Displaying similar documents to “A nonexistence result for the Kurzweil integral”

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

Branko Sarić (2010)

Czechoslovak Mathematical Journal

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Let be an interval in and let be a real valued function defined at the endpoints of and with a certain number of discontinuities within . Assuming to be differentiable on a set to the derivative , where is a subset of at whose points can take values or not be defined at all, we adopt the convention that and are equal to at all points of and show that , where denotes the total value of the integral. The paper ends with a few examples that illustrate the...

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...

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

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If is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of is where the supremum is taken over all intervals . Define the translation by . Then tends to as tends to , i.e., is continuous in the Alexiewicz norm. For particular functions, can tend to 0 arbitrarily slowly. In general, as , where is the oscillation of . It is shown that if is a primitive of then . An example shows that the function need not be in . However, if...

On Kurzweil-Stieltjes integral in a Banach space

Giselle A. Monteiro, Milan Tvrdý (2012)

Mathematica Bohemica

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In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space We extend results obtained by Štefan Schwabik and complete the theory so that it will be well applicable to prove results on the continuous dependence of solutions to generalized linear differential equations in a Banach space. By Schwabik, the integral exists if has a bounded semi-variation on and is regulated on We prove that this integral has sense also if is regulated...