Characterization of the Constant Sign of a Class of Periodic and Neumann Green’s Functions via Spectral Theory

In this paper, we characterize the regions of constant sign of the Green’s fucntions related to operator \(T_n[p,M]\,u(t)=u^{(n)}(t)+p\,u^{(n-2)}(t)+M\,u(t)\) , with n being an even number, \(n\ge 4\) , and \(p\le 0\) , coupled to periodic or Neumann boundary conditions. The results generalize the situation considered in Cabada and López-Somoza (Differ Equ Appl 14(2):335–347, 2022) for the particular case of \(p=0\) .