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arxiv: 2605.21732 · v1 · pith:5MDV5KEOnew · submitted 2026-05-20 · 🧮 math.FA · math.CA

Hybrid and Component-wise Leggett-Williams type Fixed Point Theorems in Product Spaces with Applications

Laura Mar\'ia Fern\'andez-Pardo This is my paper

Pith reviewed 2026-05-22 07:40 UTC · model grok-4.3

classification 🧮 math.FA math.CA
keywords fixed point theoremsLeggett-Williams theoremproduct spacesmultiplicity of fixed pointscoexistence fixed pointsboundary value problemsnonlinear systemspositive solutions
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The pith

Leggett-Williams type conditions applied component-wise in product spaces guarantee nine distinct fixed points including four coexistence points.

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

The paper develops fixed point theorems for operators defined on the Cartesian product of two normed linear spaces. Imposing Leggett-Williams type conditions independently on each component establishes at least nine distinct fixed points. Four of these are coexistence fixed points where both components are nontrivial. A hybrid version using Leggett-Williams conditions in one component and Krasnosel'skii compression-expansion in the other produces three fixed points, with direct application to multiple positive solutions of nonlinear second-order boundary value systems.

Core claim

Leggett-Williams type conditions in each component of the system guarantee the existence of nine distinct fixed points, of which four are coexistence fixed points with all components nontrivial. A hybrid approach combining Leggett-Williams conditions in one component with Krasnosel'skii compression-expansion conditions in the other yields three fixed points, applied to establish multiple positive solutions for nonlinear systems of second-order equations with two-point boundary conditions.

What carries the argument

Component-wise Leggett-Williams conditions on operators acting in the product of two normed spaces, without inter-component compatibility requirements.

Load-bearing premise

The operators satisfy the Leggett-Williams type conditions separately in each component of the product space.

What would settle it

An explicit operator on a product space that meets the separate Leggett-Williams conditions in each component but has only eight or fewer fixed points would disprove the nine-point multiplicity result.

Figures

Figures reproduced from arXiv: 2605.21732 by Laura Mar\'ia Fern\'andez-Pardo.

Figure 1
Figure 1. Figure 1: Illustration of the three different sets for which a fixed point exists under the assumptions of [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Localization of the nine fixed points under the assumptions of Theorem [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graphical illustration of the behavior of [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graphical illustration of h in [4, 8]. Theorem 5.4 Under the assumptions of Proposition 5.1, set p1 = m2e −1/β2 and p2 = m1e −1/β1 , where m1, m2 ∈ R satisfy (5.23) and (5.24), respectively. Then system (5.18) admits at least four solutions, each with both components nontrivial, for all β1 and β2 satisfying β1 − β1e −1/β1 > s˜(k1) f1 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
read the original abstract

In this paper, we present new multiplicity fixed point theorems for operators acting on Cartesian products of two normed linear spaces. We show that Leggett-Williams type conditions in each component of the system guarantee the existence of nine distinct fixed points, of which four of them are coexistence fixed points, i.e., points with all components nontrivial. In addition, a hybrid approach combining Leggett-Williams conditions in one component with Krasnosel'skii compression-expansion conditions in the other allows us to obtain three fixed points. As an application, we establish the existence of multiple positive solutions for nonlinear systems of second-order equations with two-point boundary conditions.

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.

Referee Report

2 major / 2 minor

Summary. The paper develops multiplicity fixed point theorems for operators on the product of two normed spaces. It claims that imposing Leggett-Williams type conditions component-wise on the coupled operators yields nine distinct fixed points (four of which are coexistence points with both components nontrivial). A hybrid version combining Leggett-Williams conditions in one component with Krasnosel'skii compression-expansion in the other yields three fixed points. The results are applied to obtain multiple positive solutions for a system of second-order boundary-value problems.

Significance. If the component-wise multiplicity arguments are rigorously justified, the work would provide a systematic way to obtain multiple coexistence solutions for coupled systems without requiring joint cone conditions on the product space. This could be useful for applications in systems of differential equations, extending classical Leggett-Williams and Krasnosel'skii theorems to product settings.

major comments (2)
  1. [§3] §3 (main multiplicity theorems): The central claim that separate Leggett-Williams conditions on each component operator suffice to produce nine simultaneous fixed points (including four coexistence points) is load-bearing. Because each operator (e.g., T1(x,y)) depends on both variables, the cone, concave functional, and expansion/compression constants for the first component are evaluated at points whose second coordinate varies over the fixed-point set of the second operator. The manuscript must explicitly verify or impose a uniformity condition ensuring the Leggett-Williams hypotheses hold simultaneously for a common pair (x*,y*); without this, the nine combinations may not be realized by a single solution pair.
  2. [Proof of Theorem 3.1] Proof of Theorem 3.1 (or equivalent): The argument appears to apply the classical Leggett-Williams theorem separately to each component and then combine the resulting fixed-point sets. This step requires showing that the intersection of the two fixed-point sets is nonempty and contains the claimed number of distinct points; the current sketch leaves open whether the dependence between components introduces additional fixed points or collapses some of the nine combinations.
minor comments (2)
  1. [§2] Notation for the product cone and the component functionals should be introduced with explicit definitions before the statements of the main theorems to avoid ambiguity when the operators are coupled.
  2. [§4] In the application section, the verification that the integral operators satisfy the Leggett-Williams constants should include explicit estimates for the Green's function and the nonlinearity bounds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify points where the presentation of the component-wise arguments requires greater explicitness to ensure rigor. We will revise the manuscript to address these issues directly.

read point-by-point responses

  1. Referee: [§3] §3 (main multiplicity theorems): The central claim that separate Leggett-Williams conditions on each component operator suffice to produce nine distinct fixed points (including four coexistence points) is load-bearing. Because each operator (e.g., T1(x,y)) depends on both variables, the cone, concave functional, and expansion/compression constants for the first component are evaluated at points whose second coordinate varies over the fixed-point set of the second operator. The manuscript must explicitly verify or impose a uniformity condition ensuring the Leggett-Williams hypotheses hold simultaneously for a common pair (x*,y*); without this, the nine combinations may not be realized by a single solution pair.

    Authors: We agree that the interdependence of the components necessitates an explicit uniformity condition. In the revised version we will add to the hypotheses of Theorem 3.1 (and its hybrid counterpart) the requirement that the cone, concave functional, and the constants r, R, c appearing in the Leggett-Williams conditions for each operator are independent of the value of the other variable, or at least hold uniformly for all values of the second variable lying in the relevant order interval determined by the fixed-point set of the companion operator. With this uniformity in place, each of the nine combinations of fixed-point indices is realized by at least one common pair (x*, y*). We will also insert a short paragraph immediately after the statement of the theorem explaining why the uniformity guarantees that the nine points are attained simultaneously rather than merely separately. revision: yes

  2. Referee: [Proof of Theorem 3.1] Proof of Theorem 3.1 (or equivalent): The argument appears to apply the classical Leggett-Williams theorem separately to each component and then combine the resulting fixed-point sets. This step requires showing that the intersection of the two fixed-point sets is nonempty and contains the claimed number of distinct points; the current sketch leaves open whether the dependence between components introduces additional fixed points or collapses some of the nine combinations.

    Authors: The referee is right that the current sketch is too terse on this point. In the revision we will expand the proof of Theorem 3.1 as follows: after invoking the classical Leggett-Williams theorem on each component (with the other variable treated as a fixed parameter), we explicitly construct the set of candidate pairs by taking the Cartesian product of the two fixed-point sets and then verify, using the uniformity condition introduced above, that every such pair is indeed a fixed point of the coupled operator (T1, T2). We will also argue that the hypotheses preclude both the appearance of extraneous fixed points outside these nine combinations and the collapse of distinct combinations into fewer points, because the component-wise cone conditions and the strict inequalities on the functionals force the fixed points to lie in distinct regions of the product cone. A detailed verification of these facts will be added to the proof. revision: yes

Circularity Check

0 steps flagged

No circularity: theorems extend established Leggett-Williams results component-wise without reduction to inputs

full rationale

The paper states that Leggett-Williams type conditions applied separately in each component of the product space guarantee nine fixed points (four coexistence). This is an extension of prior multiplicity theorems (Leggett-Williams, Krasnosel'skii) to systems of operators on Cartesian products, with no quoted self-definition, no fitted parameter renamed as prediction, and no load-bearing self-citation chain. The abstract and application to boundary-value problems treat the component-wise hypotheses as independent inputs whose joint realization follows from the product-space construction; the derivation remains self-contained against external fixed-point benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on background cone theory and fixed point theorems from the literature as standard assumptions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Leggett-Williams and Krasnosel'skii fixed point theorems hold in the individual normed spaces
    The new results extend these established theorems component-wise to the product space.
  • domain assumption The spaces admit suitable cones for applying the multiplicity conditions
    Required for the Leggett-Williams type conditions to apply in the product setting.

pith-pipeline@v0.9.0 · 5633 in / 1277 out tokens · 39133 ms · 2026-05-22T07:40:59.793887+00:00 · methodology

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