Local stability of the Pexiderized Cauchy and Jensen's equations in fuzzy spaces

The functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to the true solution of (ξ).

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms:

Let G₁ be a group and let G₂ be a metric group with the metric d(·,·). Given ε > 0, does there exist δ > 0 such that if a function h : G₁ → G₂ satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G₁, then there exists a homomorphism H : G₁ → G₂ with d(h(x), H(x)) < ε for all x ∈ G₁?

In other words, we are looking for situations when the homomorphisms are stable, i.e., if a mapping is almost a homomorphism, then there exists a true homomorphism near it. If we turn our attention to the case of functional equations, then we can ask the question: When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation.

In 1941, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings under the assumption that G₁ and G₂ are Banach spaces. In 1950, Aoki [3] provided a generalization of the Hyers' theorem for additive mappings, and in 1978, Th.M. Rassias [4] succeeded in extending the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The stability phenomenon that was introduced and proved by Th.M. Rassias is called the generalized Hyers-Ulam stability. Forti [6] and Gǎvruta [7] have generalized the result of Th.M. Rassias, which permitted the Cauchy difference to become arbitrary unbounded. 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. A large list of references can be found, for example, in [8‐29].

Following [30], we give the following notion of a fuzzy norm.

Definition 1.1. [30] 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 s, t ∈ ℝ,

(N₁) N(x, t) = 0 for all t ≤ 0;

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

(N₃) N ( c x , t ) = N ( x , t | c | ) if c ≠ 0;

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

(N₅) N(x,·) is a nondecreasing function on ℝ and lim_t→∞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.

Example 1.2. Let (X, ||·||) be a normed linear space and let α, β > 0. Then,

Example 1.3. Let (X, ||·||) be a normed linear space and let β > α > 0. Then,

Definition 1.4. Let (X, N) be a fuzzy normed space. A sequence {x _n } in X is said to be convergent if there exists x ∈ X such that lim_n→∞N(x _n - x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {x _n }, and we denote it by N - lim x _n = x.

The limit of the convergent sequence {x _n } in (X, N) is unique. Since if N - lim x _n = x and N-lim x _n = y for some x, y ∈ X, it follows from (N₄) that

for all t > 0 and n ∈ ℕ. So, N(x - y, t) = 1 for all t > 0. Hence, (N₂) implies that x = y.

Definition 1.5. Let (X, N) be a fuzzy normed space. A sequence {x _n } in X is called a Cauchy sequence if, for any ε > 0 and t > 0, there exists M ∈ ℕ such that, for all n ≥ M and p > 0,

It follows from (N₄) that every convergent sequence in a fuzzy normed space is a Cauchy sequence. If, in a fuzzy normed space, every Cauchy sequence is convergent, then the fuzzy norm is said to be complete, and the fuzzy normed space is called a fuzzy Banach space.

Example 1.6. [21] Let N : ℝ × ℝ → [0, 1] be a fuzzy norm on ℝ defined by

Recently, several various fuzzy stability results concerning a Cauchy sequence, Jensen and quadratic functional equations were investigated in [17‐20].

In 1998, Jung [16] investigated the Hyers-Ulam stability for Jensen's equation on a restricted domain. In this section, we prove a local Hyers-Ulam stability of the Pexiderized Jensen functional equation in fuzzy normed spaces.

Theorem 2.1. Let X be a normed space, (Y, N) be a fuzzy Banach space, and f, g, h : X→ Y be mappings with f(0) = 0. Suppose that δ > 0 is a positive real number, and z₀is a fixed vector of a fuzzy normed space (Z, N') such that

for all x, y ∈ X with ||x|| + ||y|| ≥ d and positive real numbers t, s. Then, there exists a unique additive mapping T : X→ Y such that

for all x ∈ X and t > 0.

Proof. Suppose that ||x|| + ||y|| < d holds. If ||x|| + ||y|| = 0, let z ∈ X with ||z|| = d. Otherwise,

It is easy to verify that

It follows from (N₄), (2.1) and (2.5) that

for all x, y ∈ X with ||x|| + ||y|| < d and positive real numbers t, s. Hence, we have

for all x, y ∈ X and positive real numbers t, s. Letting x = 0 (y = 0) in (2.6), we get

for all x, y ∈ X and positive real numbers t, s. It follows from (2.6) and (2.7) that

for all x, y ∈ X and positive real numbers t, s. Hence,

for all x, y ∈ X and positive real numbers t, s. Letting y = x and t = s in (2.8), we infer that

for all x ∈ X and positive real number t. replacing x by 2 ⁿ x in (2.9), we get

for all x ∈ X, n ≥ 0 and positive real number t. It follows from (2.10) that

for all x ∈ X, t > 0 and integers n ≥ m ≥ 0. For any s, ε > 0, there exist an integer l > 0 and t₀ > 0 such that N'(10δz₀, t₀) > 1 - ε and ∑ k = m n - 1 t 0 2 k > s for all n ≥ m ≥ l. Hence, it follows from (2.11) that

for all n ≥ m ≥ l. So { f ( 2 n x ) 2 n } is a Cauchy sequence in Y for all x ∈ X. Since (Y, N) is complete, { f ( 2 n x ) 2 n } converges to a point T(x) ∈ Y. Thus, we can define a mapping T : X → Y by T ( x ) : = N - lim n → ∞ f ( 2 n x ) 2 n. Moreover, if we put m = 0 in (2.11), then we observe that

Therefore, it follows that

for all x ∈ X and positive real number t.

Next, we show that T is additive. Let x, y ∈ X and t > 0. Then, we have

Since, by (2.8),

we get

By the definition of T, the first three terms on the right hand side of the inequality (2.13) tend to 1 as n → ∞. Therefore, by tending n → ∞ in (2.13), we observe that T is additive.

Next, we approximate the difference between f and T in a fuzzy sense. For all x ∈ X and t > 0, we have

Since T ( x ) : = N - lim n → ∞ f ( 2 n x ) 2 n, letting n → ∞ in the above inequality and using (N) and (2.12), we get (2.2). It follows from the additivity of T and (2.7) that

for all x ∈ X and t > 0. So, we get (2.3). Similarly, we can obtain (2.4).

To prove the uniqueness of T, let S : X→ Y be another additive mapping satisfying the required inequalities. Then, for any x ∈ X and t > 0, we have

Therefore, by the additivity of T and S, it follows that

for all x ∈ X, t > 0 and n ≥ 1. Hence, the right hand side of the above inequality tends to 1 as n → ∞. Therefore, T(x) = S(x) for all x ∈ X. This completes the proof.    □

The following is a local Hyers-Ulam stability of the Pexiderized Cauchy functional equation in fuzzy normed spaces.

Theorem 2.2. Let X be a normed space, (Y, N) be a fuzzy Banach space, and f, g, h : X→ Y be mappings with f(0) = 0. Suppose that δ > 0 is a positive real number, and z₀is a fixed vector of a fuzzy normed space (Z, N') such that

for all x, y ∈ X with ||x|| + ||y|| ≥ d and positive real numbers t, s. Then, there exists a unique additive mapping T : X→ Y such that

for all x ∈ X and t > 0.

Proof. For the case ||x|| + ||y|| < d, let z be an element of X which is defined in the proof of Theorem 2.1. It follows from (N₄), (2.5) and (2.14) that

for all x, y ∈ X with ||x|| + ||y|| < d and positive real numbers t, s. Hence, we have

for all x, y ∈ X and positive real numbers t, s. Letting x = 0 (y = 0) in (2.15), we get

for all x, y ∈ X and positive real numbers t, s. It follows from (2.15) and (2.16) that

for all x, y ∈ X and positive real numbers t, s. The rest of the proof is similar to the proof of Theorem 2.1, and we omit the details.    □