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On Ulam Stability of a Functional Equation in Banach Modules

Published online by Cambridge University Press:  20 November 2018

Lahbib Oubbi
Lahbib Oubbi*
Affiliation:
Department ofMathematics, École Normale Supérieure, Mohammed V University of Rabat, PO Box 5118, Takaddoum, 10105 Rabat (Morocco) e-mail: oubbi@daad-alumni.de
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Abstract

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Let $X$ and $Y$ be Banach spaces and let $f\,:\,X\,\to \,Y$ be an odd mapping. For any rational number $r\,\ne \,2$, C. Baak, D. H. Boo, and Th. M. Rassias proved the Hyers–Ulam stability of the functional equation

$$rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{x}_{j}}}}{r} \right)\,+\,\sum\limits_{\begin{smallmatrix} i\left( j \right)\,\in \left\{ 0,\,1 \right\} \\ \sum\nolimits_{j=1}^{d}{i\left( j \right)=\ell } \end{smallmatrix}}{rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{\left( -1 \right)}^{i\left( j \right)}}{{x}_{j}}}}{r} \right)}\,=\,\left( C_{d-1}^{\ell }\,-\,C_{d-1}^{\ell -1}\,+\,1 \right)\,\sum\limits_{j=1}^{d}{f\left( {{x}_{j}} \right),}$$

where $d$ and $\ell$ are positive integers so that $1\,<\,\ell \,<\,\frac{d}{2}$, and $C_{q}^{p}\,:=\frac{q!}{\left( q-p \right)!p!},\,p,\,q\,\in \,\mathbb{N}$ with $p\,\le \,q$.

In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers–Ulam stable. We thus extend and generalize Baak et al.’s result. Other questions concerning the $^{*}$-homomorphisms and the multipliers between ${{C}^{*}}$-algebras are also considered.

Keywords

39A3039B1039A0646Hxxlinear functional equationHyers-Ulam stabilityBanach modulesC*-algebra homomorphisms
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 173 - 183
Copyright
Copyright © Canadian Mathematical Society 2017

References

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