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A HYPERSTABILITY RESULT FOR THE CAUCHY EQUATION

Part of: Equations and inequalities involving nonlinear operators Nonlinear operators and their properties

Published online by Cambridge University Press:  22 August 2013

JANUSZ BRZDĘK
JANUSZ BRZDĘK*
Affiliation:
Department of Mathematics, Pedagogical University, Podchorążych 2, PL-30-084 Kraków, Poland email jbrzdek@up.krakow.pl
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Abstract

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We prove a hyperstability result for the Cauchy functional equation $f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function $f$, mapping a normed space ${E}_{1} $ into a normed space ${E}_{2} $, and for all real numbers $r, s$ with $r+ s\gt 0$ one of the following two conditions must be valid:

$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$
$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$
 In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem.

Keywords

hyperstabilityCauchy equationadditive functionrestricted domain

MSC classification

Secondary: 39B62: Functional inequalities, including subadditivity, convexity, etc. 47H10: Fixed-point theorems 39B82: Stability, separation, extension, and related topics 47H14: Perturbations of nonlinear operators 47J20: Variational and other types of inequalities involving nonlinear operators (general)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 33 - 40
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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