Simple Properties and Existence Theorem for the Henstock-Kurzweil-Stieltjes Integral of Functions Taking Values on C[a,b] Space-valued Functions

Andrew Felix Iv Suarez Cunanan,Julius Benitez

doi:10.29020/nybg.ejpam.v13i1.3626

Andrew Felix Iv Suarez Cunanan, Julius Benitez

Open Access

https://doi.org/10.29020/nybg.ejpam.v13i1.3626

Copy DOI

Abstract

Henstock--Kurzweil integral, a nonabsolute integral, is a natural extension of the Riemann integral that was studied independently by Ralph Henstock and Jaroslav Kurzweil. This paper will introduce the Henstock--Kurzweil--Stieltjes integral of $\mathcal{C}[a,b]$-valued functions defined on a closed interval $[f,g]\subseteq\mathcal{C}[a,b]$, where $\mathcal{C}[a,b]$ is the space of all continuous real-valued functions defined on $[a,b]\subseteq\mathbb{R}$. Some simple properties of this integral will be formulated including the Cauchy criterion and an existence theorem will be provided.

Highlights

The Henstock–Kurzweil–Stieltjes integral is a generalized Riemann–Stieltjes integral which has properties similar to it
Where D = {([hi−1, hi], ti)}ni=1 is a tagged division of [f, g] Notion of integrals for Banach space-valued functions like Henstock integral for Banach space-valued functions, Henstock–Stieltjes integral of real-valued functions with respect to an increasing function and Henstock–Stieltjes integral for Banach spaces were already defined by Cao [3]
We say that the function F is Henstock−Kurzweil−Stieltjes integrable with respect to H on [f, g] to S ∈ C[a, b], briefly HKS-integrable, if for any > 0, there exists a gauge δ on [f, g] such that for any δ-fine tagged division D of [f, g], we have

Summary

Introduction

The Henstock–Kurzweil–Stieltjes integral is a generalized Riemann–Stieltjes integral which has properties similar to it. In the paper [9], Ubaidillah introduce the Henstock– Kurzweil integral of functions taking values in C[a, b] through Riemann sums. D where D = {([hi−1, hi], ti)}ni=1 is a tagged division of [f, g] Notion of integrals for Banach space-valued functions like Henstock integral for Banach space-valued functions, Henstock–Stieltjes integral of real-valued functions with respect to an increasing function and Henstock–Stieltjes integral for Banach spaces were already defined by Cao [3],. Lim [7] and Tikare [8], respectively. In this paper we change the way to define the domain of the function and the integrator. We shall choose first a closed interval [f, g] as our domain and a continuous real-valued function H instead of the identity map as our integrator

Preliminaries

Simple Properties

An Existence Theorem