Constant sign Green's function for simply supported beam equation

Abstract

The aim of this paper consists on the study of the following fourth-order operator: \begin{equation*} T[M]\,u(t)\equiv u^{(4)}(t)+p_1(t)u'''(t)+p_2(t)u''(t)+Mu(t),\ t\in I \equiv [a,b] , \end{equation*} coupled with the two point boundary conditions: \begin{equation*} u(a)=u(b)=u''(a)=u''(b)=0 . \end{equation*} So, we define the following space: \begin{equation*} X=\left\lbrace u\in C^4(I) : 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*} 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 a 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.