The general linear equation on open connected sets

P. Leonetti*, J. Schwaiger

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Fix non-zero reals α1, … , αn with n≥ 2 and let K be a non-empty open connected set in a topological vector space such that ∑ inαiK⊆ K (which holds, in particular, if K is an open convex cone and α1, … , αn> 0). Let also Y be a vector space over F: = Q(α1, … , αn). 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 ∑ inαi≠ 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 languageEnglish
Pages (from-to)201-211
Number of pages11
JournalActa Mathematica Hungarica
Volume161
Issue number1
DOIs
Publication statusPublished - 1 Jun 2020

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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