On the Superstability and Stability of the Pexiderized Exponential Equation

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