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

Aneta Sikorska-Nowak

Displaying similar documents to “Integro-differential equations on time scales with Henstock-Kurzweil delta integrals”

On a generalization of Henstock-Kurzweil integrals

Jan Malý, Kristýna Kuncová (2019)

Mathematica Bohemica

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We study a scale of integrals on the real line motivated by the $M C_{α}$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.

Volterra integral equations governed by highly oscillatory functions on the time scale.

Satco, Bianca (2009)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

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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.

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 $(x, y) \in {[0, 1]}^{2}$ .

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...

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.

Differential equations in banach space and henstock-kurzweil integrals

Ireneusz Kubiaczyk, Aneta Sikorska (1999)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, using the properties of the Henstock-Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x' = f(t,x) and inclusion x' ∈ F(t,x) in a Banach space, where f is Henstock-Kurzweil integrable and satisfies some conditions.

On the existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals

Aneta Sikorska-Nowak (2004)

Annales Polonici Mathematici

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We prove some existence theorems for nonlinear integral equations of the Urysohn type $x (t) = φ (t) + λ \int_{0}^{a} f (t, s, x (s)) d s$ and Volterra type $x (t) = φ (t) + \int_{0}^{t} f (t, s, x (s)) d s$ , $t \in I_{a} = [0, a]$ , where f and φ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

Volterra integral inclusions via Henstock-Kurzweil-Pettis integral

Bianca Satco (2006)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we prove the existence of continuous solutions of a Volterra integral inclusion involving the Henstock-Kurzweil-Pettis integral. Since this kind of integral is more general than the Bochner, Pettis and Henstock integrals, our result extends many of the results previously obtained in the single-valued setting or in the set-valued case.

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...