Let be an Abelian semigroup and be an Abelian group uniquely divisible by . A well-known result of Djokovic says that a function has the form , where is symmetric and -additive and , if and only if for all .
Let now be a commutative ring with identity, uniquely divisible by , fixed; let , be -modules. One of the results of the present paper gives a characterization through a functional equation connected with the differences, of the functions of the form , where is -linear and symmetric.
The stability of this functional equation and of other similar equations is then studied in Banach spaces over non-Archimedean valued fields.