Hyers-Ulam stability of quadratic forms in 2-normed spaces

We introduce some definitions on 2-Banach spaces [1], [2]. Definition 1.1. Let X be a real linear space with dim X ≥ 2 and ‖·, ·‖ : X2 → R be a function. Then (X, ‖·, ·‖) is called a linear 2-normed space if the following conditions hold: (a) ‖x, y‖ = 0 if and only if x and y are linearly dependent, (b) ‖x, y‖ = ‖y, x‖, (c) ‖αx, y‖ = |α|‖x, y‖, (d) ‖x, y + z‖ ≤ ‖x, y‖ + ‖x, z‖ for all α ∈ R and x, y, z ∈ X. In this case, the function ‖·, ·‖ is called a 2-norm on X. Definition 1.2. Let {xn} be a sequence in a linear 2-normed space X. The sequence {xn} is said to be convergent in X if there exists an element x ∈ X such that

limm,n→∞ ‖xn − xm , y‖ = 0 for a Cauchy sequence {xn}. A 2-Banach space is defined to be a linear 2-normed space in which every Cauchy sequence is convergent.
In 1940, Ulam [4] suggested the stability problem of functional equations concerning the stability of group homomorphisms: Let a group G and a metric group H with the metric ρ be given. For each ε > 0, the question is whether or not there is a δ > 0 such that if f : The case of approximately additive mappings was solved by Hyers [5] under the assumption that G and H are Banach spaces. In 1978, Rassias [6] generalized the result of Hyers as follows: Let f : G → H be a mapping between Banach spaces and let 0 ≤ p < 1 be fixed. If f satisfies the inequality for some θ ≥ 0 and for all x, y ∈ G, then there exists a unique additive mapping A : G → H such that In Banach spaces, Bae and Park [7][8][9] investigated the stability problem of some functional equations: and The quadratic forms f 1 , f 2 , f 3 : R × R → R given by f 1 (x, y) := ax 2 + bxy + cy 2 , f 2 (x, y) := ax 2 + by 2 and f 3 (x, y) := axy are solutions of (1), (2) and (3), respectively. In 2011, Park [10] investigated approximate additive, Jensen and quadratic mappings in 2-Banach spaces. In this paper, we also investigate the stability of the functional equations (1), (2) and (3)

Results
Throughout this paper, let X be a normed space and Y a 2-Banach space.
for all x, y, z, w, u, v ∈ X. Then there exists a unique mapping F : for all x, y, u, v ∈ X.
Proof. Letting y = x and w = z in (4), we have for all x, z, u, v ∈ X. Thus we obtain for all x, z, u, v ∈ X and all j. Replacing z by y in the above inequality, we see that for all x, y, u, v ∈ X and all j. For given integers l, m(0 ≤ l < m), we get for all x, y, u, v ∈ X. By (6), the sequence for all x, y, z, w, u, v ∈ X and all j. Letting j → ∞, we see that F satisfies (1). Setting l = 0 and taking m → ∞ in (6), one can obtain the inequality (5). If G : X × X → Y is another mapping satisfying (1) and (5) for all x, y, u, v ∈ X. Hence the mapping F is the unique mapping satisfying (1), as desired.
for all x, y, z, w, u, v ∈ X. Then there exists a unique mapping F : for all x, y, u, v ∈ X.
In the case p > 2 in Theorem 2.1, one can also obtain a similar result.

Theorem 2.3.
Let p ∈ (0, 2), ε > 0 and δ, η ≥ 0, and let f : X × X → Y be a surjective mapping such that for all x, y, z, w, u, v ∈ X. Then there exists a unique mapping F : X × X → Y satisfying (2) such that for all x, y, u, v ∈ X.
Proof. Letting y = x and w = −z in (7), we obtain that for all x, z, u, v ∈ X. Putting x = 0 in (9), we get for all z, u, v ∈ X. Replacing z by −z in the above inequality, we have for all z, u, v ∈ X. By the above two inequalities, we see that for all z, u, v ∈ X. Setting y = x and w = z in (7), we have for all x, z, u, v ∈ X. Replacing z by −z in the above inequality, we see that for all x, z, u, v ∈ X. By (11) and (12), we know that for all x, z, u, v ∈ X. By (9) and (13), we get for all x, z, u, v ∈ X. By (10) and the above inequality, we have for all x, z, u, v ∈ X. Thus we obtain that 4 j ε + δ 2 (2−p)j (16‖x‖ p + 17‖z‖ p ) + 10η 4 j (‖u‖ + ‖v‖)

]︂
for all x, z, u, v ∈ X and all j. Hence, for given integers l, m (0 ≤ l < m), we see that for all x, y, z, w, u, v ∈ X. Then there exists a unique mapping F : X × X → Y satisfying (2) such that ‖f (x, y) − F(x, y), f (u, v)‖ ≤ 2 3 ε for all x, y, u, v ∈ X.