arXiv subjects

Xiaohuan Mu

Publications and source records attributed to Xiaohuan Mu.

5 recordsLinked to original sources

arXiv2026

Survey of Metric fixed point theory in random functional analysis

Based on the idea of randomizing the traditional space theory of functional analysis, random functional analysis has been developed as functional analysis over random metric spaces, random normed modules and random locally convex modules. Since these random frameworks have much more complicated algebraic, topological and geometric structures than their prototypes, the development of fixed point theory in random functional analysis had been almost stagnant before 2010. Unexpectedly, with the deep development of stable set theory fixed point theory in random functional analysis, including both its metric and topological fixed point theory, has made considerable progress in the recent 15 years. The purpose of this paper is to survey the important progress in metric fixed point theory in random functional analysis, including the random Banach contraction mapping principle and Caristi fixed point theorem on complete random metric spaces, and fixed point theorems for random nonexpansive and asymptotically nonexpansive mappings in complete random normed modules. Besides, the connections among the topics surveyed, random equations and random fixed point theorems for random operators are also briefly mentioned.

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arXiv2025

The random Kakutani fixed point theorem in random normed modules

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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arXiv2025

A new complete proof of the random Brouwer fixed point theorem and its implied consequences of unification

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of $L^{0}$-simplicial subdivisions of $L^{0}$-simplexes. Based on this, we are able to achieve a new complete proof of the random Brouwer fixed theorem in random Euclidean spaces, which can provide a solid foundation for various contemporary applications of interest. Afterward, we unify the works currently available and closely related to the random Brouwer fixed theorem: we first prove that the stochastic Brouwer fixed point theorem occurring elsewhere in stochastic analysis is equivalent to a special case of our random Brouwer fixed theorem, and then prove a general random Borsuk theorem and its equivalence with the random Brouwer fixed theorem. Finally, we conclude this paper with commentaries on recent state of study of the famous Schauder conjecture.

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arXiv2024

Common fixed point theorems for a commutative family of nonexpansive mappings in complete random normed modules

In this paper, we first introduce and study the notion of random Chebyshev centers. Further, based on the recently developed theory of stable sets, we introduce the notion of random complete normal structure so that we can prove the two deeper theorems: one of which states that random complete normal structure is equivalent to random normal structure for an $L^0$-convexly compact set in a complete random normed module; the other of which states that if $G$ is an $L^0$-convexly compact subset with random normal structure of a complete random normed module, then every commutative family of nonexpansive mappings from $G$ to $G$ has a common fixed point. We also consider the fixed point problems for isometric mappings in complete random normed modules. Finally, as applications of the fixed point theorems established in random normed modules, when the measurable selection theorems fail to work, we can still prove that a family of strong random nonexpansive operators from $(Ω,\mathcal{F},P)\times C$ to $C$ has a common random fixed point, where $(Ω,\mathcal{F},P)$ is a probability space and $C$ is a weakly compact convex subset with normal structure of a Banach space.

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arXiv2024

The relations among the notions of various kinds of stability and their applications

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application of which, it is easy to see that the notion of $d$-$σ$-stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of $σ$-stability introduced for a nonempty subset of a random normed module, as another application we give the final version of the characterization for a $d$-$σ$-stable random metric space to be stably compact. Second, we prove that an $L^{\infty}$-module is an $L^{p}$-normed $L^{\infty}$-module iff it is generated by a complete random normed module, from which it is easily seen that the gluing property of an $L^{p}$-normed $L^{\infty}$-module can be derived from the $σ$-stability of the generating random normed module, as applications the known and new basic facts of module duals for $L^{p}$-normed $L^{\infty}$-modules can be obtained, in a simple and direct way, from the theory of random conjugate spaces of random normed modules. Third, we prove that a random normed space is order complete iff it is complete with respect to the $(\varepsilon,λ)$-topology, as an application it is proved that the $d$-decomposability of an order complete random normed space is exactly its $d$-$σ$-stability. Finally, we prove that an equivalence relation on the product space $X\times B$ of a nonempty set $X$ and a complete Boolean algebra $B$ is regular iff it can be induced by a $B$-valued Boolean metric $d$ on $X$, as an application it is proved that a nonempty subset of a Boolean set $(X,d)$ is universally complete iff it is a $B$-stable set defined by a regular equivalence relation.

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