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 allm ≥ 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


Introduction and Preliminaries
In 1897, Hensel 1 has introduced a normed space which does not have the Archimedean property.
During the last three decades theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings, and superstrings 2 .Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of intuition 3-10 .
Let Ã be a field.NA3 the strong triangle inequality ultrametric ; namely, Then X, • is called a non-Archimedean space.
It follows from NA3 that for all x, y ∈ E, where and p are constants with > 0 and p < 1.Then the limit exists for all x ∈ E and L : E → E is the unique additive mapping which satisfies for all x ∈ E. Also, if for each x ∈ E the mapping f tx is continuous in t ∈ Ê, then L is Ê-linear.
Moreover, D. G. Bourgin 16 and Gȃvrut ¸a 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 Gȃvrut ¸a 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 ∈ Ê such that r p q / 1 and f satisfies the inequality for all x, y ∈ X.Then there exists a unique additive mapping L : X → Y satisfying 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 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 or there exists a natural number m 0 such that ii the sequence {T m x} is convergent to a fixed point y * of T; iii y * is the unique fixed point of T in Λ {y ∈ Ω : d T m 0 x, 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 associated with the Jensen functional equation.

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: for all x, y ∈ A.
Theorem 2.1.Let f : A → B be a function for which there exist functions ϕ, ψ : for all x, y ∈ A, then there exists a unique ring homomorphism H : A → B such that for all x ∈ A.
Proof.It follows from 2.4 that for all x, y ∈ X.By 2.6 , lim n → ∞ 1/|2| n ϕ 0, 0 0. Hence, ϕ 0, 0 0. Letting x y 0 in 2.2 , we get f 0 ≤ ϕ 0, 0 0. So f 0 0. Let us define Ω to be the set of all mappings g : A → B and introduce a generalized metric on Ω as follows: It is easy to show that Ω, d is a generalized complete metric space 44, 45 .Now we consider the function T : Ω → Ω defined by Tg x 1/2 g 2x for all x ∈ A and all g ∈ Ω.Note that for all g, h ∈ Ω,

2.9
Hence, we see that d Tg, Th ≤ Ld g, h 2.10 for all g, h ∈ Ω, that is, T is a strictly self-function of Ω with the Lipschitz constant L.
Putting y : x in 2.2 , we have Now, from the fixed point alternative, it follows that there exists a fixed point H of T in Ω such that for all x ∈ A, since lim n → ∞ d T n f, H 0. On the other hand it follows from 2.2 , 2.6 , and 2.13 that for all x ∈ A and K > 0. Again using the fixed point alternative, we get and so we conclude that

2.22
For the case ϕ x, y : δ θ x p • y p where θ, δ are nonnegative real numbers and 0 < 2p < 1 , there exists a unique ring homomorphism H : A → B satisfying for all x ∈ A.
In the following we establish the superstability of ring homomorphisms on non-Archimedean Banach algebras associated with the Jensen functional equation f x y /2 f x f y /2.Theorem 2.4.Suppose there exist functions ϕ, ψ : A × A → 0, ∞ such that there exists a constant 0 < L < 1 such that ϕ 0, 2y ≤ |2|Lϕ 0, y , ψ 2x, 2y ≤ |2| A non-Archimedean absolute value on Ã is a function | • | : Ã → Ê such that for any a, b ∈ Ã we have i |a| ≥ 0 and equality holds if and only if a 0, ii |ab| |a||b|, iii |a b| ≤ max{|a|, |b|}.Let X be a linear space over a scalar field Ã with a non-Archimedean nontrivial valuation | • |.A function • : X → Ê is a non-Archimedean norm valuation if it satisfies the following conditions: NA1 x 0 if and only if x 0; NA2 rx |r| x for all r ∈ Ã and x ∈ X;

be unbounded. Theorem 1.1 T. M. Rassias . Let f : E → E be a mapping from a normed vector space E into a Banach space E subject to the inequality
2and therefore a sequence {x m } is Cauchy in X if and only if {x m 1 − x m } converges to zero in a non-Archimedean space.By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.A non-Archimedean Banach algebra is a complete non-Archimedean algebra A which satisfies ab ≤ a b for all a, b ∈ A. For more detailed definitions of non-Archimedean Banach algebras, we can refer to 11 .
According to the fixed point alterative, since H is the unique fixed point of T in the set Λ {g ∈ Ω : d f, g < ∞},H is the unique function such that Then we can choose L |2| 2p−1 and we get the desired results.Remark 2.3.Let f : A → B be a function for which there exist functions ϕ, ψ : A × A → 0, ∞ satisfying 2.2 and 2.3 .Let 0 < L < 1 be a constant such that ϕ x/2, y/2 ≤ L/|2| ϕ x, y for all x, y ∈ A. By a similar method to the proof of Theorem 2.1, one can show that there exists a unique ring homomorphism H : A → B satisfying Corollary 2.2.Let θ, p, s be nonnegative real numbers with p, and s > 1 and 2s− 2p ≥ 1. Suppose that f : A → B is a function such that Δf x, y ≤ θ x p • y p f xy − f x f y ≤ θ x s • y s 2.19for all x, y ∈ A. Then there exists a unique ring homomorphism H : A → B satisfying

2
Lψ x, y 2.24 for all x, y ∈ A. Moreover, assume that f : A → B is a function such that for all x, y ∈ A. Then f is a ring homomorphism.