The distributional Henstock-Kurzweil integral and applications (Q344232)

The paper studies the distributional Henstock-Kurzweil integral, its properties and certain applications. After recalling some basic theory from \textit{E. Talvila} [Real Anal. Exch. 33, No. 1, 51--82 (2008; Zbl 1154.26011)], weak and strong convergence theorems are proved. Also, it is shown that the space \(\mathcal{D}_{\mathrm{HK}}\) of Henstock-Kurzweil integrable distributions can be made into an ordered Banach space with a regular cone, and a fixed-point theorem in \(\mathcal{D}_{\mathrm{HK}}\) is given. As an application, the authors prove the existence of solutions to nonlinear integral equations involving the distributional Henstock-Kurzweil integral which are not solutions in the classical sense.