# What kind of uniqueness can I conclude for solutions to a simple functional equation?

I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness.

At the most vague version, I am in the following situation. I have some particular (smooth) function $A(t_0,q_0,t_1,q_1)$, where $t_0,t_1$ are real variables and $q_0,q_1$ vary over some manifold $M$, i.e. $A: \mathbb R \times M \times \mathbb R \times M \to \mathbb R$. I have another (smooth) function $Q(t_0,q_0,t_1,q_1;t) : \mathbb R \times M \times \mathbb R \times M \times \mathbb R \to M$. I am interested in (smooth) solutions $F(t_0,q_0,t_1,q_1)$ to the "composition" formula: $$A(t_0,q_0,t_1,q_1)\,F(t_0,q_0,t,Q(t_0,q_0,t_1,q_1;t))\,F(t,Q(t_0,q_0,t_1,q_1;t),t_1,q_1) = F(t_0,q_0,t_1,q_1)$$ I know one such (nonzero) solution explicitly. I am hoping that, presumably with slightly more data (some sort of "initial data"), I can conclude that my solution is uniquely determined. So my question is:

What extra data for $F$, and what restrictions on $A$, are needed for a functional equation of the above type to have a unique solution?

For those interested, I can give you more precise information. I have some "physics" encoded by a function $S(t_0,q_0,t_1,q_1)$, which I am thinking of as a "Hamilton-Jacobi function". This function defines $Q$ via: $$\left. \frac{\partial}{\partial q} \bigl[ S(t_0,q_0,t,q) + S(t,q,t_1,q_1) \bigr] \right]_{q = Q(t_0,q_0,t_1,q_1;t)} = 0$$ and satisfies: $$S(t_0,q_0,t,q) + S(t,q,t_1,q_1) \bigr|_{q = Q(t_0,q_0,t_1,q_1;t)} = S(t_0,q_0,t_1,q_1)$$ The function $A$ is given by $$\left| \det \left[ \frac{\partial^2}{\partial q^2} \bigl[ S(t_0,q_0,t,q) + S(t,q,t_1,q_1) \bigr] \right]_{q = Q(t_0,q_0,t_1,q_1;t)} \right|^{-1/2}$$ Then the function $F$ is: $$F(t_0,q_0,t_1,q_1) = \left| \det \frac{\partial^2}{\partial q_0\partial q_1} \bigl[ S(t_0,q_0,t_1,q_1) \bigr] \right|^{1/2}$$ which as defined makes sense only on $\mathbb R^n$. Actually, $A$ makes sense as a volume form, and $F$ as a half density in each variable, and then everything transforms correctly under changes of coordinates.

Anyway, I would really like to conclude that this $F$ is the only solution, but I don't know if I can.

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Incidentally, checking the slightly more explicit case is a nice exercise, or see my papers on the arXiv :) In any case, probably I should have written the original equation I want to solve logarithmically as $a + f_0 + f_1 = f_{01}$, to make it even more clear that I do not expect there to be a unique solution --- multiply the original $F(t_0,q_0,t_1,q_1)$ by $e^{\lambda(t_1 - t_0)}$ for any $\lambda$, for example. But I do expect that there's a "small" family, like is the case for differential equations? –  Theo Johnson-Freyd May 15 '10 at 3:03

Here's a vague answer then: there's an old 1978 paper in Aequationes Mathematicae mentionning the problem of uniqueness for the functional equation $\varphi (x)=h(x,\varphi [f_1(x)],\dots, \varphi [f_n(x)] )$ under some hypotheses. This seems to be of the same form as your equation (identifying $x=(q_0,t_0,q_1,t_1)$ and so on). Maybe what you're looking for is in there, or in later papers quoting that one.