Local stability of a generalized Cauchy equation in Banach spaces

Let (G,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(G, \cdot)}$$\end{document} be a group and E be a Banach space. Assume that f:G→E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f \colon G\rightarrow E}$$\end{document} is a map such that f(G) is an open set containing 0. If there exists an ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon > 0}$$\end{document} and a p > 1 so that |‖f(x)+f(y)‖-‖f(xy)‖|≤εmin{‖f(x)+f(y)‖p,‖f(xy)‖p}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big| \|f(x) + f(y)\| - \|f(xy)\|\big| \leq \varepsilon \min \big \{\|f(x) + f(y)\|^p, \|f(xy)\|^p\big\}$$\end{document}for all x,y∈G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x, y \in G}$$\end{document}, then f is an additive map onto E. If E is a finite-dimensional Banach space, the result holds when f(G) (not necessarily open) contains 0 as an interior point.