Convergence Theorems for the Henstock—Kurzweil—Pettis Integral

We prove some convergence theorems for the Henstock-Kurzweil- Pettis and Denjoy-Pettis integrals. Since these integrals are more general than some “classical” non-absolute integrals and than the Pettis integral, we generalize well-known convergence theorems for both types of the mentioned integrals.     

关于 C-Pettis 积分的一个控制收敛定理

In this paper, we give the Riemann-type extensions of Dunford integral and Pettis integral, C-Dunford integral and C-Pettis integral. We discuss the relationship between the C-Pettis integral and Pettis integral, and prove a controlled convergence theorem for the C-Pettis integral.     

关于取值于局部凸空间中集值映射的积分

本文引进了一类集值映射的积分，研究了这种积分的若干性质．在此基础上，讨论了这种积分与抽象函数的Riemann积分、Bochner积分和Petits积分的关系．     

Some Kurzweil-Henstock-type integrals and the wide Denjoy integral

Kurzweil-Henstock integrals related to local systems and the wide Denjoy integral are discussed in the frame of their comparability and compatibility. Dedicated to the memory of Ralph Henstock (1923–2007)     

NOWHERE WEAK DIFFERENTIABILITY OF THE PETTIS INTEGRAL

Abstract

For an arbitrary infinite-dimensional Banach space 𝔛, we construct examples of strongly-measurable 𝔛-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the Lebesgue Differentiation Theorem fails rather spectacularly. We also relate the degree of nondifferentia-bility of the indefinite Pettis integral to the cotype of 𝔛, from which it follows that our examples are reasonably sharp.     

Denjoy integral and Henstock-Kurzweil integral in vector lattices,II

In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.      

Existence results for Urysohn integral inclusions involving the Henstock integral

This paper presents two existence results for Urysohn integral inclusions in Banach spaces, the set-valued integral involved being the Henstock integral. In particular, the existence of continuous solutions for integral inclusions of Volterra and of Hammerstein type is obtained. Our results extend the existence theorems given in literature in the single- or set-valued case, under Bochner or Pettis integrability hypothesis. The compactness of solutions set is also studied.     

On the Pettis Integral of Closed Valued Multifunctions

In this work, the study of Pettis integrability for multifunctions (alias set-valued maps), whose values are allowed to be unbounded, is initiated. For this purpose, two notions of Pettis integrability, and of Pettis integral, are considered and compared. The first notion is similar to that of the weak integral, already known for vector-valued functions, and is defined via support functions. The second notion resembles the classical Aumann definition using integrable selections, but it involves the Pettis integrable selections rather than the Bochner integrable ones. The above two integrals are shown to coincide in a quite general setting. Several criteria for a multifunction to be Pettis integrable (in one sense or the other) are proved. On the other hand, due to the possibility of infinite values for the support functions, we are led to introduce a more general notion of scalar integrability involving the negative part of these functions. We compare the scalar integrability of a multifunction with that of its measurable selections. We also provide some new results concerning multifunctions with bounded values and/or new proofs of already existing ones. Examples are included to illustrate the results and to introduce open problems.     

The McShane, PU and Henstock integrals of Banach valued functions

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized.     

On the Existence of Weak Solutions of Nonlinear Integral Equations in Banach Spaces

In this paper, we use the techniques of measure of weak noncompactness and Henstock–Kurzweil–Pettis integrals to discuss the existence theorem of weak solutions for a class of the nonlinear integral equations and obtain a new result, which improves and extends some relevant results for differential and integral equations in Banach spaces.     

广义函数Denjoy积分的收敛性问题

本文讨论广义函数De njoy积分的收敛性问题.首先给出了广义Denjoy可积函数空间中强收敛、弱收敛、弱~*收敛和广义函数Denjoy积分收敛的关系;证明拟一致收敛是广义函数Denjoy积分收敛的一个充分必要条件;最后指出了Denjoy可积广义函数列弱~*收敛与强收敛等价当且仅当原函数等度连续.     

On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.     

Vitali type convergence theorems for Banach space valued integrals

Let(Ω,Σ,μ)be a complete probability space and let X be a Banach space.We introduce the notion of scalar equi-convergence in measure which being applied to sequences of Pettis integrable functions generates a new convergence theorem.We also obtain a Vitali type I-convergence theorem for Pettis integrals where I is an ideal on N.     

本刊英文版Vol.29(2013),No.11论文摘要(英文)

<正>Vitali Type Convergence Theorems for Banach Space Valued Integrals Marek BALCERZAK Kazimierz MUSIAL Abstract Let(Ω,∑,μ)be a complete probability space and let X be a Banach space.We introduce the notion of scalar equi-convergence in measure which being applied to sequences of Pettis integrable functions generates a new convergence theorem.We also obtain a Vitali type I-convergence theorem for Pettis integrals where I is an ideal on N.On Universally Left-stability ofε-Isometry     

A unified elliptic-type integral and associated recurrence relations

Elliptic-type integrals have their importance or potential in certain problems in radiation physics and nuclear technology. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. Beside explicit representations of certain families of unified elliptic-type integrals, contour integral representation, various recurrence relations are derived in the present investigation. The results derived in this article are of manifold generality and provide extensions and unification to the large number of known results established earlier.     

关于Banach-值函数的Henstock积分

讨论了Banach-值函数的H enstock积分与P ettis积分的关系,证明了若f是几乎可分值且弱有界的,则f是H enstock可积的当且仅当f是P ettis可积的.     

集值映射Pettis-Aumann积分的一些性质

对集值映射的Pettis-Aumann型积分的性质作了讨论，给出了Pettis-Aumann积分型积分的Fatou引理形式以及其它一些收敛定理，探讨了带参数集值映射的积分性质。     

On the equivalence of Beukers-type and Sorokin-type multiple integrals

It is well known that a triple Beukers-type integral, as defined by G. Rhin and C. Viola, can be transformed into a suitable triple Sorokin-type integral. I will discuss possible extensions to the n-dimensional case of a similar equivalence between suitably defined Beukers-type and Sorokin-type multiple integrals, with consequences on the arithmetical structure of such integrals as linear combinations of zeta-values with rational coefficients.     

C半群的概率逼近问题

借助Pettis积分、随机过程、矩生成函数及算子值数学期望,给出了一般形式的C半群概率逼近指数公式、生成定理及其收敛速度的估计式,也从另一个角度得出C半群概率表示的Vonorovskaya型渐近公式.