Some new results on the superstability of the Cauchy equation on semigroups (Q1936958)

Let \((S,\cdot)\) be a commutative semigroup, \(f: S\to \{z\in\mathbb{C}:\;-\pi<\text{Im}\, z\leq \pi\}\). Suppose that for some mappings \(\psi: S\to\mathbb{R}^{+}\), \(\phi: S^2\to\mathbb{R}^{+}\) we have \[ |f(x\cdot y)-f(x)-f(y)|\leq\phi(x,y)\quad\text{and}\quad |f(x)|\leq\psi(x),\qquad x,y\in S. \] Assume that there exists a \(p\in S\) such that \[ \text{Re}\,f(p)>0,\quad\sum_{m=0}^{\infty}\phi(p,p^{m+1})<\infty \] and \[ \psi(x\cdot p)\leq \psi(x),\qquad x\in S. \] Then \(f\) is a Cauchy mapping, i.e. \[ f(x\cdot y)=f(x)+f(y). \] The given proof is based on the fixed-point method.