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 R onto a Banach algebra B satisfying parallel to f(x + y) - f(x) - f(y)parallel to <= epsilon and parallel to f(x . y) - f(x)f(y)parallel to <= delta for all x, y is an element of R, there exists a unique ring homomorphism h : R -> B such that parallel to f(x) - h(x)parallel to <= epsilon, x is an element of R. Moreover, b . (f(x) - h(x)) = 0, (f(x) - h(x)) . b = 0, for all x is an element of 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.