TY - JOUR
T1 - The general linear equation on open connected sets
AU - Leonetti, P.
AU - Schwaiger, J.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - 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 ∑ i≤nαiK⊆ 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 ∑ i≤nαi≠ 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.
AB - 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 ∑ i≤nαiK⊆ 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 ∑ i≤nαi≠ 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.
KW - existence and uniqueness of extension
KW - general linear equation
KW - open connected set
KW - Pexider equation
UR - http://www.scopus.com/inward/record.url?scp=85073963864&partnerID=8YFLogxK
U2 - 10.1007/s10474-019-00987-6
DO - 10.1007/s10474-019-00987-6
M3 - Article
AN - SCOPUS:85073963864
SN - 0236-5294
VL - 161
SP - 201
EP - 211
JO - Acta Mathematica Hungarica
JF - Acta Mathematica Hungarica
IS - 1
ER -