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Solutions and Green’s function of the first order linear equation with reflection and initial conditions

Alberto Cabada* and Fernando Adrián Fernández Tojo

Author Affiliations

Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia, 15706, Spain

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Boundary Value Problems 2014, 2014:99  doi:10.1186/1687-2770-2014-99

Published: 7 May 2014

Abstract

This work is devoted to the study of the existence and sign of Green’s functions for first order linear problems with constant coefficients and initial (one point) conditions. We first prove a result on the existence of solutions of nth order linear equations with involutions via some auxiliary functions to later prove a uniqueness result in the first order case. We study then different situations for which a Green’s function can be obtained explicitly and derive several results in order to obtain information as regards the sign of the Green’s function. Once the sign is known, optimal maximum and anti-maximum principles follow.

Keywords:
equations with involutions; equations with reflection; Green’s functions; maximum principles; comparison principles; periodic conditions