## Category: Paneah

### Where have all the functional equations gone (part III)

The last post ended with the following problem:

Problem: Find all continuous solutions to the following functional equation:

(FE) $f(t) = f\left(\frac{t+1}{2} \right) + f \left(\frac{t-1}{2} \right) \,\, , \,\, t \in [-1,1] .$

In the previous post I explained why all continuously differentiable solutions of the functional equation (FE) are linear, that is, of the form $f(x) = cx$, but now we remove the assumption that the solution be continuously differentiable and ask whether the same conclusion holds. I found this problem to be extremely interesting, and at this point I will only give away that I eventually solved it, but after five (!) years.

In principle, it is plausible that, when one enlarges the space of functions in which one is searching for a solution from $C^1[-1,1]$ to the much larger $C[-1,1]$, then new solutions will appear. On the other hand, the dynamical system affiliated with this problem (the dynamical space generated by the maps $\delta_1(t) = \frac{t+1}{2}$ and $\delta_2(t) = \frac{t-1}{2}$ on the space $[-1,1]$) is minimal, and therefore one expects the functional equation to be rigid enough to allow only for the trivial solutions (at least under some mild regularity assumptions). In short, a good case can be made in favor of either a conjecture that all the continuous solutions are linear or a conjecture that there might be new, nonlinear solutions.

$u_{tt} + u_{ss} - u_{xx} - u_{yy} = 0 ,$    in     $\mathbb{R}^4$,