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

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