## Abstract and Applied Analysis

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

Faruk Polat

#### 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 $||f(x+y)-f(x)-f(y)||\le \epsilon$ and $||f(x\cdot y)-f(x)f(y)||\le \delta$ for all $x,y\in \mathcal{R}$, there exists a unique ring homomorphism $h:\mathcal{R}\to \mathcal{B}$ such that $||f(x)-h(x)||\le \epsilon ,x\in \mathcal{R}$. Moreover, $b\cdot (f(x)-h(x))=0,(f(x)-h(x))\cdot b=0$, for all $x\in \mathcal{R}$ and all $b$ from the algebra generated by $h(\mathcal{R})$. In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 653508, 9 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495623

Digital Object Identifier
doi:10.1155/2012/653508

Mathematical Reviews number (MathSciNet)
MR2872311

Zentralblatt MATH identifier
1235.39030

#### Citation

Polat, Faruk. Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras. Abstr. Appl. Anal. 2012 (2012), Article ID 653508, 9 pages. doi:10.1155/2012/653508. https://projecteuclid.org/euclid.aaa/1355495623

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