The random Kakutani fixed point theorem in random normed modules
Functional Analysis
2025-10-07 v2
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).
Cite
@article{arxiv.2509.15649,
title = {The random Kakutani fixed point theorem in random normed modules},
author = {Qiang Tu and Xiaohuan Mu and Tiexin Guo and Guang Yang and Yuanyuan Sun},
journal= {arXiv preprint arXiv:2509.15649},
year = {2025}
}