Year: 2014
Type: article
Abstract: Let $(X,+)$ be an Abelian group and $E$ be a Banach space. Suppose that $f:X\rightarrow E$ is a surjective map satisfying the inequality $$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(x-y)\Vert \,|\leq {\it\varepsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(x-y)\Vert ^{p}\}\end{eqnarray}$$ for some ${\it\varepsilon}>0$ , $p>1$ and for all $x,y\in X$ . We prove that $f$ is an additive map. However, this result does not hold for $0<p\leq 1$ . As an application, we show that if $f$ is a surjective map from a Banach space $E$ onto a Banach space $F$ so that for some ${\it\epsilon}>0$ and $p>1$ $$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(u)-f(v)\Vert \,|\leq {\it\epsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(u)-f(v)\Vert ^{p}\}\end{eqnarray}$$ whenever $\Vert x-y\Vert =\Vert u-v\Vert$ , then $f$ preserves equality of distance. Moreover, if $\dim E\geq 2$ , there exists a constant $K\neq 0$ such that $Kf$ is an affine isometry. This improves a result of Vogt [‘Maps which preserve equality of distance’,Studia Math.45(1973) 43–48].
Authors: YUNBAI DONG, BENTUO ZHENG
Institutions: Wuhan Textile University, University of Memphis
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,550
OpenAlex ID: W2052006804