The noncompact Schauder fixed point theorem in random normed modules and its applications

Abstract

Random normed modules (briefly, RN modules) are a random generalization of ordinary normed spaces, whose L 0 -norm induces two kinds of most useful topologies (called the (ε, λ)-topology and the locally L 0 -convex topology). The purpose of this paper is to generalize the classical Schauder fixed point theorem to RN modules under the two kinds of topologies. Motivated by the randomized version of the classical Bolzano-Weierstrass theorem, we first systematically and deeply study the random sequential compactness under the (ε, λ)-topology and random total boundedness under the locally L 0convex topology for a σ-stable subset of a σ-stable RN module, establishing the Hausdorff theorem on their equivalence, which allows us to construct the well defined random Schauder projection and countably many decompositions of the mapping in question so that we can prove Schauder fixed point theorem in a σ-stable RN module, namely every σ-stable continuous mapping (under either of the two topologies) of a random sequentially compact closed L 0 -convex subset into itself has a fixed point. The new fixed point theorem both unifies all the random generalizations currently available of the classical Brouwer or Schauder fixed point theorem and meets the need of the future applications of RN modules to stochastic analysis and stochastic finance.

The Krein-Milman 定理 in random locally convex modules and its applications

“…正如文献 [32] 中提到的稳定拓扑与稳定紧集是文献 [6] 中条件拓扑与条件紧集等概念在传统集论 下的解释. 受文献 [33] 的启发, Guo 等 [20] 在 RN-模中提出并系统发展了随机序列紧集的理论, 成功 建立了随机序列紧 L 0 -凸集上的 Schauder 不动点定理, 并给出了条件 Nash 均衡点的存在性定理. 不 难证明 RN-模中的随机序列紧性与稳定紧性是等价的, 因此稳定紧性可看作经典拓扑学中紧性理论 的恰当推广.…”

“…本文正是在随机泛函分析上述进展的基础上进一步对 RLC-模中的稳定紧的 L 0 -凸集进行深入 研究并建立 Krein-Milman 定理. 文献 [20] 证明了 RN-模中的随机序列紧集在 (ε, λ)-拓扑下是完备 的且是 L 0 -凸紧的. 受此启发, 我们希望证明 RLC-模中的稳定紧集具有类似的性质.…”