symbol σ is overloaded: it denotes the function σ(m,n,p), its infimum σ(p), and the limit σ. Please use distinct notation, for example σ_∞ for the limit.
  3. [§2, Remark 2.3] The claimed equivalence (1)⇔(2) for arbitrary non-expansive T is false in general: a rotation of the unit circle is non-expansive with all orbits bounded but no fixed point. The remark should be restricted to the setting of Theorem 2.2 or omitted.
  4. [§3, Example 2 and references] There are several typographical issues: "ordered paires" in (2.1), "invoing" before (2.5), "sarisfies" in Example 1, and "inb-Metric Spaces" in the title. Also, reference [12] lists "E. Keeler and A. Meir"; the standard ordering is A. Meir and E. Keeler.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is self-contained, with no fitted parameter, no conclusion assumed, and no load-bearing self-citation.

full rationale

The derivation of Theorem 2.2 uses only the definitions of Leader contraction, non-expansiveness, bounded orbits, and the internally constructed quantities sigma and theta. No quantity is defined in terms of the purported fixed point, no parameter is fitted to a subset of data and then renamed a prediction, and the existence of the fixed point is never assumed during the proof. The cited inclusion Ma, MK ⊂ Le is attributed to Jachymski [10] and to Meir-Keeler and Leader, not to the author, so the argument does not rest on a self-citation chain. The proof's potentially invalid analytic step, passing from pointwise nonincrease of sigma(m,n,p) to sup sigma(m,n,p+1) ≤ inf sigma(m,n,p), is a correctness and rigor concern rather than a circularity: that step does not make the theorem equivalent to its own assumptions. The theorem would stand or fall on the validity of that inference, independently of any circular dependency. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper introduces no new entities or free parameters. Its main unproved input is the inclusion of Matkowski and Meir-Keeler contractions in the Leader class for b-metric spaces, which is cited to a metric-space source. The proof also implicitly assumes that the pointwise monotonicity σ(m,n,p+1) ≤ σ(m,n,p) can be promoted to a bound of the supremum by the infimum, an assumption that is false in general.

assumptions (2)
  • domain assumption Matkowski and Meir-Keeler contractions are contained in the class of Leader contractions in b-metric spaces.
    The paper cites [10] for this inclusion, but [10] concerns standard metric spaces; no b-metric proof is given. Proposition 2.1 and the framing of N.Le as containing MK and Ma depend on this.
  • standard math A nonexpansive map on a b-metric space is continuous.
    Used at the end of Theorem 2.2 to conclude T z = z from convergence of the orbit; this does hold because Δ(Tx, Ty) ≤ Δ(x, y).

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Cite this review

Pith. "Pith review of Fixed Points of Meir-Keeler and Leader Contractions with bounded orbits in b-Metric Spaces." pith.science (2026). https://pith.science/paper/UEFQSGUE

@misc{pith2026250609074,
  author       = {Pith},
  title        = {Pith review of: Fixed Points of Meir-Keeler and Leader Contractions with bounded orbits in b-Metric Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEFQSGUE}},
  note         = {Machine review of arXiv:2506.09074}
}
read the original abstract

We establish fixed-point theorems for Meir-Keeler-type contractions in b-metric spaces. While Lu et al. demonstrated via an explicit counterexample that classical Meir-Keeler contractions may fail to admit fixed points in this setting, we prove that a natural strengthening of the conditions yields existence results. Specifically, we show that every non-expansive Leader contraction with bounded orbits in a b-metric space possesses a fixed point. To contextualize our findings, we present a hierarchical diagram illustrating that the fixed-point theory of non-expansive Leader contractions subsumes earlier results, including Meir-Keeler contractions, the primary focus of this work. Our proofs hold in arbitrary b-metric spaces, without relying on the triangle inequality, requiring instead only the assumption of unique limits. This work not only resolves the limitation exposed by Lu et al.'s counterexample but also establishes a unifying framework for future research in the literature.

Figures

Figures reproduced from arXiv: 2506.09074 by the authors.

Figure 1
Figure 1. Hierarchy of contraction classes in b-metric spaces. Blue highlights our new class of non-expansive Leader contractions (N.Le), showing its position between Meir-Keeler (MK) and Leader (Le) contractions, while prop￾erly containing Matkowski contractions (Ma). The arrows indicate proper inclusions [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

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Works this paper leans on

22 extracted references · 22 canonical work pages

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