DOI: 10.24193/fpt-ro.2023.1.23

Authors: Xiao-Song Yang

Abstract: This note presents some new results on existence of zeros in continuous mappings for convex domains that can be regarded as generalizations of the Bolzano intermediate value theorem concerning the cube and the Borsuk-Ulam theorem for the unit ball. Based on the classic Borsuk-Ulam theorem, the main result is proved that under some antipodal-type inequality conditions any continuous map defined on a convex domain has zero point in the domain. As an application, we present a new proof of the well known Poincaré–Miranda theorem, also showing that the Borsuk-Ulam theorem implies the Poincaré–Miranda theorem.

Key Words and Phrases: Bolzano intermediate value theorem, the Borsuk-Ulam theorem, the Poincaré-Miranda theorem, continuous mappings, convex domains.

2010 Mathematics Subject Classification: 55M20, 54M25.

Published on-line: February 1st, 2023.

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