The Distributional Henstock-Kurzweil Integral and Applications: A Survey

Abstract

This study presents a summary of the current state of research on the distributional Henstock-Kurzweil integral. Basic properties such as integration by parts, Hake theorem, inner product, Hölder inequality, second mean value theorem, orderings, Banach lattice, convergence theorems, fixed point theorems, are shown. This study also summarizes its applications in integral and differential equations.

Riemann-Liouville derivative over the space of integrable distributions

“…On the other hand, in [10] and [11] it is shown that the completion of the Henstock-Kurzweil integrable functions space, HK[a, b], is isomorphic to D HK . Furthermore, in [11], [34] and [37] the following result is proved. , see [1].…”

“…From the previous proposition we obtained the following relations. Note that J n a f is a temperate distribution when f ∈ D HK and n ≥ 0, see [34], [37]. Conclusions.…”

“…Observe that if f is in L 1 [a, b], then J n a f (x) = J n a f (x), since the distributional Henstock-Kurzweil integral contains the Lebesgue integral. Via the Hölder inequality, [34] and [37], it is easy to see that J n a f is a temperate distribution for any f ∈ D HK and n ≥ 0.…”

Existence of Solutions for Functional Integral Equation Involving the Henstock-Kurzweil-Stieltjes Integral

“…dg can be identified with a Stieltjes measure and will have the effect of suddenly changing the state of the system at the points of discontinuity of g, that is, the system could be controlled by some impulsive force. The Henstock-Kurzweil-Stieltjes integral, which is a generalization of the Lebesgue-Stieltjes integral (Krejčí, 2006;Kurzweil, 1957;Lee, 1989;Schwabik & Ye, 2005), has been proved useful in the study of ordinary differential equations (Chew, 1988;Chew & Flordeliza, 1991;Heikkilä & Ye, 2012;Ye & Liu, 2016).…”

Inclusion Properties of Henstock-Orlicz Spaces

“…Many researchers have studied Henstock-Kurzweil integral, Orlicz spaces, inclusion properties, and ℋ -Orlicz spaces. Several studies on Henstock-Kurzweil integrals include discussing some properties of Henstock-Kurzweil integrable function in n-dimensional spaces (Herlinawati, 2021), the different definition of Henstock-Kurzweil integral on a closed bounded interval by using primitive (Leng & Yee, 2018), Henstock-Kurzweil integral on Riezs Spaces (Boccuto et al, 2012), generalization of this integral (Malý & Kuncová, 2019), the distributional Henstock-Kurzweil integral and also applications in integral and differential equation (Liu, 2016), the Henstock-Kurzweil transform (Sánchez-Perales et al, 2012)(Sánchez-Perales et al, 2012)(Sánchez-Perales et al, 2012)(Sánchez-Perales et al, 2012)(Sánchez-Perales et al, 2012)(Sánchez-Perales et al, 2012) (Mendoza-Torres et al, 2015Mendoza Torres et al, 2002;Sánchez-Perales et al, 2012;Talvila, 2002), and inclusion relations for several spaces such as Lebesgue spaces, Henstock-Kurzweil spaces, and bounded variation spaces (Mendoza Torres et al, 2009).…”