Constant sign Green's function for simply supported beam equation

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The aim of this paper consists on the study of the following fourth-order operator: equationEc::T4 T[M]u(t)≡ u⁽⁴⁾(t)+p₁(t)u"'(t)+p₂(t)u"(t)+Mu(t),t∈ I ≡ [a,b], equation coupled with the two point boundary conditions: equationEc::cf u(a)=u(b)=u"(a)=u"(b)=0. equation So, we define the following space: equationEc::esp X= u∈ C⁴(I) u(a)=u(b)=u"(a)=u"(b)=0. equation Here p₁∈ C³(I) and p₂∈ C²(I). By assuming that the second order linear differential equation equationEc::2or L₂ u(t)≡ u"(t)+p₁(t)u'(t)+p₂(t)u(t)=0, t∈ I, equation 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.