## Abstract

Fix non-zero reals α_{1}, … , α_{n} with n≥ 2 and let K be a non-empty open connected set in a topological vector space such that ∑ _{i}_{≤}_{n}α_{i}K⊆ K (which holds, in particular, if K is an open convex cone and α_{1}, … , α_{n}> 0). Let also Y be a vector space over F: = Q(α_{1}, … , α_{n}). We show, among others, that a function f: K→ Y satisfies the general linear equation ∀x1,…,xn∈K,f(∑i≤nαixi)=∑i≤nαif(xi)if and only if there exist a unique F-linear AX→ Y and unique b∈ Y such that f(x) = A(x) + b for all x∈ K, with b= 0 if ∑ _{i}_{≤}_{n}α_{i}≠ 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.

Original language | English |
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Pages (from-to) | 201-211 |

Number of pages | 11 |

Journal | Acta Mathematica Hungarica |

Volume | 161 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jun 2020 |

## Keywords

- existence and uniqueness of extension
- general linear equation
- open connected set
- Pexider equation

## ASJC Scopus subject areas

- Mathematics(all)