On Jensen-like form of Mikusiński's functional equation

Abstract

We consider the conditional Jensen functional equation

$$f(x+y)\ne 0 \implies 2f\big(\frac{x+y}{2}\big)=f(x)+f(y), \quad x,y\in \mathcal{G}$$

for functions \(f \colon \mathcal{G} \to\mathcal{V}\), where \((\mathcal{G},+)\) and \((\mathcal{V},+)\) are uniquely 2-divisible groups, with \((\mathcal{V},+)\) being abelian. Additionally, we investigate the hyperstability of this functional equation.