QUADRATIC ρ-FUNCTIONAL INEQUALITIES

Abstract

In this paper, we solve the quadratic ρ-functional inequalities (0.1) ${\parallel}f(x+y)+f(x-y)-2f(x)-2f(y){\parallel}$ $\leq$ ${\parallel}{\rho}(4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y)){\parallel}$, where $\rho$ is a fixed complex number with $\left|{\rho}\right|$ < 1, and (0.2) ${\parallel}4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y){\parallel}$ $\leq$ ${\parallel}{\rho}(f(x+y)+f(x-y)-2f(x)-2f(y)){\parallel}$, where ρ is a fixed complex number with |ρ| < $\frac{1}{2}$. Furthermore, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces.

1. INTRODUCTION AND PRELIMINARIES

The stability problem of functional equations originated from a question of Ulam [11] 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 [6] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [8] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gǎvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

The functional equation

is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [10] for mappings f : E1 → E2, where E1 is a normed space and E2 is a Banach space. Cholewa [3] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group.

The functional equation

is called a Jensen type quadratic equation. See [2, 4, 7, 9, 12] for more information on the stability problems of functional equations.

In Section 2, we solve the quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in complex Banach spaces.

In Section 3, we solve the quadratic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in complex Banach spaces.

Throughout this paper, let G be a 2-divisible abelian group. Assume that X is a real or complex normed space with norm ∥ · ∥ and that Y is a complex Banach space with norm ∥ · ∥.

 

2. QUADRATIC ρ-FUNCTIONAL INEQUALITY (0.1)

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

In this section, we solve and investigate the quadratic ρ-functional inequality (0.1) in complex Banach spaces.

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

for all x, y ∈ G, then f : G → Y is quadratic.

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

Letting x = y = 0 in (2.1), we get ∥2f(0)∥ ≤ |ρ|∥f(0)∥. So f(0) = 0.

Letting y = x in (2.1), we get ∥f(2x) − 4f(x)∥ ≤ 0 and so f(2x) = 4f(x) for all x ∈ G. Thus

for all x ∈ G.

It follows from (2.1) and (2.2) that

and so

for all x, y ∈ G.                                        □

We prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (2.1) in complex Banach spaces.

Theorem 2.2. Let r > 2 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 quadratic mapping h : X → Y such that

for all x ∈ X.

Proof. Letting x = y = 0 in (2.3), we get ∥2f(0)∥ ≤ |ρ|∥f(0)∥. So f(0) = 0.

Letting y = x in (2.3), we get

for all x ∈ X. So

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.6) that the sequence is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping h : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), 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 h : X → Y is quadratic.

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

which tends to zero as n → ∞ for all x ∈ X. So we can conclude that h(x) = T(x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping h : X → Y is a unique quadratic mapping satisfying (2.4).                                        □

Theorem 2.3. Let r < 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.3). Then there exists a unique quadratic mapping h : X → Y such that

for all x ∈ X.

Proof. It follows from (2.5) that

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 a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping h : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.8), we get (2.7).

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 real Banach space, then all the assertions in this section remain valid.

 

3. QUADRATIC ρ-FUNCTIONAL INEQUALITY (0.2)

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

In this section, we solve and investigate the quadratic ρ-functional inequality (0.2) in complex Banach spaces.

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

for all x, y ∈ G, then f : G → Y is quadratic.

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

Letting x = y = 0 in (3.1), we get ∥f(0)∥ ≤ |ρ|∥2f(0)∥. So f(0) = 0.

Letting y = 0 in (3.1), we get and so

for all x ∈ G.

It follows from (3.1) and (3.2) that

and so

for all x, y ∈ G.                                        □

We prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (3.1) in complex Banach spaces.

Theorem 3.2. Let r > 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that

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

for all x ∈ X.

Proof. Letting x = y = 0 in (3.3), we get ∥f(0)∥ ≤ |ρ|∥2f(0)∥. So f(0) = 0.

Letting y = 0 in (3.3), we get

for all x ∈ X. So

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.6) that the sequence is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping h : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.4).

It follows from (3.3) that

for all x, y ∈ X. So

for all x, y ∈ X. By Lemma 3.1, the mapping h : X → Y is quadratic.

Now, let T : X → Y be another quadratic mapping satisfying (3.4). Then we have

which tends to zero as n → ∞ for all x ∈ X. So we can conclude that h(x) = T(x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping h : X → Y is a unique quadratic mapping satisfying (3.4).                                         □

Theorem 3.3. Let r < 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (3.3). Then there exists a unique quadratic mapping h : X → Y such that

for all x ∈ X.

Proof. It follows from (3.5) that

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.8) that the sequence is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping h : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.8), we get (3.7).

The rest of the proof is similar to the proof of Theorem 3.2.                                         □

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