On the Stability of A Class of Functional Equations

Volume 4, Issue 5, 2003

Article 104

ON THE STABILITY OF A CLASS OF FUNCTIONAL EQUATIONS

BELAID BOUIKHALENE

DÉPARTEMENT DE MATHÉMATIQUES ET INFORMATIQUE
FACULTÉ DES SCIENCES BP 133,
14000 KÉNITRA, MOROCCO.
E-Mail: bbouikhalene@yahoo.fr

Received 20 July, 2003; Accepted 24 October, 2003.
Communicated by: K. Nikodem

ABSTRACT. In this paper, we study the Baker's superstability for the following functional equation

$\displaystyle \sum_{\varphi \in \Phi }\int_{K}f(xk\varphi (y)k^{-1})d\omega _{K}(k)$

$E\left( K\right)$

$\displaystyle =\vert\Phi \vert f(x)f(y), x,y\in G$

where is a locally compact group, is a compact subgroup of , $\omega _{K}$ is the normalized Haar measure of , $\Phi$ is a finite group of -invariant morphisms of and is a continuous complex-valued function on satisfying the Kannappan type condition, for all $x,y,z\in G$

$\displaystyle \int_{K}\int_{K}f(zkxk^{-1}hyh^{-1})d\omega _{K}(k)d\omega _{K}(h)\qquad\qquad\qquad\qquad$

(*)

$\displaystyle =\int_{K}\int_{K}f(zkyk^{-1}hxh^{-1})d\omega _{K}(k)d\omega _{K}(h).$

We treat examples and give some applications.

Key words:
Functional equation, Stability, Superstability, Central function, Gelfand pairs.

2000 Mathematics Subject Classification:
39B72.

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