A Jensen-type variant of Dhombres' functional equation and its superstability

Abstract

In this paper, we investigate a Jensen-type version of Dhombres' functional equation, namely$f\left(x\right)+f\left(y\right)\ne 0\Rightarrow 2f\left(\frac{x+y}{2}\right)=f\left(x\right)+f\left(y\right),x,y\in G$ for functions $f:G\to V$, where $(G,+)$ and $(V,+)$ are uniquely 2-divisible abelian groups. We also investigate the superstability behavior of this functional equation.