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 σ-stable T_c-upper semicontinuous mapping F:G -> 2^G \ {∅} 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 [Guo et al., Math. Ann. 391(3), 3863--3911 (2025)].

This work was supported by the National Natural Science Foundation of China (Grant Nos.12371141, 12426645, 12426654) and the Natural Science Foundation of Hunan Province of China (Grant No.2023JJ30642).