arxiv: 2604.12415 · v1 · submitted 2026-04-14 · 🧮 math.CA · cs.NA· math.NA

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An eigenvalue result for Neumann BVPs with functional terms

Giuseppe Antonio Veltri

Pith reviewed 2026-05-10 14:16 UTC · model grok-4.3

classification 🧮 math.CA cs.NAmath.NA

keywords Neumann boundary value problemseigenvalue problemsHammerstein integral equationsfunctional termsfixed-point iterationexistence and localizationparameter-dependent problemsnumerical approximation

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The pith

Neumann boundary value problems with a functional term admit localized eigenvalue-eigenfunction pairs after reformulation as Hammerstein integral equations.

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

The paper shows how to guarantee and locate solutions to parameter-dependent Neumann problems that contain an extra functional term by turning the differential equation into an equivalent Hammerstein integral equation. Once in integral form, a known existence-and-localization theorem supplies intervals that must contain the eigenvalues. The same reformulation yields a convergent fixed-point iteration that can be coded to compute both the eigenvalues and the corresponding eigenfunctions to any desired accuracy. Two pseudocodes and a working MATLAB implementation are given so that the theoretical bounds can be checked numerically and the shapes of the eigenfunctions can be plotted for fixed norm.

Core claim

By integrating the second-order differential equation twice and enforcing the Neumann boundary conditions, the original boundary-value problem is converted into a Hammerstein integral equation whose kernel depends on the eigenvalue parameter; an abstract fixed-point theorem then guarantees the existence of solutions inside explicit intervals, and the same equation admits a convergent iteration that produces the eigenfunctions.

What carries the argument

The Hammerstein integral equation obtained by double integration of the differential equation together with the Neumann conditions, which carries both the existence-localization argument and the convergent iteration.

If this is right

Eigenvalues are confined to explicit closed intervals that depend on the functional term and the boundary data.
A simple iterative scheme starting from any continuous function converges to the eigenfunction of fixed norm.
The same code produces both the eigenvalue approximation and the corresponding eigenfunction plot for any chosen parameter value.
The localization intervals can be tightened by refining the bounds used in the abstract theorem.

Where Pith is reading between the lines

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

The iteration scheme may be accelerated by Newton-type corrections once a rough eigenvalue is known.
Similar double-integration tricks could convert other linear or nonlinear boundary conditions into Hammerstein form.
The method supplies a practical way to check whether a proposed functional term produces eigenvalues inside a physically relevant window.

Load-bearing premise

The reformulated Hammerstein integral equation must satisfy every hypothesis of the abstract existence-and-localization theorem that is invoked.

What would settle it

An explicit choice of the functional term and parameter range for which the Hammerstein equation fails one of the cited hypotheses yet the original Neumann problem still possesses an eigenvalue-eigenfunction pair inside the claimed interval.

Figures

Figures reproduced from arXiv: 2604.12415 by Giuseppe Antonio Veltri.

**Figure 1.** Figure 1: Plots of Fρ and Fρ in the cases ρ = 0.1, ρ = 0.15 and ρ = 1, respectively. One can show that Fρ is a positive function independently of ρ, and this means that, to apply Theorem 2.1, we have to focus on the function Fρ. In particular, we have to search for the values of ρ such that the function Fρ achieves a positive maximum, that is an equivalent condition to (5b) in Theorem 2.1, and to find the absolute m… view at source ↗

**Figure 2.** Figure 2: Plots of Fρ and Fρ in the cases ρ = 0.1, ρ = 0.4 and ρ = 1, respectively. This inequality is equivalent to the following: ρ < 1 4 log max [0,2/3] − C D =: ρ2. Therefore, Fρ attains a positive maximum if and only if 0 < ρ < max{ρ1, ρ2} =: ρ0. Note that − A(t) B(t) ′ = 8 3 √ 3 tan π 2 t sin2 π 2 t + 1′ = 4π 3 √ 3 h sin2 π 2 t + tan2 π 2 t i ≥ 0 for every t ∈ (0, 2/3). This means that … view at source ↗

**Figure 3.** Figure 3: The plots on the left illustrate the obtained approximations of the eigenvalues along with the theoretical bounds; the plots on the right illustrate the errors of consistency of the approximations. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: The plot on the left illustrates the approximation of the two eigenfunctions corresponding to ρ = 0.5 in the case ϵ = 1; the plot on the right illustrates the approximation of the two eigenfunctions corresponding to ρ = 0.2 in the case ϵ = −1; Acknowledgements G. A. Veltri is a member of the “Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni” (GNAMPA) of the Istituto Nazio… view at source ↗

read the original abstract

We study the existence and localization of eigenvalue-eigenfunction pairs for parameter-dependent Neumann BVPs with a functional term. By reformulating the problems as a Hammerstein integral equation, we apply an existence and localization result and propose a convergent fixed-point iteration scheme. Finally, two pseudocodes and a MATLAB implementation are provided to numerically approximate the eigenvalues and validate the theoretical localization bounds. We also illustrate an approximation of the eigenfunctions for a fixed norm.

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

The paper gives a clean existence-localization result for eigenvalues of Neumann problems with functional terms by recasting them as Hammerstein equations and adding a convergent iteration plus MATLAB checks.

read the letter

The main takeaway is that this work takes parameter-dependent Neumann BVPs with a functional term, turns them into Hammerstein integral equations via the usual Green's function, applies an existing existence and localization theorem, and supplies a convergent fixed-point iteration with pseudocode and MATLAB runs to approximate the eigenvalues and eigenfunctions. That combination is what the authors deliver. The reformulation step is routine but necessary here because the functional term sits inside the nonlinearity, and the localization bounds plus the iteration scheme give something concrete that people can use. The numerical part is a plus; showing that the bounds hold in examples and providing the code makes the theoretical claim more believable than a pure existence paper would be. The abstract and structure indicate they verify the hypotheses of the cited theorem after the change of variables, so the pipeline looks internally consistent. The soft spots are limited. The result stays inside the standard template for this subfield, so the novelty is mainly in the specific application rather than a new theorem. Convergence of the iteration is claimed for the parameter ranges of interest, but readers will want to see how sensitive it is when the functional term grows or when the parameter moves near the boundary of the existence interval. The examples are illustrative, not a broad sweep, which is fine for validation but leaves open how the method behaves on harder cases. This paper is for people already working on nonlocal or functional BVPs in nonlinear analysis. A specialist who needs existence plus computable bounds for Neumann problems with this kind of term will get direct value from the reformulation and the code. It is not aimed at a wider audience. The work is coherent on its own terms and supplies both theory and numerics, so it deserves a serious referee rather than a desk reject. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper claims to study the existence and localization of eigenvalue-eigenfunction pairs for parameter-dependent Neumann BVPs with a functional term. By reformulating the problems as Hammerstein integral equations via the Green's function, it applies an existence and localization result from fixed-point theory and proposes a convergent fixed-point iteration scheme. Pseudocodes and a MATLAB implementation are provided to numerically approximate the eigenvalues, validate the theoretical localization bounds, and illustrate eigenfunction approximations for a fixed norm.

Significance. If the central results hold, the work extends eigenvalue theory for BVPs to include functional terms, combining analytical localization with a practical iterative scheme and numerical tools. This could aid in analyzing nonlocal or parameter-dependent problems in differential equations, with the numerical validation strengthening applicability.

major comments (2)

[§3] §3 (Main results): The verification that the reformulated Hammerstein integral operator satisfies all hypotheses of the cited existence and localization theorem is insufficiently detailed. In particular, the conditions on the functional term (e.g., growth, positivity, or cone-mapping properties) and the parameter range for λ need explicit checking against the theorem's assumptions, as this is load-bearing for the existence claim.
[§4] §4 (Numerical scheme): The convergence of the proposed fixed-point iteration is asserted but lacks a rigorous rate estimate or dependence on the functional term and parameter; without this, the claim that the scheme converges for the parameter ranges of interest remains unverified and affects the localization validation.

minor comments (2)

[§1] The notation distinguishing the functional term from the standard nonlinearity could be clarified in the introduction and preliminaries to prevent reader confusion.
Figure captions for the numerical examples should explicitly state the parameter values and norm used for the eigenfunction plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to improve the explicitness of the arguments where needed.

read point-by-point responses

Referee: [§3] §3 (Main results): The verification that the reformulated Hammerstein integral operator satisfies all hypotheses of the cited existence and localization theorem is insufficiently detailed. In particular, the conditions on the functional term (e.g., growth, positivity, or cone-mapping properties) and the parameter range for λ need explicit checking against the theorem's assumptions, as this is load-bearing for the existence claim.

Authors: We agree that a more detailed verification strengthens the main result. In the revised version of Section 3 we have inserted an explicit subsection that checks each hypothesis of the cited theorem against the concrete functional term: we verify the growth bound, positivity, and cone-invariance properties directly from the assumptions on the nonlinearity, and we delineate the precise interval of λ for which the operator maps the cone into itself. These verifications are now written out step by step rather than left implicit. revision: yes
Referee: [§4] §4 (Numerical scheme): The convergence of the proposed fixed-point iteration is asserted but lacks a rigorous rate estimate or dependence on the functional term and parameter; without this, the claim that the scheme converges for the parameter ranges of interest remains unverified and affects the localization validation.

Authors: We accept that an explicit convergence-rate estimate was missing. The iteration is a standard Picard iteration for the Hammerstein operator; under the contraction condition already established in Section 3 the Banach fixed-point theorem supplies a geometric rate whose constant depends on the Lipschitz modulus of the functional term and on λ. In the revised Section 4 we now derive this rate explicitly and show that it remains strictly less than one throughout the parameter interval used for the localization bounds, thereby justifying the numerical validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's core steps are: (1) reformulate the parameter-dependent Neumann BVP with functional term as a Hammerstein integral equation via the standard Green's function for the Neumann problem, (2) invoke an external existence/localization theorem for Hammerstein equations to obtain eigenvalue-eigenfunction pairs, and (3) construct a convergent fixed-point iteration whose convergence is justified by the same external theorem. No equation is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The cited existence result is treated as an independent black box; the numerical pseudocodes merely illustrate the bounds already guaranteed by the theorem. The derivation is therefore self-contained against external benchmarks and contains no reduction of the claimed result to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, no free parameters, ad-hoc axioms, or invented entities are introduced; the work relies on standard tools from integral equations and fixed-point theory.

pith-pipeline@v0.9.0 · 5358 in / 1130 out tokens · 53688 ms · 2026-05-10T14:16:27.417408+00:00 · methodology

discussion (0)

Reference graph

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