Extensions of Antipodal-Type Theorems

The paper develops the extensions of both the antipodal (Borsuk-Ulam) theorem and Browder theorem to the cases embracing star-shaped domains of the studied mappings and a multi-valued nature of the latter. To be more specific, by making use of the triangulation procedure, we spread out the antipodal and fixed-point theorems to the case of not necessarily convex (star-shaped) domains. In addition, similar extensions are obtained for multi-valued mappings defined over star-shaped sets. Moreover, a direct algorithm shaping the required connected path of the zero points of the mapping has been designed, and its convergence demonstrated.