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

Abstract

Making use of the fundamental solution of the heat equation we prove stabilities of the Pexider equation and the Pexider-exponential equation in spaces of generalized functions such as the Schwartz tempered distributions, Fourier hyperfunctions and Gelfand-Shilov generalized functions.

On a Stability of Pexiderized Exponential Equation

“…As we shall see in Section 3, the stability problem (1.2) appears when we deal with the distributional analogue of the Pexiderized version of the stability of (1.1) (see [5,6]). From now on, a function m from a semigroup S, + to the field C is said to be exponential provided m(x + y) = m(x)m(y).…”

Hyers-Ulam stability of a certain Fredholm integral equation

“…After that Aoki [2], Bourgin [4] and Rassias [24] have generalized the Hyers result. These days the Hyers-Ulam stability for different functional equations was proved by many mathematicians (see [9,14,25,32]). A generalization Ulam problem was recently proposed by replacing functional equations with differential equations.…”

Stability of Pexider Equations on Semigroup with No Neutral Element

“…If has a neutral element, we obtain the following stability theorem for Pexider-exponential equation [3].…”

“…However, in the scope of the author, no results are found. In this paper, we prove the stability theorem for the functional inequalities (3) and (6) when has no neutral element.…”