Reflection equation with piecewise constant argument

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.