ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN β-HOMOGENEOUS F-SPACES

Abstract

In this paper, we solve the additive ρ-functional inequalities (0.1) ||f(2x-y)+f(y-x)-f(x)|| $\leq$ ||${\rho}(f(x+y)-f(x)-f(y))$||, where ρ is a fixed complex number with |ρ| < 1, and (0.2) ||f(x+y)-f(x)-f(y)|| $\leq$ ||${\rho}(f(2x-y)-f(y-x)-f(x))$||, where ρ is a fixed complex number with |ρ| < $\frac{1}{2}$. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in β-homogeneous F-spaces.

1. INTRODUCTION AND PRELIMINARIES

The stability problem of functional equations originated from a question of Ulam [23] concerning the stability of group homomorphisms.

The functional equation f(x+y) = f(x) +f(y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [8] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [14] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability of quadratic functional equation was proved by Skof [22] for mappings f : E1 → E2, where E1 is a normed space and E2 is a Banach space. Cholewa [5] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. The stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 3, 4, 6, 9, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 24, 25]).

Definition 1.1. Let X be a (complex) linear space. A nonnegative valued function || · || is an F-norm if it satisfies the following conditions:

(FN1) ||x|| = 0 if and only if x = 0; (FN2) ||λx|| = ||x|| for all x ∈ X and all λ with |λ| = 1; (FN3) ||x + y|| ≤ ||x|| + ||y|| for all x, y ∈ X; (FN4) ||λnx|| → 0 provided λn → 0; (FN5) ||λxn|| → 0 provided xn → 0.

Then (X, || · ||) is called an F∗-space. An F-space is a complete F∗-space.

An F-norm is called β-homogeneous (β > 0) if ||tx|| = |t|β||x|| for all x ∈ X and all t ∈ ℂ and (X, || · ||) is called a β-homogeneous F-space (see [16]).

In Section 2, we solve the additive ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in β-homogeneous F-space.

In Section 3, we solve the additive ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in β-homogeneous F-space.

Throughout this paper, let β1, β2 be positive real numbers with β1 ≤ 1 and β2 ≤ 1. Assume that X is a β1-homogeneous F-space with norm || · || and that Y is a β2-homogeneous F-space with norm || · ||.

 

2. ADDITIVE ρ-FUNCTIONAL INEQUALITY (0.1) IN β-HOMOGENEOUS F-SPACES

Throughout this section, assume that ρ is a complex number with |ρ| < 1.

We solve and investigate the additive ρ-functional inequality (0.1) in β-homogeneous F-spaces.

Lemma 2.1. If a mapping f : X → Y satisfies

for all x, y ∈ X, then f : X → Y is additive.

Proof. Assume that f : X → Y satisfies (2.1).

Letting x = 0 and y = 0 in (2.1), we get ||f(0)|| ≤ ||ρ (f(0))|| and so f(0) = 0 with |ρ| < 1.

Letting x = 0 in (2.1), we get ||f(−y) + f(y)|| ≤ 0 and so f is an odd mapping.

Letting x = z and y = z − w in (2.1), we get

for all z, w ∈ X.

It follows from (2.1) and (2.2) that

and so f(2x − y) + f(y − x) = f(x) for all x, y ∈ X. It is easy to show that f is additive.                                                 ☐

We prove the Hyers-Ulam stability of the additive ρ-functional inequality (2.1) in β-homogeneous F-spaces.

Theorem 2.2. Let and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting x = y = 0, in (2.3), we get ||f(0)|| ≤ 0. So f(0) = 0.

Letting y = 0 in (2.3), we get

for all x ∈ X.

Letting x = 0 in (2.3), we get

for all y ∈ X.

From (2.5) and (2.6), we get

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.8) that the sequence is Cauchy for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping A : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.8), we get (2.4).

It follows from (2.3) that

for all x, y ∈ X. So

for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is additive.

Now, let T : X → Y be another additive mapping satisfying (2.4). Then we have

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T(x) for all x ∈ X. This proves the uniqueness of A, as desired.                                                 ☐

Theorem 2.3. Let and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying (2.3). Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. It follows from (2.7) that

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.10) that the sequence is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping A : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.9).

The rest of the proof is similar to the proof of Theorem 2.2.                                                 ☐

Remark 2.4. If ρ is a real number such that −1 < ρ < 1 and Y is a β-homogeneous real F-space, then all the assertions in this section remain valid.

 

3. ADDITIVE ρ-FUNCTIONAL INEQUALITY (0.2) IN β-HOMOGENEOUS F-SPACES

Throughout this section, assume that ρ is a complex number with .

We solve and investigate the additive ρ-functional inequality (0.2) in β-homogeneous F-spaces.

Lemma 3.1. If a mapping f : X → Y satisfies

for all x, y ∈ X, then f : X → Y is additive.

Proof. Assume that f : X → Y satisfies (3.1).

Letting x = y = 0 in (3.1), we get ||f(0)|| ≤ 0. So f(0) = 0.

Letting y = x in (3.1), we get ||f(2x) − 2f(x)|| ≤ 0 and so

for all x ∈ G.

Letting y = 2x in (3.1), we get ||f(3x) − f(x) − f(2x)|| ≤ 0 and from (3.2),

for all x ∈ X.

Letting y = −x in (3.1), we get ||f(x) + f(−x)|| ≤ ||ρ(f(3x) + f(−2x) − f(x))||. From (3.2) and (3.3), f(3x)+f(−2x)−f(x) = 2f(x)+2f(−x), so ||f(x) + f(−x)|| ≤ 0, and we get

for all x ∈ X. So f is an odd mapping.

Letting x = z, y = z − w in (3.1), we get

and from (3.4),

for all z, w ∈ X.

It follows from (3.1) and (3.5) that

and so f(x + y) = f(x) + f(y) for all x, y ∈ X. So f is additive.                                                 ☐

We prove the Hyers-Ulam stability of the additive ρ-functional inequality (3.1) in β-homogeneous F-spaces.

Theorem 3.2. Let and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting x = y = 0 in (3.4), we get ||f(0)|| ≤ 0. So f(0) = 0.

Letting y = x in (3.6), we get

for all x ∈ X. So

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that the sequence is Cauchy for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping A : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.9), we get (3.7).

The rest of the proof is similar to the proof of Theorem 2.2.                                                 ☐

Theorem 3.3. Let and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying (3.4). Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. It follows from (3.8) that

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.11) that the sequence is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping A : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.11), we get (3.10).

The rest of the proof is similar to the proof of Theorem 2.2.                                                 ☐

Remark 3.4. If ρ is a real number such that and Y is a β-homogeneous real F-space, then all the assertions in this section remain valid.