On generalized Hyers-Ulam stability of additive mapings on restricted domains of Banach spaces

Abstract

Let ${(G,\cdot)}$ be a group (not necessarily Abelian) with unit ${e}$ and ${E}$ be a Banach space. In this paper, we show that there exist ${\alpha(p) > 0}$ for any ${0 < p < 1}$ and ${\beta(p,\varepsilon),\gamma(p,\varepsilon) > 0}$ for any ${0 < \varepsilon < \alpha(p)}$, such that for any surjective map ${f: G\rightarrow E}$ satisfying ${\big|\|f(x) + f(y)\|-\|f(xy) \|\big|\leq\varepsilon \|f(x)+f(y)\|^p}$ for all ${x,y\in G}$, there is a unique additive ${T:G\rightarrow E}$ such that ${\|f(x)-T(x)\|\leq\gamma(p,\varepsilon)\|f(x)\|^p}$ for all ${x\in G}$ satisfying ${\|f(x)\|\geq\beta(p,\varepsilon)}$. Moreover, we have ${\lim_{\varepsilon\rightharpoonup 0} \frac{\gamma(p,\varepsilon)}{\varepsilon} < \infty.}$