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

TL;DR

This work derives an explicit Green's function $H_{m,M}$ for a second-order differential equation with reflection and a piecewise constant argument under periodic boundary conditions, and analyzes the parameter regions where $H_{m,M}$ preserves a constant sign. It connects sign properties to first Dirichlet eigenvalues of related, Dirichlet-posed problems, enabling sign-based fixed-point arguments without requiring full Green's function computation. Leveraging Krasnosel'skii's fixed-point method, the authors establish existence results for nonlinear problems, including a positive solution to a perturbed Schrödinger-type equation. The results provide a rigorous framework for nonlocal, involution-influenced boundary value problems with potential applications in quantum and optical lattice models, and offer numerical and analytical criteria to identify sign regions and eigenvalues relevant for nonlinear analysis.

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.

Figures (6)

• Figure 1: Numerical approximation of the regions in which the Green's function $H_{m,M}$ maintains a constant sign. The domain where $H_{m,M}>0$ is depicted in blue, while the domain where $H_{m,M}<0$ is depicted in red.
• Figure 2: Explicitly conjectured regions in which the Green's function $H_{m,M}$ maintains a constant sign when $T=1.6$. The domain where $H_{m,M}>0$ is depicted between the black and blue curves, while the domain where $H_{m,M}<0$ is shown between the black and red curves.
• Figure 3: Representation of the regions of constant sign for the function $H_{m,M}$ for different values of $T$. The region with a positive sign is the one between the blue and black lines, while the region with a negative sign is the one between the black and red lines.
• Figure 4: Graphical representation of the monotonic decrease of the first Dirichlet eigenvalue with respect to increasing values of $s_0$ for $T=4.8$.
• Figure 5: Representation of the eigenvalue $\lambda_1$ as a function of $T$.

Theorems & Definitions (33)

• Definition 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Proposition 4.3
• Remark 4.4
• Theorem 4.5
• proof
• Proposition 4.6
• Proposition 5.1