The general linear equation on open connected sets

Abstract

Fix non-zero reals ${\alpha }_1,\dots ,{\alpha }_n$ with $n\ge 2$ and let $K$ be a non-empty open connected set in a topological vector space such that ${\sum }_{i\le n}{\alpha }_{i}K\subseteq K$ (which holds, in particular, if $K$ is an open convex cone and ${\alpha }_1,\dots ,{\alpha }_n>0$). Let also $Y$ be a vector space over $\mathbb{F}:=\mathbb{Q}({\alpha }_1,\dots ,{\alpha }_n)$. We show, among others, that a function $f:K\to Y$ satisfies the general linear equation $$\forall x_1,\dots ,x_n\in K,f\left({\sum }_{i\le n}{\alpha }_i{x}_i\right)={\sum }_{i\le n}{\alpha }_{i}f(x_i)$$ 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 ${\sum }_{i\le n}{\alpha }_i\ne 1$. 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.

11 pages, no figures