Let (G,+) be a locally compact abelian Hausdorff group, đ is a finite automorphism group of G, Îș = cardđŠ and let ” be a regular compactly supported complex-valued Borel measure on G such that $\mu ({\rm{G}}) = {1 \over \kappa }$. We find the continuous solutions f, g : G â â of the functional equation
$$\sum\limits_{k \in {\cal K}} {\sum\limits_{\lambda \in {\cal K}} {\int_{\rm{G}} {{\rm{f}}({\rm{x}} + {\rm{k}} \cdot {\rm{y}} + } \lambda \cdot {\rm{s}}){\rm{d}}\mu ({\rm{s}}) = {\rm{g}}({\rm{y}}) + \kappa {\rm{f}}({\rm{x}}),\,{\rm{x}},{\rm{y}} \in {\rm{G}},} } $$
in terms of k-additive mappings. This equations provides a common generalization of many functional equations (quadratic, Jensenâs, Cauchy equations).