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

TL;DR

This paper demonstrates the use of Hartman-Stampacchia theorems to prove Gale-Nikaido-Debreu lemmas and establishes equivalences among these theorems and fixed-point theorems.

Contribution

It introduces a novel approach linking Hartman-Stampacchia theorems with classical fixed-point theorems and related lemmas.

Findings

Proves Gale-Nikaido-Debreu lemmas using Hartman-Stampacchia theorems

Establishes equivalences among key theorems in fixed-point theory

Creates a cycle of logical implications among these fundamental results

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.