APPROXIMATE QUADRATIC MAPPINGS IN MODULAR SPACES

In this article, we investigate an alternative generalized Hyers–Ulam stability theorem of a modified quadratic functional equation in a modular space Xρ using ∆3-condition without the Fatou property on the modular function ρ. AMS Subject Classification: 39B82, 39B72, 16W25


Introduction
In 1940, S.M. Ulam [12] raised the question concerning the stability of group homomorphisms: Let G be a group and let G ′ be a metric group with the metric d(•, •).Given ε > 0, does there exist a δ > 0 such that if a mapping f : G → G ′ satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G, then there exists a homomorphism for all x ∈ G? The case of approximately additive mappings was solved by D.H. Hyers [5] under the assumption that X and Y are Banach spaces.A generalization of Hyers' theorem was provided by Th.M. Rassias [8] in 1978 and by P. Gǎvruta [4] in 1994.The equation f (x + y) + f (x − y) = 2[f (x) + f (y)], which may be originated from the important parallelogram equality x + y 2 + x − y 2 = 2[ x 2 + y 2 ] in inner product spaces, is called a quadratic functional equation, and every solution of the quadratic functional equation is said to be a quadratic mapping.The Hyers-Ulam stability problem for the quadratic functional equation was proved by F. Skof [11] for a mapping f : X → Y , where X is a normed space and Y is a Banach space.S. Czerwik [3] proved the Hyers-Ulam stability of the quadratic functional equation with the sum of powers of norms in the sense of Th.M. Rassias approach using direct method.J.M. Rassias [6] proved the Hyers-Ulam stability of the quadratic functional equation with the product of powers of norms using direct method.On the other hand, C. Borelli and G.L. Forti [2] have proved the generalized Hyers-Ulam stability theorem of the quadratic functional equation whose quadratic difference was controlled by general functions with regularity conditions.
Next, we introduce to recall some basic definitions and remarks of modular spaces with modular functions, which are primitive notions corresponding to norms or metrics, as in the followings.A modular ρ defines a corresponding modular space, i.e., the linear space χ ρ given by χ ρ = {x ∈ χ : ρ(λx) → 0 as λ → 0}.
Let ρ be a convex modular.Then, the modular space χ ρ can be equipped with a norm called the Luxemburg norm, defined by A modular function ρ is said to satisfy the ∆ 2 -condition if there exists κ > 0 such that ρ(2x) ≤ κρ(x) for all x ∈ χ ρ .
Definition 3. Let χ ρ be a modular space and let {x n } be a sequence in χ ρ .Then, (1) They say that the modular ρ has the Fatou property if and only if ρ(x) ≤ lim inf n→∞ ρ(x n ) whenever the sequence {x n } is ρ-convergent to x.
Concerning the stability of functional equations, G. Sadeghi [10] has proved generalized Hyers-Ulam stability via the fixed point method of a generalized Jensen functional equation f (rx + sy) = rg(x) + sh(y) in convex modular spaces with the Fatou property satisfying the ∆ 2 -condition with 0 < κ ≤ 2. In the paper [13], the authors have presented the generalized Hyers-Ulam stability of quadratic functional equations via the extensive studies of fixed point theory in the framework of modular spaces whose modulars are convex, lower semicontinuous but do not satisfy any relatives of ∆ 2 -conditions.Recently, A. Zivari-Kazempour and M. Eshaghi Gordji [14] have determined the general solution of the quadratic functional equation which is equivalent to the quadratic functional equation and then they have proved its generalized Hyers-Ulam stability in normed spaces.In this article, we first present generalized Hyers-Ulam stability via direct method of the equation ( 1) in modular spaces without using the Fatou property and ∆ 3 -conditions, and then investigate alternatively generalized Hyers-Ulam stability via direct method of the equation ( 1) in modular spaces using necessarily ∆ 3 -conditions without the Fatou property.

Generalized Hyers-Ulam Stability of Equation (1)
Before making up the main subject, we use the following abbreviation for notational convenience : for all x, y, z in a linear space X.In the following theorem, we first present a generalized Hyers-Ulam stability via direct method of the equation ( 1) in modular spaces without using the Fatou property and ∆ 3 -condition.
Theorem 4. Let X be a linear space and χ ρ a ρ-complete convex modular space.Suppose that a mapping f : and φ : for all x, y, z ∈ X.Then there exists a unique quadratic mapping 9 n , x ∈ X, which satisfies the equation (1) and for all x ∈ X.
Proof.Letting y = z := x in (2), we obtain and so for all x ∈ X.By induction on n, one can prove the following functional inequality for all x ∈ X.Now, replacing x by 3 m x in (6), we have which converges to zero as m → ∞ by the assumption (3).Thus the above inequality implies that the sequence { f (3 n x) 9 n } is ρ-Cauchy for all x ∈ X and so it is convergent in χ ρ since the space χ ρ is ρ-complete.Thus, we may define a mapping F 1 : X → χ ρ as Now, we claim the mapping F 1 satisfies the equation (1).Setting (x, y, z) := (3 n x, 3 n y, 3 n z) in (2), and dividing the resulting inequality by 9 n , we get for all x, y, z ∈ X.Thus, it follows from the property ρ(αu) ≤ αρ(u), (0 for all x, y, z ∈ X and all positive integers n.Taking the limit as n → ∞, one obtains ρ( 1 15 DF 1 (x, y, z)) = 0, and so DF 1 (x, y, z) = 0 for all x, y, z ∈ X. Hence F 1 satisfies the equation ( 1) and so it is quadratic.
On the other hand, since n i=0 1 9 i+1 + 1 9 ≤ 1 for all n ∈ N, it follows from (5) and the convexity of the modular ρ that without applying the Fatou property of the modular ρ for all x ∈ X and all n ∈ N, from which we obtain the approximation of f by the quadratic mapping F 1 as follows for all x ∈ X by taking n → ∞ in the last inequality.
To show the uniqueness of F 1 , we assume that there exists a quadratic mapping G 1 : X → χ ρ which satisfies the inequality for all x ∈ X, but suppose F 1 (x 0 ) = G 1 (x 0 ) for some x 0 ∈ X.Then there exists a positive constant ε > 0 such that ε < ρ(F 1 (x 0 ) − G 1 (x 0 )).For such given ε > 0, it follows from (3) that there is a positive integer n 0 ∈ N such that Since F 1 and G 1 are quadratic mappings, we see from the equality which leads a contradiction.Hence the mapping F 1 is a unique quadratic mapping near f satisfying the approximation (4) in the modular space χ ρ .
As corollaries of Theorem 4, we obtain the following stability results of the equation ( 1), which generalize stability results in normed spaces.
Corollary 5. Suppose X is a linear space and χ ρ is a ρ-complete convex modular space.Let f : X → χ ρ be a mapping satisfying ρ(Df (x, y, z)) ≤ ε for some ε > 0 and for all x, y, z ∈ X.Then there exists a unique quadratic mapping F 1 : X → χ ρ which satisfies (1) and Proof.We see from (5) that for all x ∈ X.Thus, by the convexity of the modular ρ one has for all x ∈ X.Then it follows by induction on n > 1 that for all x ∈ X.In fact, it is true for n = 2. Assume that the inequality (12) holds true for n.Thus, using the convexity of the modular ρ, we deduce which proves (12) for n + 1.Now, replacing x by 3 −m x in (12), we have x ∈ X, which satisfies the equation (1) and for all x ∈ X, which is an implicit alternative stability result in the paper [14].
As corollaries of Theorem 8, we obtain the following stability results of the equation ( 1), which generalize stability results in normed spaces.
Remark 12.In Corollary 11, if f (0) = 0 and r i > 0 for some i = 1, 2, 3, then we find that Df (x, y, 0) = 0 for the case r 3 > 0 without loss of generality, and so f is itself quadratic in this case.