For a function $f: R^n \times R^n \times \rightarrow R^n$ analytic in a neighborhood of $(0,0,0)$, I would like to show the existence of an analytic function $\varphi$ such that $\varphi(0)=0$ and $f(x,\varphi(x),\varphi(\varphi(x)))=0$ .

This problem comes from the econ literature ($f$ is a very compact representation of a certain class of models). In this literature, I have seen several papers referencing the IFT theorem, in order to propose a solution based on the unknown coefficients method. None of these papers make the connection actually explicit. It is certainly not obvious to me, because of the function composition and also because of the fact that there are in general several solutions.

I'd be very grateful for any pointer about how to proceed or any reference to any math literature that might have adressed a similar problem.

Assuming the existence of an infinitely differentiable function one gets that $\varphi^{(1)}(0)$ must solve in $X$ the matrix quadratic equation $$f'_1 + f'_2 X + f'_3 X^2.$$ It has finitely many solutions and for a given choice of one them, one can plug it back in the original solution and solve in turn all higher order coefficients, solving a linear system in each step, exactly as in the IFT. This procedure is guaranteed to work in case $\varphi^{(1)}$ is the single convergent solution to the quadratic equation. But it doesn't guarantee the existence of a solution….