Let $I=\{x\in\mathrm{I\!R}\,|\, x>0\}$ and $n\geq 1$. Suppose $f:\,I^n\to I^n$ is continuous, strictly concave and increasing. (Herein, concavity and monotonicity are meant componentwise.) Suppose also that there exist $a,b\in I^n$ with $a<b$ such that $f(a)>a$ and $f(b)<b$. Does $f$ then have a unique fixed point in $I^n$?
The above question is related to theorem 3.3 in
Kennan, John, Uniqueness of Positive Fixed Points for Increasing Concave Functions on Rn: An Elementary Result, Review of Economic Dynamics 4, 893-899 (2001). (https://www.sciencedirect.com/science/article/abs/pii/S1094202501901334).
which states
``Suppose $f$ is an increasing and strictly concave function from $\mathrm{I\!R}^n \to\mathrm{I\!R}^n$ such that $f(0)\geq 0$, $f(a)>a$ for some positive vector $a$, and $f(b)<b$ for some vector $b>a$. Then, $f$ has a unique fixed point.´´
In the above, all inequalities are meant componentwise. I cannot apply theorem 3.3 directly, since, in my case, the function $f$ is not strictly concave $[0,+\infty[\,^n$.
Many thanks in advance for your ideas on how to answer the above question.