The general linear equation on open connected sets

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 Rado and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.