2022 International Conference on Applied Mathematics, Informatics, and Computing Sciences (AMICS 2022)

Abstract: The central problems of this talk are extensions of certain antipodal and fixed-point theorems to the cases of non-convex sets (star-shaped sets, to be more exact). The techniques used in the fixed-point theorems are widely used in Operations Research, Mathematical Programming, Theory of Games and in many other areas of Optimization Theory and its Applications. These techniques are appropriate when establishing the existence of solutions in the problems of Mathematical Programming, Convex Games, mathematical programs with equilibrium constraints (MPEC), naming only few of the areas of applications. Because of that, any extensions of the classical theorems of fixed points (like the theorem by Brouwer for the single-valued functions and the theorem by Kakutani for multi-valued functions) are very interesting, important and enjoy numerous applications.
The Fixed-Point Theorem by Brouwer and the Theorem by Borsuk-Ulam have been characterized as two powerful topological tools of a very similar structure. In the books of topology such as Krasnosel’sky (1964), Herings (1996), Yang (1999) these two theorems are listed as well as the relationships between them (in fact, the Theorem by Borsuk-Ulam implies the Fixed-Point theorem by Brouwer). The majority of the fixed-point theorems deal with the results for the functions defined over the convex sets. However, in many applied problems, the domains of the involved functions are not necessarily convex; for example, the feasible sets in bilevel programming lack this property even in linear bilevel programming problems (cf., Dempe, S. (2002).
This work presents various extensions for the Borsuk-Ulam antipodal theorem and the Browder theorem to the case when the domains of the involved applications are in the star-shaped sets (that is, are not convex) and the structures of the mappings are multi-valued. Even more, this work contains the explicit description of an algorithm of construction of the zero-path, and in this way it extends the Browder theorem.