Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Let A = d/dθ denote the generator of the rotation group in the space C(Γ), where Γ denotes the unit circle. We show that the stochastic Cauchy problem

Continuity of mappings between Fréchet spaces

For all linear and many nonlinear operators between Fréchet spaces, several continuity results are established. Especially, a new closed graph theorem is given.     

集值映射的伪(*)连续与弱(*)连续性

本文引入了集值映射的伪（＊）连续性与弱（＊）连续性概念的定义,研究了伪（＊）连续、（＊）连续及弱（＊）连续的等价关系,最后研究了乘积空间中的集值映射成为伪（＊）连续和弱（＊）连续的充要条件。     

Connection between a Cauchy system representation of Kalaba and Fourier transforms

Recently R.E. Kalaba reduced the solution of a system of integral equations to the solution of a Cauchy system. In this paper we derive the same Cauchy system via the Wiener-Hopt procedure.     

On a connection between powers of operators and fractional Cauchy problems

Many phenomena in mathematical physics and in the theory of stochastic processes are recently described through fractional evolution equations. We investigate a general framework for connections between ordinary non-homogeneous equations in Banach spaces and fractional Cauchy problems. When the underlying operator generates a strongly continuous semigroup, it is known, using a subordination argument, that the fractional evolution equation is well posed. In this case, we provide an explicit form of the solution involving special functions, one example being the Airy function.     

赋准范空间之间线性算子的五种有界性与连续性

综合散见于多种文献的不同描述,明确线性算子拓扑有界、邻域有界、范有界与强有界的定义,引进次强有界的概念,给出赋准范空间之间与赋β-范空间之间线性算子的各种有界性以及连续性之间的关系定理与反例.     

Matrix refinement equations: Continuity and smoothness

In this paper we give some criteria for the existence of compactly supported C ^k+α-solutions (k is an integer and 0 ⩽ α < 1) of matrix refinement equations. Several examples are presented to illustrate the general theory.     

Cone monotone mappings: Continuity and differentiability

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces, and we define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): K$K$-monotone dominated and cone-to-cone monotone mappings. K$K$-monotone dominated mappings naturally generalize mappings with finite variation (in the classical sense) and K$K$-monotone functions defined by Borwein, Burke and Lewis to mappings with domains and ranges of higher dimensions. First, using results of Veselý and Zají?ek, we show some relationships between these classes. Then, we show that every K$K$-monotone function f:X→R$f:X\to \mathbb{R}$, where X$X$ is any Banach space, is continuous outside of a set which can be covered by countably many Lipschitz hypersurfaces. This sharpens a result due to Borwein and Wang. As a consequence, we obtain a similar result for K$K$-monotone dominated and cone-to-cone monotone mappings. Finally, we prove several results concerning almost everywhere differentiability (also in metric and w^∗$w^{\ast }$-senses) of these mappings.     

Relationship between solutions of families of two-point boundary value problems and Cauchy problems

We carry out a qualitative analysis and suggest a method for the solution of the two-point boundary value problems A _l υ = g, x ? [0, l], l ? (0, L) = B ? R ₊; $$ \alpha v'_x \left| {_{x = + 0} = F_1 (v,l)} \right|_{x = 0} , \beta v'_x \left| {_{x = l - 0} = F_2 (v,l)} \right|_{x = l} , $$ , where α, β ? R, υ is the unknown function, g = g(x) is a given real function, F ₁(y, l) and F ₂(y, l) are known real functions defined on the sets Y ₁ × B and Y ₂ × B, respectively, where Y ₁ ∪ Y ₂ ? R, and A _l is the restriction of A corresponding to the embedding parameter l. (Here A is an operator taking an arbitrary function in the set of function classes defined in the paper to C(B ₁), where B ₁ = [0, L).) The study takes into account the dependence of solutions of various versions of these two-point boundary value problems on the parameter l. We construct algorithms for the reduction of these families of two-point boundary value problems to Cauchy problems for ordinary differential equations and integro-differential equations that contain only first derivatives of the unknown functions with respect to the parameter l.     

Approximation of semilinear fractional Cauchy problem: II

Semigroup Forum - We consider the semidiscrete approximation of the Cauchy problem $$\begin{aligned} ({\mathbf {D}}_{t}^{\alpha }u)(t) = A u(t) +f\big (t,u(t)\big ), \quad t \in (0,T],\ u(0) =...     

Parametrized multicriteria optimization: Continuity and closedness of optimal multifunctions

This paper studies continuity properties of the marginal multifunction and solution multifunction for optimization problems with respect to a preorder relation. Among others, we present a number of results which are generalizations to multicriteria problems of Theorems due to C. Berge, W. Hogan, T. Tanino, and Y. Sawaragi.     

赋范空间中凸泛函Lipschitz连续性与函数有下界的关系

关于凸函数局部有上界和函数Lipschitz连续性的等价性已经被多次研究过,但是这些研究都未曾涉及凸函数的Lipschitz连续性与函数有下界的关系.本文利用Hamel基构造了一个反例,说明了即使凸函数在全空间有下界也不能得到函数的Lipschitz连续性.接着,在空间完备的情形下,运用Baire纲理论证明了,函数在某一球型邻域内均下半连续等价于函数的Lipschitz连续性.     

Fragmentability and Continuity of Semigroup Actions

fragmentability in the sense of Jayne and Rogers and its natural uniform generalizations play a major role in this paper. Our applications show that problems concerning the continuity of induced actions have satisfactory solutions for Asplund Banach spaces X (without additional restrictions, if S is a topological group) and, moreover, for a new locally convex version of Asplund spaces introduced in the paper. The starting point of this concept was the characterization of Asplund spaces due to Namoika and Phelps in terms of fragmentability.     

Strong C-Sets and Decompositions of Continuity

In this paper, we introduce strongly semi-closed sets and strongly -closed sets and using these sets we obtain decompositions of continuity.     

Cauchy transforms of self-similar measures: the Laurent coefficients

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in . Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {a_n}_n=0^∞ of F. We give asymptotic formulas for {a_n}_n=0^∞ and for F^(k)(z) near the support of μ, hence the precise growth rates on |a_n| and |F^(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series.