ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY BANACH SPACES

Abstract

Let $M_1f(x,y):=\frac{3}{4}f(x+y)-\frac{1}{4}f(-x-y)+\frac{1}{4}(x-y)+\frac{1}{4}f(y-x)-f(x)-f(y)$, $M_2f(x,y):=2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})-f(x)-f(y)$ Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) $N(M_1f(x,y)-{\rho}M_2f(x,y),t){\geq}\frac{t}{t+{\varphi}(x,y)}$ and (0.2) $N(M_2f(x,y)-{\rho}M_1f(x,y),t){\geq}\frac{t}{t+{\varphi}(x,y)}$ in fuzzy Banach spaces, where ρ is a fixed real number with ρ ≠ 1.

1. INTRODUCTION AND PRELIMINARIES

Katsaras [14] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [11, 16, 37]. In particular, Bag and Samanta [3], following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [15]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [4].

We use the definition of fuzzy normed spaces given in [3, 19, 20] to investigate the Hyers-Ulam stability of additive ρ-functional inequalities in fuzzy Banach spaces.

Definition 1.1 ([3, 19, 20, 21]). Let X be a real vector space. A function N : X × ℝ → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ ℝ,

(N1) N(x, t) = 0 for t ≤ 0;

(N2) x = 0 if and only if N(x, t) = 1 for all t > 0;

(N3)

(N4) N(x + y, s + t) ≥ min{N(x, s),N(y, t)};

(N5) N(x, ·) is a non-decreasing function of ℝ and limt→∞ N(x, t) = 1.

(N6) for x ≠ 0, N(x, ·) is continuous on ℝ.

The pair (X, N) is called a fuzzy normed vector space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [19, 20].

Definition 1.2 ([3, 19, 20, 21]). Let (X, N) be a fuzzy normed vector space. A sequence {xn} in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N(xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn} and we denote it by N-limn→∞ xn = x.

Definition 1.3 ([3, 19, 20, 21]). Let (X, N) be a fuzzy normed vector space. A sequence {xn} in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ ℕ such that for all n ≥ n0 and all p > 0, we have N(xn+p – xn, t) > 1 – ε.

It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn} converging to x0 in X, then the sequence {f(xn)} converges to f(x0). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [4]).

The stability problem of functional equations originated from a question of Ulam [36] 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 [13] 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 [28] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

The functional equation f(x + y) + f(x – y) = 2f(x) + 2f(y) 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 [35] for mappings f : E1 → E2, where E1 is a normed space and E2 is a Banach space. Cholewa [9] 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, 5, 6, 7, 10, 17, 18, 22, 25, 26, 27, 29, 30, 31, 32, 33, 34, 38, 39]).

Park [23, 24] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces.

In Section 2, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces by using the direct method.

In Section 3, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces by using the direct method.

Throughout this paper, assume that X is a real vector space and (Y, N) is a fuzzy Banach space. Let ρ be a real number with ρ ≠ 1.

 

2. ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITY (0.1)

In this section, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces.

We need the following lemma to prove the main results.

Lemma 2.1.

(i) If an odd mapping f : X → Y satisfies

for all x, y ∈ X, then f is the Cauchy additive mapping.

(ii) If an even mapping f : X → Y satisfies f(0) = 0 and (2.1), then f is the quadratic mapping.

Proof. (i) Letting y = x in (2.1), we get f(2x) – 2f(x) = 0 and so f(2x) = 2f(x) for all x ∈ X. Thus

for all x ∈ X.

It follows from (2.1) and (2.2) that

and so

for all x, y ∈ X.

(ii) Letting y = x in (2.1), we get and so f(2x) = 4f(x) for all x ∈ X. Thus

for all x ∈ X.

It follows from (2.1) and (2.3) that

and so

for all x, y ∈ X.     □

Theorem 2.2. Let φ : X2 → [0, ∞) be a function such that

for all x, y ∈ X.

(i) Let f : X → Y be an odd mapping satisfying

for all x, y ∈ X and all t > 0. Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0, where .

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (2.5). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0, where for all x, y ∈ X.

Proof. (i) Letting y = x in (2.5), we get

and so

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (2.4) and (2.9) 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 (2.9), we get (2.6).

By (2.5),

for all x, y ∈ X, all t > 0 and all n ∈ ℕ. So

for all x, y ∈ X, all t > 0 and all n ∈ ℕ. Since for all x, y ∈ X and all t > 0,

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

(ii) Letting y = x in (2.5), we get

and so

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (2.4) and (2.11) that the sequence is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping Q : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.11), we get (2.7).

The rest of the proof is similar to the above additive case.     □

Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm || · ||.

(i) Let f : X → Y be an odd mapping satisfying

for all x, y ∈ X and all t > 0. Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0.

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (2.12). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0.

Proof. The proof follows from Theorem 2.2 by taking φ(x, y) := θ(||x||p + ||y||p) for all x, y ∈ X, as desired.     □

Theorem 2.4. Let φ : X2 → [0, ∞) be a function such that

for all x, y ∈ X.

(i) Let f : X → Y be an odd mapping satisfying (2.5). Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0, where for all x, y ∈ X.

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (2.5). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0, where for all x, y ∈ X.

Proof. (i) It follows from (2.8) that

and so

for all x ∈ X and all t > 0.

(ii) It follows from (2.10) that

and so

for all x ∈ X and all t > 0.

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

Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm || · ||.

(i) Let f : X → Y be an odd mapping satisfying (2.12). Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0.

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (2.12). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0.

Proof. The proof follows from Theorem 2.4 by taking φ(x, y) := θ(||x||p + ||y||p) for all x, y ∈ X, as desired.     □

 

3. ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITY (0.2)

In this section, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces.

Lemma 3.1.

(i) If an odd mapping f : X → Y satisfies

for all x, y ∈ X, then f is the Cauchy additive mapping.

(ii) If an even mapping f : X → Y satisfies f(0) = 0 and (3.1), then f is the quadratic mapping.

Proof. (i) Letting y = 0 in (3.1), we get

for all x ∈ X.

It follows from (3.1) and (3.2) that

and so

for all x, y ∈ X.

(ii) Letting y = 0 in (3.1), we get

for all x ∈ X.

It follows from (3.1) and (3.3) that

and so

for all x, y ∈ X.      □

Theorem 3.2. Let φ : X2 → [0, ∞) be a function such that

for all x, y ∈ X.

(i) Let f : X → Y be an odd mapping satisfying

for all x, y ∈ X and all t > 0. Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0, where for all x, y ∈ X.

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (3.5). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0, where for all x, y ∈ X.

Proof. (i) Letting y = 0 in (3.5), we get

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (3.4) and (3.9) 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 (3.9), we get (3.6).

By (3.5),

for all x, y ∈ X, all t > 0 and all n ∈ ℕ. So

for all x, y ∈ X, all t > 0 and all n ∈ ℕ. Since for all x, y ∈ X and all t > 0,

for all x, y ∈ X. By Lemma 3.1, the mapping A : X → Y is Cauchy additive.

(ii) Letting y = 0 in (3.5), we get

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (3.4) and (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 Q : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.11), we get (3.7).

The rest of the prrof is similar to the above additive case.     □

Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm || · ||.

(i) Let f : X → Y be an odd mapping satisfying

for all x, y ∈ X and all t > 0. Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0.

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (3.12). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0.

Proof. The proof follows from Theorem 3.2 by taking φ(x, y) := θ(||x||p + ||y||p) for all x, y ∈ X, as desired.     □

Theorem 3.4. Let φ : X2 → [0, ∞) be a function such that

for all x, y ∈ X.

(i) Let f : X → Y be an odd mapping satisfying (3.5). Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0, where for all x, y ∈ X.

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (3.5). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0, where for all x, y ∈ X.

Proof. (i) It follows from (3.8) that

and so

for all x ∈ X and all t > 0.

(ii) It follows from (3.10) that

and so

for all x ∈ X and all t > 0.

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

Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm || ·||.

(i) Let f : X → Y be an odd mapping satisfying (3.12). Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X.

(ii) Let f : X → Y be an even mapping satisfying f(0) = 0 and (3.12). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X.

Proof. The proof follows from Theorem 3.4 by taking φ(x, y) := θ(||x||p + ||y||p) for all x, y ∈ X, as desired.     □