Ulam’s Type Stability of Involutional-Exponential Functional Equations

Abstract

Let S be a commutative semigroup, $f,g:S\to \mathbb{C}$ and $\sigma :S\to S$ an involution. In this paper we consider the stability of involution-exponential functional equations $\left|f\left(x+\sigma y\right)-g\left(x\right)f\left(y\right)\right|\le \varphi \left(x\right)\left[\text{resp}.,\hspace{0.17em}\mathrm{ }\varphi \left(y\right)\right],\mathrm{ }|f(x+\sigma y)-f(x)g(y)|\le \varphi (x)\mathrm{ }[\text{resp}.,\hspace{0.17em}\mathrm{ }\varphi (y)]$ for all $x,y\in S$, where $\varphi :S\to {\mathbb{R}}^{+}$ satisfies the growth condition: there exists $C>1$ such that ${\text{lim}}_{k\to \mathrm{\infty }}{C}^{-k}\varphi (kx)=0$ for each $x\in S$. We also consider the stability of $L^{\mathrm{\infty }}$-version $|f(x+\sigma y)-f(x)f(y){|}_{L^{\mathrm{\infty }}({\mathbb{R}}^{2n})}\le ϵ,$ where $f:{\mathbb{R}}^n\to \mathbb{C}$ is a locally integrable function.