Kakutani–Fan–Glicksberg type results in non-separated spaces

In this paper we establish a von Neumann–Fan type intersection theorem and present several applications to the existence and structure of solutions to a Nash equilibrium problem and a von Neumann–Sion–Fan type theorem under assumptions that are less restrictive than the classical ones. This is done through analysis of the trajectories of an upper semicontinuous convex set-valued map with nonempty closed values defined on a nonempty bounded closed convex subset of a (not necessarily Hausdorff) locally convex space. We also answer Open Problem 5.10 of A.T.-M. Lau and L. Yao (2017) [16] in affirmative.