Spectral characterization of the constant sign Green’s functions for periodic and Neumann boundary value problems of even order

Abstract

In this paper we will characterize the interval of real parameters M in which the Green’s function GM , related to the operator T2n[M]u(t) := u(2n)(t)+Mu(t) coupled to periodic, u(i)(0) = u(i)(T) , i = 0, . . . ,2n −1, or Neumann, u(2i+1)(0) = u(2i+1)(T) = 0, i =0, . . . ,n−1, boundary conditions, has constant sign on its square of definition. More concisely, we will prove that the optimal values are given as the first zeros of GM(0,0) or GM(T/2,0) , depending both on the sign of GM and on the fact whether 2n is, or is not, a multiple of 4. Such values will be characterized as the eigenvalues of the operator u(2n) related to suitable boundary conditions. This characterization allows us to obtain such values without calculating the exact expression of the Green’s function.