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For a function $f: R^n \times R^n \times R^n \rightarrow R^n$ analytic in a neighborhood of $(0,0,0)$ such that $f(0,0,0)=0$, I would like to show the existence of a function $\varphi$ analytic on a neighborhood of 0 such that $\varphi(0)=0$ and $f(x,\varphi(x),\varphi(\varphi(x)))=0$ . To remain in R, one must assume that the matrix equation in X $$f'_1 + f'_2 X + f'_3 X^2=0$$ has at least one real solution.

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 (1,2) 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 $\varphi$, associated with the Taylor expansion $\sum_k^{\infty} \varphi^{(k)}x^k$ one gets that $\varphi^{(1)}(0)$ must solve in $X$ the matrix quadratic equation from above. In case this equation doesn't have any solution, there is of course no differentiable solution to the original problem. Otherwise this equation has finitely many solutions and for a given choice of one one them, one can plug it back in the original problem 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 (see 3). But it doesn't guarantee the existence of a solution….

Here are the papers referenced above:

  1. Jin, Judd (2002) Perturbation methods for general dynamic stochastic models (unpublished, link to wp)
  • there is a theorem, without the proof...
  1. J. Fernández-Villaverde, J.F. Rubio-Ramírez, F. Schorfheide Solution and Estimation Methods for DSGE Models (link)
  • mentions the fact that the radius of convergence is unknown but no proof it is not zero...
  1. Hong Lan, Alexander Meyer-Gohde (2014). Solvability of perturbation solutions in DSGE models. Journal of Economic Dynamics and Control. (link)
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  • $\begingroup$ $f : R^n \times R^n \times \to R^n$ is surely a typo; what is it supposed to be? What are some papers where you have seen such references? What is $\varphi^{(1)}$? $\endgroup$
    – LSpice
    Commented Jul 9 at 16:21
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    $\begingroup$ Thanks! I have corrected the typo and added some references. $\varphi^{(1)}$ is the first order derivative: I made that clear above. $\endgroup$
    – albop
    Commented Jul 9 at 16:55
  • $\begingroup$ This is hopeless in this generality. If $f(x,y,z)=g(x,z)$ and you make the usual extra assumptions that guarantee the local existence and uniqueness of a $\psi(x)$ such that $g(x,\psi(x))=0$, then you are still left with the rather intractable problem of finding a compositional square root of $\psi$. Compare here: mathoverflow.net/questions/17614/solving-ffx-gx $\endgroup$
    – Christian Remling
    Commented Jul 9 at 17:42
  • $\begingroup$ Well, what makes the problem more specific is that I'm looking for a local solution, in the neighborhood of a fixed point (0). I clarified the question to make it more obvious. I looked at the reference you gave (thanks!). $\endgroup$
    – albop
    Commented Jul 9 at 19:14
  • $\begingroup$ ... I looked at the reference you gave , which looks for $f$ in $f(f(x))=g(x)$. I would say that my particular usecase assumes $g(0)=0$. Also, for the quadratic equation to have a solution, one needs to assume $g'(0)>0$. These assumptions are not met by any of the counterexample I see in the link. I clarified my question by adding explicitly the assumption that the quadratic equation needs to have one solution. $\endgroup$
    – albop
    Commented Jul 9 at 19:47

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