Fixed Points of Mappings Satisfying a Weakly Contractive Type Condition

In this paper, we discuss a fixed point theorem for mappings derived by a pair of mappings satisfying weak(k, k/) contractive type condition on the tensor product spaces. Let X and Y be Banach spaces and T_1 : X γ Y → X and T_2: X γ Y → Y be two operators which satisfy weak(k, k/) contractive type condition. Using T_1 and T_2, we construct an operator T on X γ Y and show that T has a unique fixed point in a closed and bounded subset of X γ Y.We derive an iteration scheme converging to this unique fixed point of T. Conversely, using a weakly contractive mapping T, we construct a pair of mappings(T_1, T_2) satisfying weak(k, k/)contractive type condition on X γ Y and from this pair, we also obtain two self mappings S_1 and S_2 on X and Y respectively with unique fixed points.     

Vanishing line for the descent spectral sequence

We prove that if $f:X \to Y$ is a surjective, cohomologically (k - 1)-connected and proper map between locally compact spaces, then for the cohomological descent spectral sequence one has $E^{pq}_{2} = 0$ provided $q < pk$.     

On Projective Mappings

Mathematical Notes - Let X,Y be Polish spaces, and let $\mathcal{B}_k $ be the σ-algebra generated by the projective class $L_{2k + 1} $ . A mapping $f:X \mapsto Y$ is called $k$...     

Some Characterizations of Spaces with Weak Form of $cs$-Networks

In this paper, we introduce the concept of statistically sequentially quotient map:A mapping f : X → Y is statistically sequentially quotient map if whenever a convergent sequence S in Y, there is a convergent sequence L in X such that f(L) is statistically dense in S. Also, we discuss the relation between statistically sequentially quotient map and covering maps by characterizing statistically sequentially quotient map and we prove that every closed and statistically sequentially quotient image of a g-metrizable space is g-metrizable. Moreover,we discuss about the preservation of generalization of metric space in terms of weakbases and sn-networks by closed and statistically sequentially quotient map.     

Limitations on the extendibility of the Radon-Nikodym Theorem

Given two locally compact spaces and a continuous map the Banach lattice is naturally a -module. Following the Bourbaki approach to integration we define generalized measures as -linear functionals . The construction of an -space and the concepts of absolute continuity and density still make sense. However we exhibit a counter-example to the natural generalization of the Radon-Nikodym Theorem in this context.

     

cut* 空间的拓扑性质

该文引入了 cut^*空间的概念,所谓的 cut^*空间是指去掉任意一点连通,去掉任意两点不连通的连通空间.通过对其性质的讨论,得到如下主要结论: 首先得到cut^*空间中每个点非开即闭,并且cut^*空间中有无限多个闭点;其次讨论了一类特殊的 cut^*空间,即去掉一点是COTS的 cut^* 空间.指出``$X$是 cut^*空间,任意 $x\inX,X\setminus\{x\}$是不可约cut空间'这样的空间类是不存在的.在文章的最后,讨论了去掉一点是LOTS的 cut^*空间的覆盖性质,得到这样的空间是紧空间或Lindel\"of空间的结论.     

Separate continuity, joint continuity and the Lindelöf property

In this paper we prove a theorem more general than the following. Suppose that is Lindelöf and -favourable and is Lindelöf and Cech-complete. Then for each separately continuous function there exists a residual set in such that is jointly continuous at each point of .

     

Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method

In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set-valued mapping with closed graph.We establish the convergence criteria of the restricted inexact Newton-type method,which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Frechet derivative of f.Indeed,we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Frechet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically regular.An application of this method to variational inequality is given.In addition,a numerical experiment is given which illustrates the theoretical result.     

Rank of the fundamental group of any component of a function space

We compute the rank of the fundamental group of any connected component of the space for and connected, nilpotent CW complexes of finite type with finite. For the component corresponding to a general homotopy class , we give a formula directly computable from the Sullivan model for . For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of and . When both and are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of explicitly in terms of the homomorphism induced by on rational cohomology.

     

PHANTOM MAPS,COGROUPS AND THE SUSPENSION MAP

ABSTRACT

The set Ph(X, Y) of pointed homotopy classes of phantom maps from X to Y admits a natural group structure if either Y is a grouplike space or X is a cogroup. In the present paper, the group structure on Ph(X,Y) is examined in the second case. (The first case was examined in an earlier paper.) The results in the two cases are similar—for instance, the group structure turns out to be abelian, divisible and independent of the grouplike structure on Y or the cogroup structure on X—but the techniques used to establish the results differ substantially in the two cases.

In addition, a study of the map g*: Ph(X,Y₁) → Ph(X,Y₂) induced by a map g: Y₁ → Y₂ of grouplike spaces is initiated. A particularly interesting special case of this situation is the suspension map Ph(X, Y) → Ph(X, ΩσY) ? Ph(σX, σY) with Y a grouplike space.     

A counterexample related to topological sums

In this paper we construct compact metric spaces , which are topologically distinct but whose topological sums and are homeomorphic.

     

形式三角矩阵环的PS性质和CESS性质

设$R$是环. 称右$R$-模$M$是PS-模,如果$M$具有投射的socle. 称$R$是PS-环,如果$R_R$是PS-模. 称$M$是CESS-模,如果$M$的任意具有基本socle的子模是$M$的某个直和因子的基本子模.本文给出了形式三角矩阵环 $T=\left( \begin{array}{cc} A & 0 \\     

Banach-Stone theorem for Banach lattice valued continuous functions

Let and be compact Hausdorff spaces, be a Banach lattice and be an AM space with unit. Let be a Riesz isomorphism such that if and only if for each . We prove that is homeomorphic to and is Riesz isomorphic to . This generalizes some known results.

     

Multiplicatively spectrum-preserving maps of function algebras

Let be a compact Hausdorff space and a function algebra. Assume that is the maximal ideal space of . Denoting by the spectrum of an , which in this case coincides with the range of , a result of Molnár is generalized by our Main Theorem: If is a surjective map with the property for every pair of functions , then there exists a homeomorphism such that

for every and every with .

     

周期吸附系统的分布混沌

由一个紧致度量空间X以及连续映射f:X→X所组成的偶对(X,f)称之为一个动力系统.若存在f的不动点p以及另一周期点q,使得对于任一非空开集U(?)X,都有∪_(n=0)~∞f~n(U)含有p和q,则称(X,f)是一个周期吸附系统,其中f~i表示f的i次迭代.本文指出:若(X,f)是一个周期吸附系统并且X是自密的,则存在一个f的分布混沌集D,使得D与每一非空开集之交都包含着一个Cantor集.     

A morphism of intersection homology induced by an algebraic map

Let be a map of algebraic varieties. Barthel, Brasselet, Fieseler, Gabber and Kaup have shown that there exists a homomorphism of intersection homology groups compatible with the induced homomorphism on cohomology. The crucial point in the argument is reduction to the finite characteristic. We give an alternative and short proof of the existence of a homomorphism . Our construction is an easy application of the Decomposition Theorem.

     

一列连续函数的遍历优化

设$T:X\rightarrow X$是紧度量空间$X$上的连续映射, $\mathcal{F}=\{f_n\}_{n\geq 1}$是$X$上的一族连续函数. 如果 $\mathcal{F}$是渐近次可加的, 那么$\sup\limits_{x\in \mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac 1 n f_n (x)=\sup\limits_{x\in X} \limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x) =\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n (x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, 其中$\mathcal{M}_T$表示$T$-\!\!不变的Borel概率测度空间, $\mathrm{Reg}(\mathcal{F},T)$ 表示函数族$\mathcal{F}$的正规点集, $\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int f_n \mathrm{d}\mu$. 这把Jenkinson, Schreiber 和 Sturman 等人的一些结果推广到渐近次可加势函数, 并且给出了次可加势函数从属原理成立的充分条件, 最后给出了 一些相关的应用.     

$c$-半层空间与几乎完全正则空间一点注记

In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces(CSS) by semi-continuous functions. We mainly show that:(1)Let X be a space. Then the following statements are equivalent:(i) X is almost completely regular.(ii) Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.(iii) If g, h : X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f : X → I such that g ≤ f ≤ h;and(2) Let X be a space. Then the following statements are equivalent:(a) X is CSS;(b) There is an operator U assigning to a decreasing sequence of compact sets(Fj)j∈N,a decreasing sequence of open sets(U(n,(Fj)))n∈N such that(b1) Fn■U(n,(Fj)) for each n ∈ N;(b2)∩n∈NU(n,(Fj)) =∩n∈NFn;(b3) Given two decreasing sequences of compact sets(Fj)j∈N and(Ej)j∈N such that Fn■Enfor each n ∈ N, then U(n,(Fj))■U(n,(Ej)) for each n ∈ N;(c) There is an operator Φ : LCL(X, I) → USC(X, I) such that, for any h ∈ LCL(X, I),0 Φ(h) h, and 0 Φ(h)(x) h(x) whenever h(x) 0.     

Invertibility preserving linear maps and algebraic reflexivity of elementary operators of length one

Let and be real or complex Banach spaces. We show that a surjective linear map preserving invertibility in both directions is either of the form or the form , where , , , and are bounded invertible linear operators. As an application we improve a result of Larson and Sourour on algebraic reflexivity of elementary operators of length one.