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

Abstract

We reformulate several well-known functional equations including Cauchy equation, Pexider equation, Jensen equation, Pompeiu equation and their generalized forms as equations for Gevrey distributions and show that every solution of each functional equation in the space of Gevrey distributions is Gevrey differentiable. Mathematics Subject Classification (1991). Primary 39B22; Secondary 46F05.

Generalized Pompeiu equation in distributions

“…In the previous papers [2][3][4][5][6]8,9,13] several functional equations have been studied in the space of Schwartz distributions. Making use of the Schwartz theory of distributions one can differentiate freely the underlying unknown functions with local integrability assumed, which is one of powerful advantages of the Schwartz theory and is applied to solve some classical functional equations [2][3][4]8,9,13].…”

“…where S(x, y) = x + y + x y, P 1 (x, y) = x, P 2 (x, y) = y, x, y ∈ I , and u • S, v • P 1 , w • P 2 are the pullbacks of u, v, w in D (I ) by S, P 1 and P 2 , respectively, and ⊗ denotes the tensor product of distributions [11,6,14]. As a result, every nontrivial solution u, v, w in D (I ) of Eq.…”

Hyers-Ulam stability theorems for Pexider equations in the space of Schwartz distributions

“…M. Rassias [15] firstly generalized the above result and since then, stability problems of many other functional equations have been investigated [3], [8], [12], [13], [15], [17]. Functional equations have been studied in the spaces of generalized functions such as Schwartz distributions, Gevrey distributions and Fourier hyperfunctions [1], [2], [6], [7]. But there are few results on stabilities for functional equations in the sense of distributions.…”

Some functional equations in the spaces of generalized functions

“…In [7] many functional equations have been studied in the space of Gevrey distributions and it was shown that every solution in the space of Gevrey distributions of such equations is a smooth function. In this paper, making use of the fundamental solution of the heat equation, we find the exact solution of the following functional equations in the spaces of generalized functions such as the Schwartz distributions and the Sato hyperfunctions:…”

“…Note that the above equations themselves make no sense in the spaces of generalized functions. Making use of the tensor product and pullback of generalized functions as in [2] and [7] we extend the above functional equations to the spaces of generalized functions. Let A, B, P 1 and P 2 be the functions…”