Convergence analysis of iterative sequences for a pair of mappings in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space E. Let S ： C→ C be a quasi-nonexpansive mapping, let T ： C→C be an asymptotically demicontractive and uniformly Lipschitzian mapping, and let F ：= {x ∈C ： Sx = x and Tx = x}≠Ф Let {xn}n≥0 be the sequence generated irom an arbitrary x0∈Cby xn＋i=（1-cn）Sxn＋cnT^nxn, n≥0.We prove the necessary and sufficient conditions for the strong convergence of the iterative sequence {xn} to an element of F. These extend and improve the recent results of Moore and Nnoli.     

A construction of weakly inverse semigroups

Let S° be an inverse semigroup with semilattice biordered set E° of idempotents and E a weakly inverse biordered set with a subsemilattice Ep = { e ∈ E ｜ arbieary f ∈ E, S（f , e） loheain in w（e）} isomorphic to E° by θ：Ep→E°. In this paper, it is proved that if arbieary f, g ∈E, f ←→ g→→ f°θD^s° g°θand there exists a mapping φ from Ep into the symmetric weakly inverse semigroup P J（E∪ S°） satisfying six appropriate conditions, then a weakly inverse semigroup ∑ can be constructed in P J（S°）, called the weakly inverse hull of a weakly inverse system （S°, E, θ, φ） with I（∑） ≌ S°, E（∑） ∽- E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

Gap Property of Bi-Lipschitz Constants of Bi-Lipschitz Automorphisms on Self-similar Sets

For a given self-similar set E ∪→ R^d satisfying the strong separation condition, let Aut（E） be the set of all bi-Lipschitz automorphisms on E. The authors prove that {f ∈ Aut（E） ： blip（f） = 1} is a finite group, and the gap property of bi-Lipschitz constants holds, i.e., inf{blip（f） ≠ 1： f ∈ Aut（E）} 〉 1, where lip（g） =sup x,y∈E x≠y ｜g（x）-g（y）｜/｜x-y｜ and blip（g） =max（lip（g）, liP（g^-1））.     

RESEARCH ANNOUNCEMENTS On a Problem Given by D.R.Smart

In [1] Section 5.2, D.R. Smart gave a problem: Does every shrinking mappingof the closed unit ball in a Banach space have a fixed point? In this paper, we givea negative answer to this problem by constructing a counter-example. Definition Let (X,d) be a metric space and T a mapping of X into X. Wecall T a shrinking mapping if d(Tx,Tg)相似文献   

全矩阵代数的抛物子代数上具有可导性的非线性映射

Let F be a field of characteristic 0, Mn（F） the full matrix algebra over F, t the subalgebra of Mn（F） consisting of all upper triangular matrices. Any subalgebra of Mn（F） containing t is called a parabolic subalgebra of Mn（F）. Let P be a parabolic subalgebra of Mn（F）. A map φ on P is said to satisfy derivability if φ（x·y） = φ（x）·y＋x·φ（y） for all x,y ∈ P, where φ is not necessarily linear. Note that a map satisfying derivability on P is not necessarily a derivation on P. In this paper, we prove that a map φ on P satisfies derivability if and only if φ is a sum of an inner derivation and an additive quasi-derivation on P. In particular, any derivation of parabolic subalgebras of Mn（F） is an inner derivation.     

Strong commutativity-preserving generalized derivations on semiprime rings

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving （scp） on S, if [f（x）, f（y）] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.     

Limit Behaviors of Composition Convergent Sequences of Probability Measures on Compact L-X Semigroups

Let S be a compact second countable Hausdorff topological semigroup. Denote byP(S) the set of all regular probability measures defined on S, and by E(S) the setof all idempotents in S. S is called a L-X semigroup if ex =xe for every e∈E(S)and x∈S. In this paper, the limit behaviors of composition convergent sequences ofprobability measures defined on compact L-X semigroups are studied. Since compact     

Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces

Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive. Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.     

一类变换半群的正则元和Green关系

设T_X为X上的全变换半群,E为X上的等价关系,令T_E(X)={f∈T_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是T_X的子半群,如果X是一个全序集,E是X上的一个凸等价关系,设OP_E(X)为T_E(X)中所有保向映射作成的半群。对于有限全序集X上一类特殊的凸等价关系E,本文刻画了半群OP_E(X)的正则元的特征,并且描述了这个半群上的Green关系。     

Green's Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation

Let ${\cal T}_X$ be the full transformation semigroup on the set $X$, \[ T_{E}(X)=\{f\in {\cal T}_X\colon \ \forall(a,b)\in E,(f(a),f(b))\in E\} \] be the subsemigroup of ${\cal T}_X$ determined by an equivalence $E$ on $X$. In this paper the set $X$ under consideration is a totally ordered set with $mn$ points where $m\geq 2$ and $n\geq 3$. The equivalence $E$ has $m$ classes each of which contains $n$ consecutive points. The set of all order preserving transformations in $T_{E}(X)$ forms a subsemigroup of $T_E(X)$ denoted by \[ {\cal O}_{E}(X)=\{f\in T_{E}(X)\colon \ \forall\, x, y\in X, \ x\leq y \mbox{ implies } f(x)\leq f(y)\}. \] The nature of regular elements in ${\cal O}_{E}(X)$ is described and the Green's equivalences on ${\cal O}_{E}(X)$ are characterized completely.     

On the Rank of the Semigroup TE(X)

${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5.     

$C(X)$上的开点与紧开拓扑

In this note we define a new topology on C(X),the set of all real-valued continuous functions on a Tychonoff space X.The new topology on C(X) is the topology having subbase open sets of both kinds:[f,C,ε[={g E C(X):|f(x)-g(x)| ε for every x∈C} and[U,r]~-={g∈C(X):g~(-1)(r)∩U≠φ},where f∈C(X),C∈KC(X)={nonempty compact subsets of X},ε 0,while U is an open subset of X and r∈R.The space C(X) equipped with the new topology T_(kh) which is stated above is denoted by C_(kh)(X).Denote X_0={x∈X:x is an isolated point of X} and X_c={x∈X:x has a compact neighborhood in X}.We show that if X is a Tychonoff space such that X_0=X_c,then the following statements are equivalent:(1) X_0 is G_δ-dense in X;(2) C_(kh)(X) is regular;(3) C_(kh)(X) is Tychonoff;(4) C_(kh)(X) is a topological group.We also show that if X is a Tychonoff space such that X_0=X_c and C_(kh)(X) is regular space with countable pseudocharacter,then X is σ-compact.If X is a metrizable hemicompact countable space,then C_(kh)(X) is first countable.     

齐型空间上的Toeplitz型算子

设X是齐型空间.设T_(j,1)和T_(j,2)是具有非光滑核的奇异积分算子,或者是±II(I是恒等算子).令Toeplitz型算子T_b=■T_(j,1)M_T_(j,2),其中M_bf(x)=b(x)f(x).研究了当b∈BMO(X)时,T_b(f)在加权情况下的有界性,以及当b∈BMO(X)时,与经典Carderon-Zygmund算子相联的T_b(f)在Morrey空间上的有界性.     

Regularity and Green's Relations for Semigroups of Transformations Preserving Orientation and an Equivalence

Let be the full transformation semigroup on a set X. For a non-trivial equivalence E on X, let

位势型算子的加权不等式

Let Ф be a non-negative locally integrable function on R^n and satisfy some weak growth conditions, define the potential type operator TФ by TФf（x）=∫R^n Ф（x-y）f（y）dy. The aim of this paper is to give several strong type and weak type weighted norm inequalities for the potential type operator TФ.     

On Certain Finite Semigroups of Order-decreasing Transformations I

Let $D_n $ (${\cal O}_n$) be the semigroup of all finite order-decreasing (order-preserving) full transformations of an $n$-element chain, and let $D(n,r) = \{\alpha\in D_n: |\mbox{Im}\alpha| \leq r\}$ (${\cal C}(n,r) = D(n,r)\cap {\cal O}_n)$ be the two-sided ideal of $D_n $ ($D_n \cap {\cal O}_n$). Then it is shown that for $r \geq 2$, the Rees quotient semigroup $DP_r(n)= D(n,r) / D(n,r-1)$ (${\cal C}P_r(n)= {\cal C}(n,r)/{\cal C} (n,r-1)$) is an ${\cal R}$-trivial (${\cal J}$-trivial) idempotent-generated 0*-bisimple primitive abundant semigroup. The order of ${\cal C}P_r(n)$ is shown to be $1+ \left(\begin{array}{c} n-1 \\ r-1 \end{array} \right) \left(\begin{array}{c} n \\ r \end{array} \right)/(n-r+1)$. Finally, the rank and idempotent ranks of ${\cal C}P_r(n)\,(r<n)$ are both shown to be equal to $\left(\begin{array}{c} n-1 \\ r-1 \end{array} \right)$.     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.