# Solving multivariable functional equation

Solve this functional equation: $$f(s,t)=4f(s,u)f(u,t)-f(s,u)-f(u,t)+\frac{1}{2}, \ \ \mbox{for any} \ \ 0 \leq s

I have found only one constant solution that is $$f(s,t)=\frac{1}{4}$$, and then got stuck. Thanks in advance.

• What function space do you work in?
– Emil
Mar 23 '20 at 12:36
• Could you tell the context of this issue ? Mar 23 '20 at 12:38
• @Emil The function can be any function from $[0,+\infty)^2$ to the set of real numbers. Mar 23 '20 at 13:04

The solution you found suggests a change of variables. Define $$g(s, t) = 4f(s, t) - 1$$. Then the functional equation is just $$g(s, t) = g(s, u) g(u, t) \qquad (0 \leq s < u < t).$$ There are rather a lot of solutions to this.
Focusing on the case $$g > 0$$, we may write $$g(s, t) = \exp m(s, t)$$, and then the functional equation becomes $$m(s,t) = m(s,u) + m(u, t),$$ which is more or less just the definition of a finitely additive signed Borel measure on $$[0, \infty)$$. For example, for any $$f \in L^1$$ there is a solution defined by $$m(s, t) = \int_s^t f$$. There are plenty of more exotic solutions.
In general, the sign and magnitude parts of the problem for $$g$$ separate. Both parts of the problem have lots of exotic solutions.