Stability of Pexider Equations on Semigroup with No Neutral Element

Abstract

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 fx+y-gx-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 ft+s-gth(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 ft+s=gth(s) for all t,s∈S, a closed form of which is not yet known.