Stability of Pexider Equations on Semigroup with No Neutral Element

Abstract EN

Let S be a commutative semigroup with no neutral element, Y a Banach space, and ℂ the set of complex numbers.

In this paper we prove the Hyers-Ulam stability for Pexider equation f x + y - g x - h ( y ) ≤ ϵ for all x , y ∈ S , where f , g , h : S → Y .

Using Jung’s theorem we obtain a better bound than that usually obtained.

Also, generalizing the result of Baker (1980) we prove the superstability for Pexider-exponential equation f t + s - g t h ( s ) ≤ ϵ for all t , s ∈ S , where f , g , h : S → ℂ .

As a direct consequence of the result we also obtain the general solutions of the Pexider-exponential equation f t + s = g t h ( s ) for all t , s ∈ S , a closed form of which is not yet known.