I need to construct a continuous function $F:\mathbb R\to \mathbb R$ with $F(0)=0$ and $F(x)=\frac32 F\left(\sqrt[3]{x^2}-1\right)$ for every $x\in\mathbb R$.
I found $F(x)\equiv0$ is a solution. Moreover, if F is a polynomial, I could prove that $F$ must be zero function. My question is: Is there another such function?
I think it must be have, but I could not construct them.
Playing around.
Putting $x^3$ for $x$, this becomes $2f(x^3) = 3f(x^2-1) $.
As Clever Sea wrote, $f$ can't be a polynomial since, if the polynomial is of degree $n$, the left side is of degree $3n$ and the right is of degree $2n$.
Putting $x = -1$, $2f(-1) = 3f(0) = 0$ so $f(-1) = 0$.
Putting $x = 1$, $2f(1) = 3f(0) = 0$ so $f(1) = 0$.
If $x_0$ is the real root of $x^3 = x^2-1$, then $f(x_0) = 0$ otherwise $f$ would not be continuous at $x_0$.
Putting $-x$ for $x$, $f(-x^3) = f(x^2-1) =f(x^3) $ so $f$ is an even function.
$x^3$ is increasing for all $x \ne 0$ and $x^2-1$ is decreasing for $x < 0$.
If $f(a) \ne 0$, there will be a neighborhood of $a$ where $f$ is of constant sign.
It's late, and I don't see how to proceed from here, so I hope this will be of use to others.
Bye.