HYERS-ULAM-RASSIAS STABILITY OF FUNCTIONAL EQUATION IN NAB-SPACES

In this paper, using direct method we investigate the Hyers-Ulam- Rassias stability of an additive functional equation in non-Archimedean Banach spaces (briefly, NAB-spaces).


Introduction and Preliminaries
A classical question in the theory of functional equations is the following: When is it true that a function which approximately satisfies a functional equation D must be close to an exact solution of D?.
If the problem accepts a solution, we say that the equation D is stable.The first stability problem concerning group homomorphisms was raised by Ulam [26] in 1940.
We are given a group G and a metric group G ′ with metric d(., .).Given ε > 0, dose there exist a δ > 0 such that if f : G → G ′ satisfies d(f (xy), f (x)f (y)) < δ, for all x, y ∈ G, then a homomorphism h : G → G; exists with d(f (x), h(x)) < ε for all x ∈ G?.
In 1978, Th.M. Rassias [18] formulated and proved the following theorem, which implies Hyers's Theorem as a special case.Suppose that E and F are real normed spaces with F a complete normed space, f : E → F is a mapping such that for each fixed x ∈ E the mapping t → f (tx) is continuous on R, and let there exist ε ≥ 0 and p ∈ [0, 1) such that for all x, y ∈ E Then there exists a unique linear mapping T : The case of the existence of a unique additive mapping had been obtained by T. Aoki [2], as it is recently noticed by Lech Maligranda.However, Aoki [2] had claimed the existence of a unique linear mapping, that is not true because he did not allow the mapping f to satisfy some continuity assumption.Th.M. Rassias [18], who independently introduced the unbounded Cauchy difference was the first to prove that there exists a unique linear mapping T satisfying In 1990, Th.M. Rassias [19] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1.In 1991, Z. Gajda [8] following the same approach as in Th.M. Rassias [24], gave an affirmative solution to this question for p > 1.It was proved by Z. Gajda [8] , as well as by Th.M. Rassias and P. Šemrl [20] that one can not prove a Th.M. Rassias type theorem when p = 1.In 1994, P. Gǎvruta [9] provided a further generalization of Th.M. Rassias theorem in which he replaced the bound ε(||x|| p + |||y|| p ) by a general control function ψ(x, y) for the existence of a unique linear mapping.
The functional equation is called the quadratic functional equation.In particular, every solution of the quadratic functional equation is said to be a quadratic mapping.A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [25] for mappings f : X → Y , where X is a normed space and Y is a Banach space.Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group.In [7], Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation.
During the last decades several stability problems of functional equations have been investigated by a number of mathematicians( [1]- [5], [11]- [24]).Definition 1.1.By a non-Archimedean field we mean a field K equipped with a function(valuation) |.| : K → [0, ∞) such that for all r, s ∈ K, the following conditions hold: Definition 1.2.Let X be a vector space over a scalar field K with a non-Archimedean non-trivial valuation Due to the fact that converges to zero in a non-Archimedean space.By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.Theorem 1.1.Let (X,d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipschitz constant L < 1.
Then, for all x ∈ X, either for all nonnegative integers n or there exists a positive integer n 0 such that: In this paper, we prove the generalized Hyers-Ulam stability of the following functional equation in non-Archimedean normed spaces.In the rest of the paper let |2| = 1.

Non-Archimedean Stability of Eq. (1.3): A Direct Method
Throughout this section, using direct method we prove the generalized Hyers-Ulam stability of composite functional equation (1.3) in non-Archimedean spaces.
Theorem 2.1.Let G is an additive semigroup and X is a complete non-Archimedean space.Assume that ϕ : G 2 → [0, +∞) be a function such that exists.Suppose that f : G → X be a mapping satisfying the inequality for all x, y ∈ G. Then the limit exist for all x ∈ G and A : G → X is an additive mapping satisfying Then A is the unique mapping satisfying (2.4).
Proof.Putting y = x in (2.3), we have Replacing x by 2 n x in (2.5), we get It follows from (2.1) and (2.6) that the sequence f (2 n x) Using induction we see that Indeed, (2.7) holds for n = 1 by (2.5).Let, (2.7) holds for n, so by (2.6), we obtain So for all n ∈ N and all x ∈ G, (2.7) holds.By taking n to approach infinity in (2.8), one obtains (2.4).
If L is another mapping satisfies (2.4) , then for x ∈ G, we get Therefore A = L.This completes the proof.
for all t ≥ 0. Let δ > 0 and f : G → X is a mapping satisfying the inequality for all x, y ∈ G. Then the limit exists for all x ∈ G and A : G → X is a unique additive mapping such that for all x ∈ G.
exists for all x ∈ G. On the other hand Applying Theorem 2.1, then we get the desired result.
Theorem 2.2.Let G is an additive semigroup and X is a complete non-Archimedean space.Assume that ϕ : G 2 → [0, +∞) be a function such that exists.Suppose that f : G → X is a mapping satisfying the inequality (2.3).Then the limit exist for all x ∈ G and A : G → X is an additive mapping satisfying

13)
Then A is the unique mapping satisfying (2.12).
Proof.By (??), we have for all x ∈ G. Replacing x by x 2 n in (2.14) , we get (2.15) It follows from (2.10) and (2.15) that the sequence k+1 ; p ≤ k < n − 1 for all x ∈ G all non-negative integer n, p with n > p ≥ 0. Letting p = 0 and passing the limit n → ∞ in the last inequality, we obtain (2.12).The rest of the proof is similar to the proof of Theorem 2.1.for all t ≥ 0. Let δ > 0 and f : G → X is a mapping satisfying the inequality (2.9).Then the limit A(x) = lim n→∞ 2 n f x 2 n exists for all x ∈ G and A : G → X is a unique additive mapping such that exists for all x ∈ G. Applying Theorem 2.2, then we get the desired result.

Example 1 . 1 .
Fix a prime number p.For any nonzero rational number x, there exists a unique integer n x ∈ Z such that x = a b p nx , where a and b are integers not divisible by p. Then |x| p := p −nx defines a non-Archimedean norm on Q.The completion of Q with respect to the metric d(x, y) = |x − y| p is denoted by Q p which is called the p-adic number field.In fact, Q p is the set of all formal series x = ∞ k≥nx a k p k where |a k | ≤ p − 1 are integers.The addition and multiplication between any two elements of Q p are defined naturally.The norm | ∞ k≥nx a k p k | p = p −nx is a non-Archimedean norm on Q p and it makes Q p a locally compact filed.Definition 1.4.Let X be a set.A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions: (a) d(x, y) = 0 if and only if x = y for all x, y ∈ X; (b) d(x, y) = d(y, x) for all x, y ∈ X; (c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
(a) d(J n x, J n+1 x) < ∞ for all n 0 ≥ n 0 ; (b) the sequence {J n x} converges to a fixed point y * of J; (c) y * is the unique fixed point of J in the set Y = {y ∈ X : d(J n 0 x, y) < ∞}; (d) d(y, y * ) ≤ 1 1−L d(y, Jy) for all y ∈ Y .