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ON HYPERSTABILITY OF ADDITIVE MAPPINGS ONTO BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices Nonlinear operators and their properties

Published online by Cambridge University Press:  30 December 2014

YUNBAI DONG  and
BENTUO ZHENG
YUNBAI DONG*
Affiliation:
School of Mathematics and Computer, Wuhan Textile University, Wuhan 430073, China email baiyunmu301@126.com
BENTUO ZHENG
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, USA email bzheng@memphis.edu
*
✉baiyunmu301@126.com
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Abstract

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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].

Keywords

hyperstabilityBanach spacesadditive mappingsfunctional equations

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 46B04: Isometric theory of Banach spaces 47H14: Perturbations of nonlinear operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 278 - 285
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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