On topological existence theorems and applications to optimization-related problems

In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization‐related problems as examples of applications of these results: quasivariational inclusions, Stampacchia‐type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.