ijpam.eu GENERALIZED HYERS–ULAM STABILITY OF REFINED QUADRATIC FUNCTIONAL EQUATIONS

In this paper, we give a general solution of a refined quadratic functional equation and then investigate its generalized Hyers–Ulam stability in quasi-normed spaces and in non-Archimedean normed spaces. AMS Subject Classification: 39B82, 39B62


Introduction
In [28], S.M. Ulam proposed the stability problem for functional equations concerning the stability of group homomorphisms.A functional equation is called stable if any approximate solution to the functional equation is near a true solution of that functional equation.In [12], D.H. Hyers considered the case of approximate additive mappings with the Cauchy difference controlled by a positive constant in Banach spaces.D.G. Bourgin [5] and T. Aoki [2] treated this problem for approximate additive mappings controlled by unbounded functions.In [23], Th.M. Rassias provided a generalization of Hyers' theorem for linear mappings which allows the Cauchy difference to be unbounded.In 1994, P. Gǎvruta [9] generalized these theorems for approximate additive mappings controlled by the unbounded Cauchy difference with regular conditions.During the last three decades a number of papers and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability and generalized Hyers-Ulam stability to a number of functional equations and mappings [1,6,8,13,22].
A stability problem of Ulam for the quadratic functional equation was first proved by F. Skof for mapping f : , where E 1 is a normed space and E 2 is a Banach space [26].In the paper [7], S. Czerwik proved the Hyers-Ulam-Rassias stability of the quadratic functional equation (1).In particular, J.M. Rassias [19,20,21] solved the stability problem of Ulam for the Euler-Lagrange type quadratic functional equation for fixed real numbers r, s with r = 0, s = 0.In particular, P.L. Kannappan [15] introduced the following functional equation and proved that a function on a real vector space is a solution of (3) if and only if there exist a symmetric biadditive function B and an additive function A such that f (x) = B(x, x) + A(x).In [3], the authors proved the generalized Hyers-Ulam stability of the functional equation which is equivalent to the quadratic equation (1).Recently, Zivari-Kazempour and M. Eshaghi Gordji [29] proved the general solution of the following quadratic functional equation and investigated the Hyers-Ulam stability of the equation ( 5) in Banach space.
In this paper, we consider the following functional equation where n is any fixed nonzero integer, and then investigate the Hyers-Ulam stability of the equation (6).

General Solution
First, we remark that the equation ( 6) is equivalent to (4) for the case n = −1 [3], and the equation ( 6) is trivially equivalent to (3) for the case n = 1.Thus we give the general solution of the equation ( 6) for the case n = −1, 0, 1 in the following Theorem 2.3.
Lemma 1.Let X and Y be vector spaces and f : X → Y be an even function satisfying the functional equation (6).Then f is quadratic.
Associating the last equation with (8), we get f ≡ 0.
Theorem 3. Let f : X → Y be a function satisfying the functional equation ( 6).Then f is quadratic and so ( 6) is equivalent to (1) for the case n = −1, 0, 1.
is even and is odd.Thus one can easily find that f e and f o satisfy the equation ( 6).Therefore, f o ≡ 0 and so f = f e is quadratic and so the equation is equivalent to (1).

The Hyers-Ulam Stability in Quasi-Banach Spaces
In this section, we investigate the generalized Hyers-Ulam stability problem for the functional equation ( 6) in quasi-Banach space.First, we introduce some basic information concerning quasi-Banach spaces which are referred in [4] and [25].Let X be a linear space.A quasi-norm is a real-valued function on X satisfying the following: (i) x ≥ 0 for all x ∈ X, and x = 0 if and only if x = 0; (ii) λx = |λ| x for any scalar λ and all x ∈ X; (iii) There is a constant M ≥ 1 such that x + y ≤ M ( x + y ) for all x, y ∈ X.
The pair (X, The smallest possible M is called the modulus of concavity of the quasi-norm • .A quasi-Banach space is a complete quasi-normed space.A quasi-norm • is called a q-norm(0 < q ≤ 1) if x + y q ≤ x q + y q for all x, y ∈ X.In this case, a quasi-Banach space is called a q-Banach space.Let X be a quasi-Banach space.Given a q-norm, the formula d(x, y) := x − y q gives us a translation invariant metric on X.By Aoki-Rolewicz Theorem [25] (see also [4]), each quasi-norm is equavalent to some q-norm.Since it is much easier to work with q-norms than quasi-norms, here and subsequently, we restrict our attention mainly to q-norms.Moreover, generalized stability theorems of functional equations in quasi-Banach spaces have been investigated by a lot of authors [14,18,27].Now we introduce an abbreviation D n f for a given mapping f : X → Y as follows: for all x, y, z ∈ X, where n = −1, 0, 1 is a fixed integer.From now on, let X be a normed linear space with norm • and Y be a q-Banach space with norm • .In this part, by using an direct method, we prove the stability theorem of the equation (6).
for all x, y, z ∈ X. Suppose that a mapping f : X → Y with f (0) = 0 satisfies the inequality for all x, y, z ∈ X.Then there exists a unique quadratic mapping for all x ∈ X.
Proof.Replacing (x, y, z) by (x, 0, 0) in ( 24), we have for all x ∈ X. Replacing x by n k x in (26) and then dividing both sides by n 2k+2 , we get for all x ∈ X and all integers k ≥ 0. Then for any integers m, k with m ≥ k ≥ 0, we obtain for all x ∈ X.Thus the sequence is Cauchy by (23).Since Y is complete, this sequence converges for all x ∈ X.So one can define a mapping It follows from ( 23) and ( 28) that for all x, y, z ∈ X.Hence, the mapping Q satisfies ( 6) and so it is quadratic.Putting k := 0 and letting m go to infinity in (27), we see that (25) holds.For the uniqueness of Q, assume that there exists a quadratic mapping Q ′ : X → Y satisfying the inequality (25).Then, we find that for all x ∈ X, which proves the uniqueness.
Corollary 7. Let α, a 1 , a 2 , a 3 be positive real numbers such that either a i > 2 or a i < 2 for all i ∈ {1, 2, 3}.Suppose that a mapping f : X → Y with f (0) = 0 satisfies the inequality for all x, y, z ∈ X.Then there exists a unique quadratic mapping

The Hyers-Ulam Stability in Non-Archimedean Spaces
Hensel [11] has introduced a normed space which does not have the non-Archimedean spaces property.During the last three decades, the theory of non-Archimedean spaces has gain the interest of physicists for their research in problems coming from quantum physics, p-adic strings and superstrings [16].
A Let X be a vector space over a field K with a non-Archimedean valuation | • |.A function • : X → [0, ∞) is said to be a non-Archimedean norm on X if it satisfies the following conditions In this case (X, • ) is called a non-Archimedean normed space.Because of the fact a sequence {x m } is Cauchy in the non-Archimedean normed space if and only if {x m+1 − x m } converges to zero with respect to the non-Archimedean norm.By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.[10,24]).Let X be a vector space and Y be a non-Archimedean Banach space.In the following, we now prove the generalized Hyers-Ulam stability of quadratic functional equation ( 6) over the non-Archimedean space.As corollaries, we obtain especially stability result over the p-adic field Q p .To avoid trivial case, we assume |n| < 1.
lim j→∞ |n| 2j ψ(n −j x, n −j y, n −j z) = 0, resp for all x, y, z ∈ X and the limit Ψ(x) ≡ lim k→∞ max |n| 2j ψ(n −j x, 0, 0) : 1 ≤ j ≤ k , resp exists for each x ∈ X. Suppose that a mapping f : X → Y with f (0) = 0 satisfies the inequality D n f (x, y, z) ≤ ψ(x, y, z), resp , for all x, y, z ∈ X.Then there exists a quadratic mapping Proof.Replacing (x, y, z) by (x, 0, 0) in (31), we have for all x ∈ X. Replacing x by n k x in (34) and then dividing both sides by |n| 2k+2 , we get for all x ∈ X.It follows from ( 35) and ( 29) that the sequence f for all x ∈ X.Using induction, one can show that for all k ∈ N and all x ∈ X.By taking k to approach infinity in (36) and using (30), one obtains (32).Replacing x, y and z by n 2k x,n 2k y and n 2k z, respectively, in (31), we get for all x, y, z ∈ X. Taking the limit as k → ∞ and using Theorem 3, we conclude that Q is quadratic.Moreover, to prove the uniqueness, we assume that there exists a quadratic mapping Q ′ : X → Y satisfying (32) and (33).
Then we figure out : m ≤ j < m + k} = 0 for all x ∈ X.This completes the proof.
Then there exists a unique quadratic mapping Q : X → Y such that for all x ∈ X.
Corollary 12. Let r = 2 and ε, θ be positive numbers, where ε = 0 if r > 2. Suppose that a mapping f : Q p → Q p with f (0) = 0 satisfies the inequality valuation is a function | • | from a field K to [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| • |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|, ∀r, s ∈ K.A field K is called a valued field if K equips with a valuation.The usual absolute values of R and C are examples of valuations.Alternatively, if the triangle inequality is replaced by the strong triangle inequality |r + s| ≤ max {|r|, |s|}, ∀r, s ∈ K, then the valuation | • | is called a non-Archimedean valuation, and the field is called a non-Archimedean field.Cleary |1| = |−1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | • | taking everything except for 0 into 1 and |0| = 0. Definition 8.

Example 9 .
Let p be a prime number.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 −nx is a non-Archimedean norm on Q p and it makes Q p a locally compact field (see
|D p f (x, y, z)| p ≤ ε + θ |x| r p + |y| r p + |z| r p (x, y, z ∈ Q p ).Then there exists a unique quadratic mappingQ : Q p → Q p such that |f (x) − Q(x)| p ≤ ε + p r θ|x| r p if r < 2 p 2 θ|x| r p if r > 2 for all x ∈ Q p .