Relationship of Green's functions related to Hill's equation coupled to different boundary conditions

Abstract

In this paper, we deduce several properties of Green's functions related to Hill's equation coupled to various boundary value conditions. In particular, the idea is to study Green's functions of the second order differential operator coupled to the Neumann, Dirichlet, periodic and mixed boundary conditions, by expressing Green's function of a given problem as a linear combination of Green's functions of the other problems. This will allow us to compare different Green's functions when their sign is constant. Finally, such properties of Green's function of the linear problem will be fundamental to deduce the existence of solutions to the nonlinear problem. The results are derived from the fixed point theory applied to the related operators defined on suitable cones in Banach spaces.