On Gajda’s type quadratic equation on a locally compact abelian group.

Let ( G , + ) be a locally compact abelian Hausdorff group, and let μ be a regular compactly supported complex-valued Borel measure on G such that μ ( G ) = 1 2 . We find the continuous solutions f , g : G → C of the functional equation ∫ G { f ( x + y − t ) + f ( x − y + t ) } d μ ( t ) = f ( x ) + g ( y ) , x , y ∈ G , in terms of quadratic and additive functions. This equation provides a common generalization of many functional equations (quadratic, Jensen’s, Drygas’ equations...). [ABSTRACT FROM AUTHOR]