On a generalization of Henstock-Kurzweil integrals

Jan Malý; Kristýna Kuncová

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Displaying similar documents to “On a generalization of Henstock-Kurzweil integrals”

The $L^{r}$ Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

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We present a method of integration along the lines of the Henstock-Kurzweil integral. All $L^{r}$ -derivatives are integrable in this method.

Convergence of ap-Henstock-Kurzweil integral on locally compact spaces

Hemanta Kalita, Ravi P. Agarwal, Bipan Hazarika (2025)

Czechoslovak Mathematical Journal

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We introduce an ap-Henstock-Kurzweil type integral with a non-atomic Radon measure and prove the Saks-Henstock type lemma. The monotone convergence theorem, $μ_{ap}$ -Henstock-Kurzweil equi-integrability, and uniformly strong Lusin condition are discussed.

Integro-differential equations on time scales with Henstock-Kurzweil delta integrals

Aneta Sikorska-Nowak (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we prove existence theorems for integro - differential equations $x^{Δ} (t) = f (t, x (t), \int ₀^{t} k (t, s, x (s)) Δ s)$ , t ∈ Iₐ = [0,a] ∩ T, a ∈ R₊, x(0) = x₀ where T denotes a time scale (nonempty closed subset of real numbers R), Iₐ is a time scale interval. Functions f,k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and...

Role of the Harnack extension principle in the Kurzweil-Stieltjes integral

Umi Mahnuna Hanung (2024)

Mathematica Bohemica

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In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with...

Integration and decompositions of weak $^{*}$ -integrable multifunctions

Kazimierz Musiał (2025)

Czechoslovak Mathematical Journal

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Conditions guaranteeing Pettis integrability of a Gelfand integrable multifunction and a decomposition theorem for the Henstock-Kurzweil-Gelfand integrable multifunctions are presented.

A full characterization of multipliers for the strong $ρ$ -integral in the euclidean space

Lee Tuo-Yeong (2004)

Czechoslovak Mathematical Journal

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We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $ρ$ -integral, introduced by Jarník and Kurzweil. Let $(𝒮_{ρ} (E), ∥ \cdot ∥)$ be the space of all strongly $ρ$ -integrable functions on a multidimensional compact interval $E$ , equipped with the Alexiewicz norm $∥ \cdot ∥$ . We show that each element in the dual space of $(𝒮_{ρ} (E), ∥ \cdot ∥)$ can be represented as a strong $ρ$ -integral. Consequently, we prove that $f g$ is strongly $ρ$ -integrable on $E$ for each strongly $ρ$ -integrable function $f$ if and only if $g$ is...

Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion

Tuo-Yeong Lee (2005)

Mathematica Bohemica

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It is shown that a Banach-valued Henstock-Kurzweil integrable function on an $m$ -dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function $f {[0, 1]}^{2} ⟶ ℝ$ and a continuous function $F {[0, 1]}^{2} ⟶ ℝ$ such that $(¶) \int_{0}^{x} \{(¶) \int_{0}^{y} f (u, v) d v\} d u = (¶) \int_{0}^{y} \{(¶) \int_{0}^{x} f (u, v) d u\} d v = F (x, y)$ for all $(x, y) \in {[0, 1]}^{2}$ .

Cauchy's residue theorem for a class of real valued functions

Branko Sarić (2010)

Czechoslovak Mathematical Journal

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Let $[a, b]$ be an interval in $ℝ$ and let $F$ be a real valued function defined at the endpoints of $[a, b]$ and with a certain number of discontinuities within $[a, b]$ . Assuming $F$ to be differentiable on a set $[a, b] ∖ E$ to the derivative $f$ , where $E$ is a subset of $[a, b]$ at whose points $F$ can take values $\pm \infty$ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to $0$ at all points of $E$ and show that $𝒦ℋ -vt \int_{a}^{b} f = F (b) - F (a)$ , where $𝒦ℋ -vt$ denotes the total value of the integral. The paper ends with a few examples that illustrate the...

A general integral

R. Estrada, J. Vindas

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We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if f is locally distributionally integrable over the real line and ψ ∈ (ℝ) is a test function, then fψ is distributionally integrable, and the formula $⟨, ψ ⟩ = () \int_{- \infty}^{\infty} f (x) ψ (x) d x$ , defines a distribution...