{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:45:17Z","timestamp":1753886717082,"version":"3.41.2"},"reference-count":10,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,10,12]],"date-time":"2011-10-12T00:00:00Z","timestamp":1318377600000},"content-version":"vor","delay-in-days":284,"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2011,1]]},"abstract":"In 1883\u20131884, Henri Poincar\u00e9 announced the result about the structure of the set of zeros of function f<\/jats:italic>\u2009 : \u2009I<\/jats:italic>n<\/jats:italic><\/jats:sup> \u2192 R<\/jats:italic>n<\/jats:italic><\/jats:sup>, or alternatively the existence of solutions of the equation f<\/jats:italic>(x<\/jats:italic>) = 0. In the case n<\/jats:italic> = 1 the Poincar\u00e9 Theorem is well known Bolzano Theorem. In 1940 Miranda rediscovered the Poincar\u00e9 Theorem. Except for few isolated results it is essentially a non\u2010algorithmic theory. The aim of this article is to introduce an algorithmical proof of the Theorem \u201cOn the existence of a chain\u201d and for n<\/jats:italic> = 3 an algorithmical proof of the Bolzano\u2010Poincar\u00e9 Theorem and to show the equivalence of Poincar\u00e9, Brouwer and \u201cOn the existence of a chain\u201d theorems.<\/jats:p>","DOI":"10.1155\/2011\/793848","type":"journal-article","created":{"date-parts":[[2011,10,12]],"date-time":"2011-10-12T19:02:31Z","timestamp":1318446151000},"update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Bolzano\u2010Poincar\u00e9 Type Theorems"],"prefix":"10.1155","volume":"2011","author":[{"given":"Przemys\u0142aw","family":"Tkacz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marian","family":"Turza\u0144ski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2011,10,12]]},"reference":[{"volume-title":"Kalejdoskop Matematyczny","year":"1954","author":"Steinhaus H.","key":"e_1_2_4_1_2"},{"volume-title":"Mathematical Snapshot","year":"1969","author":"Steinhaus H.","key":"e_1_2_4_2_2"},{"key":"e_1_2_4_3_2","first-page":"57","article-title":"A discrete form of Jordan curve theorem","author":"Sur\u00f3wka W.","year":"1993","journal-title":"Annales Mathematicae Silesianae"},{"key":"e_1_2_4_4_2","series-title":"Mathematical World","doi-asserted-by":"crossref","DOI":"10.1090\/mawrld\/002","volume-title":"Fixed Points","author":"Shaskin Y. 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