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

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