The Bolzano–Poincaré–Miranda theorem – discrete version

In 1883–1884, Henri Poincaré [4,5] published the result about the structure of the set of zeros of function f:In→Rn. In the case n=1 the Poincaré theorem is well known as the Bolzano theorem. In 1940 Miranda [3], (for more informations see Kulpa, 1997 [2], ) rediscovered the Poincaré theorem and proved that the Bolzano–Poincaré–Miranda theorem and the Brouwer fixed point theorem are equivalent. The same type theorem was published in 1938 by Eilenberg and Otto [1], and is known as the theorem on partitions and is used as a characterization of the covering dimension. Except for few isolated results (Scarf, 1973/2000 [6]) it is essentially a non-algorithmic theory. The aim of this article is to present a discrete version of the Bolzano–Poincaré–Miranda theorem and show that this discrete version could be the main tool in proving fixed point theorems.