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 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).     

关于函数y=f(x)与y=|f(x)|

本文对函数 y=f ( x)与 y=|f ( x) |的部分性质进行了比较和分析 ,指出了它们性质上的区别与联系 .     

Über die Funktionalgleichungf(f(x)*f(y)) = f(x*y)

Ohne Zusammenfassung     

OSCILLATORY BEHAVIOUR OF SOLUTIONS OF y"+P(x)y=f(x)

This paper is a study of the oscillatory and asymptotic behaviour of solutions of the second order nonhomogeneous linear differential equation y"+P(x)y=f(x), and the associated homogeneous equation. Conditions are determined, under which the nonhomogeneous equation is oscillatory if and only if the homogeneous equation is oscillatory.     

On the functional equation f(x+g(y))-f(y+g(y))=f(x)-f(y) on groups

We solve the equation

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

Aequationes mathematicae -