I am reading a paper on multiple solutions for boundary value problems of fourth-order differential systems. In the paper, there is a nonlinear term $f\in C\left[(0,1)\times \mathbb{R}^+\times \mathbb{R}, \mathbb{R}\right]$, which is assumed to be semipositone. I found a definition of semipositione online, which says: A (nonlinear) function on the reals is called positone if it is continuous, positive and monotone. A function on the reals is called semipositone if it is positone expect at zero where it is not positive.


I am confused because if the function is negative a zero it will also be negative in a small enough neighbourhood of zero by dint of its continuity. Could you please provide me with a good definition of semipositone?


Thank you in advance.



What are semipositone functions?
The continuous function $f:[0,\infty)\rightarrow\mathbb{R}$ is called *semipositone* if it is monotonically increasing, $f(0)<0$ and $f(u)>0$ for some $u>0$. So indeed, $f$ is negative in a neighborhood of the origin. If instead $f(0)>0$ the function is called *positone*. The case $f(0)=0$ does not have a special name.


The word positone was coined by Keller and Cohen in their [1967 paper](https://authors.library.caltech.edu/100708/1/16087.pdf).


![](https://i.stack.imgur.com/KmVlz.png)
