Does any research mathematics involve solving functional equations?

In additive combinatorics, one often seeks to count patterns such as an arithmetic progression $a, a+r, \ldots, a+(k-1)r$. When doing so, one is naturally led to expressions such as $$ {\bf E}_{a,r \in G} f_0(a) f_1(a+r) \ldots f_{k-1}(a+(k-1)r)$$ for some finite abelian group $G$ and some complex-valued functions $f_0,\ldots,f_{k-1}$. If these functions are bounded in magnitude by $1$, then the above expression is also bounded in magnitude by one. When does equality hold? Precisely when one has a functional equation $$ f_0(a) f_1(a+r) \ldots f_{k-1}(a+(k-1)r) = c$$ for some constant $c$ of magnitude $1$. One can solve this functional equation, and discover that each $f_j$ must take the form $f_j(a) = e^{2\pi i P_j(a)}$ for some polynomial $P_j: G \to {\bf R}/{\bf Z}$ of degree at most $k-2$. This observation can be viewed as the starting point for the study of Gowers uniformity norms, and one can perform a similar analysis to start understanding many other patterns in additive combinatorics.

In ergodic theory, cocycle equations, of which the coboundary equation

$$ \rho(x) = F(T(x)) - F(x)$$

is the simplest example, play an important role in the study of extensions of dynamical systems and their cohomology. Despite the apparently algebraic nature of such equations, though, one often solves these equations instead by analytic means (and in particular, not by IMO techniques), for instance by using the spectral theory or mixing properties of the shift $T$, and exploiting the measurability or regularity properties of $\rho$ or $F$. (The solving of such equations, incidentally, is a crucial aspect of the ergodic theory analogue of the study of the Gowers uniformity norms mentioned earlier, as developed by Host-Kra and Ziegler.)

Returning to the more "contrived" functional equations of Olympiad type, note that such equations usually use (a) the additive structure of the domain and range, (b) the multiplicative structure of the domain and range, and (c) the fact that the domain and range are identical (so that one can perform compositions such as $f(f(x))$). In most mathematical subjects, at least one of these features is absent or irrelevant, which helps explain why such equations are relatively rare in research mathematics. For instance, in many branches of analysis, the range of functions (typically ${\bf R}$ or ${\bf C}$) usually has no natural reason to be identified with the domain of functions (which may ``accidentally'' be ${\bf R}$ or ${\bf C}$, but is often more naturally viewed in a more general category, such as that of measure spaces, topological spaces, or manifolds), so (c) is usually absent. Conversely, in dynamics, (c) is prominent, but (a) and (b) are not. The only fields that come to my mind that naturally exhibit all three of (a), (b), (c) (without also automatically exhibiting much richer algebraic structure, such as ring homomorphism structure) are complex dynamics, universal algebra, and certain types of cryptography, but I don't have enough experience in these fields to actually provide some interesting examples.

I don't know if this actually counts (since I don't know if this functional equation is actually useful...), but in the paper

R.M. Dicker, A set of independent axioms for a field and a condition for a group to be the multiplicative group of a field, Proc. London Math. Soc., 18, 1968, p. 114-124

you can find the following theorem:

Let $A$ be an abelian group (written multiplicatively). Adjoin a new element $0$ to get the set $F$. Assume that there is a function $f : F \to F$ such that for all $x,y \in F$ with $y \neq 0$:

1) $f(0) = 1$

2) $f(f(x))=x$

3) $f(f(x) f(y)) = y f(x f(y^{-1}))$

Then $F$ can be made into a field such that $A = F^*$ and $f(x) = 1 - x$.

Here is another example from algebra which can be found in the article

Zoran Sunik, An Ideal Functional Equation with a Ring, Mathematics Magazine 77 (2004) 4, 310--313.

If $R$ is an integral domain, then the maps $f : R \to R$ solving the functional equation

$f(xz - y) f(x) f(y) + 3 f(0) = 1 + 2 f(0)^2 + f(x) f(y)$

are exactly the characteristic functions of ideals of $R$.

Also functional equations describing subrings and prime ideals are mentioned there.

It is used a lot in information theory: See this.