On an Asymptotic Behavior of Exponential Functional Equation

The stability problems of the exponential (functional) equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved: Let X be a real (or complex) normed space. A mapping f : X . . is exponential if and only if f(x + y) . f(x)f(y) . 0 as .x. + .y. . . under some suitable conditions.