# 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.

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## Acknowledgement

The authors are grateful to the anonymous referee for suggestions that helped improving the overall presentation of the article.

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Correspondence to P. Leonetti.