| Popis: |
Let (S, +) be an abelian semigroup, let (H, +) be an abelian group which is uniquely 2-divisible, and let ϕ be an endomorphism of S. We find the solutions f, h : S → H of each of the functional equations f(x+y)+f(x+ϕ(y))=h(x)+f(y)+f∘ϕ(y), x,y∈S,f(x+y)+f(x+ϕ(y))=h(x)+2f(y), x,y∈S,\matrix{ {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + f\left( y \right) + f \circ \varphi \left( y \right),\,x,y \in S,} \hfill \cr {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + 2f\left( y \right),\,x,y \in S,} \hfill \cr } in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations. |
