Values In Banach Spaces Research Articles

Bimeasures in Banach spaces

In this paper we study integration with respect to a bimeasure with finite semivariation. The bimeasures as well as the functions to be integrated, take on their values in Banach spaces.

Coagulation-Fragmentation Processes

We study the well-posedness of coagulation-fragmentation models with diffusion for large systems of particles. The continuous and the discrete case are considered simultaneously. In the discrete situation we are concerned with a countable system of coupled reaction-diffusion equations, whereas the continuous case amounts to an uncountable system of such equations. These problems can be handled by interpreting them as abstract vector-valued parabolic evolution equations, where the dependent variables take values in infinite-dimensional Banach spaces. Given suitable assumptions, we prove existence and uniqueness in the class of volume preserving solutions. We also derive sufficient conditions for global existence.

On the convergence and representation of random fuzzy number integrals

In this paper, a new kind of convergence, w-s convergence is introduced for certain type of fuzzy numbers taking values in separable reflexive Banach spaces. The convergence theorem of random fuzzy number integrals in the w-s sense is given and a condition, under which a fuzzy number function can be represented by random fuzzy number integrals, is obtained.

Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces

In the present paper we consider the transition semigroup Pt related to some stochastic reaction-diAusion equations with the non- linear term f having polynomial growth and satisfying some dis- sipativity conditions. We are proving that it has a regularizing eAect in the Banach space of continuous functions COOU, where O R d is a bounded open set. In L 2 OOU the only result proved is the strong Feller property, for da 1. Here we are able to prove that if f 2 C 1 ORU and d 3, then Ptu2 C 1 b OCOOUU for any u2 BbOCOOUU and t > 0. An important application is to the study of the ergodic properties of the system. These results are also of interest for some problem in sto- chastic control.

The evolution of core collections can be described via Banach space valued stochastic processes

Core collections, as e.g., the set of source journals selected by the Institute for Scientific Information, vary from year to year: most of them stay, some leave, and others enter. In this paper, we show that the mechanism of this process can be revealed by using stochastic processes with values in an infinite dimensional Banach space. More concretely we show that the evolution can be described by a quasi-martingale with values in the Hubert space L 2. Criteria for the convergence of these processes are given, leading to a stable limit set of core items, for t (time) high.

Liapunov–Richter theorem in B-convex spaces

Nonatomic finitely additive correspondences with values in B-convex Banach spaces necessarily have the property that the closure of the range is convex. For this one assumes that the correspondences are of bounded variation.

SIMPLY REGULATED FUNCTIONS AND SEMIVARIATION IN UNIFORMLY CONVEX SPACES

In the last two decades the Riemann generalized integral, having values in Banach spaces, has been increasingly studied. The development in this area concerns mainly with the Henstock-Kurzweil (see e.g.[1], [2]), and the Dushnik and the Young integrals (see e.g.[3]). Recently it has appeared in the literature many proper applications in this field (proper here considered in the sense that the results are not disguises of an essentially finite dimensional frame) (see e.g.[4],[5]). In this paper we give a result having its roots in this frame.

Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces

Pointwise and mean ergodic theorems under dierent assumptions for subadditive super- stationary families of random sets whose values are weakly (or strongly) compact convex subsets of a separable Banach space are presented. The results generalize the results of [14], where random sets in Rd are considered. Techniques used here are inspired by [3].

Laplace Transforms and Integrated, Regularized Semigroups in Locally Convex Spaces

In this paper, we consider the vector-valued Laplace transforms,r-times (r∈[0,∞)) integrated semigroups and regularized semigroups in the context of sequentially complete locally convex spaces. Our theorems develop the corresponding results in [1,11], including the well known integrated version of the classical Widder's representation theorem of Laplace transforms for functions taking values in Banach spaces. Moreover, we study a class of differential operators on certain function spaces. Optimal conditions, making them the generators of integrated or regularized semigroups, are obtained. We finally show some applications to abstract Cauchy problems.

Φ-INEQUALITIES AND LAWS OF LARGE NUMBERS OF HARDY MARTINGALE TRANSFORMS

Using stopping time method we proved the Φ-inequalities, pointwise convergence, strong and weak laws of large numbers of Hardy martingale transforms with values in complex Banach spaces, and applying them to give several characterizations of AUMD spaces.

Stochastic integration with respect to a stochastic integral

In this paper we prove first the property of integration with respect to a measure defined by density,h(fm) = (hf)mor a measure mand functions f,h, taking values in Banach spaces. Then we use this result to prove the similar “associativity” property of the stochastic integralL.(K-X)= (LK) Xfor processes X,K,Ltaking values in Banach spaces

The Law of the Iterated Logarithm for m -Dependent Banach Space Valued Random Variables

A theorem on the law of the iterated logarithm is established for m-dependent B-valued random variables. The conditions in our theorem match their independent analogues and appear as necessary or minimal for the results. According to an example given in the paper, the situation we face is much different from the finite dimensional case and therefore, so is the way we solve the problems.

Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients

We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.

Abstract Perron-Stieltjes integral

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite4). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In \cite3 Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig's integral the Dushnik interior integral presents the background. \endgraf It should be mentioned that abstract Stieltjes integration was recently used by O. Diekmann, M. Gyllenberg and H. R. Thieme in \cite1 and \cite2 for describing the behaviour of some evolutionary systems originating in problems concerning structured population dynamics.

SMOOTHING OF ABSTRACT FUNCTIONS

This paper investigates the problem of approximating uniformly continuous maps by smoother maps. Here the uniformly continuous maps are defined on subsets of Banach spaces and take values in other Banach spaces. Bibliography: 11 titles.

Laws of Large Numbers in Banach Spaces of Type $(F,F_1 )$

This paper gives strong and weak laws of large numbers for random elements with values in Banach spaces of type $(F,F_1 )$. The known laws of large numbers in Banach spaces of stable and Rademacher type p are special cases of these results. Characterizations of spaces of type $(F,F_1 )$ are given in terms of these laws.

The generalized Yosida–Hewitt theorem

The Yosida–Hewitt (YH, for short) theorem [YH] has many versions and generalizations in diverse settings, e.g. functionals on vector lattices and spaces of vector-valued functions, measures with values in Banach spaces, topological groups and vector lattices, etc. In this paper we derive a very general form of the YH theorem dealing with the much more general case of operators acting in vector lattices (VLs, for short) and Banach spaces (BSs, for short). A unified approach to all settings mentioned above may be founded on decompositions for operators in VLs.

Some Bifurcation Results Including Banach Space Valued Parameters

Some Bifurcation Results Including Banach Space Valued Parameters

Canonical Factorization of Gaussian Covariance Operators and Some of Its Applications

This paper contains a systematic approach to the analysis of Gaussian random elements with values in Banach spaces. This approach is based on the idea of canonical factorization of Gaussian covariance operators.

On LIL behaviour for moving averages of some infinitely divisible random measures

We study LIL's for moving averages of Banach space valued (deterministic) functions wrt. homogeneous infinitely divisible independently scattered random measures on metrizable LCA topological groups.