On the Superstability and Stability of the Pexiderized Exponential Equation

Abstract

The main purpose of this paper is to establish some new results on<br />the superstability and stability via a fixed point approach for<br />the Pexiderized exponential equation, i.e.,<br />$$|f(x+y)-g(x)h(y)|leq psi(x,y),$$<br />where $f$, $g$ and $h$ are three functions from an arbitrary<br />commutative semigroup $S$ to an arbitrary unitary complex Banach<br />algebra and also $psi: S^{2}rightarrow [0,infty)$ is a<br />function. Furthermore, in connection with the open problem of Th.<br />M. Rassias and our results we generalized the theorem of Baker,<br />Lawrence, Zorzitto and theorem of L. Sz$acute{e}$kelyhidi.