d’Alembert functional equations in distributions

Abstract

Abstract.In this paper we shall develop a method to define and solve the D'Alembert functional equation in distributions. We shall also show that for regular distributions (i.e., locally integrable functions) the distributional solution reduces to the classical one.

Stability of a Jensen type equation in the space of generalized functions

“…After the work of Hyers, the stability problems of various functional equations, such as Pexider equation [10], D'Alembert functional equation [7], Quadratic equation [3], and so on, have been investigated by a number of authors.…”

Reformulation of some functional equations in the space of Gevrey distributions and regularity of solutions

“…Many well-known functional equations have been studied in the Schwartz distributions (see [2], [3], [4], [5], [10] and [11] etc.). Actually it was shown that most of these functional equations in the distributions reduce to the classical ones.…”

“…When we reformulate the above equations in the Gevrey distributions we do not follow the method using operators in [2], [3], [4] and [10], [11], but use fundamental properties of generalized functions such as the tensor product and the pullback of Gevrey distributions. In fact, this method gives essentially the same formulation as in [2], [3], [4] and [10], [11], but the final form of each equation looks like the classical one.…”

“…In fact, this method gives essentially the same formulation as in [2], [3], [4] and [10], [11], but the final form of each equation looks like the classical one.…”

“…This method turns out to be quite different from and much easier than the way used in [1], [2], [3] and [10], [11]. Actually, in these papers differential operators were applied to get the regularity of solutions.…”