Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations

Abstract

The aim of this article is to extend the applications of a fixed point method to provide the intuitionistic fuzzy versions of Hyers-Ulam stability for the generalized forms of cubic and quartic functional equations. This way shows that the concept of stability is related to some fixed point of a suitable operator.

Keywords

Cubic functional equation Hyers-Ulam stability intuitionistic fuzzy normed space quartic functional equation

1 Introduction

The notion of fuzzy sets which is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering has been introduced by Zadeh [40] in 1965. After that, fuzzy theory has become very active area of research and a lot of developments have been made in the theory of fuzzy sets to find the fuzzy analogues of the classical set theory; for example see [27]. Some of them are fuzzy linear systems which have many applications in science, such as control problems, information, physics, statistics, engineering, economics, finance and even social sciences; for instance, see [1, 2]. Also, a fuzzy version of measures, namely, intuitionistic fuzzy measures can be found in [7].

The concept of intuitionistic fuzzy normed spaces, initially has been introduced by Saadati and Park in [33]. Then, Saadati et al. have obtained a modified case of intuitionistic fuzzy normed spaces by improving the separation condition and strengthening some conditions in the definition of [34].

In [37], Ulam proposed the general Ulam stability problem: “When is it true that by slightly changing the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In [17], Hyers gave the first affirmative answer to the question of Ulam for additive functional equations on Banach spaces. On the other hand, Cădariu and Radu noticed that a fixed point alternative method is very important for the solution of the Ulam problem. In other words, they employed this fixed point method to the investigation of the Cauchy functional equation [14] and for the quadratic functional equation [13] (for more applications of this method, see [10 , 16]). The generalized Hyers-Ulam stability of different functional equations in intuitionistic fuzzy normed spaces has been studied by a number of the authors (see [22 , 38])

In 2001, J.M. Rassias [28], introduced the cubic functional equation

$\begin{matrix} f (x + 2 y) - 3 f (x + y) + 3 f (x) - f (x - y) \\ = 6 f (y) \end{matrix}$ (1.1) and established the solution of the Ulam-Hyers stability problem for these cubic mappings. It is easy to see that the function f (x) = ax³ satisfies (1.1). Thus, every solution of the cubic functional Equation (1.1) is said to be a cubic function. The following alternative cubic functional equation

$\begin{matrix} f (2 x + y) + f (2 x - y) \\ = 2 f (x + y) + 2 f (x - y) + 12 f (x) \end{matrix}$ (1.2) has been introduced by Jun and Kim in [24]. They found out the general solution and established the Hyers-Ulam stability for the functional Equation (1.2). They also [25] introduced the cubic functional equation

$\begin{matrix} f (x + 2 y) + f (x - 2 y) \\ = 4 f (x + y) + 4 f (x - y) - 6 f (x) \end{matrix}$ (1.3) and proved the Hyers-Ulam stability problem for (1.3). Since f (2x) =8f (x), the last functional equation is equal to the following

$\begin{matrix} f (x + 2 y) + f (x - 2 y) = 2 f (x) - f (2 x) \\ + 4 f (x + y) + 4 f (x - y) . \end{matrix}$ (1.4) Some generalized cubic functional Equations of (1.2) and (1.4) have been introduced in [19, 23], respectively. In [11], Bodaghi et al. introduced a generalization of the cubic functional Equation (1.4) as follows:

$\begin{matrix} f (x + ny) + f (x - ny) \\ = 2 (2 cos (\frac{n π}{2}) + n^{2} - 1) f (x) \\ - \frac{1}{2} (cos (\frac{n π}{2}) + n^{2} - 1) f (2 x) \\ + n^{2} {f (x + y) + f (x - y)} \end{matrix}$ (1.5) for an integer n ≥ 1. They determined the general solution and proved the Hyers-Ulam stability problem for the Equation (1.5). The stability of the Equation (1.5) in various spaces is studied in [9].

The quartic functional equation

$\begin{matrix} f (x + 2 y) + f (x - 2 y) + 6 f (x) \\ = 4 f (x + y) + 4 f (x - y) + 24 f (y) \end{matrix}$ (1.6) was introduced by Rassias [29]. The fact that every solution of (1.6) is even implies that it can be written as follows:

$\begin{matrix} f (2 x + y) + f (2 x - y) = 24 f (x) \\ - 6 f (y) + 4 f (x + y) + 4 f (x - y) . \end{matrix}$ (1.7) Lee et al. [20] investigated the general solution and the Hyers-Ulam stability of (1.7). Lee and Chung [21] considered the following quartic functional equation, which is a generalization of (1.7),

$\begin{matrix} f (mx + y) + f (mx - y) \\ = 2 m^{2} (m^{2} - 1) f (x) - 2 (m^{2} - 1) f (y) \\ + m^{2} f (x + y) + m^{2} f (x - y) \end{matrix}$ (1.8) for fixed integer m with m ≠ 0, ± 1. Lastly, Kang [18] obtained the general form of a quartic functional equation as follows:

$\begin{matrix} f (mx + ny) + f (mx - ny) \\ = m^{2} n^{2} {f (x + y) + f (x - y)} \\ + 2 m^{2} (m^{2} - n^{2}) f (x) \\ - 2 n^{2} (m^{2} - n^{2}) f (y) \end{matrix}$ (1.9) for fixed integers m and n such that m ≠ 0, n ≠ 0, m ± n ≠ 0 (for the correction of some details in [18] see [8]).

In this paper we firstly introduce the following functional equation which is somewhat different from (1.5):

$\begin{matrix} f (mx + y) + f (mx - y) \\ = m {f (x + y) + f (x - y)} \\ + 2 m (m^{2} - 1) f (x) \end{matrix}$ (1.10) where m is an integer and m ≥ 2. Obtaining the general solution of (1.10), we show that it is a cubic functional equation. Then, by combining the Equations (1.5) and (1.10), we consider the following new cubic functional equation which is the generalization from of the previous versions:

$\begin{matrix} f (mx + ny) + f (mx - ny) \\ = {mn}^{2} {f (x + y) + f (x - y)} \\ + 2 m (m^{2} - n^{2}) f (x) \end{matrix}$ (1.11) where m, n are integer numbers with m ≥ 2. Note that when m = 2, n = 1 or m = 1, n = 2, we have the Equations (1.2) and (1.3), respectively. Finally, we determine some stability results concerning the functional Equations (1.9) and (1.11) in the setting of intuitionistic fuzzy normed spaces by a fixed point alternative method. We also study the intuitionistic fuzzy continuity through the existence of a certain solution of a fuzzy stability problem for approximately cubic and quartic functional equations.

2 Solution of Equation (1.11)

To achieve our aim in this section, we need the following lemma.

Lemma 2.1. Let X and Y be real vector spaces. If a function f : X ⟶ Y satisfies the functional Equation (1.10) for all integers m ≥ m₀, where m₀ is a fixed positive integer, then f satisfies f (kx) = k³f (x) for all positive integers k.

Proof. Letting x = y = 0 in (1.10), we get f (0) =0. Once more, by putting x = 0 in (1.10), we have f (- y) = - f (y) for all y ∈ X. Repalcing (x, y) by (x, mx) in (1.10), we have $\begin{matrix} f (2 mx) + f (0) & = & m {f ((m + 1) x) + f ((m - 1) x)} \\ + 2 m (m^{2} - 1) f (x) (x \in X) . \end{matrix}$ By assumption and the obtained results, we have f ((m - 1) x) = (m - 1) ³f (x) for all x ∈ X. Now, let k be a positive integer number. Interchanging (x, y) into (x, (m + k) x) in (1.10), we have $\begin{matrix} f ((2 m + k) x + f (kx)) \\ = m {f ((m + k + 1) x) - f ((m + k - 1) x)} \\ + 2 m (m^{2} - 1) f (x) (x \in X) . \end{matrix}$ It now follows from the hypothesis and the above equality that f (kx) = k³f (x) for all x ∈ X. □

Theorem 2.2.LetX and Y be real vector spaces. Then, for the mapping f : X ⟶ Y the following statements are equivalent:

f satisfies the functional Equation (1.2);

f satisfies the functional Equation (1.4);

f satisfies the functional Equation (1.10);

f satisfies the functional Equation (1.5).

Proof. (i)⇒(ii) Assume that f : X ⟶ Y satisfies the functional Equation (1.2). Putting x = y = 0 in (1.2), we have f (0) =0. Letting x = 0 in (1.2), we get f (- y) = - f (y) for all y ∈ X. Easily, we have f (2x) =8f (x) for all x ∈ X. Replacing y by 2y in (1.2) and using the last equality, we get (1.4).

(ii)⇒(i) The proof is similar to the former part.

(i)⇒(iii) Replacing x by x + y and x - y in (1.2), respectively, and applying again (1.2), we obtain $\begin{matrix} f (3 x + y) + f (3 x - y) \\ = 3 {f (x + y) + f (x - y)} + 48 f (x) . \end{matrix}$ Similar to the above, we can deduce that $\begin{matrix} f (4 x + y) + f (4 x - y) \\ = 4 {f (x + y) + f (x - y)} + 120 f (x) . \end{matrix}$ Using the above method, we get $\begin{matrix} f (mx + y) + f (mx - y) \\ = a_{m} {f (x + y) + f (x - y)} + b_{m} f (x) \end{matrix}$ where ${\begin{matrix} a_{m} = 2 a_{m - 1} - a_{m - 2}, a_{2} = 2, a_{3} = 3 \\ b_{m} = 12 a_{m - 1} + 2 b_{m - 1} - b_{m - 2}, \\ b_{2} = 12, b_{3} = 48 . \end{matrix}$ Solving the above recurrence equations, we have $a_{m} = m and b_{m} = 2 m (m^{2} - 1)$ for all x, y ∈ X and all positive integers m ≥ 2.

(iii)⇒(i) Suppose that f satisfies the functional Equation (1.10) for any positive integer m ≥ m₀. So f satisfies (1.10) for every positive integer k, in particular for k = m (m - 1). Hence for each x, y ∈ X, we have

$\begin{matrix} f (m (m - 1) x + y) + f (m (m - 1) x - y) \\ = m {f ((m - 1) x + y) + f ((m - 1) x - y)} \\ + 2 m (m^{2} - 1) f ((m - 1) x) \end{matrix}$ (2.12) for all x, y ∈ X. On the other hand,

$\begin{matrix} f ((m^{2} - m) x + y) + f (m^{2} - m) x - y) \\ = (m^{2} - m) {f (x + y) + f (x - y)} \\ + 2 (m^{2} - m) ((m^{2} - m)^{2} - 1) f (x) \end{matrix}$ (2.13) for all x, y ∈ X. Using (2.12) and (2.13), we get

$\begin{matrix} m {f ((m - 1) x + y) + f ((m - 1) x - y)} \\ = (m^{2} - m) {f (x + y) + f (x - y)} \\ + 2 (m^{2} - m) ((m^{2} - m)^{2} - 1) f (x) \\ - 2 m (m^{2} - 1) f ((m - 1) x) \end{matrix}$ (2.14) for all x, y ∈ X. By Lemma 2.1, we have f ((m - 1) x) = (m - 1) ³f (x) for all x ∈ X. The last equality and (2.14) imply that $\begin{matrix} f ((m - 1) x + y) + f ((m - 1) x - y) \\ = (m - 1) {f (x + y) + f (x - y)} \\ + 2 (m - 1) ((m - 1)^{2} - 1) f (x) \end{matrix}$ for all x, y ∈ X.

(ii)⇔(iv) It is proved in [11, Theorem 2.1]

Remark 2.3. Recall that the Equation (1.11) is cubic and by Theorem 2.2 it is equivalent to the Equation (1.3). Since f (2x) =8f (x), this cubic equation is equal to $\begin{matrix} f (x + my) + f (x - my) \\ = m^{2} {f (x + y) + f (x - y)} - 2 (m^{2} - 1) f (x) . \end{matrix}$ for all integer numbers n ≥ 1.

Corollary 2.4. The functional Equation (1.11) is cubic and hence, every solution of the functional Equation (1.11) is also a cubic mapping.

Proof. The result immediately follows from Theorem 2.2 and Remark 2.3.□

3 Intuitionistic fuzzy stability of (1.9)and (1.11)

In this section, we restate the usual terminology, notations and conventions of the theory of intuitionistic fuzzy normed space, as in [26 , 35]. In general, the definition of an intuitionistic fuzzy set is given in [6] for the first time. And the research in this area has been described in details in [3] and [4].

Let ≤_L be a order relation on the set $L = {(x_{1}, x_{2}) : (x_{1}, x_{2}) \in [0, 1]^{2}, x_{1} + x_{2} \leq 1}$ defined by $(x_{1}, x_{2}) \leq_{L} (y_{1}, y_{2}) \Leftrightarrow x_{1} \leq y_{1}, y_{2} \leq x_{2}$ for all (x₁, x₂) , (y₁, y₂) ∈ L. It is easy to check that the pair (L, ≤ _L) is a complete lattice (see also [30] and [35]). We denote the units of L by 0_L = (0, 1) and 1_L = (1, 0). Recall that the above order relation is a well-known definition due to Atanassov [5].

Definition 3.1. Let U be a non-empty set called the universe. An L-fuzzy set in U is defined as a mapping $F : U ⟶ L$ . For each u in U, $F (u)$ represents the degree (in L) to which u is an element of $F$ . An intuitionistic fuzzy set $F_{μ, ν}$ in a universal set U is an object $F_{μ, ν} = {(μ_{F} (u), ν_{F} (u)) : u \in U}$ , where $μ_{F} (u)$ and $ν_{F} (u)$ belong to [0, 1] for all u ∈ U with $μ_{F} (u) + ν_{F} (u) \leq 1$ . The numbers $μ_{F} (u)$ and $ν_{F} (u)$ are called the membership degree and the non-membership degree, respectively, of u in $F_{μ, ν}$ .

Definition 3.2. A triangular norm (t-norm) on L is a mapping $T : L \times L ⟶ L$ satisfying the following conditions:

$T (x, 1_{L}) = x$ (boundary condition) (x ∈ L);

$T (x, y) = T (y, x)$ (commutativity) (x, y ∈ L);

$T (x, T (y, z)) = T (T (x, y), z)$ (associativity) (x, y, z ∈ L);

x₁ ≤ _Ly₁ and $x_{2} \leq_{L} y_{2} \Rightarrow T (x_{1}, x_{2}) \leq_{L} T (y_{1}, y_{2})$ (monotonicity) (x₁, x₂, y₁, y₂ ∈ L).

A t-norm $T$ on L is said to be continuous if,for any x, y ∈ L and any sequences {x_n} and {y_n} which converge to x and y, respectively, then $lim_{n \to \infty} T (x_{n}, y_{n}) = T (x, y) .$

Example 3.3. Let x = (x₁, x₂) , y = (y₁, y₂) ∈ L. Then $T (x, y) = (x_{1} y_{1}, \min {x_{2} + y_{2}, 1})$ and $M (x, y) = (\min {x_{1}, y_{1}}, \max {x_{2}, y_{2}})$ are continuous t-norm [38].

Here, we define a sequence $T^{n}$ , recursively by $T^{1} = T$ and $\begin{matrix} T^{n} (x^{(1)}, x^{(1)}, \dots, x^{(n + 1)}) \\ = T (T^{n - 1} (x^{(1)}, x^{(1)}, \dots, x^{(n)}), x^{(n + 1)}) \end{matrix}$ for all n ≥ 2 and x^(j) ∈ L.

Definition 3.4. A negator on L is a decreasing mapping $N : L ⟶ L$ satisfying $N (0_{L}) = 1_{L}$ and $N (1_{L}) = 0_{L}$ . If $N (N (x)) = x$ , for all x ∈ L, then $N$ is called an involutive negator. A negator on [0, 1] is a decreasing mapping $N : L ⟶ L$ satisfying $N (0) = 1$ and $N (1) = 0$ . The standard negator on [0, 1] is defined by $N_{s} (x) = 1 - x$ for all x ∈ [0, 1].

The following defnitions of an intuitionistic fuzzy normed space is taken from [31].

Definition 3.5. Let $L = (L, \leq_{L})$ . Let X be a vector space, $T$ be a continuous t-norm on L and $P$ be an L-fuzzy set on X × (0, ∞) satisfying the following conditions:

$0 <_{L} P (x, t)$ ;

$P (x, t) = 1_{L}$ if and only if x = 0;

$P (α x, t) = P (x, \frac{t}{| α |})$ for all α ≠ 0;

$T (P (x, t), P (y, s)) \leq_{L} P (x + y, t + s)$ ;

The map $P (x, \cdot) : (0, \infty) ⟶ L$ is continuous;

$lim_{t \to 0} P (x, t) = 0_{L}$ and $lim_{t \to \infty} P (x, t) = 1_{L}$ ;

for all x, y ∈ X and all t, s > 0. Then the triple

(X, P, T)

is called an L-fuzzy normed space. In this case

P

is called

L

-fuzzy norm (briefly, L-fuzzy norm). If

P = P_{μ, ν}

is an intuitionistic fuzzy set, then the triple

(X, P_{μ, ν}, T)

is said to be an intuitionistic fuzzy normed space (briefly, IFN-space). In this case,

P_{μ, ν}

is called an intuitionistic fuzzy norm on X (Some example of IFN-spaces are provided in [38, 39]).

Note that, if $P$ is an L-fuzzy norm on X, then the following statements hold:

$P (x, t)$ is nondecreasing with respect to t for all x ∈ X;

$P (x - y, t) = P (y - x, t)$ for all x, y ∈ X and t > 0.

Example 3.6. [38] Let (X, ∥ · ∥) be a normed space. Let $T (x, y) = (x_{1} y_{1}, \min {x_{2} + y_{2}, 1})$ for all x = (x₁, x₂) , y = (y₁, y₂) ∈ L and μ, ν be membership and non-membership degree, respectively, of an intuitionistic fuzzy set defined by $P_{μ, ν} (x, t) = (μ (x, t), ν (x, t)) = (\frac{t}{t + ∥ x ∥}, \frac{∥ x ∥}{t + ∥ x ∥})$ for all $t \in ℝ^{+} .$ Then $(X, P_{μ, ν}, T)$ is an IFN-space.

Definition 3.7. Let $(X, P_{μ, ν}, T)$ be an IFN-space.

A sequence {x_n} in $(X, P_{μ, ν}, T)$ is said to be convergent to a point x if $P_{μ, ν} (x_{n} - x, t) \to 1_{L}$ as n→ ∞ for all t > 0;

A sequence {x_n} in $(X, P_{μ, ν}, T)$ is called a Cauchy sequence if, for every t > 0 and 0 < ɛ < 1, there exists a positive integer N such that $(N_{s} (ɛ), ɛ) \leq_{L} P_{μ, ν} (x_{n} - x_{m}, t)$ for all m, n > N, where N_s is the standard negator;

$(X, P_{μ, ν}, T)$ is said to be complete if and only if every Cauchy sequence in $(X, P_{μ, ν}, T)$ is convergent to a point in $(X, P_{μ, ν}, T)$ . A complete intuitionistic fuzzy normed space is called an intuitionistic fuzzy Banachspace.

We bring the following theorem which a result in fixed point theory [15]. This result plays a fundamental role to arrive our purpose in this paper (an extension of the result was given in [36]).

Theorem 3.8. (The fixed point alternative theorem) Let (Δ, d) be a complete generalized metric space and $J : Δ ⟶ Δ$ be a mapping with Lipschitz constant L < 1. Then, for each element α ∈ Δ, either $d (J^{n} α, J^{n + 1} α) = \infty$ for all n ≥ 0, or there exists a natural number n₀ such that

$d (J^{n} α, J^{n + 1} α) < \infty$ for all n ≥ n₀;

the sequence ${J^{n} α}$ is convergent to a fixed point β^* of $J$ ;

β^* is the unique fixed point of $J$ in the set $Δ_{1} = {β \in Δ : d (J^{n_{0}} α, β) < \infty}$ ;

$d (β, β^{*}) \leq \frac{1}{1 - L} d (β, J β)$ for all β ∈ Δ₁.

Let m and n be integers such that m, n ≥ 2. For notational convenience, given a function f : X ⟶ Y, we define the difference operator

$D_{m, n} f (x, y) : = f (mx + ny) + f (mx - ny)$

-mn² {f (x + y) + f (x - y)} -2m (m² - n²) f (x)

for all x, y ∈ X.

Here and subsequently, we assume that $T$ is one of t-norms defined in Example 3.3. In the upcoming theorem, we prove the generalized Ulam-Hyers stability of the Equation (1.11) in intuitionistic fuzzy normed spaces, based on Theorem 3.8.

Theorem 3.9.Let j ∈ {1, - 1} be fixed and let α be a positive real number with α ≠ m³. Let X be a linear space and let $(Z, P_{μ, ν}^{'}, T^{'})$ be an intuitionistic fuzzy normed space. Let φ : X × X ⟶ Z be a mapping such that $P_{μ, ν}^{'} (α^{j} φ (x, y), t) \leq_{L} P_{μ, ν}^{'} (φ (m^{j} x, m^{j} y), t)$ (3.15) for all x ∈ X and t > 0. If $(Y, P_{μ, ν}, T)$ is a complete intuitionistic fuzzy normed space and f : X ⟶ Y is a mapping such that $P_{μ, ν}^{'} (φ (x, y), t) \leq_{L} P_{μ, ν} (D_{m, n} f (x, y), t)$ (3.16) for all x, y ∈ X and t > 0, then there exists a unique cubic mapping $C : X ⟶ Y$ such that $P_{μ, ν}^{'} (φ (x, 0), 2 | m^{3} - α | t) \leq_{L} P_{μ, ν} (f (x) - C (x), t)$ (3.17) for all x ∈ X and t > 0.

Proof. For the case j = 1, we consider α < m³ and for j = -1, regard α > m³. Putting y = 0 in (3.16), we get $P_{μ, ν}^{'} (φ (x, 0), t) \leq_{L} P_{μ, ν} (2 f (mx) - 2 m^{3} f (x), t)$ (3.18) for all x ∈ X and t > 0. Thus

$\begin{matrix} P_{μ, ν}^{'} (φ (m^{\frac{1}{2} (j - 1)} x, 0), 2 m^{\frac{3}{2} (j + 1)} t) \\ \leq_{L} P_{μ, ν} (m^{- 3 j} f (m^{j} x) - f (x), t) \end{matrix}$ (3.19) for all x ∈ X and t > 0. We consider the set Ω = {g : X ⟶ Y} and introduce the generalized metric on X as follows: $\begin{matrix} d (g, h) : = \inf {K \in (0, \infty) : \\ P_{μ, ν}^{'} (φ (m^{\frac{1}{2} (j - 1)} x, 0), 2 t) \\ \leq_{L} P_{μ, ν} (g (x) - h (x)), Kt), \forall x \in X, t > 0} . \end{matrix}$ if there exist such constant K, and d (g, h) =∞, otherwise. It is easy to check that d is a complete metric (see also [12]). Define the mapping $J : Ω ⟶ Ω$ by $Jg (x) = m^{- 3 j} g (m^{j} x)$ for all x ∈ X. Given g, h ∈ Ω. Let δ be an arbitrary constant with d (g, h) < δ. Then $\begin{matrix} P_{μ, ν}^{'} (φ (m^{\frac{1}{2} (j - 1)} x, 0), 2 t) \\ \leq_{L} P_{μ, ν} (g (x) - h (x), δ t) \end{matrix}$ for all x ∈ X and t > 0. So $\begin{matrix} P_{μ, ν}^{'} (φ (m^{\frac{1}{2} (j - 1)} x, 0), 2 {(\frac{m^{3}}{α})}^{j} t) \\ \leq_{L} P_{μ, ν} (g (m^{j} x) - h (m^{j} x), m^{3 j} δ t) \\ = P_{μ, ν} (Jg (x) - Jh (x), δ t) \end{matrix}$ for all x ∈ X and t > 0. Hence, $d (J g, J h) \leq {(\frac{α}{m^{3}})}^{j} d (g, h)$ for all g, h ∈ Ω. Thus $J$ is a strictly contractive mapping of Ω with the Lipschitz constant ${(\frac{α}{m^{3}})}^{j}$ . It follow from (3.19) that $d (f, J f) \leq m^{- \frac{3}{2} (j + 1)}$ . By Theorem 3.8, there exists a mapping $C : X ⟶ Y$ satisfying:

(1) $C$ is a unique fixed point of $J$ in the set Ω₁ = {g ∈ ω : d (f, g) < ∞}, which is satisfied $C (m^{j} x) = m^{3 j} C (x)$ (3.20) for all x ∈ X. In other words, there exists a K > 0 with $\begin{matrix} P_{μ, ν}^{'} (φ (m^{\frac{1}{2} (j - 1)} x, 0), 2 t) \\ \leq_{L} P_{μ, ν} (f (x) - C (x), Kt) \end{matrix}$ for all x ∈ X and t > 0.

(2) $d (J^{k} f, C) \to 0$ as k→ ∞. This implies that $C (x) = lim_{k \to \infty} J^{k} f (x) = lim_{k \to \infty} m^{- 3 jk} f (m^{jk} x)$ (3.21) for all x ∈ X.

(3) For every f ∈ Ω, we have $d (f, C) \leq \frac{1}{1 - {(\frac{α}{m^{3}})}^{j}} d (Jf, f) .$ Since, $d (Jf, f) \leq m^{- \frac{3}{2} (j + 1)}$ , we have $d (f, C) \leq \frac{α^{- \frac{1}{2} (j - 1)}}{| m^{3} - α |}$ . The last inequality implies that

$\begin{matrix} P_{μ, ν}^{'} (φ (m^{\frac{1}{2} (j - 1)} x, 0), 2 t) \\ \leq_{L} P_{μ, ν} (f (x) - C (x), \frac{α^{- \frac{1}{2} (j - 1)}}{| m^{3} - α |} t) \end{matrix}$ (3.22) for all x ∈ X and t > 0. The relations (3.15) and (3.22) show that the inequality (3.17) holds. Now, replace m^jkx and m^jky by x and y in (3.16), respectively, $\begin{matrix} P_{μ, ν}^{'} (φ (x, y), \frac{m^{3 jk}}{α^{jk}} t) \\ \leq_{L} P_{μ, ν}^{'} (D_{m, n} f (m^{jk} x, m^{jk} y), m^{3 jk} t) \\ \leq_{L} P_{μ, ν} (m^{- 3 jk} D_{m, n} f (m^{jk} x, m^{jk} y), t) \end{matrix}$ for all x, y ∈ X and t > 0. Letting k tends to infinity and using (3.21), we see that $C$ is a cubic mapping.□

Corollary 3.10. Let λ be a nonnegative real number with λ : = r + s ≠ 3, X be a normed space with norm ∥· ∥, $(Z, P_{μ, ν}^{'}, T)$ be an intuitionistic fuzzy normed space, $(Y, P_{μ, ν}, T)$ be a complete intuitionistic fuzzy normed space, and let z₀ ∈ Z. If f : X ⟶ Y is a mapping such that

$\begin{matrix} P_{μ, ν}^{'} ((∥ x ∥^{r} ∥ y ∥^{s} + ∥ x ∥^{λ} + ∥ y ∥^{λ}) z_{0}, t) \\ \leq_{L} P_{μ, ν} (D_{m, n} f (x, y), t) \end{matrix}$ for all x, y ∈ X and t > 0, then there exists a unique cubic mapping $C : X ⟶ Y$ such that

$\begin{matrix} P_{μ, ν}^{'} (∥ x ∥^{λ} z_{0}, 2 | m^{3} - m^{λ} | t) \\ \leq_{L} P_{μ, ν} (C (x) - f (x), t) \end{matrix}$ for all x ∈ X and t > 0.

Proof. Defining φ (x, y) : = (∥ x ∥ ^r ∥ y ∥ ^s + ∥ x ∥ ^λ + ∥ y ∥ ^λ) z₀ and applying Theorem 3.9, we get the desired result.□ The idea of th following example is taken from [39, Example 3.7] which provides an illustration.

Example 3.11. Let (A, ∥ · ∥) be a Banach algebra and let $M (x, y)$ be the continuous t-norm defined in Example 3.3. Then $(A, P_{μ, ν}, M)$ is an IFN-space for which $P_{μ, ν}$ is the intuitionistic fuzzy set defined in Example 3.6. Define f : A ⟶ A via f (x) = x³ + ∥ x ∥ x₀, where x₀ is a unit vector in A. An easy computation shows that $\begin{matrix} ∥ D_{m, n} f (x, y) ∥ \\ \leq 2 m (m^{2} + 2 n^{2} + 1) ∥ x ∥ + 2 n (mn + 1) ∥ y ∥ \end{matrix}$ for all x, y ∈ A. Thus $\begin{matrix} P_{μ, ν} ([2 m (m^{2} + 2 n^{2} + 1) ∥ x ∥ + 2 n (mn + 1) ∥ y ∥] x_{0}, t) \\ \leq_{L} P_{μ, ν} (D_{m, n} f (x, y), t) \end{matrix}$ for all x, y ∈ A and t > 0. Consider φ : A × A ⟶ A defined through $\begin{matrix} φ (x, y) & = & [2 m (m^{2} + 2 n^{2} + 1) ∥ x ∥ \\ + 2 n (mn + 1) ∥ y ∥] x_{0} \end{matrix}$ for all x, y ∈ A. We have $P_{μ, ν} (m^{j} φ (x, y), t) \leq_{L} P_{μ, ν} (φ (m^{j} x, m^{j} y), t)$ for all x, y ∈ A and t > 0 in which j ∈ {1, - 1}. Therefore, all the conditions of Theorem 3.9 hold when α = m ≠ m³. It implies that f can be approximated by a cubic mapping. In fact there exists a unique cubic mapping $C : X ⟶ Y$ such that $\begin{matrix} P_{μ, ν}^{'} (∥ x ∥ x_{0}, 4 (m^{4} - m^{2}) (m^{2} + 2 n^{2} + 1) t) \\ \leq_{L} P_{μ, ν} (C (x) - f (x), t) \end{matrix}$ for all x, y ∈ A and t > 0.

In the sequal, for the sake of convenience, let us denote $\begin{matrix} Q_{m, n} f (x, y) : = f (mx + ny) + f (mx - ny) \\ - m^{2} n^{2} {f (x + y) + f (x - y)} \\ - 2 m^{2} (m^{2} - n^{2}) f (x) + 2 n^{2} (m^{2} - n^{2}) f (y) \end{matrix}$ for all x, y ∈ X.

We have the following result which is analogous to Theorem 3.9 for the quartic functional Equation (1.9).

Theorem 3.12.Let j ∈ {1, - 1} be fixed and let α be a positive real number with α ≠ m⁴. Let X be a linear space and let $(Z, P_{μ, ν}^{'}, T^{'})$ be an intuitionistic fuzzy normed space. Let φ : X × X ⟶ Z be a mapping such that $P_{μ, ν}^{'} (α^{j} φ (x, y), t) \leq_{L} P_{μ, ν}^{'} (φ (m^{j} x, m^{j} y), t)$ (3.23) for all x ∈ X and t > 0. If $(Y, P_{μ, ν}, T)$ is a complete intuitionistic fuzzy normed space and f : X ⟶ Y is a mapping such that $P_{μ, ν}^{'} (φ (x, y), t) \leq_{L} P_{μ, ν} (Q_{m, n} f (x, y), t)$ (3.24) for all x, y ∈ X and t > 0, then there exists a unique quartic mapping $Q : X ⟶ Y$ such that $Λ (x, | m^{4} - α | t) \leq_{L} P_{μ, ν} (Q (x) - f (x), t)$ (3.25) for all x ∈ X and t > 0, where

$\begin{matrix} Λ (x, t) : = T (P_{μ, ν}^{'} (φ (0, 0), \frac{1 - (m^{2} - n^{2})^{2}}{n^{2} (m^{2} - n^{2})} t), \\ P_{μ, ν}^{'} (φ (x, 0), t)) . \end{matrix}$ (3.26)

Proof. For the case j = 1, we consider α < m⁴ and for j = -1, regard α > m⁴. Putting x = y = 0 in (3.24), we have $P_{μ, ν}^{'} (φ (0, 0), t) \leq_{L} P_{μ, ν} (2 λ_{m, n} f (0), t)$ for all x ∈ X and t > 0 in which λ_m,n = 1 - (m² - n²) ². Thus

$\begin{matrix} P_{μ, ν}^{'} (φ (0, 0), \frac{t}{n^{2} (m^{2} - n^{2})}) \\ \leq_{L} P_{μ, ν} (2 n^{2} (m^{2} - n^{2}) λ_{m, n} f (0), t) \end{matrix}$ (3.27) for all x ∈ X and t > 0. Letting y = 0 in (3.24), we get $\begin{matrix} P_{μ, ν}^{'} (φ (x, 0), t) \leq_{L} P_{μ, ν} (2 f (mx) \\ - 2 m^{4} f (x) + 2 n^{2} (m^{2} - n^{2}) f (0), t) \end{matrix}$ for all x ∈ X and t > 0. The above inequality implies that

$\begin{matrix} P_{μ, ν}^{'} (φ (x, 0), \frac{t}{λ_{m, n}}) \\ \leq_{L} P_{μ, ν} (2 λ_{m, n} f (mx) - 2 m^{4} λ_{m, n} f (x) \\ + 2 n^{2} (m^{2} - n^{2}) λ_{m, n} f (0), t) \end{matrix}$ (3.28) for all x ∈ X and t > 0. It follows from (3.27) and (3.28) that $\begin{matrix} T (P_{μ, ν}^{'} (φ (0, 0), \frac{t}{2 n^{2} (m^{2} - n^{2})}), \\ P_{μ, ν}^{'} (φ (x, 0), \frac{t}{2 λ_{m, n}})) \\ \leq_{L} P_{μ, ν} (2 λ_{m, n} f (mx) - 2 m^{4} λ_{m, n} f (x), t) \end{matrix}$ for all x ∈ X and t > 0. Hence $Λ (x, t) \leq_{L} P_{μ, ν} (f (mx) - m^{4} f (x), t)$ (3.29) for all x ∈ X and t > 0, where Λ (x, t) is defined in (3.26). Thus

$\begin{matrix} Λ (m^{\frac{1}{2} (j - 1)} x, m^{2 (j + 1)} t) \\ \leq_{L} P_{μ, ν} (m^{- 4 j} f (m^{j} x) - f (x), t) \end{matrix}$ (3.30) for all x ∈ X and t > 0. We take the set Ω = {g : X ⟶ Y} and define the generalized metric on X as follows: $\begin{matrix} d (g, h) : = \inf {K \in (0, \infty) : Λ (m^{\frac{1}{2} (j - 1)} x, t) \\ \leq_{L} P_{μ, ν} (g (x) - h (x)), Kt), \forall x \in X, t > 0} \end{matrix}$ if there exist such constant K, and d (g, h) =∞, otherwise. d is a complete metric (see the proof of Theorem 3.9). Define the mapping $J : Ω ⟶ Ω$ by $Jg (x) = m^{- 4 j} g (m^{j} x)$ for all x ∈ X. Given g, h ∈ Ω. Let δ be an arbitrary constant with d (g, h) < δ. Then $Λ (m^{\frac{1}{2} (j - 1)} x, t) \leq_{L} P_{μ, ν} (g (x) - h (x), δ t)$ for all x ∈ X and t > 0. So $\begin{matrix} Λ (m^{\frac{1}{2} (j - 1)} x, {(\frac{m^{4}}{α})}^{j} t) \\ \leq_{L} P_{μ, ν} (g (m^{j} x) - h (m^{j} x), m^{4 j} δ t) \\ = P_{μ, ν} (Jg (x) - Jh (x), δ t) \end{matrix}$ for all x ∈ X and t > 0. So, $d (J g, J h) \leq {(\frac{α}{m^{4}})}^{j} d (g, h)$ for all g, h ∈ Ω. Therefore, $J$ is a strictly contractive mapping of Ω with the Lipschitz constant ${(\frac{α}{m^{4}})}^{j}$ . It follow from (3.30) that $d (f, J f) \leq m^{- 2 (j + 1)}$ . Theorem 3.8 shows there exists a mapping $Q : X ⟶ Y$ which is a unique fixed point of $J$ in the set Ω₁ = {g ∈ ω : d (f, g) < ∞}. Indeed, $Q (m^{j} x) = m^{4 j} Q (x) (x \in X) .$ (3.31) In other words, there exists a K > 0 with $Λ (m^{\frac{1}{2} (j - 1)} x, t) \leq_{L} P_{μ, ν} (f (x) - Q (x), Kt)$ for all x ∈ X and t > 0. Also $Q (x) = lim_{k \to \infty} J^{k} f (x) = lim_{k \to \infty} m^{- 4 jk} f (m^{jk} x)$ for all x ∈ X. On the other hand, for each f ∈ Ω, we have $d (f, C) \leq \frac{1}{1 - {(\frac{α}{m^{4}})}^{j}} d (Jf, f)$ . Therefore, $d (f, C) \leq \frac{α^{- \frac{1}{2} (j - 1)}}{| m^{4} - α |}$ . This inequality necessitates that

$\begin{matrix} Λ (m^{\frac{1}{2} (j - 1)} x, t) \\ \leq_{L} P_{μ, ν} (f (x) - Q (x), \frac{α^{- \frac{1}{2} (j - 1)}}{| m^{4} - α |} t) \end{matrix}$ (3.32) for all x ∈ X and t > 0. It follows from (3.23) and (3.32) that the relation (3.25) is true. Now, replace m^jkx and m^jky by x and y in (3.24), respectively, $\begin{matrix} P_{μ, ν}^{'} (φ (x, y), \frac{m^{4 jk}}{α^{jk}} t) \\ \leq_{L} P_{μ, ν}^{'} (Q_{m, n} f (m^{jk} x, m^{jk} y), m^{4 jk} t) \\ \leq_{L} P_{μ, ν} (m^{- 4 jk} Q_{m, n} f (m^{jk} x, m^{jk} y), t) \end{matrix}$ for all x, y ∈ X and t > 0. Taking k tends to infinity and using (3.280), we see that $Q$ is a quartic mapping. □

The next result is a direct consequence of Theorem 3.12. We omit its proof.

Corollary 3.13.Let λ be a nonnegative real number with λ : = r + s ≠ 4, X be a normed space with norm∥· ∥, $(Z, P_{μ, ν}^{'}, T)$ be an intuitionistic fuzzy normed space, $(Y, P_{μ, ν}, T)$ be a complete intuitionistic fuzzy normed space, and let z₀ ∈ Z. If f : X ⟶ Y is a mapping such that $\begin{matrix} P_{μ, ν}^{'} ((∥ x ∥^{r} ∥ y ∥^{s} + ∥ x ∥^{λ} + ∥ y ∥^{λ}) z_{0}, t) \\ \leq_{L} P_{μ, ν} (Q_{m, n} f (x, y), t) \end{matrix}$ for all x, y ∈ X and t > 0, then there exists a unique quartic mapping $Q : X ⟶ Y$ such that $P_{μ, ν}^{'} (∥ x ∥^{λ} z_{0}, | m^{4} - m^{λ} | t) \leq_{L} P_{μ, ν} (Q (x) - f (x), t)$ for all x ∈ X and t > 0.

Footnotes

Acknowledgments

The author expresses his sincere thanks to the reviewer for the careful and detailed reading of the manuscript and suggesting some related references.

References

Abbasbandy

and Allahviranloo

, The adomian decomposition method applied to The fuzzy system of the second kind, Int J Uncer Fuzzy Know Based Syst14(1) (2006), 101–110.

Abbasbandy

, Allahviranloo

and Ezzaty

, A metod for solving general fuzzy linear systems,Los Angeles, J Fuzzy Math15(4) (2007).

Atanassov

, On intuitionistic fuzzy sets theory, Springer, Berlin, 2012.

Atanassov

, Intuitionistic fuzzy sets, Springer, Heidelberg, 1999.

Atanassov

, Intuitionistic fuzzy sets, Fuzzy sets and Systems20(1) (1986), 87–96.

Atanassov

, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, Deposed in Central Sci-Techn Library of Bulg Acad of Sci (1983), 1697/84. in Bulg.

Ban

, Intuitionistic Fuzzy Measures. Theory and Applications. Nova Sci. Publishers, New York, 2006.

Bodaghi

, Stability of a quartic functional equation, The Scientific World Journal (2014) 9. Art. ID 752146. doi: 10.1155/2014-752146

Bodaghi

, Ulam stability of a cubic functional equation in various spaces, Mathematica55(2) (2013), 125–141.

10.

Bodaghi

, Alias

I.A.

and Ghahramani

M.H.

, Ulam stability of a quartic functional equation, Abst Appl Anal (2012) 9. Art. ID 232630, doi: 10.1155/2012-232630

11.

Bodaghi

, Moosavi

S.M.

and Rahimi

, The generalized cubic functional equation and the stability of cubic Jordan *-derivations, Ann Univ Ferrara59 (2013), 235–250.

12.

Cădariu

and Radu

, Fixed points and stability for functional equations in probabilistic metric and random normed spaces, Fixed Point Theory and Applications2009 (2009), 18. Article ID 589143.

13.

Cădariu

and Radu

, Fixed points and the stability of quadratic functional equations, An Univ Timişoara, Ser Mat Inform41 (2003), 25–48.

14.

Căadariu

and Radu

, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math Ber346 (2004), 43–52.

15.

Diaz

J.B.

and Margolis

, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull Amer Math Soc74 (1968), 305–309.

16.

Eshaghi Gordji

, Bodaghi

and Park

, A fixed point approach to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras, UPB Sci Bull, Series A73(2) (2011), 65–73.

17.

Hyers

D.H.

, On the stability of the linear functional equation, Proc Nat Acad Sci USA27 (1941), 222–224.

18.

Kang

, On the stability of generalized quartic mappings in quasi-β-normed spaces, Journal of Inequalities and Applications Volume 201011. Article ID 198098. doi: 10.1155/2010-198098.

19.

Koh

and Kang

, On the stability of a generalized cubic functional equation, Bull Korean Math Soc45(4) (2008), 739–748.

20.

Lee

, Im

and Hwang

, Quartic functional equations, J Math Anal Appl307 (2005), 387–394.

21.

Lee

and Chung

, Stability of quartic functional equations in the spaces of generalized functions, Adv Difference Equ2009 (2009). Art. ID 838347.

22.

Mursaleen

and Mohiuddine

S.A.

, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals42 (2009), 2997–3005.

23.

Najati

, The generalized Hyers-Ulam-Rassias stability stability of a cubic functional equation, Turk J Math31 (2007), 395–408.

24.

Jun

K.W.

and Kim

H.M.

, The generalized Hyers-Ulam-Russias stability of a cubic functional equation, J Math Anal Appl274(2) (2002), 267–278.

25.

Jun

K.W.

and Kim

H.M.

, On the Hyers-Ulam-Rassias stability of a general cubic functional equation, Math Inequ Appl6(2) (2003), 289–302.

26.

Park

J.H.

, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals22 (2004), 1039–1046.

27.

Peeva

and Kyosev

, Fuzzy Relational Calculus-Theory, Applications and Software. World Scientific, New Jersey, 2004.

28.

Rassias

J.M.

, Solution of the Ulam stability problem for cubic mappings, Glasnik Matematicki Serija III36(1) (2001), 63–72.

29.

Rassias

J.M.

, Solution of the Ulam stability problem for quartic mappings, Glasnik Matematicki Series III34(2) (1999), 243–252.

30.

Saadati

, A note on “Some results on the IF-normed spaces”, Chaos, Solitons and Fractals41 (2009), 206–213.

31.

Saadati

, Cho

Y.J.

and Vahidi

, The stability of the quartic functional equation in various spaces, Computers and Mathematics with Application60 (2010), 1994–2002.

32.

Saadati

and Park

, Non-archimedean L-fuzzy normed spaces and stability of functional equations, Computers and Mathematics with Application60 (2010), 2488–2496.

33.

Saadati

and Park

J.H.

, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and Fractals27 (2006), 331–344.

34.

Saadati

, Sedghi

and Shobe

, Modified intuitionistic fuzzy metric spaces and some fixed point theorems, Chaos, Solitons and Fractals38 (2008), 36–47.

35.

Saadati

, Razani

and Adibi

, A common fixed point theorem in L-fuzzy metric spaces, Chaos, Solitons and Fractals33 (2007), 354–363.

36.

Turinici

, Sequentially iterative processes and applications to Volterra functional equations, Annales Univ, Mariae-Curie Sklodowska (Sect A)32 (1978), 127–134.

37.

Ulam

S.M.

, Problems in Modern Mathematics, Chapter VI, Science Ed., Wiley, New York, 1940.

38.

T.Z.

, Rassias

M.J.

and Xu

W.X.

, Stability of a general mixed additive-cubic functional equation in non-archimedean fuzzy normed spaces, Journal of Mathematical Physics 09350851 (2010), 19.

39.

T.Z.

, Rassias

M.J.

, Xu

W.X.

and Rassias

J.M.

, A fixed point approach to the intuitionistic fuzzy stability of quintic and sextic functional equations, Iranian J Fuzzy Sys9(5) (2012), 21–40.

40.

Zadeh

L.A.

, Fuzzy sets, Inform Control8 (1965), 338–353.