Functional Inequalities Associated with Additive Mappings

Abstract

The functional inequality $\Vert f\left(x\right)+2f\left(y\right)+2f\left(z\right)\Vert \le \Vert 2f\left(x/2+y+z\right)\Vert +\varphi \left(x,y,z\right)\left(x,y,z\in G\right)$ is investigated, where $G$ is a group divisible by $2,f:G\to X$ and $\varphi :G^3\to \left[0,\infty \right)$ are mappings, and $X$ is a Banach space. The main result of the paper states that the assumptions above together with (1) $\varphi \left(2x,-x,0\right)=0=\varphi \left(0,x,-x\right)$ $\left(x\in G\right)$ and (2) ${\lim }_{n\to \infty }\left(1/2^n\right)\varphi \left(2^{n+1}x,2^{n}y,2^{n}z\right)=0$, or ${\lim }_{n\to \infty }{2}^n\varphi \left(x/2^{n-1},y/2^n,z/2^n\right)=0\left(x,y,z\in G\right)$, imply that $f$ is additive. In addition, some stability theorems are proved.