Gauge Integral

Roland Coghetto

Displaying similar documents to “Gauge Integral”

The origin and developments of Kurzweil's generalized Riemann integral

Jean Mawhin (2025)

Czechoslovak Mathematical Journal

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The paper describes to origin and motivation of Kurzweil in introducing a Riemann-type definition for generalized Perron integrals and his further contributions to the topics.

Substitution formulas for the Kurzweil and Henstock vector integrals

Márcia Federson (2002)

Mathematica Bohemica

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Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the Generalized Riemann Integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach-space valued context, the Kurzweil integral properly contains that of Henstock. In the present paper, we consider abstract vector integrals of Kurzweil and...

A constructive integral equivalent to the integral of Kurzweil

Márcia Federson (2002)

Czechoslovak Mathematical Journal

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We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.

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

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.