Stability of generalized quadratic functional equation on a set of measure zero

Abstract In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equation where E is a real (or complex) vector space. This result was used to demonstrate the Hyers-Ulam stability on a set of Lebesgue measure zero for the same functional equation.


Introduction
The concept of the stability for functional equations was introduced for the first time by S.M. Ulam in 1940 [33]. Ulam started the stability by the following question Given a group G, a metric group (G , d), a number δ > 0 and a mapping f : G → G which satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G, does there exist an homomorphism h : G → G and a constant γ > 0, depending only on G and G such that d(f (x), h(x)) ≤ γδ for all x in G?
In 1941, Ulam's problem for the case of approximately additive mappings was solved by D.H. Hyers [16] on Banach spaces. In 1950 T. Aoki [3] provided a generalization of the Hyers theorem for additive mappings and in 1978 Th.M. Rassias [30] generalized the Hyers theorem for linear mappings by considering an unbounded Cauchy difference. For more information on the concept of the stability of functional equations see, for example [6,12,15,17,18,19,22,24].
The stability problems of several functional equations on a restricted domain have been extensively investigated by a number of authors, for example [5,11,13,21,29,31].
It is very natural to ask if the restricted domain D = {(x, y) ∈ E 2 : x + y ≥ d} can be replaced by a much smaller subset Ω ⊂ D, i.e. a subset of measure zero in a measurable space E.
In 2013, J. Chung in [10] answered to this question by considering the stability of the Cauchy functional equation In 2014, J. Chung and J.M. Rassias [11] proved the stability of the quadratic functional equation in a set of measure zero. In 2015, M. Almahalebi in [2], proved the Hyers-Ulam stability for the Drygas functional equation for all (x, y) ∈ Ω, where Ω ⊂ R 2 is of Lebesgue measure zero.
Throughout this paper, let E be a real (or complex) vector space and F be a real (or complex) Banach space.
Our aim is to prove the Hyers-Ulam stability on a set of Lebesgue measure zero for the K-quadratic functional equation where f : E → F are applications and K is a finite subgroup of the group of automorphisms of E and card K = L. These results are applied to the study of an asymptotic behaviour of this functional equation.

Notations and preliminary results
In this section, we need to introduce some notions and notations. A function A : E → F between two vector spaces E and F is said to be additive if A(x + y) = A(x) + A(y) for all x, y ∈ E. In this case, it is easily seen that A(rx) = rA(x) for all x ∈ E and all r ∈ Q.
Generalized quadratic functional equation

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Let k ∈ N * \ {1} and A : E k → F be a function, then we say that A is k-additive if it is additive with respect to each variable. In addition, we say that A is symmetric if . , x k ∈ E and σ is a permutation of (1, 2, . . . , k).
Let k ∈ N * \{1} and A : E k → F be symmetric and k-additive and let In this way, a function A k : E → F which satisfies for all λ ∈ Q and x ∈ E A k (λx) = λ k A k will be called a rational-homogeneous form of degree k (assuming Let F E denote the vector space (over a field K) consisting of all maps from E into F . For h ∈ E, define the linear difference operator ∆ h on F E by

Y. Aribou, H. Dimou, Ab. Chahbi and S. Kabbaj
Then Since for all β ∈ K, we study two different cases.
By a similar calculation to the previous case, we obtain Consequently, we get Now, in view of (2), (3) and (4) we have Generalized quadratic functional equation which ends the proof.
Then there exists a unique (GP) function p : E → F of degree at most L such that p is a solution of (1) and Proof. According to (5), we have Replacing u by u + v, we get Using (6) and (7), we obtain Then by [19,Theorem II] there exists a (GP) function q : E → F of degree at most L such that For 0 ≤ k ≤ L, there is a rational-homogeneous form A k : E → F of degree k such that [154] Y. Aribou, H. Dimou, Ab. Chahbi and S. Kabbaj By (5) and (8), we get for all x, y ∈ E. Now (9) says, in light of (10), that for all x, y ∈ E, Replacing x by rx and y by ry in (11), where r ∈ Q, we conclude that for all x, y ∈ E. By continuity, (12) holds for all real r and all x, y ∈ E. Now suppose that φ : F → R is a continuous linear functional. Then by (12), we get for all x, y ∈ E and all r ∈ R.
Since a real polynomial function is bounded if and only if it is constant, from the last inequality we surmise that, for 1 ≤ j ≤ L, Generalized quadratic functional equation

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for all x, y ∈ E. As φ : F → R is arbitrary continuous linear functional, by the Hahn-Banach theorem, Let p(x) = q(x) − q(0), then p is a (GP) function of degree at most L and by (13) it is a solution of equation (1) k∈K Finally, by using (8) and (14), we get Let p be another (GP) function solution of (1) of degree at most L such that A similar proof to that in [19, Theorem III] yields p = p .
Then there exists a unique (GP) function p : E → F of degree at most L such that p is a solution of (1) and Proof. By posing f = f − f (0), it is easy to show that f satisfies First, we observe that From the above inequalities (16) and (17), we have By Theorem 3.2, the result follows.

Stability of equations (1) on a set of measure zero
Let E be a vector space and F be a real (or complex) Banach space. For given x, y, t ∈ E and a finite subgroup K of the group of automorphisms of E, we define Let Ω ⊂ E 2 . Throughout this section we assume that Ω satisfies the condition: For given x, y ∈ E there exists t ∈ E such that P x,y,t ⊂ Ω. (C) In the following, we prove the Hyers-Ulam stability theorem for the generalized quadratic functional equation (15) in Ω.
Since Ω satisfies (C), for given x, y ∈ E, there exists t ∈ E such that Thus, we have Generalized quadratic functional equation

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Using the triangle inequality, we get This implies that Next, according to Theorem 3.3, there exists a unique generalized polynomial (GP) q : E → F of degree at most L such that This completes the proof.

Corollary 4.3
Let ε ≥ 0 be fixed. Suppose that f : E → F satisfies the functional inequality where σ : E → F is an involution. Then there is a unique quadratic Q : E → F , and an additive A : Proof. By taking L = 2 and K = {I, σ} in Theorem 4.1, there exists a unique generalized polynomial (GP) of degree at most 2 which is a solution of the following functional equation We use [Theorem 6, [25]] to complete the proof.

Applications
In this section, we construct a set of measure zero satisfying the condition (C) for E = R. From now on, we identify R 2 with C. Using K = {I}, respectively K = {I, −I} for R. The following lemma is a crucial key of our construction [28,Theorem 1.6]. The set R of real numbers can be partitioned as R = F ∪ K, where F is of the first Baire category, i.e. F is a countable union of nowhere dense subsets of R, and K is of Lebesgue measure zero.
The following lemma was proved by J. Chung and J.M. Rassias in [10] and [11].
Lemma 5.2 Let K be a subset of R of measure zero such that K = R\{K} is of first Baire category. Then, for any countable subsets U ⊂ R, V ⊂ R\{0} and M > 0, there exists t ≥ M such that In the following theorem, we give the construction of a set Ω of Lebesgue measure zero.
where K is a subset of R of measure zero such that K = R\{K} is of the first Baire category. Then Ω satisfies the condition (C) and is of two-dimensional Lebesgue measure zero.
Proof. By the construction of Ω, the condition (C) is equivalent to the fact that for every x, y ∈ R, there exists t ∈ R such that exp −π 6 i P x,y,t ⊂ K × K.
The inclusion (19) is equivalent to S x,y,t := It is easy to check that the set S x,y,t is contained in a set of form U + tV . We consider two cases In this case we find the functional equation of Cauchy Generalized quadratic functional equation In this case we find the quadratic functional equation By Lemma 5.2, for given x, y ∈ R and M > 0 there exists α ≥ M such that Thus, Ω satisfies (C). This completes the proof.
As a consequence of Theorem 4.1 and corollary, we obtain the asymptotic behaviour of f satisfying the asymptotic condition,then there exists a sequence δ n monotonically decreasing to 0 such that Corollary 5.5 Suppose that f : R → F satisfies the condition (20). Then, f is a generalized quadratic functional.
Proof. The condition (20) implies that, for each n ∈ N, there exists d n > 0 such that for all (x, y) ∈ Ω dn . From previous corollary, Ω dn := {(x, y) ∈ Ω : |x| + |y| ≥ d n } satisfies (C). Thus, by Theorem 4.1, there exists a unique generalized polynomial for all x ∈ R. By replacing n ∈ N by m ∈ N in (22) and using the triangle inequality, we have for all x ∈ R. For all n 1 , n ∈ N and x ∈ R, we have necessarily q n = q n1 + q n (0) − q n1 (0). Since q n (0) = q n1 (0) = 0, we have in (23) q n = q n1 . Now, letting n → ∞ in (22), we get the result. This completes the proof. If we define Ω ⊂ R 2n as an appropriate rotation of 2n-product K 2n of K, then Ω has 2n-dimensional measure zero and satisfies (C). Consequently, we obtain the following.
Corollary 5.6 Let F be a Banach space. Suppose that f : R n → F satisfies the functional inequality k∈K f (x + k · y) − Lf (x) − Lf (y) ≤ ε for all (x, y) ∈ Ω. Then there exists a unique quadratic mapping q : R n → F such that for all x ∈ R n .