关于集值系统x∈F(x,y),y∈G(x,y)解的存在性

Let (E, ‖·‖) be a uniformly convex Banach space, X a nonempty compact and convex subset of E. Let F be a closed mapping of X×X into 2^X, G a mapping of X×X into C(X). It is shown that if for any f∈C(X), x∈X, F(x,f(x)) is a closed and convex subset of X, and G(x,f(x)) is a continuous function and for any x,y₁,y₂∈X,H(x,G(x,y₁),G(x,y₂))≤‖y₁-y₂‖ then there exist X₀,y₀∈X such that X₀∈F(x₀,y₀) and y₀∈G(x₀, y₀).     

关于非线性泛函的一个问题

Suppose that H is a Hilbert space, D is a convex closed set inis a functioal, f(x)=1/2‖x‖²-g(x). Suppose that the minimum of f(x) withrespect to D is attained at x₀∈D and f(x) has a bounded linear Gateaux differential at x₀. In this paper we prove that f(x₀) is a critical value of f(x) when and only when g′(x₀)∈I_D(x₀)={(1-λ)x₀+λy|y∈D.λ≥0}.     

关于Banach空间光滑性的两点注记

In this paper, We show that a real or complex Banach space is smooth at x₀(∈S(X)) iff, for any y∈S(X).(?)where X_x0,y= span {x₀, y} . For a real Banach space X, we obtain that X is F-differentia ble at x₀(∈S(X)) iff (?) Remark. The sufficient conditions of the conclusion have been proved by Yu Xintai [2].     

关于反向适应序列的一些结果

Let X be a Banach space, (x_n, F_n, n<- 1) a X-valued adapted sequence on probability space (Q, F, P) . Let T be all stopping times with respect to(F_n,n < - 1) . (x_n, F_n,n< - 1) is called a T- uniform amart if there exists a t₀∈T such that for each t∈T with t0,E‖x_t‖<∞ and if (?)=0.In this paper we prove that.     

赋范线性空间的第二对偶空间

Theorem. A mapping x^** of X^* into sealar field Φ is a linear functional if and only if there exists a μ∈ba(S) such that X^**(X^*)=∫_sx^*(s)μ(ds),x^*∈X^* Theorem. F∈ba(S.Σ)^* if and only if there exists a λ∈ba (T) (T is the closed unit sphere of the space B(S,Σ))such that F(μ)=∫(∫_st(s)μ(ds))λ(dt),μ∈ba(S,Σ).     

一类L-Fuzzy拓扑空间——1.L-Fuzzy点的邻近构造

Suppose (L₀,≤) is a poset (i.e. partially ordered set) with an order reversing involution λ→λ′ and has an all element 1. Let L={b|b is a subset of L₀ and satisfies the condition: if λ∈b,μ≤λ then μ∈b} then (L,?,∩,∪) is a complete and distributive lattice. L is called the lattice induced from poset L₀. There is a mapping α→a′={λ∈L₀|λ′(?)a} of a lattice L onto L. The mapping is clearly order reversing involution.     

正规算子和亚正规算子的一些特征

In this paper it is proved that if T is a bounded linear operator on a Hilbert space H and λ(?)c1(W(T)), where cl(W(T)) is the closure of W(T)={(Tx, x); x∈H, ‖x‖ =1}, then T is normal iff U_λ=(T-λ)^*-1 (T-λ) is hyponormal and T is hyponormal iff U_λ=(T-λ)^*-1 (T-λ) is normaliod.     

Co-RS-紧拓扑

A topoogy R(τ) is constructed from a given topology τ on a set X. R(τ) is coarser than τ, and the following are some results based on this topology:1. Continuity and RS-continuity are equivalent if the codomain is retopologized by R(τ).2. Each topological space has a coarser extremely disconnected topology.3. The class of semi-open sets with respect to R(τ) is a topology.4. T₂ and semi-T₂ properties are equivalent on a space whose topology is R(τ).5. Minimal R₀-spaces are RS-compact.6. Maximal a-compact spaces are RS-compact.     

Two Notes on Top ological Groups

In this short paper, we firstly give a short proof of Birkhoff-Kakutani Theorem by Moore metrizable Theorem. Then we prove that G is a topological group if it is a paratopological group which is a dense G_δ-set in a locally feebly compact regular space X.     

Semiamart与Gaft类的收敛性

Let (Ω,F,P) be a probability space,(F_n) be an increasing sequence of sub-σ-fields of F, a sequence (x_n) of integrable random variables adapted to (F_n), if (x_n) is progressive uniformly integrable or(x_n)is a L¹-bounded Semiamart, then ( 1 ) x_m-x_n→0 (Pr.), ( 2 ) E (x_m|F_n)-x_n→0 (pr.) are equivalent is proved in this paper .