Extension of a generalized Pexider equation

Abstract:

The equations $k(s+t)=\ell (s)+n(t)$ and $k(s+t)=m(s)n(t)$, called Pexider equations, have been completely solved on $\mathbb {R}^2.$ If they are assumed to hold only on an open region, they can be extended to $\mathbb {R}^2$ (the second when $k$ is nowhere 0) and solved that way. In this paper their common generalization $k(s+t)=\ell (s)+m(s)n(t)$ is extended from an open region to $\mathbb {R}^2$ and then completely solved if $k$ is not constant on any proper interval. This equation has further interesting particular cases, such as $k(s+t)=\ell (s)+m(s)k(t)$ and $k(s+t)=k(s)+m(s)n(t),$ that arose in characterization of geometric and power means and in a problem of equivalence of certain utility representations, respectively, where the equations may hold only on an open region in $\mathbb {R}^2.$ Thus these problems are solved too.