Functional Inequalities Associated with Additive Mappings

Abstract

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