Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras - Faruk Polat (Abstract and Applied Analysis)
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.