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# On Ulam Stability of a Functional Equation in Banach Modules

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Published:2016-09-20
Printed: Mar 2017
• Lahbib Oubbi,
Department of Mathematics, Ecole Normale Supérieure , Mohammed V University of Rabat, PO Box 5118, Takaddoum, 10105 Rabat (Morocco)
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## Abstract

Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping. For any rational number $r \ne 2$, C. Baak, D. H. Boo, and Th. M. Rassias have proved the Hyers-Ulam stability of the following functional equation: \begin{align*} r f \left(\frac{\sum_{j=1}^d x_j}{r} \right) & + \sum_{\substack{i(j) \in \{0,1\} \\ \sum_{j=1}^d i(j)=\ell}} r f \left( \frac{\sum_{j=1}^d (-1)^{i(j)}x_j}{r} \right) = (C^\ell_{d-1} - C^{\ell -1}_{d-1} + 1) \sum_{j=1}^d f(x_j) \end{align*} where $d$ and $\ell$ are positive integers so that $1 \lt \ell \lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(q-p)!p!}$, $p, q \in \mathbb{N}$ 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. Different questions concerning the *-homomorphisms and the multipliers between C*-algebras are also considered.
 Keywords: linear functional equation, Hyers-Ulam stability, Banach modules, C*-algebra homomorphisms.
 MSC Classifications: 39A30 - Stability theory 39B10 - unknown classification 39B1039A06 - Linear equations 46Hxx - Topological algebras, normed rings and algebras, Banach algebras {For group algebras, convolution algebras and measure algebras, see 43A10, 43A20}

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