{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T18:03:54Z","timestamp":1775239434265,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T00:00:00Z","timestamp":1657756800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T00:00:00Z","timestamp":1657756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2022,9]]},"DOI":"10.1007\/s10957-022-02067-2","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T16:03:02Z","timestamp":1657814582000},"page":"1072-1080","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Direct Proof of the Gale\u2013Nikaido\u2013Debreu Lemma Using Sperner\u2019s Lemma"],"prefix":"10.1007","volume":"194","author":[{"given":"Thanh","family":"Le","sequence":"first","affiliation":[]},{"given":"Cuong","family":"Le Van","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9037-3019","authenticated-orcid":false,"given":"Ngoc-Sang","family":"Pham","sequence":"additional","affiliation":[]},{"given":"Cagri","family":"Saglam","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,7,14]]},"reference":[{"key":"2067_CR1","volume-title":"Infinite Dimensional Analysis: A Hitchhiker\u2019s Guide","author":"CD Aliprantis","year":"2006","unstructured":"Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker\u2019s Guide, 3rd edn. Springer, Berlin (2006)","edition":"3"},{"key":"2067_CR2","doi-asserted-by":"publisher","first-page":"265","DOI":"10.2307\/1907353","volume":"22","author":"KJ Arrow","year":"1954","unstructured":"Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265\u2013290 (1954)","journal-title":"Econometrica"},{"key":"2067_CR3","doi-asserted-by":"crossref","unstructured":"Ben-El-Mechaiekh, H., Bich, P., Florenzano, M.: General equilibrium and fixed point theory: a partial survey. CES Working Paper Series (2009)","DOI":"10.1007\/s11784-009-0129-2"},{"key":"2067_CR4","volume-title":"Espaces Topologiques et Fonctions Multivoques","author":"C Berge","year":"1959","unstructured":"Berge, C.: Espaces Topologiques et Fonctions Multivoques. Dunod, Paris (1959)"},{"key":"2067_CR5","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/BF01456931","volume":"71","author":"LEJ Brouwer","year":"1911","unstructured":"Brouwer, L.E.J.: Uber Abbildung von Mannigfaltigkeiten. Math. Ann. 71, 97\u2013115 (1911)","journal-title":"Math. Ann."},{"key":"2067_CR6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511625756","volume-title":"Fixed Point Theorems with Applications to Economics and Game Theory","author":"KC Border","year":"1985","unstructured":"Border, K.C.: Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, Cambridge (1985)"},{"issue":"1","key":"2067_CR7","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/BF01449883","volume":"64","author":"C Carath\u00e9odory","year":"1907","unstructured":"Carath\u00e9odory, C.: Uber den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64(1), 95\u2013115 (1907). (in German)","journal-title":"Math. Ann."},{"key":"2067_CR8","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1016\/S0021-9800(67)80062-0","volume":"2","author":"DIA Cohen","year":"1967","unstructured":"Cohen, D.I.A.: On Sperner lemma. J. Comb. Theory 2, 585\u2013587 (1967)","journal-title":"J. Comb. Theory"},{"issue":"10","key":"2067_CR9","doi-asserted-by":"publisher","first-page":"886","DOI":"10.1073\/pnas.38.10.886","volume":"38","author":"G Debreu","year":"1952","unstructured":"Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. 38(10), 886\u2013893 (1952)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"11","key":"2067_CR10","doi-asserted-by":"publisher","first-page":"876","DOI":"10.1073\/pnas.42.11.876","volume":"42","author":"G Debreu","year":"1956","unstructured":"Debreu, G.: Market equilibrium. Proc. Natl. Acad. Sci. 42(11), 876\u2013878 (1956)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"2067_CR11","volume-title":"Theory of Value: An Axiomatic Analysis of Economic Equilibrium","author":"G Debreu","year":"1959","unstructured":"Debreu, G.: Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Wiley, New York (1959)"},{"key":"2067_CR12","volume-title":"Handbook of Mathematical Economics","author":"G Debreu","year":"1982","unstructured":"Debreu, G.: Existence of competitive equilibrium. In: Arrow, K., Kirman, A. (eds.) Handbook of Mathematical Economics, vol. II. North-Holland, Amsterdam (1982)"},{"key":"2067_CR13","doi-asserted-by":"publisher","DOI":"10.2307\/j.ctt5vjv6v","volume-title":"Finding Equilibrium: Arrow, Debreu, McKenzie and the Problem of Scientific Credit","author":"T Duppe","year":"2014","unstructured":"Duppe, T., Weintraud, E.R.: Finding Equilibrium: Arrow, Debreu, McKenzie and the Problem of Scientific Credit. Princeton University Press, Princeton (2014)"},{"key":"2067_CR14","unstructured":"Florenzano, M.: L\u2019\u00c9quilibre \u00e9conomique g\u00e9n\u00e9ral transitif et intransitif: problemes d\u2019existence. Monographies du S\u00e9minaire d\u2019\u00c9conom\u00e9trie, Editions du CNRS (1981)"},{"key":"2067_CR15","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1016\/0304-4068(82)90022-2","volume":"9","author":"M Florenzano","year":"1982","unstructured":"Florenzano, M.: The Gale\u2013Nikaido\u2013Debreu lemma and the existence of transitive equilibrium with or without the free-disposal assumption. J. Math. Econ. 9, 113\u2013134 (1982)","journal-title":"J. Math. Econ."},{"key":"2067_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5315-8","volume-title":"General Equilibrium Analysis: Existence and Optimality Properties of Equilibria","author":"M Florenzano","year":"2003","unstructured":"Florenzano, M.: General Equilibrium Analysis: Existence and Optimality Properties of Equilibria. Springer, Berlin (2003)"},{"key":"2067_CR17","doi-asserted-by":"crossref","unstructured":"Florenzano, M.: Two lemmas that changed general equilibrium theory. CES Working Paper Series (2009)","DOI":"10.1016\/j.geb.2009.04.009"},{"key":"2067_CR18","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/0165-1765(86)90213-2","volume":"22","author":"M Florenzano","year":"1986","unstructured":"Florenzano, M., Le Van, C.: A note on the Gale\u2013Nikaido\u2013Debreu lemma and the existence of general equilibrium. Econ. Lett. 22, 107\u2013110 (1986)","journal-title":"Econ. Lett."},{"key":"2067_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-56522-9","volume-title":"Finite Dimensional Convexity and Optimization","author":"M Florenzano","year":"2001","unstructured":"Florenzano, M., Le Van, C.: Finite Dimensional Convexity and Optimization. Springer, Berlin (2001)"},{"key":"2067_CR20","doi-asserted-by":"publisher","first-page":"155","DOI":"10.7146\/math.scand.a-10436","volume":"3","author":"D Gale","year":"1955","unstructured":"Gale, D.: The law of supply and demand. Math. Scand. 3, 155\u2013169 (1955)","journal-title":"Math. Scand."},{"key":"2067_CR21","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1016\/0304-4068(75)90009-9","volume":"2","author":"D Gale","year":"1975","unstructured":"Gale, D., Mas-Colell, A.: An equilibrium existence theorem for a general model without ordered preferences. J. Math. Econ. 2, 9\u201315 (1975)","journal-title":"J. Math. Econ."},{"key":"2067_CR22","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1016\/0304-4068(79)90015-6","volume":"6","author":"D Gale","year":"1975","unstructured":"Gale, D., Mas-Colell, A.: Corrections to an equilibrium existence theorem for a general model without ordered preferences. J. Math. Econ. 6, 297\u2013298 (1975)","journal-title":"J. Math. Econ."},{"key":"2067_CR23","first-page":"437","volume-title":"Introduction \u00e0 la Th\u00e9orie des Fonctions d\u2019une Variable","author":"J Hadamard","year":"1910","unstructured":"Hadamard, J.: Note sur quelques applications de l\u2019indice de Kronecker. In: Tannery, J. (ed.) Introduction \u00e0 la Th\u00e9orie des Fonctions d\u2019une Variable, vol. 2, 2nd edn., pp. 437\u2013477. A. Hermann & Fils, Paris (1910)","edition":"2"},{"issue":"3","key":"2067_CR24","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1215\/S0012-7094-41-00838-4","volume":"8","author":"S Kakutani","year":"1941","unstructured":"Kakutani, S.: A generalization of Brouwer\u2019s fixed point theorem. Duke Math. J. 8(3), 457\u2013459 (1941)","journal-title":"Duke Math. J."},{"key":"2067_CR25","doi-asserted-by":"publisher","first-page":"590","DOI":"10.1257\/jel.20191247","volume":"59","author":"MA Khan","year":"2021","unstructured":"Khan, M.A.: On the finding of an equilibrium: Duppe\u2013Weintraud and the problem of scientific credit. J. Econ. Lit. 59, 590\u2013633 (2021)","journal-title":"J. Econ. Lit."},{"key":"2067_CR26","doi-asserted-by":"publisher","first-page":"132","DOI":"10.4064\/fm-14-1-132-137","volume":"14","author":"B Knaster","year":"1929","unstructured":"Knaster, B., Kuratowski, K., Mazurkiewicz, S.: A Ein Beweis des Fixpunktsatzes fur n-Dimensionale Simplexe. Fund. Math. 14, 132\u2013137 (1929)","journal-title":"Fund. Math."},{"issue":"1","key":"2067_CR27","doi-asserted-by":"publisher","first-page":"143","DOI":"10.7146\/math.scand.a-10463","volume":"4","author":"HW Kuhn","year":"1956","unstructured":"Kuhn, H.W.: A note on \u201cthe law of supply and demand\u2019\u2019. Math. Scand. 4(1), 143\u2013146 (1956)","journal-title":"Math. Scand."},{"key":"2067_CR28","doi-asserted-by":"publisher","first-page":"1238","DOI":"10.1073\/pnas.61.4.1238","volume":"61","author":"HW Kuhn","year":"1968","unstructured":"Kuhn, H.W.: Simplicial approximations of fixed points. Proc. Natl. Acad. Sci. U. S. A. 61, 1238\u20131242 (1968)","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2067_CR29","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/BF00935276","volume":"37","author":"C Le Van","year":"1982","unstructured":"Le Van, C.: Topological degree and the Sperner lemma. J. Optim. Theory Appl. 37, 371\u2013377 (1982)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2067_CR30","doi-asserted-by":"publisher","first-page":"361","DOI":"10.2307\/1969615","volume":"63","author":"E Michael","year":"1956","unstructured":"Michael, E.: Continuous selections. I. Ann. Math. Second Ser. 63(2), 361\u2013382 (1956)","journal-title":"Ann. Math. Second Ser."},{"key":"2067_CR31","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511794216","volume-title":"Game Theory","author":"M Maschler","year":"2013","unstructured":"Maschler, M., Solan, E., Zamir, S.: Game Theory. Cambridge University Press, Cambridge (2013)"},{"key":"2067_CR32","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1111\/j.1467-999X.1956.tb00099.x","volume":"8","author":"H Nikaido","year":"1956","unstructured":"Nikaido, H.: On the classical multilateral exchange problem. Metroeconomica 8, 135\u2013145 (1956)","journal-title":"Metroeconomica"},{"issue":"3","key":"2067_CR33","first-page":"187","volume":"27","author":"S Park","year":"1999","unstructured":"Park, S.: Ninety years of the Brouwer fixed point theorem. Vietnam J. Math. 27(3), 187\u2013222 (1999)","journal-title":"Vietnam J. Math."},{"key":"2067_CR34","unstructured":"Park, S., Jeong, K.S.: The proof of the Sperner lemma from the Brouwer fixed point theorem. https:\/\/www.researchgate.net\/publication\/264969230 (2003)"},{"key":"2067_CR35","volume-title":"The Computation of Economic Equilibria","author":"H Scarf","year":"1973","unstructured":"Scarf, H., Hansen, T.: The Computation of Economic Equilibria. Yale University Press, New Haven (1973)"},{"key":"2067_CR36","volume-title":"Handbook of Mathematical Economics","author":"H Scarf","year":"1982","unstructured":"Scarf, H.: The computation of equilibrium prices: an exposition. In: Arrow, K., Kirman, A. (eds.) Handbook of Mathematical Economics, vol. II. North-Holland, Amsterdam (1982)"},{"key":"2067_CR37","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/BF02940617","volume":"6","author":"E Sperner","year":"1928","unstructured":"Sperner, E.: Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes. Abh. Math. Semin. Univ. Hambg. 6, 265\u2013272 (1928)","journal-title":"Abh. Math. Semin. Univ. Hambg."},{"issue":"10","key":"2067_CR38","doi-asserted-by":"publisher","first-page":"930","DOI":"10.2307\/2589747","volume":"106","author":"FE Su","year":"1999","unstructured":"Su, F.E.: Rental Harmony: Sperner lemma in Fair Division. Am. Math. Mon. 106(10), 930\u2013942 (1999)","journal-title":"Am. Math. Mon."},{"key":"2067_CR39","first-page":"59","volume":"13","author":"H Uzawa","year":"1962","unstructured":"Uzawa, H.: Walras existence theorem and Brouwer\u2019s fixed point theorem. Econ. Stud. Q. 13, 59\u201362 (1962)","journal-title":"Econ. Stud. Q."},{"key":"2067_CR40","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/0097-3165(74)90031-4","volume":"17","author":"M Yoseloff","year":"1974","unstructured":"Yoseloff, M.: Topological proofs of some combinatorial theorems. J. Comb. Theory Ser. A 17, 95\u2013111 (1974)","journal-title":"J. Comb. Theory Ser. A"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02067-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-022-02067-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02067-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T14:25:41Z","timestamp":1658845541000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-022-02067-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,14]]},"references-count":40,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,9]]}},"alternative-id":["2067"],"URL":"https:\/\/doi.org\/10.1007\/s10957-022-02067-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,14]]},"assertion":[{"value":"23 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 July 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}