Generalized Hyers-ulam Stability of Quadratic Functional Equations: a Fixed Point Approach

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation f2x y 4fx fy fx y − fx − y in Banach spaces.


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 partial answer to the question of Ulam for Banach spaces.Hyers' theorem was generalized by Aoki 3 for additive mappings and by Rassias 4 for linear mappings by considering an unbounded Cauchy difference.The paper of Rassias 4 has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations.A generalization of the Rassias theorem was obtained by Gȃvrut ¸a 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.
The functional equation We recall the following theorem by Diaz and Margolis.
Theorem 1.1 see 18 .Let X, d be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 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 this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of the following quadratic functional equation: in Banach spaces.Throughout this paper, assume that X is a normed vector space with norm • and that Y is a Banach space with norm • .
In 1996, Isac and Rassias 19 were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.

Fixed points and generalized Hyers-Ulam stability of quadratic functional equations
For a given mapping f : X → Y , we define for all x, y ∈ X.
for all x, y ∈ X.It follows from 2.2 and 2.4 that for all x, y ∈ X.So the mapping f : for all x, y ∈ X, the mapping f : X → Y satisfies 2.2 .
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation Cf x, y 0.

Fixed Point Theory and Applications
Theorem 2.2.Let f : X → Y be a mapping with f 0 0 for which there exists a function ϕ : X 2 → 0, ∞ such that there exists an L < 1 such that ϕ x, 0 ≤ 4Lϕ x/2, 0 for all x ∈ X, and Cf x, y ≤ ϕ x, y 2.9 for all x, y ∈ X.Then there exists a unique quadratic mapping Q : X → Y satisfying 2.2 and for all x ∈ X.
Proof.Consider the set and introduce the generalized metric on S as follows: Letting y 0 in 2.9 , we get for all x ∈ X. Hence d f, Jf ≤ 1/4.By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following.1 Q is a fixed point of J, that is, for all x ∈ X.The mapping Q is a unique fixed point of J in the set This implies that Q is a unique mapping satisfying 2.17 such that there exists K ∈ 0, ∞ satisfying This implies that the inequality 2.10 holds.It follows from 2.8 , 2.9 , and 2.20 that for all x, y ∈ X.So CQ x, y 0 for all x, y ∈ X.By Proposition 2.1, the mapping Q : X → Y is quadratic.
Therefore, there exists a unique quadratic mapping Q : X → Y satisfying 2.2 and 2.10 , as desired.
Corollary 2.3.Let p < 2 and θ ≥ 0 be real numbers, and let f : X → Y be a mapping such that Cf x, y ≤ θ x p y p 2.23 for all x, y ∈ X.Then there exists a unique quadratic mapping Q : X → Y satisfying (2.2) and for all x ∈ X.
Proof.The proof follows from Theorem2.2 by taking ϕ x, y : θ x p y p 2.25 for all x, y ∈ X.Then we can choose L 2 p−2 and we get the desired result.

Fixed Point Theory and Applications
Remark 2.4.Let f : X → Y be a mapping for which there exists a function ϕ : X 2 → 0, ∞ satisfying 2.9 and f 0 0 such that for all x, y ∈ X.By a similar method to the proof of Theorem 2.2, one can show that if there exists an L < 1 such that ϕ x, 0 ≤ 1/4 Lϕ 2x, 0 for all x ∈ X, then there exists a unique quadratic mapping Q : X → Y satisfying 2.2 and for all x ∈ X.
For the case p > 2, one can obtain a similar result to Corollary 2.3 Theorem 2.5.Let f : X → Y be an even mapping f 0 0 for which there exists a function ϕ : X 2 → 0, ∞ satisfying 2.8 and 2.9 such that there exists an L < 1 such that ϕ x, −x ≤ 4Lϕ x/2, −x/2 for all x ∈ X.Then there exists a unique quadratic mapping Q : X → Y satisfying 2.2 and for all x ∈ X.
Proof.Consider the set and introduce the generalized metric on S as follows: Therefore, there exists a unique quadratic mapping Q : X → Y satisfying 2.2 and 2.28 , as desired.
Corollary 2.6.Let p < 1 and θ ≥ 0 be real numbers, and let f : X → Y be an even mapping such that Cf x, y ≤ θ• x p • y p 2.41 for all x, y ∈ X.Then there exists a unique quadratic mapping Q : X → Y satisfying 2.2 and 2.42 for all x ∈ X.
Fixed Point Theory and Applications still true if the relevant domain X is replaced by an Abelian group.Czerwik 8 proved the generalized Hyers-Ulam stability of the quadratic functional equation and Park 9 proved the generalized Hyers-Ulam stability of the quadratic functional equation in Banach modules over a C * -algebra.The stability problems of several functional equations have been extensively 1 is called a quadratic functional equation.In particular, every solution of the quadratic functional equation is said to be a quadratic function.A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof 6 for mappings f : X → Y , where X is a normed space and Y is a Banach space.Cholewa 7 noticed that the theorem of Skof is By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following.1 Q is a fixed point of J, that is, for all x ∈ X. Hence d f, Jf ≤ 1/4.forallx ∈ X.3 d f, Q ≤ 1/ 1 − L d f, Jf, which implies the inequality for all x, y ∈ X.So CQ x, y 0 for all x, y ∈ X.By Proposition 2.1, the mapping Q : X → Y is quadratic.