FUZZY STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION WITH THE FIXED POINT ALTERNATIVE

Abstract

In [41], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed positive integer l holds for all x₁, ⋯ , x_2l ∈ V . For the above equality, we can define the following functional equation Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces.

1. INTRODUCTION AND PRELIMINARIES

The stability problem of functional equations originated from a question of Ulam [51] concerning the stability of group homomorphisms. Hyers [13] 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 Th.M. Rassias [40] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th.M. Rassias’ approach.

The functional equation

f(x + y) + f(x − y) = 2f(x) + 2f(y)

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [50] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [9] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [14, 16, 17, 31, 32, 33, 34, 35, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49]).

In [33], Park, Lee and Shin proved that an even mapping f : V → W satisfies the functional equation (0.1) if and only if the even mapping f : V → W is quadratic. Moreover, they proved the Hyers-Ulam stability of the quadratic functional equation (0.1) in real Banach spaces.

Katsaras [18] 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, 20, 52]. In particular, Bag and Samanta [2], following Cheng and Mordeson [7], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [19]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3].

We use the definition of fuzzy normed spaces given in [2, 24, 25] to investigate a fuzzy version of the Hyers-Ulam stability for the functional equation (0.1) in the fuzzy normed vector space setting.

Definition 1.1 ([2, 24, 25, 26]). 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)if c ≠= 0; (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 [24, 27].

Definition 1.2 ([2, 24, 25, 26]). 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 ([2, 24, 25]). 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 [3]).

Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies

(1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory.

Theorem 1.4 ([4, 10]). Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either

d(Jnx, Jn+1x) = ∞

for all nonnegative integers n or there exists a positive integer n0 such that

(1), ∀n≥ n0; (2) the sequence {Jnx} converges to a fixed point y* of J; (3) y* is the unique fixed point of J in the set Y = {y ∈ X | d(Jn0x, y) < ∞}; (4) for all y ∈ Y .

In 1996, G. Isac and Th.M. Rassias [15] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 27, 29, 30, 36]).

Starting with the paper [24], the stability of some functional equations in the framework of fuzzy normed spaces or random normed spaces has been investigated (see e.g., [21, 22, 23, 24, 25, 26, 27, 28]).

This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces for an odd case. In Section 3, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces for an even case.

Throughout this paper, assume that X is a vector space and that (Y, N) is a fuzzy Banach space. Let l be a fixed positive integer.

 

2. HYERS-ULAM STABILITY OF THE FUNCTIONAL EQUATION (0.1): AN ODD CASE

For a given mapping f : X → Y , we define

for all x1, ⋯ , x2l ∈ X.

Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation Cf(x1, ⋯ , x2l) = 0 in fuzzy Banach spaces: an odd case.

Theorem 2.1. Let φ : X2l → [0,∞) and be functions such that there exists an L < 1 with

for all x1, ⋯ , x2l ∈ X. Let f : X → Y be an odd mapping satisfying

for all x1, ⋯ , x2l ∈ 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.

Proof. Letting x1 = ⋯ = xl = x and xl+1 = ⋯ = x2l = 0 in (2.1), we get

for all x ∈ X.

Consider the set

and introduce the generalized metric on S:

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete. (See the proof of Lemma 2.1 of [22].)

Now we consider the linear mapping J : S → S such that

for all x ∈ X.

Let g, h ∈ S be given such that d(g, h) = ε. Then

for all x ∈ X and all t > 0. Hence

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that

for all g, h ∈ S.

It follows from (2.3) that

By Theorem 1.4, there exists a mapping A : X → Y satisfying the following:

(1) A is a fixed point of J, i.e.,

for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set

This implies that A is a unique mapping satisfying (2.4) such that there exists a µ ∈ (0, ∞) satisfying

for all x ∈ X;

(2) d(Jnf, A) → 0 as n → ∞. This implies the equality

for all x ∈ X;

(3), which implies the inequality

This implies that the inequality (2.2) holds.

By (2.1),

for all x1, ⋯ , x2l ∈ X, all t > 0 and all n ∈ ℕ. So

for all x1, ⋯ , x2l ∈ X, all t > 0 and all n ∈ ℕ. Since

for all x1, ⋯ , x2l ∈ X and all t > 0,

for all x1, ⋯ , x2l ∈ X and all t > 0. Thus CA(x1, ⋯ , x2l) = 0. Since A is odd, it follows from Lemma 2.1 of [35] that the mapping A : X → Y is additive, as desired.              ☐

Corollary 2.2. Let θ ≥ 0 and let p be a real number with p> 1. Let X be a normed vector space with norm ∥ · ∥. Let f : X → Y be an odd mapping satisfying

for all x1, ⋯ , x2l ∈ 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.

Proof. The proof follows from Theorem 2.1 by taking

for all x1, ⋯ , x2l ∈ X. Then we can choose L = 21−p and we get the desired result.              ☐

Theorem 2.3. Let φ : X2l → [0,∞) and be functions such that there exists an L < 1 with

for all x1, ⋯ , x2l ∈ X. Let f : X → Y be an odd mapping satisfying (2.1). Then exists for each x ∈ X and defines an additive mapping A : X → Y such that

for all x ∈ X and all t > 0.

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1.

Consider the linear mapping J : S → S such that

for all x ∈ X.

It follows from (2.3) that

for all x ∈ X and all t > 0. Thus

for all x ∈ X and all t > 0. So .

The rest of the proof is similar to the proof of Theorem 2.1.               ☐

Corollary 2.4. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm ∥ · ∥. 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.

Proof. The proof follows from Theorem 2.3 by taking

for all x1, ⋯ , x2l ∈ X. Then we can choose L = 2p−1 and we get the desired result.               ☐

 

3. HYERS-ULAM STABILITY OF THE FUNCTIONAL EQUATION (0.1): AN EVEN CASE

In this section, using the fixed point method, we prove the Hyers-Ulam stability of the functional equation Cf(x1, ⋯ , x2l) = 0 in fuzzy Banach spaces: an even case.

Theorem 3.1. Let φ : X2l → [0,∞) and be functions such that there exists an L < 1 with

for all x1, ⋯ , x2l ∈ X. Let f : X → Y be an even mapping satisfying f(0) = 0 and (2.1). 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. Letting x1 = ⋯ = xl = x and xl+1 = ⋯ = x2l = 0 in (2.1), we get

for all x ∈ X.

Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that

for all x ∈ X.

Let g, h ∈ S be given such that d(g, h) = ε. Then

for all x ∈ X and all t > 0. Hence

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that

for all g, h ∈ S.

It follows from (3.2) that

By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following:

(1) Q is a fixed point of J, i.e.,

for all x ∈ X. Since f : X → Y is even, Q : X → Y is an even mapping. The mapping Q is a unique fixed point of J in the set

This implies that Q is a unique mapping satisfying (3.3) such that there exists a µ ∈ (0, ∞) satisfying

for all x ∈ X;

(2) d(Jnf, Q) → 0 as n → ∞. This implies the equality

for all x ∈ X;

(3) , which implies the inequality

This implies that the inequality (3.1) holds.

The rest of the proof is similar to the proof of Theorem 2.1.               ☐

Corollary 3.2. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm ∥ · ∥. 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.

Proof. The proof follows from Theorem 3.1 by taking

for all x1, ⋯ , x2l ∈ X. Then we can choose L = 22−p and we get the desired result.               ☐

Similarly, we can obtain the following. We will omit the proof.

Theorem 3.3. Let φ : X2l → [0,∞) and be functions such that there exists an L < 1 with

for all x1, ⋯ , x2l ∈ X. Let f : X → Y be an even mapping satisfying f(0) = 0 and (2.1). Then exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

for all x ∈ X and all t > 0.

Corollary 3.4. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm ∥ · ∥. 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.

Proof. The proof follows from Theorem 3.3 by taking

for all x1, ⋯ , x2l ∈ X. Then we can choose L = 2p−2 and we get the desired result.               ☐