On additive functions satisfying some relations

Abstract. We prove that if an additive commutative semigroup \({\mathbb G}\) (with identity element 0) and \({\mathbb G}\)-valued completely additive functions \(f_0,~f_1,~f_2\) satisfy the relation \(f_0(n)+f_1(2n+1)+f_2(n+2)=0\) for all \(n\in{\mathbb N}\), then \(f_0(n)=f_1(2n+1)=f_2(n)=0\) for all \(n\in{\mathbb N}\). The same result is proved when the relation \(f_0(n)+f_1(2n-1)+f_2(n+2)=0\) holds for all \(n\in {\mathbb N}\).