Solutions of the first order linear equation with reflection and general linear conditions

TL;DR

This paper investigates first order linear equations with reflection and general boundary conditions, providing explicit solutions, existence criteria, and positivity estimates, with applications illustrated through examples.

Contribution

It introduces explicit Green's functions for problems with antiperiodic conditions and establishes existence, uniqueness, and positivity criteria for solutions under general linear conditions.

Findings

Explicit Green's function for antiperiodic boundary conditions

Sufficient conditions for existence and uniqueness

Positivity estimates for solutions

Abstract

This work is devoted to the study of first order linear problems with involution and general linear conditions. We first study the problem in the case of antiperiodic boundary conditions, giving an explicit Green's function for it. Then we move forward to more general linear boundary conditions, focusing on sufficient conditions for existence and uniqueness of solution. At the end of the paper we give estimates that ensure the positivity of the solution in the general problems and illustrate these applications with examples.