Constant sign Green's function for simply supported beam equation

The aim of this paper consists on the study of the following fourthorder operator: T [M ]u(t) ≡ u(t)+p1(t)u(t)+p2(t)u(t)+M u(t) , t ∈ I ≡ [a, b] , (1) coupled with the two point boundary conditions: u(a) = u(b) = u(a) = u(b) = 0 . (2) So, we define the following space: X = { u ∈ C(I) | u satisfies boundary conditions (2) } . (3) Here p1 ∈ C(I) and p2 ∈ C(I). By assuming that the second order linear differential equation L2 u(t) ≡ u(t) + p1(t)u(t) + p2(t)u(t) = 0 , t ∈ I, (4) is disconjugate on I, we characterize the parameter’s set where the Green’s function related to operator T [M ] in X is of constant sign on I × I. Such characterization is equivalent to the strongly inverse positive (negative) character of operator T [M ] on X and comes from the first eigenvalues of operator T [0] on suitable spaces.