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arxiv: 1907.09378 · v2 · pith:VNUJOTHWnew · submitted 2019-07-22 · 🧮 math.FA

On an equation characterizing multi-cubic mappings and its stability and hyperstability

Abasalt Bodaghi , Behrouz Shojaee This is my paper

Pith reviewed 2026-05-24 17:47 UTC · model grok-4.3

classification 🧮 math.FA MSC 39B5239B8247H10
keywords multi-cubic mappingsfunctional equationHyers-Ulam stabilityhyperstabilityfixed point methodstability of functional equationsseveral variables
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The pith

Mappings cubic in each of several variables obey a functional equation that is Hyers-Ulam stable and hyperstable.

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

The paper introduces mappings of n variables that remain cubic when any n-1 variables are fixed. It first shows that any such mapping satisfies a specific functional equation obtained by expanding the cubic condition in each variable. The central technical step applies a fixed-point theorem in a complete metric space of all mappings equipped with a distance scaled by the size of the perturbation. This yields both ordinary Hyers-Ulam stability and the stronger hyperstability conclusion that the only mappings satisfying a vanishing perturbation condition are the exact multi-cubic ones.

Core claim

An n-variable mapping is multi-cubic precisely when it satisfies the expanded functional equation obtained by writing the cubic identity separately in each coordinate; the same equation is stable in the Hyers-Ulam sense because a suitable operator on the space of mappings is contractive with respect to a metric that records the deviation from the equation, and the resulting fixed point is exactly multi-cubic. The same argument immediately implies hyperstability.

What carries the argument

Fixed-point theorem applied to the complete metric space of mappings whose distance is defined by the supremum of the normalized deviation from the multi-cubic equation.

If this is right

  • Any mapping whose deviation from the multi-cubic equation is bounded by a suitable function of the variables admits a unique nearby exact multi-cubic mapping.
  • If the deviation tends to zero faster than a prescribed rate, the mapping must already be exactly multi-cubic.
  • The same fixed-point construction works for any n, so the stability result scales directly with the number of variables.
  • Hyperstability follows at once from the contraction mapping argument without additional hypotheses.

Where Pith is reading between the lines

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

  • The same technique can be tested on multi-quadratic or multi-additive mappings by writing the analogous expanded equations.
  • The result supplies a concrete way to decide whether a numerically observed mapping is close to a multi-cubic one by checking the size of the functional-equation residual.
  • Because the proof uses only completeness of the codomain and a contraction estimate, it applies verbatim to Banach spaces over any scalar field.

Load-bearing premise

The space of all mappings from the domain to the codomain, equipped with the metric that measures deviation from the functional equation, must be complete and the operator that rearranges the equation must have a fixed point under the given bound on the perturbation.

What would settle it

An explicit mapping that stays within a small but positive distance of satisfying the multi-cubic equation for all tuples yet lies at a positive distance from every exact multi-cubic mapping would falsify the stability claim.

read the original abstract

In this paper, we introduce $n$-variables mappings which are cubic in each variable. We show that such mappings satisfy a functional equation. The main purpose is to extend the applications of a fixed point method to establish the Hyers-Ulam stability for the multi-cubic mappings. As a consequence, we prove that a multi-cubic functional equation can be hyperstable.

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

0 major / 3 minor

Summary. The paper defines multi-cubic mappings (cubic in each variable separately) on vector spaces, derives the functional equation that characterizes them, and applies a fixed-point theorem in a suitable complete metric space to prove Hyers-Ulam stability of the equation; as a corollary it obtains hyperstability.

Significance. If the derivations hold, the work supplies a concrete extension of fixed-point stability techniques to the multi-variable cubic setting, giving both a characterizing equation and explicit stability/hyperstability results that fit the existing literature on Hyers-Ulam stability for polynomial functional equations.

minor comments (3)
  1. The abstract states that the mappings 'satisfy a functional equation' but does not display the equation itself; the introduction or §2 should state the precise multi-cubic equation (presumably something of the form f(x+y+...) + ... = 8f(x) + ... or the standard cubic form) so that the stability statement is immediately readable.
  2. The fixed-point argument is described as 'direct application'; the manuscript should verify explicitly that the operator T defined on the space of mappings satisfies the contraction condition (e.g., d(Tf,Tg) ≤ k d(f,g) with k<1) under the stated perturbation hypotheses, citing the precise metric and the value of k.
  3. Notation for the n-variable case (e.g., the multi-index or the separate cubic conditions) should be introduced once and used consistently; a short table or displayed equation summarizing the cubic condition in each variable would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on multi-cubic mappings, their characterizing equation, and the fixed-point approach to Hyers-Ulam stability and hyperstability. The recommendation of minor revision is noted; however, no specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines multi-cubic mappings, derives the functional equation they satisfy from first principles by direct substitution into the multi-variable cubic condition, and then invokes an external fixed-point theorem (standard in the Hyers-Ulam literature) to obtain stability and hyperstability. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or definitional tautology; the fixed-point application is a direct, non-circular use of a known contraction-mapping result on a suitably metrized function space. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard tools from functional analysis; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The underlying space is a complete normed vector space allowing application of fixed point theorems for stability.
    Implicit in the use of fixed point methods for Hyers-Ulam stability.

pith-pipeline@v0.9.0 · 5581 in / 944 out tokens · 34179 ms · 2026-05-24T17:47:23.741084+00:00 · methodology

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Reference graph

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