HYERS-ULAM STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS ON DIVISIBLE SQUARE-SYMMETRIC GROUPOID

Let (X, ⋄) be a divisible square-symmetric groupoid, and (Y, ∗, d) a complete metric divisible square-symmetric groupoid. In this paper, we obtain the Hyers-Ulam stability problem of functional inequality d(f(x⋄y)∗f(x⋄y), σ∗(f(x)∗f(y))) ≤ ε(x, y) for approximate mapping f : X → Y of functional equation h(x ⋄ y) ∗ h(x ⋄ y) = σ∗(h(x) ∗ h(y)) differing by ε : X → [0,∞). AMS Subject Classification: 39B52, 39B72, 47H09


Introduction
In 1940, S.M. Ulam [19] proposed the Hyers-Ulam stability problem for functional equations concerning the stability of group homomorphisms : Under what condition does there exist a homomorphism near an approximately homomorphism?
In 1941, D.H. Hyers [9] considered the case of approximate additive mappings with the Cauchy difference controlled by a positive constant in Banach spaces.Thereafter we usually say that the equation E 1 (h) = E 2 (h) has the Hyers-Ulam stability if for an approximate solution f of this equation, i.e., for a function f with |E 1 (f ) − E 2 (f )| ≤ δ, there exists a function g such that E 1 (g) = E 2 (g) and |f (x) − g(x)| ≤ ε.
D.G. Bourgin [3] and T. Aoki [2] treated this problem for approximate additive mappings controlled by unbounded function.In [17], Th.M. Rassias provided a generalization of Hyers' theorem for linear mappings which allows the Cauchy difference to be unbounded.In 1994, P. Gǎvruta [8] 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 [1,5,7,10,16].
The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is said to von-Nuemann functional equation.In particular, every solution of the von-Nuemann functional equation is said to be a quadratic function.The Hyers-Ulam stability of von-Nuemann functional equation was proved by Skof [18] for mapping f : E 1 → E 2 , where E 1 is a normed space and E 2 is a Banach space.For more information of stability of von-Neumann functional equation, we refer [4,6,12,15].
Let (X, ⋄), (Y, * ) be groupoids with binary operations.If the binary operation ⋄ satisfies the following identity, the operation ⋄ will be called square-symmetric.Note that the square-symmetric of ⋄ implies that σ ⋄ (x) := x ⋄ x is an endomorphism.Thus we remark that for all n ∈ N. It is an crucial relation when the mapping σ n ⋄ is an automorphism of (X, ⋄), that is, when σ ⋄ is bijective.Then, for each y ∈ X the equation σ ⋄ (x) = y can be solved uniquely with respect to x ∈ X.Such an element x is denoted by x = σ −1 ⋄ (y).Then it is easy to see that σ −1 ⋄ : X → X is an endomorphism.A binary operation ⋄ such that σ ⋄ is an automorphism of (X, ⋄) is called divisible, the corresponding groupoid is said to be a divisible groupoid.
We remember that the Lipschitz modulus Lip(φ) for a function φ : Y → Y on the space Y with metric d defined by Then it follows that Lip(φ • ψ) ≤ Lip(φ)Lip(ψ).
Over the setting of square-symmetric groupoid, G.H. Kim [11] obtain the Hyers-Ulam stability of the following functional inequality d(f This paper aims to investigate the Hyers-Ulam stability for the following functional equation of f : X → Y : A particular case of ( 1) is the von-Nuemann functional equation when X, Y are semigroups with the operation +.Thus we will have the generalized stability results of von-Neumann functional equation.

The Hyers-Ulam Stability of Functional Equation (1) via Direct Method
Theorem 1.Let (X, ⋄) be a divisible square-symmetric groupoid and let (Y, * , d) be a complete metric divisible square-symmetric groupoid.Assume that a mapping f : for all x, y ∈ X with f (id X ) = id Y and ε : for all x, y ∈ X.Then for every x ∈ X, converges and h : X → Y is the solution of (1) such that for all x ∈ X.Moreover, if {Lip(σ −2n * )Lip(σ 2n * )} n≥1 is bounded then h is the only solution of (1) such that the mapping for all positive integer n ∈ N. First, we would like to prove that for each fixed x ∈ X the sequence {f n (x)} is convergent.Replacing x and y by σ n ⋄ (x) in equality (2), we get Thus we obtain It produce the following inequality that for positive integers m, n with m, n > 0, which shows that the sequence {f n (x)} is Cauchy and thus converges in Y .Since Y is complete, we can define a function h : X → Y as follows for all x ∈ X.Now, for n ∈ N, we have for all x ∈ X. Taking n → ∞, we get (4) for the limit function h of {f n }.
Next, we wil prove that h satisfies (1).Replacing x and y by σ n ⋄ (x) and σ n ⋄ (y) in ( 2) and using the endomorphism of σ ⋄ , we get for all x, y ∈ X.Since σ * is an automorphism of (Y, * ), σ −2n * is also automorphism for all n.Therefore, for all x, y ∈ X. Letting n → ∞ in the last inequality, we get that h satisfies (4).Finally we complete the proof by showing the uniqueness.Assume that h ′ : X → Y is an arbitrary solution of (1) such that the mapping x → d(f (x), h ′ (x)) is bounded with bound in (4).Putting x = y in (1), we have that Using induction, we get Hence, for every x ∈ X, we get by (4), Since {Lip(σ −2n * )Lip(σ 2n * )} is bounded, taking the limit as n → ∞ in the above inequality, we obtain h ′ (x) = h(x) for all x ∈ X.This completes the proof.Theorem 2. Let (X, ⋄) be a divisible square-symmetric groupoid and let (Y, * , d) be a complete metric divisible square-symmetric groupoid.Assume that a mapping f : for all x, y ∈ X with f (id X ) = id Y and ε : for all x, y ∈ X.Then for every x ∈ X, converges and h : X → Y is the solution of (1) such that for all x ∈ X.

Moreover, if {Lip(σ −2n
for all positive integer n ∈ N. Let x ∈ X be fixed.Replacing x and y by σ −n ⋄ (x) in (7), we obtain Thus we figure out by ( 10) for all x ∈ X.Hence, it follows by the above convergence that for all n, m ∈ N which shows that the sequence {f n (x)} is Cauchy in Y. Therefore, a mapping h : X → Y given by ( 8) is well defined.For all n ∈ N, we have Letting n → ∞, we obtain (8) for the limit function h of {f n }.Now we prove that h satisfies (1).Putting σ −n ⋄ (x) and σ −n ⋄ (y) into x and y in (7) and using the endomorphism of σ n ⋄ , we obtain Whence, taking the limit as n → ∞ in the last inequality, we get that f is a homomorphism, that is, it satisfies (1).The proof of the uniqueness of h is now based on the identity In fact, let {Lip(σ 2n * )Lip(σ −2n * )} be bounded and assume that f ′ is another solution of (1) such that the mapping x → d(f (x).h ′ (x)) is bounded with bound (9).Then for every x ∈ X we get by ( 9) and ( 10) )} is bounded.This proves the uniqueness of h.The proof of the theorem is complete.

The Hyers-Ulam Stability of Functional Equation (1) via Fixed Point Theorem
In this section, we will mention about the stability of the functional equation (1) using another method.First, we give the definition of a generalized metric on a set X. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following properties (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ Y .
In [13], Margolis and Diaz constructed the variable method using a fixed point theory, which is variously applied to the theory of functional equations.
Theorem 3. (The Alternative Fixed Point [13]) 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 d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n 0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n 0 ; (2) the sequence {J n x} converges to a fixed point y * of J; (3) y * is the unique fixed point of J in the set Y = {y ∈ X | d(J n 0 x, y) < ∞}; Using the alternative fixed point theorem, we investigate the stability of the functional equation (1).Theorem 4. Let (X, ⋄) be a divisible square-symmetric groupoid and let (Y, * , d) be a complete metric divisible square-symmetric groupoid and L ∈ (0, 1).Assume that a mapping f : for all x, y ∈ X with f (id X ) = id Y and ε : for all x, y ∈ X.Then for every x ∈ X, converges and H : X → Y is the solution of (1) such that for all x ∈ X.
Proof.Consider the set and introduce a generalized metric on S : where, as usual, inf ∅ = +∞.It is easy to show that (S, d ′ ) is complete (see the proof of Lemma 2.1 of [14]) Now we consider the linear mapping T : S → S such that By Theorem 3, there exists a mapping H : X → Y satisfying the following : (1) H is a fixed point of T , i.e., for all x ∈ X.The mapping H is a unique fixed point of T in the set This means that H is a unique fixed point satisfying ( 14).