Spectral characterization of the constant sign Green's functions for periodic and Neumann boundary value problems of even order
ISSN: 1847-120X, 1848-9605
Ano de publicación: 2022
Volume: 14
Número: 2
Páxinas: 335-347
Tipo: Artigo
Outras publicacións en: Differential Equations & Applications
Resumo
In this paper we will characterize the interval of real parameters M in which theGreen’s function GM , related to the operator T2n[M]u(t) := u(2n)(t) + M u(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. Suchvalues will be characterized as the eigenvalues of the operator u(2n) related to suitable boundaryconditions. This characterization allows us to obtain such values without calculating the exactexpression of the Green’s function.