On the Stability of Quadratic Functional Equations

The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki 3 for additive mapping and by Th. M. Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias 4 has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. Th. M. Rassias 5 during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda 6 , following the same approach as in 4 , gave an affirmative solution to this question for p > 1. It was shown by Gajda 6 as well as by Rassias and Šemrl 7 that one cannot prove a Th.M. Rassias’ type theorem when p 1. J. M. Rassias 8 , following the spirit of the innovative approach of Th. M. Rassias 4 for the unbounded Cauchy difference, proved a similar stability theorem in which he replaced the factor ‖x‖p ‖y‖p by ‖x‖p · ‖y‖q for p, q ∈ R with p q / 1. The functional equation f x y f x − y 2f x 2f y 1.1


Introduction
The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki 3 for additive mapping and by Th. M. Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias 4 has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. Th. M. Rassias 5 during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda 6 , following the same approach as in 4 , gave an affirmative solution to this question for p > 1. It was shown by Gajda 6 as well as by Rassias andŠemrl 7 that one cannot prove a Th.M. Rassias' type theorem when p 1. J. M. Rassias 8 , following the spirit of the innovative approach of Th. M. Rassias 4 for the unbounded Cauchy difference, proved a similar stability theorem in which he replaced the factor x p y p by x p · y q for p, q ∈ R with p q / 1. The functional equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A Hyers-Ulam-Rassias stability problem for the quadratic functional equation was proved by Skof 9 for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa 10 noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. In 11 , Czerwik proved the Hyers-Ulam-Rassias stability of the quadratic functional equation. Several functional equations have been investigated in 12-17 . Throughout this paper, assume that k is a fixed positive integer.
In this paper, we solve the functional equation and prove the Hyers-Ulam-Rassias stability of the functional equation 1.2 in Banach spaces.

Hyers-Ulam-Rassias stability of the quadratic functional equation
for all x, y ∈ X. So the mapping f : x 2f y for all x, y ∈ X. We prove 2.1 for k j by induction on j. For the case j 1, 2.1 holds by the assumption. For the case j 2, since for all x, y ∈ X, then 2.1 holds.
Jung Rye Lee et al. 3 Assume that 2.1 holds for j n − 2 and j n − 1 2 < n ≤ k . By the assumption, for all x, y ∈ X, 2.1 holds for j n. Hence the mapping f : X → Y satisfies 2.1 for j k.
From now on, assume that X is a normed vector space with norm || · || and that Y is a Banach space with norm · .
For a given mapping f : for all x, y ∈ X. Now we prove the Hyers-Ulam-Rassias stability of the quadratic functional equation Df x, y 0.
for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that for all x ∈ X.
Proof. Letting y 0 in 2.9 , we get for all x ∈ X. Hence which tends to zero as n → ∞ for all x ∈ X. So we can conclude that Q x T x for all x ∈ X. This proves the uniqueness of Q. So there exists a unique quadratic mapping Q : X → Y satisfying 2.10 . for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that for all x ∈ X.
Proof. The proof follows from Theorem 2.2 by taking ϕ x, y : θ ||x|| p ||y|| p 2.19 for all x, y ∈ A.
Theorem 2.4. Let f : X → Y be a mapping with f 0 0 for which there exists a function ϕ : X 2 → 0, ∞ satisfying 2.9 such that ϕ x, y : for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that for all x ∈ X.
Jung Rye Lee et al.

5
Proof. It follows from 2.11 that for all x ∈ X.
Hence for all x, y ∈ A.
From now on, assume that k 2.
Theorem 2.6. Let f : X → Y be a mapping with f 0 0 for which there exists a function ϕ : X 2 → 0, ∞ satisfying 2.9 such that ϕ x, y : for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that for all x ∈ X.