STABILITY OF A CUBIC FUNCTIONAL EQUATION IN 2-NORMED SPACES

Abstract

In this paper, we prove the generalized Hyers-Ulam stability of the following cubic funtional equation f(2x + y) + f(2x - y) = 2f(x + 2y) - 4f(x + y) + 18f(x) - 12f(y) by the direct method in 2-normed spaces.

1. Introduction and preliminaries

Gähler [4,5] has introduced the concept of 2-normed spaces and Gähler and White [16] introduced the concept of 2-Banach spaces. Lewandowska published a series of papers on 2-normed sets and generalized 2-normed spaces [10,11]. Recently, Park [12] investigated approximate additive mappings, approximate Jensen mappings and approximate quadratic mappings in 2-Banach spaces.

We list some definitions related to 2-normed spaces.

Definition 1.1. Let X be a linear space over ℝ with dim X > 1 and let ∥· , ·∥ : X × X → ℝ be a function satisfying the following properties :

(1) ∥x, y∥ = 0 if and only if x and y are linearly dependent,

(2) ∥x, y∥ = ∥y, x∥,

(3) ∥ax, y∥ = |a|∥x, y∥, and

(4) ∥x, y + z∥ ≤ ∥x, y∥ + ∥x, z∥

for all x, y, z ∈ X and a ∈ ℝ. Then the function ∥· , ·∥ is called a 2-norm on X and (X, ∥· , ·∥) is called a 2-normed space.

Let (X, ∥· , ·∥) be a 2-normed space. Suppose that x ∈ X and ∥x, y∥ = 0 for all y ∈ X. Suppose that x ≠ 0. Since dim X > 1, choose y ∈ X such that {x, y} is linearly independent and so by (1) in Definition1.1, we have ∥x, y∥ ≠ 0, which is a contradiction. Hence we have the following lemma.

Lemma 1.2. Let (X, ∥· , ·∥) be a 2-normed space. If x ∈ X and ∥x, y∥ = 0 for all y ∈ X, then x = 0.

Definition 1.3. A sequence {xn} in a 2-normed space (X, ∥· ; ·∥) is called a 2-Cauchy sequence if

for all x ∈ X.

Definition 1.4. A sequence {xn} in a 2-normed space (X, ∥· , ·∥) is called 2-convergent if

for all y ∈ X and for some x ∈ X. In case, {xn} said to be converge to x and denoted by xn → x as n → ∞ or limn→∞ xn = x.

A 2-normed space (X, ∥· , ·∥) is called a 2-Banach space if every 2-Cauchy sequence in X is 2-convergent. Now, we state the following results as lemma [12].

Lemma 1.5. Let (X, ∥· , ·∥) be a 2-normed space. Then we have the following :

(1) |∥x, z∥ − ∥y, z∥| ≤ ∥x − y, z∥ for all x, y, z ∈ X,

(2) if∥x, z∥ = 0 for all z ∈ X, then x = 0, and

(3) for any 2-convergent sequence {xn} in X,

for all z ∈ X.

In 1940, S.M.Ulam [15] proposed the following stability problem :

“Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exists a constant c > 0 such that if a mapping f : G1 → G2 satisfies d(f(xy); f(x)f(y)) < 0 for all x, y ∈ G1, then there exists a unique homomorphism h : G1 → G2 with d(f(x); h(x)) < δ for all x ∈ G1?”

In the next year, D. H. Hyers [7] gave a partial solution of Ulam’s problem for the case of approximate additive mappings. Subsequently, his result was generalized by T. Aoki [1] for additive mappings and by TH. M. Rassias [14] for linear mappings, to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of funtional equations have been extensively investigated by a number of mathematicians.

Rassias [13] introduced the cubic functional equation

and Jun and Kim [8] introduced the following cubic funtional equation

In this paper, we inverstigate the following cubic funtional equation

which is a linear combination of (1) and (2) and proved the generalized Hyers-Ulam stability of (3) in 2-normed spaces.

 

2. Stability of (3) from a normed space to a 2-Banach space

Thoughout this section, (X, ∥·∥) or simply X is a real normed space and (Y, ∥·,·∥) or simply Y is a 2-Banach space. We start the following theorem.

Theorem 2.1. A mapping f : X → Y satisfies (3) if and only if f is cubic.

Proof. Suppose that f satisfies (3). Letting x = y = 0 in (3), we have f(0) = 0 and letting y = 0 in (3), we have

for all x ∈ X. Letting x = 0 in (3), by (4), we have f(y) = −f(−y) for all y ∈ X and so f is odd. Letting y = −y in (3), we have

for all x, y ∈ X and by (3) and (5), we have

for all x, y ∈ X. Hence by (3) and (6), we have

for all x, y ∈ X. Interching x and y in (7), since f is odd, f satisfies (2) and hence f is cubic.

For any function f : X → Y, we define the difference operator Df : X × X → Y by

Now we prove the generalized Hyers-Ulam stability of (3).

Theorem 2.2. Let ε ≥ 0, p and q be positive real numbers with p + q < 3 and r > 0. Suppose that f : X → Y is a function such that

for all x, y ∈ X and z ∈ Y. Then there exists a unique cubic function C : X → Y satisfying (3) and

for all x ∈ X and z ∈ Y.

Proof. Letting x = y = 0 in (8), we have ∥2f(0); z∥ = 0 for all z ∈ Y and by the definition of 2-norm, we have f(0) = 0. Putting y = 0 in (8), we have

for all x ∈ X and z ∈ Y and so

for all x ∈ X and z ∈ Y. Replacing x by 2x in (11), we get

for all x ∈ X and z ∈ Y. By (11) and (12), we get

for all x ∈ X and z ∈ Y. By induction on n, we can show that

for all x ∈ X and z ∈ Y. For m, n ∈ ℕ with n < m and x ∈ X, by (13), we have

Since is a 2- Cauchy sequence in Y for all x ∈ X. Since Y is a 2-Banach space, the sequence is a 2-convergent in Y for all x ∈ X and so we can define a mapping C : X → Y as

for all x ∈ X. By (14), we have

for all x ∈ X and z ∈ Y and by Lemma 1.5 , we have

for all x ∈ X and z ∈ Y. Next we will show that C satisfies (3). By (8), we have

for all z ∈ Y, because p < 3, q < 3, p + q < 3 and so DC(x, y) = 0 for all x, y ∈ X. By Theoem 2.1, C is cubic.

To show that C is unique, suppose there exists another cubic function C′ : X → Y which satisfies (3) and (9). Since C and C′ are cubic, for all x ∈ X. It follows that

So ∥ C′(x)−C(x), z ∥= 0 for all z ∈ Y and hence C′(x) = C(x) for all x ∈ X.

Related with Theorem 2.2, we can also the following theorem.

Theorem 2.3. Let ε ≥ 0, p and q be positive real numbers with p, q > 3 and r > 0. Suppose that f : X → Y is a function satisfying (8). Then there exists a unique cubic function C : X → Y satisfying (3) and

for all x ∈ X and z ∈ Y.

Proof. Letting x = y = 0 in (8), we have ∥2f(0), z∥ = 0 for all z ∈ Y and so we have f(0) = 0. Putting y = 0 and replacing x by in (8), we get

for all x ∈ X and z ∈ Y and so

for all x ∈ X and z ∈ Y. Replacing x by in (16), we get

for all x ∈ X and z ∈ Y. By (16) and (17), we get

for all x ∈ X and z ∈ Y. By induction on n, we can show that

for all x ∈ X and z ∈ Y. For m, n ∈ ℕ with n < m and x ∈ X, by (18), we have

and since p > 3, is a 2- Cauchy sequence in Y for all x ∈ X. Since Y is a 2-Banach space, the sequence is a 2-convergent in Y for all x ∈ X. Define C : X → Y as

for all x ∈ X. By (18), we have

for all x ∈ X and z ∈ Y and by Lemma 1.5, we have

for all x ∈ X and z ∈ Y. Next we will show that C satisfies (3).

for all z ∈ Y, because p, q > 3 and so DC(x, y) = 0 for all x, y ∈ X. By Theoem 2.1, C is cubic.

To show that C is unique, suppose there exists another cubic function C′ : X → Y which satisfies (3) and (15). Since C and C′ are cubic, for all x ∈ X. It follows that

So ∥ C′(x)−C(x), z ∥= 0 for all z ∈ Y and hence C′(x) = C(x) for all x ∈ X.

 

3. Stability of (3) from a 2- normed space to a 2-Banach space

In this section, we study similar problems of (3). Let (X, ∥· , ·∥) be a 2-normed space and (Y, ∥· , ·∥) a 2- Banach space.

Theorem 3.1. Let ε ≥ 0 and p and q be positive real numbers with p + q < 3. Suppose that f : X → Y is a function such that

for all x, y ∈ X and z ∈ Y. Then there exists a unique cubic function C : X → X satisfying (3) and

for all x ∈ X and z ∈ Y.

Proof. Letting x = y = 0 in (19). We have ∥2f(0), z∥ = 0 for all z ∈ Y, so we have f(0) = 0. Putting y = 0 in (19), we have

for all x ∈ X and z ∈ Y. Therefore

for all x ∈ X and z ∈ Y. Replacing x by 2x in (21), we get

for all x ∈ X and z ∈ Y. By induction on n, we can show that

for all x ∈ X and z ∈ Y. For m, n ∈ ℕ with n < m and x ∈ X, by (22), we get

for all x ∈ X and z ∈ Y. Since p < 3, is a 2- Cauchy sequence in Y for all x ∈ X. Since Y is a 2-Banach space, the sequence is a 2-convergent in Y for all x ∈ X. Define C : X → Y as

for all x ∈ X and by Lemma 1.5 and (22), we have (20). Next we show that C satisfies (3). By (19), we have

for all z ∈ Y and so DC(x, y) = 0 for all x, y ∈ X. By Theoem 2.1, C is cubic.

To show that C is unique, suppose that there exists another cubic function C′ : X → Y which satisfies (3) and (20). Since C and C′ are cubic, for all x ∈ X. Since p < 3,

So ∥ C′(x)−C(x), z ∥= 0 for all z ∈ Y and hence C′(x) = C(x) for all x ∈ X.

Similar to Theorem 3.1, we have the following theorem.

Theorem 3.2. Let (X, ∥· , ·∥) be a 2- Banach space. Let ε ≥ 0, p and q be positive real numbers with p, q > 3. Suppose that f : X → X is a function satisfying (19). Then there exists a unique cubic function C : X → X satisfying (3) and

for all x, z ∈ X.