Reflection equation with piecewise constant argument

TL;DR

This paper investigates linear and nonlinear nonlocal differential equations with reflection and piecewise constant arguments, deriving explicit solutions, parameter conditions for Green's function sign, and proving solution existence using monotone methods.

Contribution

It introduces a novel approach to analyze equations with involution and piecewise constant arguments, including explicit solutions and conditions for Green's function positivity.

Findings

Explicit Green's function for the linear problem derived.

Conditions identified for the Green's function to have a constant sign.

Existence of solutions for nonlinear problems established using monotone methods.

Abstract

In this work, we study nonlocal differential equations with particular focus on those with reflection in their argument and piecewise constant dependence. The approach entails deriving the explicit expression of the solution to the linear problem by constructing the corresponding Green's function, as well as developing a novel formula to delineate the set of parameters involved in the analyzed equations for which the Green's function exhibits a constant sign. Furthermore, we demonstrate the existence of solutions for nonlinear problems through the utilisation of the monotone method. The aforementioned methodology is specifically applied to the linear problem with periodic conditions $v'(t) + mv(-t) + Mv([t]) = h(t)$ for $t \in [-T,T]$, proving several existence results for the associated nonlinear problem and precisely delimiting the region where the Green's function $H_{m,M}$ has a constant sign on its domain of definition. The equations studied have the potential to be applied in fields such as biomedicine or quantum mechanics. Furthermore, this work represents a significant advance, as it is, as far the authors know, the first time that equations with involution and piecewise constant arguments have been studied together.