In a little calculation I'm doing for fun, I've come across this equation involving a function of two arguments which is nested on the right side:
$$f(t_1 + t_2, K) = f\bigl(t_2, f(t_1, K)\bigr)$$
I'm looking for functions $f$ which satisfy this equation for all $t_1,t_2 \ge 0$, $K > 0$, given the condition $f(0, K) = K$ and that $f$ is (EDIT:) piecewise continuously differentiable in $t$.
If I assume that the solution takes the form $f(t, K) = Kg(t)$ then I can reduce this to Cauchy's functional equation and from there demonstrate that the only solution is $f(t, K) = K e^{at}$ for any constant $a$. But I would like to relax that assumption if possible; I'm curious about other possible forms, where $f(t, K)$ may have a different dependence on $K$. Is there a method I could use to find other such solutions if they exist, or to prove that they don't?
Algebraically, your equation is equivalent to saying that $f$ defines a continuous group homomorphism from $(\mathbb R,+)$ to the group of invertible (and continuous?) functions $\mathbb R_+\to\mathbb R_+$ under function composition.
The arithmetic structure of $\mathbb R_+$ does not matter -- only the topology (and smoothness) does. So you can take your solution and conjugate it with any smooth bijection $g: \mathbb R_+\to\mathbb R_+$: $$f_g(t,K) = g(g^{-1}(K)e^t)$$
(Hmm, missed the $t\ge 0$, so it only has to be a monoid homomorphism. This allows even more solution, such as Didier's $at+K$).