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.
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
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
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
and proved the Hyers-Ulam stability problem for (1.3). Since f (2x) =8f (x), the last functional equation is equal to the following
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:
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
was introduced by Rassias [29]. The fact that every solution of (1.6) is even implies that it can be written as follows:
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),
for fixed integer m with m ≠ 0, ± 1. Lastly, Kang [18] obtained the general form of a quartic functional equation as follows:
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):
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:
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.
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
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
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:
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
Similar to the above, we can deduce that
Using the above method, we get
where
Solving the above recurrence equations, we have
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
for all x, y ∈ X. On the other hand,
for all x, y ∈ X. Using (2.12) and (2.13), we get
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
for all x, y ∈ X.
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
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.□
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
defined by
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 0L = (0, 1) and 1L = (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 . For each u in U, represents the degree (in L) to which u is an element of . An intuitionistic fuzzy set in a universal set U is an object , where and belong to [0, 1] for all u ∈ U with . The numbers and are called the membership degree and the non-membership degree, respectively, of u in .
Definition 3.2. A triangular norm (t-norm) on L is a mapping satisfying the following conditions:
(boundary condition) (x ∈ L);
(commutativity) (x, y ∈ L);
(associativity) (x, y, z ∈ L);
x1 ≤ Ly1 and (monotonicity) (x1, x2, y1, y2 ∈ L).
A t-norm on L is said to be continuous if,for any x, y ∈ L and any sequences {xn} and {yn} which converge to x and y, respectively, then
Example 3.3. Let x = (x1, x2) , y = (y1, y2) ∈ L. Then and are continuous t-norm [38].
Here, we define a sequence , recursively by and
for all n ≥ 2 and x(j) ∈ L.
Definition 3.4. A negator on L is a decreasing mapping satisfying and . If , for all x ∈ L, then is called an involutive negator. A negator on [0, 1] is a decreasing mapping satisfying and . The standard negator on [0, 1] is defined by for all x ∈ [0, 1].
The following defnitions of an intuitionistic fuzzy normed space is taken from [31].
Definition 3.5. Let . Let X be a vector space, be a continuous t-norm on L and be an L-fuzzy set on X × (0, ∞) satisfying the following conditions:
;
if and only if x = 0;
for all α ≠ 0;
;
The map is continuous;
and ;
for all x, y ∈ X and all t, s > 0. Then the triple is called an L-fuzzy normed space. In this case is called -fuzzy norm (briefly, L-fuzzy norm). If is an intuitionistic fuzzy set, then the triple is said to be an intuitionistic fuzzy normed space (briefly, IFN-space). In this case, is called an intuitionistic fuzzy norm on X (Some example of IFN-spaces are provided in [38, 39]).
Note that, if is an L-fuzzy norm on X, then the following statements hold:
is nondecreasing with respect to t for all x ∈ X;
for all x, y ∈ X and t > 0.
Example 3.6. [38] Let (X, ∥ · ∥) be a normed space. Let 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
for all Then is an IFN-space.
Definition 3.7. Let be an IFN-space.
A sequence {xn} in is said to be convergent to a point x if as n→ ∞ for all t > 0;
A sequence {xn} in is called a Cauchy sequence if, for every t > 0 and 0 < ɛ < 1, there exists a positive integer N such that for all m, n > N, where Ns is the standard negator;
is said to be complete if and only if every Cauchy sequence in is convergent to a point in . 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 be a mapping with Lipschitz constant L < 1. Then, for each element α ∈ Δ, either for all n ≥ 0, or there exists a natural number n0 such that
for all n ≥ n0;
the sequence is convergent to a fixed point β* of ;
β* is the unique fixed point of in the set ;
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
-mn2 {f (x + y) + f (x - y)} -2m (m2 - n2) f (x)
for all x, y ∈ X.
Here and subsequently, we assume that 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 be an intuitionistic fuzzy normed space. Let φ : X × X ⟶ Z be a mapping such that
for all x ∈ X and t > 0. If is a complete intuitionistic fuzzy normed space and f : X ⟶ Y is a mapping such that
for all x, y ∈ X and t > 0, then there exists a unique cubic mapping such that
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
for all x ∈ X and t > 0. Thus
for all x ∈ X and t > 0. We consider the set Ω = {g : X ⟶ Y} and introduce the generalized metric on X as follows:
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 by for all x ∈ X. Given g, h ∈ Ω. Let δ be an arbitrary constant with d (g, h) < δ. Then
for all x ∈ X and t > 0. So
for all x ∈ X and t > 0. Hence, for all g, h ∈ Ω. Thus is a strictly contractive mapping of Ω with the Lipschitz constant . It follow from (3.19) that . By Theorem 3.8, there exists a mapping satisfying:
(1) is a unique fixed point of in the set Ω1 = {g ∈ ω : d (f, g) < ∞}, which is satisfied
for all x ∈ X. In other words, there exists a K > 0 with
for all x ∈ X and t > 0.
(2) as k→ ∞. This implies that
for all x ∈ X.
(3) For every f ∈ Ω, we have
Since, , we have . The last inequality implies that
for all x ∈ X and t > 0. The relations (3.15) and (3.22) show that the inequality (3.17) holds. Now, replace mjkx and mjky by x and y in (3.16), respectively,
for all x, y ∈ X and t > 0. Letting k tends to infinity and using (3.21), we see that 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 ∥· ∥, be an intuitionistic fuzzy normed space, be a complete intuitionistic fuzzy normed space, and let z0 ∈ Z. If f : X ⟶ Y is a mapping such that
for all x, y ∈ X and t > 0, then there exists a unique cubic mapping such that
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 be the continuous t-norm defined in Example 3.3. Then is an IFN-space for which 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
for all x, y ∈ A. Thus
for all x, y ∈ A and t > 0. Consider φ : A × A ⟶ A defined through
for all x, y ∈ A. We have
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 such that
for all x, y ∈ A and t > 0.
In the sequal, for the sake of convenience, let us denote
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 be an intuitionistic fuzzy normed space. Let φ : X × X ⟶ Z be a mapping such that
for all x ∈ X and t > 0. If is a complete intuitionistic fuzzy normed space and f : X ⟶ Y is a mapping such that
for all x, y ∈ X and t > 0, then there exists a unique quartic mapping such that
for all x ∈ X and t > 0, where
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
for all x ∈ X and t > 0 in which λm,n = 1 - (m2 - n2) 2. Thus
for all x ∈ X and t > 0. Letting y = 0 in (3.24), we get
for all x ∈ X and t > 0. The above inequality implies that
for all x ∈ X and t > 0. It follows from (3.27) and (3.28) that
for all x ∈ X and t > 0. Hence
for all x ∈ X and t > 0, where Λ (x, t) is defined in (3.26). Thus
for all x ∈ X and t > 0. We take the set Ω = {g : X ⟶ Y} and define the generalized metric on X as follows:
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 by for all x ∈ X. Given g, h ∈ Ω. Let δ be an arbitrary constant with d (g, h) < δ. Then
for all x ∈ X and t > 0. So
for all x ∈ X and t > 0. So, for all g, h ∈ Ω. Therefore, is a strictly contractive mapping of Ω with the Lipschitz constant . It follow from (3.30) that . Theorem 3.8 shows there exists a mapping which is a unique fixed point of in the set Ω1 = {g ∈ ω : d (f, g) < ∞}. Indeed,
In other words, there exists a K > 0 with
for all x ∈ X and t > 0. Also
for all x ∈ X. On the other hand, for each f ∈ Ω, we have . Therefore, . This inequality necessitates that
for all x ∈ X and t > 0. It follows from (3.23) and (3.32) that the relation (3.25) is true. Now, replace mjkx and mjky by x and y in (3.24), respectively,
for all x, y ∈ X and t > 0. Taking k tends to infinity and using (3.280), we see that 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∥· ∥, be an intuitionistic fuzzy normed space, be a complete intuitionistic fuzzy normed space, and let z0 ∈ Z. If f : X ⟶ Y is a mapping such that
for all x, y ∈ X and t > 0, then there exists a unique quartic mapping such that
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.
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