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T -equivalences generated by shape function on the real line

Dug Hun Hong (2003)

Kybernetika

This paper is devoted to give a new method of generating T-equivalence using shape function and finding the exact calculation formulas of T-equivalence induced by shape function on the real line. Some illustrative examples are given.

The ap-Denjoy and ap-Henstock integrals

Jae Myung Park, Jae Jung Oh, Chun-Gil Park, Deuk Ho Lee (2007)

Czechoslovak Mathematical Journal

In this paper we define the ap-Denjoy integral and show that the ap-Denjoy integral is equivalent to the ap-Henstock integral and the integrals are equal.

The AP-Denjoy and AP-Henstock integrals revisited

Valentin A. Skvortsov, Piotr Sworowski (2012)

Czechoslovak Mathematical Journal

The note is related to a recently published paper J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis...

The Basic Existence Theorem of Riemann-Stieltjes Integral

Kazuhisa Nakasho, Keiko Narita, Yasunari Shidama (2016)

Formalized Mathematics

In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15],...

The Borel structure of some non-Lebesgue sets

Don L. Hancock (2004)

Colloquium Mathematicae

For a given function in some classes related to real derivatives, we examine the structure of the set of points which are not Lebesgue points. In particular, we prove that for a summable approximately continuous function, the non-Lebesgue set is a nowhere dense nullset of at most Borel class 4.

The continuous solutions of a generalized Dhombres functional equation

L. Reich, Jaroslav Smítal, M. Štefánková (2004)

Mathematica Bohemica

We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J J is a given increasing homeomorphism of an open interval J ( 0 , ) and f ( 0 , ) J is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under ϕ and which contains in its interior no fixed point except for 1 . They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...

The converse of the Hölder inequality and its generalizations

Janusz Matkowski (1994)

Studia Mathematica

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ʃ Ω x y d μ ϕ - 1 ( ʃ Ω ϕ x d μ ) ψ - 1 ( ʃ Ω ψ x d μ ) for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist...

The converse problem for a generalized Dhombres functional equation

L. Reich, Jaroslav Smítal, M. Štefánková (2005)

Mathematica Bohemica

We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J J is a given homeomorphism of an open interval J ( 0 , ) and f ( 0 , ) J is an unknown continuous function. A characterization of the class 𝒮 ( J , ϕ ) of continuous solutions f is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when ϕ is increasing. In the present paper we solve the converse problem, for which continuous maps f ( 0 , ) J , where J is an interval, there is an increasing homeomorphism ϕ of J such that f 𝒮 ( J , ϕ ) . We...

The Covering Principle for Darboux Baire 1 functions

Piotr Szuca (2007)

Fundamenta Mathematicae

We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected G δ subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].

The Denjoy extension of the Riemann and McShane integrals

Jae Myung Park (2000)

Czechoslovak Mathematical Journal

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval a , b into a Banach space X . It is shown that a Denjoy-Bochner integrable function on a , b is Denjoy-Riemann integrable on a , b , that a Denjoy-Riemann integrable function on a , b is Denjoy-McShane integrable on a , b and that a Denjoy-McShane integrable function on a , b is Denjoy-Pettis integrable on a , b . In addition, it is shown that for spaces that do not contain a copy of c 0 , a measurable Denjoy-McShane integrable...

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