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

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.