Generalized stability of Isometries on real Banach spaces
Bull. Korean Math. Soc. 2006 Vol. 43, No. 2, 309-318
Published online June 1, 2006
Eun Hwi Lee and Dal-Won Park
Jeonju University, Kongju National University
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 $\|x\|\leq \delta$.
Keywords : $(\varepsilon, p)$-isometry, isometry, real Banach spaces
MSC numbers : 39B72
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