# Constant sign solution for simply supported beam equation with non-homogeneous boundary conditions

## TL;DR

This paper investigates the sign properties of solutions to a fourth-order beam equation with non-homogeneous boundary conditions, establishing equivalences and conditions for solutions to maintain a constant sign.

## Contribution

It introduces new equivalence results linking inverse positivity of the operator under different boundary conditions and provides conditions ensuring constant sign solutions.

## Key findings

- Equivalence between inverse positivity with non-homogeneous and homogeneous boundary conditions.
- Conditions under which the problem admits a unique constant sign solution.
- Results demonstrating the operator's strongly inverse positive or negative character.

## Abstract

The aim of this paper is to study the following fourth-order operator: T[p,c]\,u(t)\equiv u^{(4)}(t)-p\,u"(t)+c(t)\,u(t)\,,\quad t\in I\equiv [a,b]\,, coupled with the non-homogeneous simply supported beam boundary conditions: u(a)=u(b)=0\,,\quad u"(a)=d_1\leq0\,,\ u"(b)=d_2\leq 0\,. First, we prove a result which makes an equivalence between the strongly inverse positive (negative) character of this operator with the previously introduced boundary conditions and with the homogeneous boundary conditions, given by: T[p,c]\,u(t)=h(t)(\geq0)\,, u(a)=u(b)=u"(a)=u"(b)=0\,, Once that we have done that, we prove several results where the strongly inverse positive (negative) character of $T[p,c]$ it is ensured. Finally, there are shown a couple of result which say that under the hypothesis that $h>0$, we can affirm that the problem for the homogeneous boundary conditions has a unique constant sign solution.