Non-Archimedean Hyers-Ulam Stability of an Additive-Quadratic Mapping

1 Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran 2 Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece 3 Department of Mathematics, Islamic Azad University, Firoozabad Branch, Firoozabad, Iran 4 Department of Mathematics Education, College of Education, Mokwon University, Daejeon 302-729, Republic of Korea


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 must be close to an exact solution of the equation?"If the problem accepts a solution, we say that the equation is stable.The first stability problem concerning group homomorphisms was raised by Ulam 1 in 1940.In the next year, Hyers 2 gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces.In 1978, Th.M. Rassias 3 proved a generalization of Hyers' theorem for linear mappings.Furthermore, in 1994, a generalization of the Th.M. Rassias' theorem was obtained by Gȃvrut ¸a 4 by replacing the bound x p y p by a general control function φ x, y .In 1983, the Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof 5 for mappings f : X → Y , where X is a normed space and Y is a Banach space.In 1984, Cholewa 6 noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik 7 proved the Hyers-Ulam stability of the quadratic functional equation.The reader is referred to 2-4, 6-48 and references therein for detailed information on stability of functional equations.
In 1897, Hensel 24 has introduced a normed space which does not have the Archimedean property.It turned out that non-Archimedean spaces have many nice applications see 16,[26][27][28]37 .Definition 1.2.Let X be a vector space over a scalar field K with a non-Archimedean nontrivial valuation | • |.A function • : X → R is a non-Archimedean norm valuation if it satisfies the following conditions: 3 the strong triangle inequality ultrametric , namely, x y ≤ max x , y , x,y ∈ X.

1.1
Then X, • is called a non-Archimedean space, due to the fact that One recalls a fundamental result in fixed point theory.
Theorem 1.5 see 7, 17 .Let X, d be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1.Then for each given element x ∈ X, either for all nonnegative integers n or there exists a positive integer n 0 such that 2 the sequence {J n x} converges to a fixed point y * of J; In 1998, D. H. Hyers, G. Isac and Th.M. Rassias 25 provided applications of stability theory of functional equations for the proof of new fixed point theorems with applications.By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors see 15, 39, 40, 42 .This paper is organized as follows.In Section 2, using fixed point method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation: where x, y, z ∈ X, in non-Archimedean normed space.In Section 3, using direct method, we prove the Hyers-Ulam stability of the additive-quadratic functional equation 1.4 in non-Archimedean normed spaces.

Stability of the Functional Equation 1.4 : A Fixed Point Approach
In this section, we deal with the stability problem for the quadratic functional equation 1.4 .
Theorem 2.1.Let X be a non-Archimedean normed space and Y a complete non-Archimedean space.Let ϕ : X 3 → 0, ∞ be a function such that there exists an α < 1 with for all x, y, z ∈ X.Let f : X → Y be an odd mapping satisfying for all x, y, z ∈ X.Then there exists a unique additive mapping A : X → Y such that for all x ∈ X.
Proof.Note that f 0 0 and f −x −f x for all x ∈ X since f is an odd mapping.Putting y z 0 in 2.2 and replacing x by 2x, we get for all x ∈ X. Putting y x and z 0 in 2.2 , we have for all x ∈ X.By 2.4 and 2.5 , we get max ϕ 2x, 0, 0 , ϕ x, x, 0 .

2.6
Consider the set S : {h : X → Y } and introduce the generalized metric on S: where, as usual, inf φ ∞.It is easy to show that S, d is complete see 31 .Now we consider the linear mapping J : S → S such that for all x ∈ X.
Let g, h ∈ S be given such that d g, h λ.Then By Theorem 1.5, there exists a mapping A : X → Y satisfying the following.
1 A is a fixed point of J, that is, This implies that A is the unique mapping satisfying 2.12 such that there exists a μ ∈ 0, for all x ∈ X.
This implies that the inequalities 2.3 holds.
It follows from 2.1 and 2.2 that for all x, y, z ∈ X. Hence A : X → Y satisfying 1.4 .It follows from 2.1 and 2.6 that α n max ϕ 2x, 0, 0 , ϕ x, x, 0 0 2.16 for all x ∈ X.So A 2x 2A x for all x ∈ X. Hence A : X → Y is additive and we get the desired result.Corollary 2.2.Let θ be a positive real number and q a real number with 0 < q < 1.Let f : X → Y be an odd mapping satisfying 2.17 for all x, y, z ∈ X.Then there exists a unique additive mapping A : X → Y such that for all x ∈ X.
Proof.The proof follows from Theorem 2.1 by taking ϕ x, y, z θ x q y q z q for all x, y, z ∈ X.Then we can choose α |2| 1−q and we get the desired result.
Theorem 2.3.Let X be a non-Archimedean normed space and Y a complete non-Archimedean space.Let ϕ : X 3 → 0, ∞ be a function such that there exists an α < 1 with for all x, y, z ∈ X.Let f : X → Y be an odd mapping satisfying 2.2 .Then there exists a unique additive mapping A : X → Y such that for all x ∈ X.
Proof.Let S, d be the generalized metric space defined in the proof of Theorem 2.1.Now we consider the linear mapping J : S → S such that Jg x : 2g x/2 for all x ∈ X.
Replacing x by x/2 in 2.6 and using 2.19 , we have The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.Let θ be a positive real number and q a real number with q > 1.Let f : X → Y be an odd mapping satisfying 2.17 .Then there exists a unique additive mapping A : X → Y such that for all x ∈ X.
Proof.The proof follows from Theorem 2.3 by taking ϕ x, y, z θ x q y q z q for all x, y, z ∈ XThen we can choose α |2| q−1 and we get the desired result.
Theorem 2.5.Let X be a non-Archimedean normed space and Y a complete non-Archimedean space.Let ϕ : X 3 → 0, ∞ be a function such that there exists an α < 1 with for all x, y, z ∈ X.Let f : X → Y be an even mapping satisfying f 0 0 and 2.2 .Then there exists a unique quadratic mapping for all x ∈ X.
Proof.Putting y x and z 0 in 2.2 , we have for all x ∈ X. Substituting y z 0 and then replacing x by 2x in 2.2 , we obtain By 2.25 and 2.26 , we get

2.27
Consider the set S * {g : X → Y ; g 0 0} and the generalized metric d * in S * defined by where, as usual, inf φ ∞.It is easy to show that S * , d * is complete see 31 .Now we consider the linear mapping J : S * , d * → S * , d * such that Jg x : 1/4 g 2x for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.6.Let θ be a positive real number and q a real number with q > 1.Let f : X → Y be an even mapping satisfying f 0 0 and 2.17 .Then there exists a unique quadratic mapping for all x ∈ X.
Proof.The proof follows from Theorem 2.5 by taking ϕ x, y, z θ x q y q z q for all x, y, z ∈ X.Then we can choose α |4| q−1 and we get the desired result.Theorem 2.7.Let X be a non-Archimedean normed space and Y a complete non-Archimedean space.Let ϕ : X 3 → 0, ∞ be a function such that there exists an α < 1 with for all x, y, z ∈ X.Let f : X → Y be an even mapping satisfying f 0 0 and 2.2 .Then there exists a unique quadratic mapping Q : X → Y such that for all x ∈ X.
Proof.It follows from 2.27 that

2.32
The rest of the proof is similar to the proof of Theorems 2.1 and 2.5.
Corollary 2.8.Let θ be a positive real number and q a real number with 0 < q < 1.Let f : X → Y be an even mapping satisfying f 0 0 and 2.17 .Then there exists a unique quadratic mapping for all x ∈ X.
Proof.The proof follows from Theorem 2.7 by taking ϕ x, y, z θ x q y q z q for all x, y, z ∈ X.Then we can choose α |4| 1−q and we get the desired result.
Let f : X → Y be a mapping satisfying f 0 0 and 1.4 .Let f e x : Then f e is an even mapping satisfying 1.4 and f o is an odd mapping satisfying 1.4 such that f x f e x f o x .So we obtain the following.
Theorem 2.9.Let X be a non-Archimedean normed space and Y a complete non-Archimedean space.Let ϕ : X 3 → 0, ∞ be a function such that there exists an α < 1 with for all x, y, z ∈ X.Let f : X → Y be a mapping satisfying f 0 0 and 2.2 .Then there exist an additive mapping A : X → Y and a quadratic mapping

2.35
for all x ∈ X.

Stability of the Functional Equation 1.4 : A Direct Method
In this section, using direct method, we prove the Hyers-Ulam stability of the additive-quadratic functional equation 1.4 in non-Archimedean space.
Theorem 3.1.Let G be an additive semigroup and X a non-Archimedean Banach space.Assume that ζ : for all x, y, z ∈ G. Suppose that, for any x ∈ G, the limit exists and f : G → X is an odd mapping satisfying

3.3
Then the limit exists for all x ∈ G and defines an additive mapping A : G → X such that then A is the unique additive mapping satisfying 3.5 .
Proof.By 2.21 , we know that for all x ∈ G. Replacing x by x/2 n in 3.7 , we obtain Thus, it follows from 3.1 and 3.8 that the sequence 3.9 By induction on n, one can show that for all n ≥ 1 and x ∈ G. Indeed, 3.10 holds for n 1 by 3.7 .Now, if 3.10 holds for n, then by 3.8 , we have

3.11
By taking n → ∞ in 3.10 and using 3.2 , one obtains 3.5 .By 3.1 and 3.3 , we get for all x, y, z ∈ X.Therefore, the mapping A : G → X satisfies 1.4 .
To prove the uniqueness property of A, let L be another mapping satisfying 3.5 .Then we have for all x ∈ G. Therefore, A L. This completes the proof.Corollary 3.2.Let ξ : 0, ∞ → 0, ∞ be a function satisfying for all t ≥ 0. Assume that κ > 0 and f : G → X is a mapping with f 0 0 such that for all x, y, z ∈ G. Then there exists a unique additive mapping A : G → X such that for all x, y, z ∈ G.The last equality comes from the fact that |2|ξ |2| −1 < 1.On the other hand, it follows that exists for all x ∈ G. Also, we have lim

3.19
Thus, applying Theorem 3.1, we have the conclusion.This completes the proof.
Theorem 3.3.Let G be an additive semigroup and X a non-Archimedean Banach space.Assume that ζ : for all x, y, z ∈ G. Suppose that, for any x ∈ G, the limit exists and f : G → X is an odd mapping satisfying 3.3 .Then the limit A x : lim n → ∞ f 2 n x /2 n exists for all x ∈ G and then A is the unique mapping satisfying 3.22 .
Proof.By 2.6 , we get for all x ∈ G. Replacing x by 2 n x in 3.24 , we obtain Thus it follows from 3.20 and 3.25 that the sequence {f 2 n x /2 n } n≥1 is convergent.Set

3.26
On the other hand, it follows from 3.25 that for all x ∈ G and p, q ≥ 0 with q > p ≥ 0. Letting p 0, taking q → ∞ in the last inequality, and using 3.21 , we obtain 3.22 .The rest of the proof is similar to the proof of Theorem 3.1.This completes the proof.
Theorem 3.4.Let G be an additive semigroup and X a non-Archimedean Banach space.Assume that ζ : for all x, y, z ∈ G. Suppose that, for any x ∈ G, the limit exists and f : G → X is an even mapping satisfying f 0 0 and 3.3 .Then the limit A x : lim n → ∞ 4 n f x/2 n exists for all x ∈ G and defines a quadratic mapping Q : G → X such that f x − Q x X ≤ Θ x .

3.30
Moreover, if then Q is the unique additive mapping satisfying 3.30 .
Proof.It follows from 2.27 that Replacing x by x/2 n in 3.32 , we have 3.33 It follows from 3.28 and 3.32 that the sequence {4 n f x/2 n } n≥1 is Cauchy sequence.The rest of the proof is similar to the proof of Theorem 3.1.
Similarly, we can obtain the following.We will omit the proof.exists and f : G → X is an even mapping satisfying f 0 0 and 3.3 .Then the limit Q x : lim n → ∞ f 2 n x /4 n exists for all x ∈ G and for all x ∈ G.Moreover, if x, 2 k x, 0 ; j ≤ k < n j 0, 3.37 then Q is the unique mapping satisfying 3.36 .
Let f : X → Y be a mapping satisfying f 0 0 and 1.4 .Let f e x : f x f −x /2 and f o x f x − f −x /2.Then f e is an even mapping satisfying 1.4 and f o is an odd mapping satisfying 1.4 such that f x f e x f o x .So we obtain the following.