Stability and Superstability of Ring Homomorphisms on Non-Archimedean Banach Algebras

and Applied Analysis 3 Moreover, D. G. Bourgin 16 and Găvruţa 17 have considered the stability problem with unbounded Cauchy differences see also 18–23 . On the other hand, J. M. Rassias 24–29 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 30 . Theorem 1.2 J. M. Rassias 24 . 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. Bourgin 16, 31 is the first mathematician dealing with stability of ring homomorphism f xy f x f y . The topic of approximate homomorphisms was studied by a number of mathematicians, see 32–37 and references therein. A function f : A → A is a ring homomorphism or additive homomorphism if f is an additive function satisfying f ( xy ) f x f ( y ) 1.8 for all x, y ∈ A. Now we will state the following notion of fixed point theory. For the proof, refer to 38 , see also 39, chapter 5 . For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to 40, 41 . In 2003, Radu 42 proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative see also 43–45 . Let X, d be a generalized metric space. An operator T : X → X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that d Tx, Ty ≤ Ld x, y for all x, y ∈ X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz. Theorem 1.3 cf. 38, 42 . Suppose that one is given a complete generalized metric space Ω, d and a strictly contractive mapping T : Ω → Ω with Lipschitz constant L. Then for each given x ∈ Ω, either d ( Tx, T 1x ) ∞ ∀m ≥ 0 1.9 or there exists a natural number m0 such that 4 Abstract and Applied Analysis i d Tx, T 1x < ∞ for all m ≥ m0, ii the sequence {Tmx} is convergent to a fixed point y∗ of T ; iii y∗ is the unique fixed point of T in Λ {y ∈ Ω : d T0x, y < ∞}; iv d y, y∗ ≤ 1/ 1 − L d y, Ty for all y ∈ Λ. Recently, the first author of the present paper 4 established the stability of ring homomorphisms on non-Archimedean Banach algebras. In this paper, using fixed point methods, we prove the generalized Hyers-Ulam stability of ring homomorphisms on non-Archimedean Banach algebras. Moreover, we investigate the superstability of ring homomorphisms on non-Archimedean Banach algebras associatedwith the Jensen functional equation. 2. Approximation of Ring Homomorphisms in Non-Archimedean Banach Algebras Throughout this section we suppose that A, B are two non-Archimedean Banach algebras. For convenience, we use the following abbreviation for a given function f : A → B: Δf ( x, y ) f ( x y ) − f x − f(y) 2.1 for all x, y ∈ A. Theorem 2.1. Let f : A → B be a function for which there exist functions φ, ψ : A ×A → 0,∞ such that ∥ ∥Δf ( x, y )∥ ∥ ≤ φ(x, y), 2.2 ∥ ∥f ( xy ) − f x f(y)∥∥ ≤ ψ(x, y) 2.3 for all x, y ∈ A. If there exists a constant 0 < L < 1 such that φ ( 2x, 2y ) ≤ |2|Lφ(x, y) ψ ( 2x, 2y ) ≤ |2|2Lψ(x, y) 2.4 for all x, y ∈ A, then there exists a unique ring homomorphismH : A → B such that ∥ ∥f x −H x ∥ ≤ 1 |2| 1 − L φ x, x , 2.5