My first time on this forum. I have the following system of equations.
- A constant $x_{min} > 0$.
- Two functions $A(x)$ and $B(x)$ defined on $x \geq x_{min}$.
- $B(x)$ is strictly increasing, continuous, and $B(x) \geq x$.
- $A(x)$ is continuous and $A(x) \in [x_{min},x]$.
I want to show that there exists $x^*$ such that $A(B(x^*)) = x^*$. Is there any fixed-point theorem I can use for this purpose? Thank you for you replies.
If $A(x)=\sqrt{x}$, $B(x)=x^4$ and $x_{min}=2$ you have that $A(B(x))=x^2\neq x,\forall x\in [2,+\infty)$, meaning you cannot prove the statement, since it is not generally true.