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arXiv · 2509.15649

The random Kakutani fixed point theorem in random normed modules

Qiang Tu·Xiaohuan Mu·Tiexin Guo·Guang Yang·Yuanyuan Sun
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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).

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Qiang Tu, Xiaohuan Mu, Tiexin Guo, Guang Yang, Yuanyuan Sun. 2025-09-19. The random Kakutani fixed point theorem in random normed modules. https://arxiv.org/abs/2509.15649

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