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arxiv: 2604.05603 · v1 · submitted 2026-04-07 · 🧮 math.FA

Hartman-Stampacchia theorem, Gale-Nikaido-Debreu lemma, and Brouwer and Kakutani fixed-point theorems

Pascal Gourdel (PSE , CES) , Cuong Le Van (CNRS , PSE , Ngoc-Sang Pham (EM Normandie) , Cuong Tran Viet (UP1 UFR02 This is my paper

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classification 🧮 math.FA
keywords Hartman-Stampacchia theoremGale-Nikaido-Debreu lemmaBrouwer fixed-point theoremKakutani fixed-point theoremequivalence of theoremsvariational inequalitiescomplementarity problems
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The pith

The Hartman-Stampacchia theorem is equivalent to the Gale-Nikaidô-Debreu lemma and to the Brouwer and Kakutani fixed-point theorems.

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

The paper demonstrates that these results can each be derived from the others, forming a closed loop of logical implications. It begins by taking the Hartman-Stampacchia theorem as the base and deriving the Gale-Nikaidô-Debreu lemma from it, then shows the remaining directions that close the cycle with the classical fixed-point theorems. A reader would care because each of these statements is routinely invoked to guarantee the existence of solutions in optimization, game theory, and equilibrium models; once any one is available, the others follow without separate proofs.

Core claim

The paper uses the Hartman-Stampacchia theorems as the primary tool to prove the Gale-Nikaidô-Debreu lemmas and establishes a cycle of equivalences among the Hartman-Stampacchia theorems, the Gale-Nikaidô-Debreu lemmas, and Kakutani and Brouwer fixed-point theorems.

What carries the argument

The cycle of equivalences that reduces each of the four statements to the others under the maintained hypotheses of convexity, compactness, and continuity or upper hemicontinuity.

If this is right

  • A direct proof of any one theorem immediately supplies proofs for the Gale-Nikaidô-Debreu lemma and both fixed-point theorems.
  • Existence results in economic equilibrium models or variational problems can invoke whichever statement is simplest to verify in the given setting.
  • Standard textbooks that list these results separately can be shortened by presenting only one and deriving the rest.

Where Pith is reading between the lines

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

  • Applied researchers could routinely check the fixed-point version in finite-dimensional cases and switch to the Hartman-Stampacchia form only when the problem is posed as a variational inequality.
  • Similar reduction arguments might link these statements to other existence theorems such as the Knaster-Kuratowski-Mazurkiewicz lemma if the same compactness and convexity hypotheses are kept.
  • In settings where one theorem has already been generalized to non-compact or non-convex domains, the cycle would automatically transport those generalizations to the other three results.

Load-bearing premise

The sets involved remain convex and compact while the maps remain continuous or upper hemicontinuous.

What would settle it

A single convex compact set together with a continuous map for which the Hartman-Stampacchia theorem holds but the Brouwer fixed-point theorem fails would break the claimed cycle.

Figures

Figures reproduced from arXiv: 2604.05603 by CES), Cuong Le Van (CNRS, Cuong Tran Viet (UP1 UFR02, Ngoc-Sang Pham (EM Normandie), Pascal Gourdel (PSE, PSE.

Figure 1
Figure 1. Figure 1: The cycle of equivalences: Hartman–Stampacchia theorem (Theorem 2.1, HS1), generalized version of Hartman–Stampacchia for correspondence (Theorem 2.2, HS2), GND lemma, Kakutani and Brouwer fixed-point theorems. incomplete financial assets. Our paper contributes to this literature because the GND lemma is key for equilibrium existence. There have been many efforts to provide a proof of the existence result … view at source ↗
Figure 2
Figure 2. Figure 2: Illustrate the vector a. Proof of Claim Since P is not a subspace, there is some x ∈ P, but x /∈ −P. Define y to be the orthogonal projection of x onto −P. Let a = y + (−x). Since x /∈ −P and −P is closed, it follows that a 6= 0N . On one hand, because −P is a convex cone, y and −x belong to −P, hence a belongs −P. On the other hand, by the choice of y, hy − x, y − yi ≤ 0 for all y ∈ −P [PITH_FULL_IMAGE:f… view at source ↗
Figure 3
Figure 3. Figure 3: Illustrate the variable λa(x). ||x + λa(x)(x − a)|| = 1. (5) This leads the quadratic equation: ||x − a||2λ 2 a (x) + 2hx, x − aiλa(x) + ||x||2 − 1 = 0. (6) [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
read the original abstract

This paper uses the Hartman-Stampacchia theorems as the primary tool to prove the Gale-Nikaid{\^o}-Debreu lemmas. It also establishes a cycle of equivalences among the Hartman-Stampacchia theorems, the Gale-Nikaid{\^o}-Debreu lemmas, and Kakutani and Brouwer fixed-point theorems.

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 / 2 minor

Summary. The manuscript uses the Hartman-Stampacchia theorem as the base to derive the Gale-Nikaidô-Debreu lemma and closes a cycle of equivalences among the Hartman-Stampacchia theorems, the Gale-Nikaidô-Debreu lemmas, and the Brouwer and Kakutani fixed-point theorems, all under the standard hypotheses of convexity, compactness, and continuity/upper hemicontinuity.

Significance. If the derivations are complete and gap-free, the work supplies a unified logical framework connecting variational inequalities, equilibrium existence, and fixed-point theory. This could streamline textbook presentations and support extensions in nonlinear analysis and mathematical economics. The choice of Hartman-Stampacchia as the primitive avoids the circularity that often appears in such equivalence cycles.

minor comments (2)
  1. The title refers to the 'Hartman-Stampacchia theorem' (singular) while the abstract and body discuss 'theorems' (plural); add a clarifying sentence in §1 on which variants are treated.
  2. Notation for the dual pairing and the sets K and C is introduced without an explicit list of standing assumptions; insert a short 'Notation and assumptions' paragraph before the first theorem statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The referee's summary accurately reflects the manuscript's approach of taking the Hartman-Stampacchia theorem as the primitive result to derive the Gale-Nikaidô-Debreu lemma while closing the equivalence cycle with the Brouwer and Kakutani theorems under standard convexity, compactness, and continuity assumptions. We provide a point-by-point response below.

read point-by-point responses

  1. Referee: The manuscript uses the Hartman-Stampacchia theorem as the base to derive the Gale-Nikaidô-Debreu lemma and closes a cycle of equivalences among the Hartman-Stampacchia theorems, the Gale-Nikaidô-Debreu lemmas, and the Brouwer and Kakutani fixed-point theorems, all under the standard hypotheses of convexity, compactness, and continuity/upper hemicontinuity.

    Authors: We appreciate this precise summary of our contributions. The manuscript indeed establishes the cycle by deriving the Gale-Nikaidô-Debreu lemma directly from the Hartman-Stampacchia theorem and then showing the remaining equivalences, thereby avoiding circularity. All proofs are carried out under the stated hypotheses, and we believe the derivations are complete and gap-free as presented in the full text. revision: no

Circularity Check

0 steps flagged

No significant circularity in the equivalence cycle

full rationale

The paper derives the Gale-Nikaidô-Debreu lemma from the Hartman-Stampacchia theorem and then closes a cycle of equivalences with the Brouwer and Kakutani fixed-point theorems. All steps rely on standard implications under the retained hypotheses of convexity, upper hemicontinuity, and compactness; none of the derivations reduces by construction to a fitted parameter, self-definition, or a load-bearing self-citation whose content is itself unverified. The Hartman-Stampacchia theorem functions as an independent base rather than a result defined in terms of the others, so the claimed cycle consists of ordinary logical equivalences already present in the variational-inequality literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests entirely on the standard axioms and hypotheses of the cited classical theorems (convexity, continuity, compactness in topological vector spaces) without introducing new free parameters or postulated entities.

axioms (1)
  • domain assumption Convexity of the underlying sets and continuity/upper hemicontinuity of the maps involved
    These are the usual hypotheses required for Hartman-Stampacchia, Gale-Nikaido-Debreu, and Kakutani/Brouwer theorems to hold.

pith-pipeline@v0.9.0 · 5385 in / 1134 out tokens · 51604 ms · 2026-05-10T19:28:42.434695+00:00 · methodology

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