EXPONENTIAL STABILITY OF LINEAR SYSTEMS IN BANACH SPACES

In this paper the author proves a new fundamental lemma of Hardy-Lebesgne class $\[{H^2}(\sigma )\]$ and by this lemma obtains some fundamental results of exponential stability of $\[{C_0}\]$-semigroup of bounded linear operators in Banach spaces. Specially, if $\[{\omega _s} = \sup \{ {\mathop{\rm Re}\nolimits} \lambda ;\lambda \in \sigma (A) < 0\} \]$ and $\[\sup \{ \left\| {{{(\lambda - A)}^{ - 1}}} \right\|;{\mathop{\rm Re}\nolimits} \lambda \ge \sigma \} < \infty \]$ , where \[\sigma \in ({\omega _s},0)\]) and A is the infinitesimal generator of a $\[{C_0}\]$-semigroup in a Banach space $X$, then $\[(a)\int_0^\infty {{e^{ - \sigma t}}\left| {f({e^{tA}}x)} \right|} dt < \infty \]$, $\[\forall f \in {X^*},x \in X\]$; (b) there exists $\[M > 0\]$ such that $\[\left\| {{e^{tA}}x} \right\| \le N{e^{\sigma t}}\left\| {Ax} \right\|\]$, $\[\forall x \in D(A)\]$; (c) there exists a Banach space $\[\hat X \supset X\]$ such that $\[\left\| {{e^{tA}}x} \right\|\hat x \le {e^{\sigma t}}\left\| x \right\|\hat x,\forall x \in X.\]$.     

算子方程组的正解及在半正微分边值系统的应用

证明了半正算子方程组{x=λK1F1(x,y),y=λk2f2(x,y)正解的存在性结果,其中λ>0为参数,P为实Banach空间E中一个完全锥,K1,K2:P→P为线性全连续算子,F1,F2:P→E为连续有界算子.作为应用,给出了一类半正微分边值系统正解存在性的结果.     

非单特征值的广义分歧定理

讨论了抽象算子方程F(λ,u)=0的局部分歧问题,其中F:R×X→Y是一个C~2微分映射,λ是参数,X,Y为Banach空间.利用Lyapunov-Schmidt约化过程及偏导算子F_u(λ~*,O)的有界线性广义逆,在dim N(F_u(λ~*,0))≥codim R(F_u(λ~*,O))=1的条件下,证明了一个广义跨越式分歧定理.当参数空间的维数等于值域余维数时,应用同样的方法又得到了多参数方程的抽象分歧定理.     

Several Abstract Iterative Schemes for Solving the Bifurcation at Simple Eigenvalues

In this paper we consider the nonlinear operator equation $\lambda x=Lx+G(\lambda,x)$ where $L$ is a closed linear operator of $X-›X, X$ is a real Banach Space, with a simple eigenvalue $\lambda_0\neq 0$. We discretize its Liapunov-Schmidt bifurcation equation instead of the original nonlinear operator equation and estimate the approximating order of our approximate solution to the genuine solution. Our method is more convenient and more accurate. Meanwhile we put forward several abstract Newton-type iterative schemes, which are more efficient for practical computation, and get the result of their super-linear convergence.     

Interpolating hereditarily indecomposable Banach spaces

The following dichotomy is proved.

Every Banach space either contains a subspace isomorphic to , or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space.

In the particular case of , it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator admits a factorization through a H.I. space. The same result holds for every strictly singular operator .

Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.     

三角Banach代数的Lie弱顺从性

设$\mathcal {A,\ B}$ 是含单位元的Banach代数, $\mathcal M$ 是一个Banach $\mathcal {A,\ B}$-双模. $\mathcal {T}=\left ( \begin{array}{cc} \mathcal {A} & \mathcal M \\ & \mathcal {B} \\ \end{array} \right )$按照通常矩阵加法和乘法,范数定义为$\|\left( \begin{array}{cc} a & m \\ & b\\ \end{array} \right)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$,构成三角Banach 代数.如果从$\mathcal T$到其$n$次对偶空间$\mathcal T^{n}$上的Lie导子都是标准的,则称$\mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$\mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的.     

关于 ``Banach spaces failing the almost isometric universal extension property'' 的一个注记

D.M. Speegle 在文献[1] 中给出了具有常数 $\alpha$的性质${\cal A}$ 的定义,并且证明了任意无限维的可分一致光滑Banach空间都具有这样的性质,而且常数 $\alpha\in [0,1)$.本文给出了一个使得无限维可分Banach空间具有这种性质的充分条件,以及几个关于文献[1] 的注解.     

$B$值混合随机变量的强大数定律

本文在Banach空间$B$是$p$可光滑($1相似文献   

On the uniform Kreiss resolvent condition

Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition

$\parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M}{{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots ,$

implies the uniform Kreiss resolvent condition

$\left\| {\sum\limits_{k = 0}^n {\frac{{T^k }}{{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L}{{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots .$

We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T ^m for each integer m ? 2.

     

On Some Applications of Geometry of Banach Spaces and Some New Results Related to the Fixed Point Theory in Orlicz Sequence Spaces

We present some applications of the geometry of Banach spaces in the approximation theory and in the theory of generalized inverses. We also give some new results, on Orlicz sequence spaces, related to the fixed point theory. After a short introduction, in Section 2 we consider the best approximation projection from a Banach space $X$ onto its non-empty subset and proximinality of the subspaces of order continuous elements in various classes of Köthe spaces. We present formulas for the distance to these subspaces of the elements from the outside of them. In Section 3 we recall some results and definitions concerning generalized inverses of operators (metric generalized inverses and Moore-Penrose generalized inverses). We also recall some results on the perturbation analysis of generalized inverses in Banach spaces. The last part of this section concerns generalized inverses of multivalued linear operators (their definitions and representations). The last section starts with a formula for modulus of nearly uniform smoothness of Orlicz sequence spaces $\ell^\Phi$equipped with the Amemiya-Orlicz norm. From this result a criterion for nearly uniform smoothness of these spaces is deduced. A formula for the Domínguez-Benavides coefficient $R(a,l_\Phi)$ is also presented, whence a sufficient condition for the weak fixed point property of the space $\ell^\Phi$is obtained.     

Z-P-S空间中的若干概率分析问题

基于Menger概率线性赋范空间,提出了Z-P-S空间这一新概念,研究了Z-P-S空间(E,F,△)中算子A的固有值与固有元问题,建立了紧连续算子A具有大于1的固有值λ且在■D上存在对应于λ的固有元的四个充分条件．     

自伴算子空间上保持Jordan积投影的线性映射

Let Bs(H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dim H ≥ 2.It is proved that a linear surjective map on Bs (H) preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator U on H such that (X)=λU XU,X∈Bs(H) for some constant λ with λ∈{1,1}.     

巴拿赫空间中半群指数吸引子的存在性

We construct an exponential attractor for semigroup in Banach space by using ω-limit compactness method and provide a new method to prove the existence of an exponential attractor in uniformly convex Banach space. As a simple application, we prove the existence of an exponential attractor for reaction diffusion equations.     

Propagation of normality along regular analytic Jordan arcs in analytic functions with values in a complex unital Banach algebra with continuous involution

Globevnik and Vidav have studied the propagation of normality from an open subset of a region of the complex plane for analytic functions with values in the space of bounded linear operators on a Hilbert space . We obtain a propagation of normality in the more general setting of a converging sequence located on a regular analytic Jordan arc in the complex plane for analytic functions with values in a complex unital Banach algebra with continuous involution. We show that in this more general setting, the propagation of normality does not imply functional commutativity anymore as it does in the case studied by Globevnik and Vidav. An immediate consequence of the Propagation of Normality Theorem is that the further generalization given by Wolf of Jamison's generalization of Rellich's theorem is equivalent to Jamison's result. We obtain a propagation property within Banach subspaces for analytic Banach space-valued functions. The propagation of normality differs from the propagation within Banach subspaces since the set of all normal elements does not form a Banach subspace.

     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

自反代数上的局部Lie $n$-导子

令$\mathcal{L}$是一个满足$X_{-} \neq X$和$(0)_{+} \neq(0)$的Banach空间$X$上的子空间格.我们证明了从${\rm Alg}\,L$映到$B(X)$中的每个局部Lie $n$-导子是一个Lie $n$-导子.     

Shift Generated Haar Spaces on Compact Domains in the Complex Plane

Haar spaces are certain finite-dimensional subspaces of $\cc(K)$, where $K$ is a compact set and $\cc(K)$ is the Banach space of continuous functions defined on $K$ having values in $\C$. We characterize those Haar spaces which are generated by shifts applied to a single, analytic function for $K\subset\C$. This means that an arbitrary finite number of shifts generates Haar spaces by forming linear hulls. We have to distinguish two cases: (a) $K\not=\overline{K^\circ}$; (b) $K=\overline{K^\circ}$. It turns out that, in case (a), an analytic Haar space generator for dimensions one and two is already a universal Haar space generator for all dimensions. The geometrically simplest case that, in case (b), $K$ is convex with smooth boundary turns out to be the most difficult case. There is one numerical example in which the entire function $f:=1/\Gamma$ is interpolated in a shift generated Haar space of dimension four.     

The Cayley Transform of the Generator of a Bounded C<Subscript>0</Subscript>-Semigroup

Let A be the generator of a uniformly bounded C₀-semigroup on the Banach space X. We present sufficient conditions on the resolvent , under which the Cayley transform is a power-bounded operator, i.e., .     

On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$

We are concerned with the derivation of Poincaré-Friedrichs type inequalities in the broken Sobolev space $W^{2,1}$($Ω$; $\mathcal{T}_h$) with respect to a geometrically conforming, simplicial triagulation $\mathcal{T}_h$ of a bounded Lipschitz domain $Ω$ in $\mathbb{R}^d$ , $d$ $∈$ $\mathbb{N}$. Such inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as ${\rm C}^0$ Discontinuous Galerkin (${\rm C}^0$${\rm DG}$) approximations of minimization problems in the Sobolev space $W^{2,1}$($Ω$), or more generally, in the Banach space $BV^2$($Ω$) of functions of bounded second order total variation. As an application, we consider a ${\rm C}^0$${\rm DG}$ approximation of a minimization problem in$BV^2$($Ω$) which is useful for texture analysis and management in image restoration.