An iterative method for solving a system of functional equations (Q1333222)

The authors solve the system of functional equations \[ \begin{alignedat}{2} & g(\tau+ x)= g(\tau) g(x)= g(x) g(\tau), && \quad x[0, 1-\tau)\\ & g(\xi+ x)= g(\xi) g(x)= g(x) g(\xi), &&\quad x[0, 1- \xi)\tag{S}\\ & g(x) g(\xi- x)= g(\xi), && \quad x[0, \xi],\end{alignedat} \] where \(g: [0, 1)\to G\) is an unknown function, \(G\) is a nontrivial group and \(\tau, \xi\in (0, 1)\) are fixed. The description of general solutions of the system (S) without any restriction on the values \(\tau\), \(\xi\) takes up 35 pages. First, the system (S) is reduced to a normalized form and then it is solved in the easiest cases. Next, the remaining cases are reduced to a sort of ``canonical'' form. An iterative method to solve these cases is developed. Moreover, some conditions on \(\xi\) and \(\tau\) under which the solutions of (S) satisfy the additional requirement are given.