Generalized stability of Isometries on real Banach Spaces

Abstract

Let $X$ and $Y$ be real Banach spaces and $\varepsilon>0$, $p>1$. Let $f : X \rightarrow Y$ be a bijective mapping with $f(0)=0$ satisfying $$ \Bigl|\|f(x)-f(y)\|-\|x-y\|\Bigr|\leq \varepsilon\|x-y\|^p$$ for all $x \in X$ and, let $f^{-1} : Y \rightarrow X$ be uniformly continuous. Then there exist a constant $\delta>0$ and $N(\varepsilon, p)$ such that $\displaystyle{\lim_{\epsilon \rightarrow 0}N(\varepsilon, p){\hskip-0.05cm}={\hskip-0.05cm}0}$ and a unique surjective isometry $I : X \rightarrow Y$ satisfying $\|f(x)-I(x)\|\leq N(\varepsilon, p)\|x\|^p$ for all $x \in X$ with $\|x\|\leq \delta$.