Differential and integral equations with Henstock-Kurzweil integrable functions

In this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing Henstock-Kurzweil integrable functions.     

Henstock-Kurzweil可积函数空间的紧性特征

本文研究Henstock-Kurzweil可积(HK可积)函数空间中的一个经典问题.文章通过研究分布Henstock-Kurzweil积分(DHK积分)的性质,给出了该问题的否定答案.进一步,利用收敛性获得了函数HK可积的一个充分必要条件.最后,在上述结论的基础上刻画了HK可积函数空间的紧性.所得结果丰富和推广了HK可...     

Bounded linear functionals on the space of Henstock-Kurzweil integrable functions

Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.     

ON THE CONSTRUCTION OF MAJOR AND MINOR FUNCTIONS

In this paper,we construct directly absolutely continuous major and minor functions of a function which is Lebesque integrable ,and we also construct directly continuous major and minor functions of a function which is Henstock-Kurzweil integrable.     

Vector Valued Martingale-Ergodic and Ergodic-Martingale Theorems

We prove martingale-ergodic and ergodic-martingale theorems for vector-valued Bochner integrable functions. We obtain dominant and maximal inequalities. We also prove weighted and multiparameter martingale-ergodic and ergodic-martingale theorems.     

Henstock-Kurzweil 可积函数的Laplace 变换的反演定理

该文建立了Henstock-Kurzweil 可积函数的 Laplace变换, 讨论了其基本性质及解析性质, 得到Henstock-Kurzweil可积意义下的反演公式, 并给出反例说明这一结果不能改进     

Compact operators and integral equations in the $${\cal H}{\cal K}$$ ℋ K space

Czechoslovak Mathematical Journal - The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Fréchet spaces $${\cal H}{\cal K}$$ (X). In...     

Functions not first-return integrable

Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0,1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C^∞ on (0,1] yet fails to be first-return integrable.     

Equations containing locally Henstock-Kurzweil integrable functions

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.     

Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II

This paper is a continuation of the paper [T.Y. Lee, Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral, J. Math. Anal. Appl. 298 (2004) 677-692], in which we proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. Let f be Henstock-Kurzweil integrable on a compact interval . For a given compact interval , set

     

Convergence theorems for HL integrable vector-valued functions with applications

In this paper we prove dominated and monotone convergence theorems for HL integrable Banach-valued functions. These results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of Fredholm type in ordered Banach spaces involving Kurzweil integrals or improper integrals. Results are used also to solve concrete second-order functional boundary value problems involving discontinuities and singularities.     

A continous version of Orlicz-Pettis theorem via vector-valued Henstock-Kurzweil integrals

We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.     

On Integrable Functions in Complete Bornological Locally Convex Spaces

A Lebesgue-type integration theory in complete bornological locally convex topological vector spaces was introduced by the first author in [17]. In this paper we continue developing this integration technique and formulate and prove some theorems on integrable functions as well as some convergence theorems. An example of Dobrakov integral in non-metrizable complete bornological locally convex spaces is given.     

On generalizations of a theorem of Ferenc Lukács

A theorem of Ferenc Lukács determines the jumps of a periodic Lebesgue integrable function f at each point of discontinuity of the first kind in terms of the partial sums of the conjugate Fourier series of f. The aim of this note is to prove analogous theorems for functions and series, introduced by Taberski ([10], [11]).     

Two convergence theorems for Henstock-Kurzweil integrals and their applications to multiple trigonometric series

We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R.P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.     

On the convergence properties of measurable multifunctions with values in a separable Banach space

In this paper we prove some convergence theorems for Banach space valued multifunctions. First we consider the notion of weak convergence of sets and prove a weak completeness and a weak compactness result of the Dunford-Pettis type for weakly compact, convex valued integrable multifunctions. Then we consider set valued martingales and establish two convergence theorems. One using the Kuratowski-Mosco mode of convergence and for the other the Hausdorff mode.     

Analytic geometry and semi-classical analysis

We study deformation theory for quantum integrable systems and prove several theorems concerning the Gevrey convergence and the unicity of perturbative expansions.     

Decomposability in the space of HKP‐integrable functions

In this paper we introduce the notion of decomposability in the space of Henstock‐Kurzweil‐Pettis integrable (for short HKP‐integrable) functions. We show representations theorems for decomposable sets of HKP‐integrable or Henstock integrable functions, in terms of the family of selections of suitable multifunctions.     

Spaces of integrable functions with respect to a vector measure and factorizations through and Hilbert spaces

We use the integration structure of the spaces of scalar integrable functions with respect to a vector measure to provide factorization theorems for operators between Banach function spaces through Hilbert spaces. A broad class of Banach function spaces can be represented as spaces of scalar integrable functions with respect to a vector measure, but this representation (the vector measure) is not unique. Since our factorization depends on the vector measure that is used for the representation we also give a characterization of those vector measures whose corresponding spaces of integrable functions coincide.     

Kantorovich–Wright integral and representation of quasi-Banach lattices

The purpose of this paper is two-fold: first, to outline a purely order-based integral of the type of the Kantorovich–Wright integral of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a K _σ-space (that is, a Dedekind σ-complete vector lattice) and, secondly, prove new theorems on the representation of Dedekind complete vector lattices and quasi-Banach lattices in the form of lattices of functions integrable or “weakly” integrable with respect to an appropriate vector measure. In particular, it is shown that, in studying quasi-Banach lattices, when the duality method does not apply, the Kantorovich–Wright integral is more flexible than the Bartle–Dunford–Schwartz integral.