Ulam’s Type Stability of Involutional-Exponential Functional Equations

Abstract

Let S be a commutative semigroup, f,g:S→C and σ:S→S an involution. In this paper we consider the stability of involution-exponential functional equations fx+σy-gxfy≤ϕxresp., ϕy, |f(x+σy)-f(x)g(y)|≤ϕ(x) [resp., ϕ(y)] for all x,y∈S, where ϕ:S→R+ satisfies the growth condition: there exists C>1 such that limk→∞C-kϕ(kx)=0 for each x∈S. We also consider the stability of L∞-version |f(x+σy)-f(x)f(y)|L∞(R2n)≤ϵ, where f:Rn→C is a locally integrable function.