Hartman-Stampacchia theorems, Gale-Nikaido-Debreu lemmas, and Brouwer/Kakutani fixed-point theorems are equivalent via a cycle of implications.
Journal of Mathematica l Analysis and Applications 457, 1353–1369 (2018)
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Hartman-Stampacchia theorem, Gale-Nikaido-Debreu lemma, and Brouwer and Kakutani fixed-point theorems
Hartman-Stampacchia theorems, Gale-Nikaido-Debreu lemmas, and Brouwer/Kakutani fixed-point theorems are equivalent via a cycle of implications.