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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 793848, 9 pages
http://dx.doi.org/10.1155/2011/793848
Research Article

The Bolzano-Poincaré Type Theorems

College of Science, Cardinal Stefan Wyszyński, University in Warsaw, ul. Dewajtis 5, 01-815 Warszawa, Poland

Received 24 February 2011; Revised 7 May 2011; Accepted 30 May 2011

Academic Editor: Giuseppe Marino

Copyright © 2011 Przemysław Tkacz and Marian Turzański. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In 1883–1884, Henri Poincaré announced the result about the structure of the set of zeros of function 𝑓 ∢ 𝐼 𝑛 β†’ 𝑅 𝑛 , or alternatively the existence of solutions of the equation 𝑓 ( π‘₯ ) = 0 . In the case 𝑛 = 1 the Poincaré Theorem is well known Bolzano Theorem. In 1940 Miranda rediscovered the Poincaré Theorem. Except for few isolated results it is essentially a non-algorithmic theory. The aim of this article is to introduce an algorithmical proof of the Theorem “On the existence of a chain” and for 𝑛 = 3 an algorithmical proof of the Bolzano-Poincaré Theorem and to show the equivalence of Poincaré, Brouwer and “On the existence of a chain” theorems.