Constant sign Green's function for simply supported beam equation

The aim of this paper consists on the study of the following fourth-order operator: \begin{equation}\label{Ec::T4} T[M],u(t)\equiv u^{(4)}(t)+p_1(t),u"‘(t)+p_2(t),u"(t)+M,u(t),,\ t\in I \equiv [a,b],, \end{equation} coupled with the two point boundary conditions: \begin{equation}\label{Ec::cf} u(a)=u(b)=u"(a)=u"(b)=0,. \end{equation} So, we define the following space: \begin{equation}\label{Ec::esp} X=\left\lbrace u\in C^4(I)\quad\mid\quad u(a)=u(b)=u"(a)=u"(b)=0 \right\rbrace ,. \end{equation} Here $p_1\in C^3(I)$ and $p_2\in C^2(I)$. By assuming that the second order linear differential equation \begin{equation}\label{Ec::2or} L_2, u(t)\equiv u"(t)+p_1(t),u’(t)+p_2(t),u(t)=0,,\quad t\in I, \end{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 $I\times I$. 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.