Nearly Ring Homomorphisms and Nearly Ring Derivations on Non-Archimedean Banach Algebras

and Applied Analysis 3 Moreover, Bourgin 15 and Găvruţa 16 have considered the stability problem with unbounded Cauchy differences see also 17–27 . On the other hand, J. M. Rassias 28–33 considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Găvruţa 34 . This stability phenomenon is called the Ulam-Găvruta-Rassias stability see also 35 . Theorem 1.2 J. M. Rassias 28 . Let X be a real normed linear space and Y a real complete normed linear space. Assume that f : X → Y is an approximately additive mapping for which there exist constants θ ≥ 0 and p, q ∈ such that r p q / 1 and f satisfies the inequality ∥ ∥f ( x y ) − f x − f(y)∥∥ ≤ θ‖x‖p∥∥y∥∥q 1.6 for all x, y ∈ X. Then there exists a unique additive mapping L : X → Y satisfying ∥ ∥f x − L x ∥ ∥ ≤ θ |2r − 2| ‖x‖ r 1.7 for all x ∈ X. If, in addition, f : X → Y is a mapping such that the transformation t → f tx is continuous in t ∈ for each fixed x ∈ X, then L is an -linear mapping. Very recently, Ravi et al. 36 in the inequality 1.6 replaced the bound by a mixed one involving the product and sum of powers of norms, that is, θ{‖x‖p‖y‖p ‖x‖2p ‖y‖2p }. For more details about the results concerning such problems and mixed product-sum stability J. M.-Rassias Stability the reader is referred to 37–49 . Khodaei and T. M. Rassias 50 have established the general solution and investigated the Hyers-Ulam-Rassias stability of the following n-dimensional additive functional equation: