Systems of generalized translation equations on a restricted domain

Summary.

In this paper continuous real valued functions that are defined on a connected and open subset of the plane and strictly monotonic in one of their variables are considered. It is proved that such a function F satisfies the functional equations

$F(x + t, y) = {\Psi}_{1} (F(x, y), t)\quad \textrm{and}\quad F(x, y + s) = {\Psi}_{2} (F(x, y), s)$

(with some real valued functions ${\Psi}_{1}, {\Psi}_{2}$ ) if, and only if, F can be represented in the form

$F(x, y) = f (ax + by)$

with a strictly monotonic function f of a single variable and real numbers a, b.