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

Similarity:

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 ± ∞ 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 ∫ 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...

The L r Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

Similarity:

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

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

Similarity:

If f is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of f is ∥ f ∥ = sup I | ∫ I f | where the supremum is taken over all intervals I ⊂ ℝ . Define the translation τ x by τ x f ( y ) = f ( y - x ) . Then ∥ τ x f - f ∥ tends to 0 as x tends to 0 , i.e., f is continuous in the Alexiewicz norm. For particular functions, ∥ τ x f - f ∥ can tend to 0 arbitrarily slowly. In general, ∥ τ x f - f ∥ ≥ osc f | x | as x → 0 , where osc f is the oscillation of f . It is shown that if F is a primitive of f then ∥ τ x F - F ∥ ≤ ∥ f ∥ | x | . An example shows that the function y ↦ τ x F ( y ) - F ( y ) need not be in L 1 . However, if...

On Kurzweil-Stieltjes integral in a Banach space

Giselle A. Monteiro, Milan Tvrdý (2012)

Mathematica Bohemica

Similarity:

In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space X . 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 ∫ a b d [ F ] g exists if F : [ a , b ] → L ( X ) has a bounded semi-variation on [ a , b ] and g : [ a , b ] → X is regulated on [ a , b ] . We prove that this integral has sense also if F is regulated...

Henstock-Kurzweil integral on BV sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

Similarity:

The generalized Riemann integral of Pfeffer (1991) is defined on all bounded BV subsets of ℝ n , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of σ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of BV sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...