随机赋范模中的非紧Schauder不动点定理及其应用
The noncompact Schauder fixed point theorem in random normed modules and its applications
AI总结：
本文在随机赋范模中引入随机序列紧集，证明非紧Schauder不动点定理，并应用于条件信息下Nash均衡点的存在性。
摘要：
受经典Bolzano--Weierstrass定理的随机化版本的启发，本文首先在随机赋范模中引入随机序列紧集的概念，并系统地发展了相关理论。基于这些发展，我们证明了相应的Schauder不动点定理：设$E$为随机赋范模，$G$为$E$中的随机序列紧$L^0$--凸集，则从$G$到$G$的每个$\sigma$--稳定连续映射都有不动点，该定理统一了此前Schauder不动点定理的所有随机推广。作为该定理的应用之一，我们证明了条件信息背景下Nash均衡点的存在性。需要指出的是，本文的主要挑战在于克服非紧性，因为随机序列紧集通常是非紧的。
英文摘要：
Motivated by the randomized version of the classical Bolzano--Weierstrass theorem, in this paper we first introduce the notion of a random sequentially compact set in a random normed module and develop the related theory systematically. From these developments, we prove the corresponding Schauder fixed point theorem: let $E$ be a random normed module and $G$ a random sequentially compact $L^0$--convex set of $E$, then every $σ$--stable continuous mapping from $G$ to $G$ has a fixed point, which unifies all the previous random generalizations of the Schauder fixed point theorem. As one of the applications of the theorem, we prove the existence of Nash equilibrium points in the context of conditional information. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a random sequentially compact set is generally noncompact.
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