| Random functional analysis in the context of this thesis refers to the functional analysis on random normed modules and random locally convex modules,which are the random generalizations of classical normed spaces and locally convex spaces,respectively.Random functional analysis has not only achieved profound theoretical developments but also become a powerful tool for studying dynamic risk measures,dynamic Nash equilibrium and stochastic differential equations.However,there are still many unresolved problems in the random nonlinear functional analysis.This thesis is dedicated to the study of several important problems among them.The main contents are as follows:The first work of this thesis is to establish a noncompact Dotson fixed point theorem in random normed modules by using the theory of random sequential compactness:if C is a σ-stable random sequentially compact L0star-shaped subset of a random normed module E,then every random nonexpansive mapping T:C→C has a fixed point.Furthermore,inspired by Smoluk’s classical result,we obtain an existence result for the best approximation point in random normed modules:let E be a random normed module and T:E→E a random nonexpansive mapping with a fixed point u and C a Tε,λ-closed,σ-stable and T-invariant subset of E such that T(C)ε,λ is random sequentially compact,then the set of best approximations PC(u)is nonempty.In addition,we also get an existence result for invariant approximations(namely the best approximation of the fixed point is also a fixed point)in random normed modules.Second,we deeply study stably compact sets of a random locally convex module by proving that they are complete with respect to the(ε,λ)topology Tε,λ and characterizing them in the way that each σ-stable family of σ-stable Tε,λ-closed subsets of them with the finite intersection property has a nonempty intersection.Based on the preliminaries,we further give the Weierstrass theorem for a proper lower semicontinuous L0-valued σ-stable function defined on a stably compact set,which implies that a stably compact L0-convex set must be L0-convexly compact.Then,we introduce the notion of an L0-extreme point for an L0-convex set and prove the corresponding Krein-Milman theorem for an L0-convexly compact set.Besides,some interesting connection and comparison between the generalized Krein-Milman theorem and the classical one for a compact convex set in a locally convex space are also given.Moreover,as an application of the generalized KreinMilman theorem,we prove that a proper,lower semicontinuous,σ-stable and L0-quasiconvex function on an L0-convexly compact set of a random locally convex module can attain its minimum value,and further if in addition,f is L0(F)-valued and L0-affine,then f can attain its minimum value at some L0extreme point of the L0-convexly compact set.Finally,we prove the following approximation result for random δ-nearsurjective ε-isometries between random normed modules:let(E,‖·‖)and(F,‖·‖)be two Tε,λ-complete random normed modules and f:E→F a stable random δ-nearsurjective ε-isometry with f(0)=0,where ε,δ∈L+0(F),then there exists a surjective L0-linear random isometry U between E and F such that ‖f(x)-U(x)‖≤4ε for all x∈E.Furthermore,making use of the above result and the relations between random normed modules and classical normed spaces,we give the approximation result for continuous nonlinear random operators:let(X,‖·‖)and(Y,‖·‖)be two real separable Banach spaces,(Ω,F,P)a probability space,ε0 and δ0 two nonnegative random variables and f:Ω×X→Y a random operator such that f(ω,·):X→Y is a continuous δ0(ω)-nearsurjective ε0(ω)-isometry and f(ω,0)=0 for any ω∈Ω,then there exist a linear and almost everywhere isometric random operator U:Ω×X→Y and Ω0∈F with P(Ω0)=1 such that ‖f(ω,x)-U(ω,x)‖≤4ε0(ω),?(ω,x)∈Ω0×X.To the best of our knowledge,it is the first time that linear and almost everywhere isometric random operators are used to approximate continuous nonlinear random operators in the theory of random operators. |
