On a General Conditional Cauchy Functional Equation

Let (G,+) be an abelian group and Y  a linear space over the field \Bbb{F}\in\{\Bbb{R}, \Bbb{C}\}. In this paper, we investigate the conditional Cauchy functional equation \[f(x+y)\neq af(x)+bf(y)\quad\Rightarrow \quad f(x+y)=f(x)+f(y),\quad x,y\in G,\] for functions f:G\to Y, where  a,b\in \Bbb{F} are fixed constants. The general solution and stability of this functional equation are described.

کلیدواژه ها:

Abelian group ، Conditional functional equation ، Mikusiński’s functional equation ، stability