arXiv subjects

Erxin Zhang

Publications and source records attributed to Erxin Zhang.

5 recordsLinked to original sources

arXiv2019

$L^0$--convex compactness and its applications to random convex optimization and random variational inequalities

In 2010, Gordan Žitković introduced the notion of convex compactness for a convex subset of a linear topological space and gave some important applications to both nonlinear analysis and mathematical economics in [ Gordan Žitković, Convex compactness and its applications, Math. Finance Econom. 3(1) (2010) 1--12 ]. Motivated by Gordan Žitković's idea, in this paper we introduce the notion of $L^0$--convex compactness for an $L^0$--convex subset of a topological module over the topological algebra $L^0(\mathcal{F},K)$, where $L^0(\mathcal{F},K)$ is the algebra of equivalence classes of random variables from a probability space $(Ω,\mathcal{F},P)$ to the scalar field $K$ of real numbers or complex numbers, endowed with the topology of convergence in probability. This paper continues to develop the theory of $L^0$--convex compactness by establishing various kinds of characterization theorems for $L^0$--convex subsets of a class of important topological modules--complete random normed modules, in particular, we make use of the theory of random conjugate spaces to give a characterization theorem of James type for a closed $L^0$--convex subset of a complete random normed module. As applications, we successfully generalize some basic theorems of classical convex optimization and variational inequalities from a convex function on a reflexive Banach space to an $L^0$--convex function on a random reflexive random normed module. Since the usual weak compactness method fails in the random setting of this paper and in particular, since the difficulties caused by the partial order structure of the range of an $L^0$--valued function also frequently occurs in the study of problems involved in this paper, we are forced to discover a series of new skills to meet the needs of this paper.

math.FA↗

arXiv2019

On $d$--$σ$--stability in random metric spaces and its applications

In 2010, the first author of this paper introduced the notion of $σ$--stability for a nonempty subset of an $L^0(\mathcal{F},K)$--module in [T.X. Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal. 258(2010), 3024--3047], this kind of $σ$--stability is purely algebraic and leads to a series of deep developments of random normed modules and random locally convex modules. Motivated by this, A. Jamneshan, M. Kupper and J. M. Zapata recently introduced another kind of $σ$--stability for a nonempty subset of a random metric space $(E,d)$, called $d$--$σ$--stability since it depends on the random metric $d$. $d$--$σ$--stability coincides with the previous $σ$--stability in the case of random normed modules, which motivates us in this paper to generalize the precise form of Ekeland's variational principle from a complete random normed module to a complete $d$--$σ$--stable random metric space. Besides, this paper also utilize $d$--$σ$--stability to generalize Nadler's fixed point theorem for a multivalued contraction mapping from a complete metric space to a complete random metric space. To our surprise, our simple fixed point theorem, however, can derive the known basic fixed point theorems of contraction type for both random operators and $σ$--stable mappings on a complete random normed module. A lot of examples shows the study of random metric spaces is more complicated than that of random normed modules.

math.FA↗

arXiv2019

$L^0$--convex compactness and random normal structure in $L^0(\mathcal{F},B)$

Let $(B,\|\cdot\|)$ be a Banach space, $(Ω,\mathcal{F},P)$ a probability space and $L^0(\mathcal{F},B)$ the set of equivalence classes of strong random elements (or strongly measurable functions) from $(Ω,\mathcal{F},P)$ to $(B,\|\cdot\|)$. It is well known that $L^0(\mathcal{F},B)$ becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue--Bochner function spaces and random functional analysis. Let $V$ be a closed convex subset of $B$ and $L^0(\mathcal{F},V)$ the set of equivalence classes of strong random elements from $(Ω,\mathcal{F},P)$ to $(B,\|\cdot\|)$, the central purpose of this paper is to prove the following two results: (1). $L^0(\mathcal{F},V)$ is $L^0$--convexly compact if and only if $V$ is weakly compact; (2). $L^0(\mathcal{F},V)$ has random normal structure if $V$ is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules and Banach space techniques, can be applied in the other related aspects.

math.FA↗

arXiv2018

Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

This paper first proves two fixed point theorems in complete random normed modules, which are respectively the random generalizations of the classical Banach's contraction mapping principle and Browder--Kirk's fixed point theorem. As applications, the first is used to give the existence and uniqueness of solutions to various kinds of backward stochastic equations under $L^0$--Lipschitz assumptions and the second is used to establish the existence of solutions to backward stochastic equations of nonexpansive type.

math.FA↗

arXiv2017

On random convex analysis

Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is convex analysis over random locally convex modules. Since random locally convex modules have the more complicated topological and algebraic structures than ordinary locally convex spaces, establishing random convex analysis will encounter harder mathematical challenges than classical convex analysis so that there are still a lot of fundamentally important unsolved problems in random convex analysis. This paper is devoted to solving some important theoretic problems. First, we establish the inferior limit behavior of a proper lower semicontinuous $L^0$--convex function on a random locally convex module endowed with the locally $L^0$--convex topology, which makes perfect the Fenchel--Moreau duality theorem for such functions. Then, we investigate the relations among continuity, locally $L^0$--Lipschitzian continuity and almost surely sequent continuity of a proper $L^0$--convex function. And then, we establish the elegant relationships among subdifferentiability, Gâteaux--differentiability and Fréchét--differentiability for a proper $L^0$--convex function defined on random normed modules. At last, based on the Ekeland's variational principle for a proper lower semicontinuous $\bar{L}^0$--valued function, we show that $\varepsilon$--subdifferentials can be approximated by subdifferentials. We would like to emphasize that the success of this paper lies in simultaneously considering the $(\varepsilon, λ)$--topology and the locally $L^0$--convex topology for a random locally convex module.

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