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In the present paper, we investigate the existence of solutions to second order nonlinear boundary value problems (BVPs) involving the distributional Henstock-Kurzweil integral. The present results in this article are generalizations of previous results in the literature.

This paper is devoted to the study of existence and dependence of solutions of the first-order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The methods used are mainly the method of upper and lower solutions and a fixed point theorem.

In this paper, we study a special Banach lattice D HK , which is induced by the distributional Henstock-Kurzweil integral, and discuss its lattice properties. We show that D HK is an AM-space with the Archimedean property and the Dunford-Pettis property but it is not Dedekind complete. We also present two fixed point theorems in D HK . Meanwhile, two examples are worked out to demonstrate the results.

AbstractThe purpose of this paper is to study the existence of periodic solutions and the topological structure of the solution set of first-order differential equations involving the distributional Henstock–Kurzweil integral. The distributional Henstock–Kurzweil integral is a general integral, which includes the Lebesgue and Henstock–Kurzweil integrals. The main results extend some previously known results in the literature.