Extended Antipodal Theorems
Since 1909 when Brouwer proved the first fixed-point theorem named after him, the fixed-point results in various settings play an important role in the optimization theory and applications. This technique has proven to be indispensable for the proofs of multiple results related to the existence of solutions to numerous problems in the areas of optimization and approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, mathematical economics, etc. It is also worthwhile to mention that the majority of problems of finding solutions (zero-points) of functions (operators) can be easily reduced to that of discovering of fixed points of properly modified mappings. Not only theoretical but also practical (algorithmic) developments are based on the fixed-point theory. For instance, the well-known simplicial (triangulation) algorithms help one to find the desired fixed points in a constructive way. That approach allows one to investigate the solvability of complicated problems arising in theory and applications. In this paper, making use of the triangulation technique, we extend some antipodal and fixed-point theorems to the case of nonconvex, more exactly, star-shaped sets. Also, similar extensions are made for set-valued mappings defined over star-shaped sets.