The general linear equation on open connected sets

Abstract

Fix non-zero reals $\alpha _1$, $\ldots ,$$\alpha _n$ with $n\ge 2$ and let $K$ be a non-empty open connected set in a topological vector space such that $\sum _{i\le n}\alpha _iK\subseteq K$ (which holds, in particular, if $K$ is an open convex cone and $\alpha _1,\ldots ,\alpha _n>0$). Let also $Y$ be a vector space over $\mathbb{F}\it :=\mathbb{Q} \it (\alpha _1,\ldots ,\alpha _n)$. We show, among others, that a function $f : K\rightarrow Y$ satisfies the general linear equation

$$\forall x_1,\ldots, x_n \in K, \quad f\big (\sum _{i\le n}\alpha _i x_i\big )=\sum _{i\le n}\alpha _i f(x_i)$$

if and only if there exist a unique $\mathbb{F}\it$-linear $A X \rightarrow Y$ and unique $b\in Y$ such that $f(x)=A(x)+b$ for all $x \in K$, with $b=0$ if $\sum _{i\le n}\alpha _i\ne 1$. The main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.