Abstract and Applied Analysis
Volume 2012 (2012), Article ID 653508, 9 pages
http://dx.doi.org/10.1155/2012/653508
Research Article

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

Faruk Polat

Department of Mathematics, Firat University, 23119 Elazig, Turkey

Received 7 June 2011; Revised 6 October 2011; Accepted 18 October 2011

Academic Editor: Jean Michel Combes

Copyright © 2012 Faruk Polat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Badora (2002) proved the following stability result. Let 𝜀 and 𝛿 be nonnegative real numbers, then for every mapping 𝑓 of a ring ℛ onto a Banach algebra ℬ satisfying | | 𝑓 ( 𝑥 + 𝑦 ) − 𝑓 ( 𝑥 ) − 𝑓 ( 𝑦 ) | | ≤ 𝜀 and | | 𝑓 ( 𝑥 ⋅ 𝑦 ) − 𝑓 ( 𝑥 ) 𝑓 ( 𝑦 ) | | ≤ 𝛿 for all 𝑥 , 𝑦 ∈ ℛ , there exists a unique ring homomorphism ℎ ∶ ℛ → ℬ such that | | 𝑓 ( 𝑥 ) − ℎ ( 𝑥 ) | | ≤ 𝜀 , 𝑥 ∈ ℛ . Moreover, 𝑏 ⋅ ( 𝑓 ( 𝑥 ) − ℎ ( 𝑥 ) ) = 0 , ( 𝑓 ( 𝑥 ) − ℎ ( 𝑥 ) ) ⋅ 𝑏 = 0 , for all 𝑥 ∈ ℛ and all 𝑏 from the algebra generated by ℎ ( ℛ ) . In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.