On a quadratic type functional equation on locally compact abelian groups

Let (G,+) be a locally compact abelian Hausdorff group, is a finite automorphism group of G, κ = card and let µ be a regular compactly supported complex-valued Borel measure on G such that μ(G)=1κ$\mu ({\rm{G}}) = {1 \over \kappa }$. We find the continuous solutions f, g : G → ℂ of the functional equation ∑k∈∑λ∈∫Gf(x+k⋅y+λ⋅s)dμ(s)=g(y)+κf(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)

Stability of generalized quadratic functional equation on a set of measure zero

In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equation ∑ k ∈ K f(x+ k.y)= Lf(x)+ Lf(y), x,y ∈ E, where E is a real (or complex) vector space. This result was used to demonstrate the Hyers-Ulam stability on a set of Lebesgue measure zero for the same functional equation