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 $\mathbb{F}:=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$. We show, among others, that a function $f: K\to Y$ satisfies the general linear equation $$ \textstyle \forall x_1,\ldots,x_n \in K,\,\,\,\,\, f\left(\sum_{i\le n}\alpha_i x_i\right)=\sum_{i\le n}\alpha_i f(x_i) $$ if and only if there exist a unique $\mathbb{F}$-linear $A:X\to Y$ and unique $b\in Y$ such that $f(x)=A(x)+b$ for all $x \in K$, with $b=0$ if $\sum_{i\le n}\alpha_i\neq 1$. The main tool of the proof is a general version of a result Rad\'{o} and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.