On a quadratic type functional equation on locally compact abelian groups

Abstract

Abstract Let (G,+) be a locally compact abelian Hausdorff group, 𝓀 is a finite automorphism group of G, κ = card𝒦 and let µ be a regular compactly supported complex-valued Borel measure on G such that μ ( G ) = 1 κ $\mu ({\rm{G}}) = {1 \over \kappa }$ . We find the continuous solutions f, g : G → ℂ of the functional equation ∑ k ∈ 𝒦 ∑ λ ∈ 𝒦 ∫ G f ( x + k ⋅ y + λ ⋅ s ) d μ ( s ) = g ( y ) + κ f ( x ) ,   x , y ∈ G , $$\sum\limits_{k \in {\cal K}} {\sum\limits_{\lambda \in {\cal K}} {\int_{\rm{G}} {{\rm{f}}({\rm{x}} + {\rm{k}} \cdot {\rm{y}} + } \lambda \cdot {\rm{s}}){\rm{d}}\mu ({\rm{s}}) = {\rm{g}}({\rm{y}}) + \kappa {\rm{f}}({\rm{x}}),\,{\rm{x}},{\rm{y}} \in {\rm{G}},} } $$ in terms of k-additive mappings. This equations provides a common generalization of many functional equations (quadratic, Jensen’s, Cauchy equations).