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On Ulam Stability of a Functional Equation in Banach Modules


Published:20160920
Printed: Mar 2017
Lahbib Oubbi,
Department of Mathematics, Ecole Normale SupĂ©rieure , Mohammed V University of Rabat, PO Box 5118, Takaddoum, 10105 Rabat (Morocco)
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 HyersUlam 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_{d1}  C^{\ell 1}_{d1} + 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 $C^p_q := \frac{q!}{(qp)!p!}$,
$p, q \in \mathbb{N}$ with $p \le q$.
In this note we solve this equation for arbitrary nonzero scalar
$r$ and show that it is actually HyersUlam 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.
MSC Classifications: 
39A30, 39B10, 39A06, 46Hxx show english descriptions
Stability theory unknown classification 39B10 Linear equations Topological algebras, normed rings and algebras, Banach algebras {For group algebras, convolution algebras and measure algebras, see 43A10, 43A20}
39A30  Stability theory 39B10  unknown classification 39B10 39A06  Linear equations 46Hxx  Topological algebras, normed rings and algebras, Banach algebras {For group algebras, convolution algebras and measure algebras, see 43A10, 43A20}
