## Advances in Differential Equations

- Adv. Differential Equations
- Volume 22, Number 5/6 (2017), 403-432.

### Constant sign Green's function for simply supported beam equation

Alberto Cabada and Lorena Saavedra

#### 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.

#### Article information

**Source**

Adv. Differential Equations Volume 22, Number 5/6 (2017), 403-432.

**Dates**

First available in Project Euclid: 18 March 2017

**Permanent link to this document**

http://projecteuclid.org/euclid.ade/1489802456

**Subjects**

Primary: 35J08: Green's functions 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 34B05: Linear boundary value problems 34B18: Positive solutions of nonlinear boundary value problems

#### Citation

Cabada, Alberto; Saavedra, Lorena. Constant sign Green's function for simply supported beam equation. Adv. Differential Equations 22 (2017), no. 5/6, 403--432. http://projecteuclid.org/euclid.ade/1489802456.