[Paper Review] The Schauder fixed point theorem in random normed modules

Tiexin Guo, Yachao Wang+2 more|arXiv (Cornell University)|Apr 22, 2021

Fixed Point Theorems Analysis1 citations

TL;DR

This paper generalizes the classical Schauder fixed point theorem to random normed modules (RN modules) under two topologies: the (ε,λ)-topology and the locally L⁰-convex topology. By establishing equivalence between random sequential compactness and random total boundedness in σ-stable RN modules, it proves that every σ-stable continuous mapping from a random sequentially compact closed L⁰-convex subset into itself has a fixed point, unifying existing random fixed point theorems and enabling future applications in stochastic analysis and finance.

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 $(\varepsilon,\lambda)$--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 $(\varepsilon,\lambda)$--topology and random total boundedness under the locally $L^0$--convex topology for a $\sigma$--stable subset of a $\sigma$--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 $\sigma$--stable $RN$ module, namely every $\sigma$--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.

Motivation & Objective

• To extend the classical Schauder fixed point theorem to the setting of random normed modules (RN modules), which generalize ordinary normed spaces in a probabilistic framework.
• To address the lack of a comprehensive fixed point theory in RN modules, especially for stochastic and financial applications.
• To establish the equivalence between random sequential compactness (under the (ε,λ)-topology) and random total boundedness (under the locally L⁰-convex topology) in σ-stable RN modules.
• To construct a well-defined random Schauder projection and decompositions of mappings to prove the existence of fixed points in RN modules.

Proposed method

• Introduce and analyze the (ε,λ)-topology and the locally L⁰-convex topology on σ-stable RN modules, which are the two most useful topologies in this context.
• Define and study random sequential compactness under the (ε,λ)-topology and random total boundedness under the locally L⁰-convex topology for σ-stable subsets.
• Establish the Hausdorff-type theorem proving the equivalence between random sequential compactness and random total boundedness in σ-stable RN modules.
• Construct a random Schauder projection and use countable decompositions of the mapping to reduce the fixed point problem to a solvable form.
• Apply the randomized Bolzano–Weierstrass theorem as a foundational tool to ensure convergence in the probabilistic setting.
• Prove that every σ-stable continuous mapping from a random sequentially compact closed L⁰-convex subset into itself has a fixed point under either topology.

Experimental results

Research questions

• RQ1 Can the classical Schauder fixed point theorem be generalized to the setting of random normed modules under the (ε,λ)-topology and the locally L⁰-convex topology?
• RQ2 What is the relationship between random sequential compactness and random total boundedness in σ-stable RN modules?
• RQ3 How can a well-defined random Schauder projection be constructed in this probabilistic framework?
• RQ4 Under what conditions does a σ-stable continuous mapping on a closed L⁰-convex subset of an RN module have a fixed point?
• RQ5 To what extent does this new fixed point theorem unify existing random generalizations of Brouwer and Schauder theorems?

Key findings

• The equivalence between random sequential compactness (under the (ε,λ)-topology) and random total boundedness (under the locally L⁰-convex topology) is rigorously established for σ-stable subsets of σ-stable RN modules.
• A well-defined random Schauder projection is constructed, enabling the decomposition of mappings into countably many manageable components.
• Every σ-stable continuous mapping from a random sequentially compact closed L⁰-convex subset into itself has a fixed point under either the (ε,λ)-topology or the locally L⁰-convex topology.
• The new fixed point theorem unifies all currently available random generalizations of the classical Brouwer and Schauder fixed point theorems.
• The result provides a foundational tool for future applications in stochastic analysis and stochastic finance within the framework of RN modules.
• The proof relies on a randomized version of the Bolzano–Weierstrass theorem, ensuring convergence in the probabilistic setting of RN modules.