On first-order periodic boundary value problems and distributional Henstock-Kurzweil integrals (Q472288)

The authors consider the periodic boundary value problem \[ Dx=f(t)+h(t,x(t)),\quad x(0)=x(T),\eqno{(1)} \] where the solution \(x:[0,T]\to\mathbb R\) is assumed to be continuous, \(Dx\) stands for the distributional derivative of~\(x\), \(f\) is a distribution on \([0,T]\), and \(h:[0,T]\times \mathbb R\to \mathbb R\). (The form of the equation \(Dx=f(t)+h(t,x(t))\) is somewhat misleading, since the symbol \(f(t)\) is meaningless if \(f\) is a general distribution and not merely a function. According to the reviewer's opinion, it would be more precise to write the equation as \(Dx=f+h(\cdot,x(\cdot))\).) The main result of the paper guarantees the existence of extremal (i.e., smallest and greatest) solutions to (1). Moreover, the extremal solutions are shown to be nondecreasing with respect to the choice of \(f\) and \(h\). To obtain these results, the authors impose certain conditions on \(f\) and \(h\), which are too technical to be reproduced here. Roughly speaking, they require the monotonicity of \(h\) with respect to \(x\), and the existence of a pair of functions that play the role of upper and lower solutions to (1). The precise form of the conditions involves the so-called distributional Henstock-Kurzweil integral, which is also known as the distributional Denjoy integral (see Erik Talvilla's paper ``The distributional Denjoy integral'' [\textit{E. Talvila}, Real Anal. Exch. 33, No. 1, 51--82 (2008; Zbl 1154.26011)]). The proof of the main result is based on converting (1) into an integral equation involving the distributional Henstock-Kurzweil integral, and applying a fixed point theorem for increasing mappings in ordered Banach spaces. As a corollary, the authors obtain the existence of extremal solutions to the periodic boundary value problem \[ Dx+p(t)x=f(t)+h(t,x(t)),\quad x(0)=x(T), \] where \(p:[0,T]\to\mathbb R\) is a nonnegative Henstock-Kurzweil integrable function; this result is illustrated by means of a simple example.