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I have asked this question here (*), but there are no answer.

Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left[0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no linear solution.

Determinate the solution of $Eq$ for $f \in C^\infty (\mathbb R)$, where $f^2(x)=f \circ f (x)$ and $f^0(x)=x$.

Reference: https://artofproblemsolving.com/community/c6h3164353_functional_equation

I have asked this question here (*), but there are no answer.

Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no linear solution.

Determinate the solution of $Eq$ for $f \in C^\infty (\mathbb R)$, where $f^2(x)=f \circ f (x)$ and $f^0(x)=x$.

Reference: https://artofproblemsolving.com/community/c6h3164353_functional_equation

I have asked this question here (*), but there are no answer.

Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left]0,+\infty\right[$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no linear solution.

Determinate the solution of $Eq$ for $f \in C^\infty (\mathbb R)$.

PS : $f^2(x)=f \circ f (x)$ and $f^0(x)=x$

(*) : https://artofproblemsolving.com/community/c6h3164353_functional_equation

I have asked this question here (*), but there are no answer.

Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left[0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no linear solution.

Determinate the solution of $Eq$ for $f \in C^\infty (\mathbb R)$, where $f^2(x)=f \circ f (x)$ and $f^0(x)=x$.

Reference: https://artofproblemsolving.com/community/c6h3164353_functional_equation