The random Kakutani fixed point theorem in random normed modules

Abstract: Based on the recently developed theory of random sequential compactness, we prove the random Kakutani fixed point theorem in random normed modules: if G is a random sequentially compact L0-convex subset of a random normed module, then every -stable Tc-upper semicontinuous mapping F:G to 2G such that F(x) is closed and L0-convex for each x in G, has a fixed point. This is the first fixed point theorem for set-valued mappings in random normed modules, providing a random generalization of the classical Kakutani fixed point theorem as well as a set-valued extension of the noncompact Schauder fixed point theorem established in Math. Ann. 391(3), 3863--3911 (2025).