Second order periodic boundary value problems with reflection and piecewise constant arguments

TL;DR

This paper studies second-order differential equations with reflection and piecewise constant arguments under periodic boundary conditions, constructing Green's functions, analyzing their sign properties, and applying fixed point methods to prove solution existence.

Contribution

It introduces a Green's function construction for these equations, analyzes its sign behavior, and applies Krasnosel'skii's method to establish existence results, including positive solutions.

Findings

Green's function constructed and analyzed for sign properties

Parameter regions identified where Green's function is positive or negative

Existence of solutions proved for nonlinear and perturbed Schrödinger equations

Abstract

In this paper, we analyze a second-order differential equation with a piecewise constant argument and reflection coupled to periodic boundary conditions. Our main contribution is the construction of the related Green's function and a detailed analysis of its properties. In particular, we determine the region in which the Green's function has constant sign, depending on the parameters $m$ and $M$ on which it depends. In some cases, we are able to characterize these parameter values in terms of the first eigenvalue related to suitable Dirichlet problems. Building in these results, we apply the Krasnosel'skii method to establish the existence of solutions for different nonlinear problems, and prove the existence of a positive solution of a perturbed Schrodinger equation.