Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras

Abstract

Badora (2002) proved the following stability result. Let $\epsilon$ and $\delta$ be nonnegative real numbers, then for every mapping $f$ of a ring $\mathcal{R}$ onto a Banach algebra $\mathcal{B}$ satisfying $\left|\right|f(x+y)-f\left(x\right)-f\left(y\right)\left|\right|\le \epsilon$ and $\left|\right|f(x\cdot y)-f\left(x\right)f\left(y\right)\left|\right|\le \delta$ for all $x,y\in \mathcal{R}$, there exists a unique ring homomorphism $h:\mathcal{R}\to \mathcal{B}$ such that $\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \epsilon ,x\in \mathcal{R}$. Moreover, $b\cdot \left(f\right(x)-h(x\left)\right)=0,\left(f\right(x)-h(x\left)\right)\cdot b=0$, for all $x\in \mathcal{R}$ and all $b$ from the algebra generated by $h\left(\mathcal{R}\right)$. In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.