On the functional equation $$f(x+y)=f(x)+f(y)$$f(x+y)=f(x)+f(y)

Monatshefte für Mathematik - An extension of the linearity of the continuous solutions of the functional equation $$f(x+y)=f(x)+f(y)$$ to measurable functions is presented.     

On the system of functional equationsf(x + y) = f(x) + f(y) andf(xy) = p (x)f(y) + q(y)f(x)

Aequationes mathematicae -     

The Functional Equationf(x+y−xy)+f(xy)=f(x)+f(y)

Aequationes mathematicae -     

On the functional equation x + f(y + f(x))=y + f(x + f(y))

For an abelian group (G, + ,0) we consider the functional equation $$f : G \to G, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G), \quad\quad\qquad (1)$$ most times together with the condition $$f(0) = 0.\qquad\qquad\qquad\qquad\qquad (0)$$ Our main question is whether a solution of ${(1) \wedge (0)}$ must be additive, i.e., an endomorphism of G. We shall answer this question in the negative (Example 3.14) Rätz (Aequationes Math 81:300, 2011).     

On two functional equations connected with the equationϕ(x+y)+ϕ(x−y)=2ϕ(x)+2ϕ(y)

Aequationes mathematicae -     

A characterization of the identity function with equation f(p+q+r)=f(p)+f(q)+f(r)

Let f(n) be a multiplicative function such that there exists a prime p ₀ at which f does not vanish. In this paper, we prove that if f satisfies the equation f(p+q+r)=f(p)+f(q)+f(r) for all primes p, q and r, then f(n)=n for all integers n≥1.     

The matrix equationa(x∘y)=a(x)+a(x)a(y)+a(y)

Aequationes mathematicae -