Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations

Abstract

In this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

On The Generalisation of Henstock-Kurzweil Fourier Transform

“…In Definition 3.4 of [5], Denjoy-Perron type integrals on unbounded intervals are defined and then it is proved that it is equivalent to Henstock-Kurzweil integral on unbounded intervals (Theorem 3.2 of [5]). Similarly, we shall define the Laplace integral on unbounded intervals and establish its properties.…”

“…From Section 3 of [6] and Definition 3.4, Theorem 3.2 of [5], it is evident that Denjoy-Perron integral or Henstock-Kurzweil integral on unbounded intervals is a particular case of Laplace integral; however, the following example will ensure that LP(R) \ HK(R) is non-empty.…”

Sturm–Liouville Differential Equations Involving Kurzweil–Henstock Integrable Functions

“…A particular characteristic of these coefficients is that they are not square Lebesgue integrable. The study of differential equations involving integrable Henstock-Kurzweil functions has been developed by several authors, for example, [1][2][3][4][5][6][7]. In [8], Pérez et al introduced the KH-Sobolev space on bounded intervals and guaranteed the existence and uniqueness of the solution to some boundary value problems involving Kurzweil-Henstock integrable functions on [0, 1].…”