Stability of a more general cubic functional equation in Felbin’s type fuzzy normed linear spaces 1

1 Introduction

    The stability problem of functional equations originates from the fundamental question: When is it true that a mathematical object satisfying a certain property approximately must be close to an object satisfying the property exactly?

In connection with the above question, in 1940, Ulam [36] raised the following question concerning the stability of group homomorphisms: Let (G₁, ∗) be a group and (G₂, ◊ , d) be a metric group with the metric d (· , ·). Given ɛ > 0, does there exist a δ (ɛ) >0 such that if a mapping h : G₁ → G₂ satisfies the inequality d ( h ( x ∗ y ) , h ( x ) ◊ h ( y ) ) < δ for all x, y ∈ G₁, then there is a homomorphism H : G₁ → G₂ with d (h (x) , H (x)) < ɛ for all x ∈ G₁ ?

The first partial solution to the question of Ulam for Banach spaces was provided by Hyers [14]. He proved that f is a mapping between Banach spaces satisfying ∥f (x + y) - f (x) - f (y) ∥ ≤ ɛ for some fixed ɛ > 0, then there exists the unique additive mapping A such that ∥f (x) - A (x) ∥ ≤ ɛ. Actually, the additive mapping A is constructed directly from the given function f and it is the most powerful tool to study the stability of several functional equations. This method is called a direct method. Hyers’ result was generalized by Aoki [1] for additive mappings and by Rassias [27] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias’ result was obtained by Găvruţă [10] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. Instead of the direct method, Cǎdariu and Vadu [4] introduced a novel method for proving the stability of functional equations via the fixed point technique. They observed that the existence of a solution A of the functional equation and the estimation of the difference with the given mapping f can be obtain from the fixed point alternative. This method is called a fixed point method. Since then, the stability of several functional equations has been extensively investigated by a number of authors using the direct and fixed point methods, and there are many interesting results concerning this problem (see [5, 35] and references therein).

In [17], Jun and Kim considered the following functional equation

f ( 2 x + y ) + f ( 2 x - y ) = 2 f ( x + y ) + 2 f ( x - y ) + 12 f ( x ) .(1.1) It is easy to see that the function f (x) = cx³ satisfies the functional Equation (1.1) and this equation is called a cubic functional equation. Every solution of this cubic functional equation is said to be a cubic mapping. Jun and Kim [17] determined the general solutions of the functional Equation (1.1) and then proved its generalized Hyers-Ulam stability in Banach spaces. In [34], Sahoo also determined the solutions f : G → H of (1.1) and its pexider generalization when G and H are uniquely divisible abelian groups using a method completely different from Jun and Kim [17]. In 2007, Chu and Kang [6] proved that the following functional equation

f ( x + 2 y ) + f ( x - 2 y ) + f ( 2 x ) = 2 f ( x ) + 4 f ( x + y ) + 4 f ( x - y )(1.2) is equivalent to the functional Equation (1.1). In 2004, Sahoo [33] determined the solutions f : G → H of the functional equation f ( x + 2 y ) + f ( x - 2 y ) + 6 f ( x ) = 4 f ( x + y ) + 4 f ( x - y ) , where G and H are divisible abelian groups such that multiplication by 2 be surjective in G and the multiplication by 6 be bijective in H. It is easy to see that if (1.2) holds for all x, y ∈ G, then by putting y = 0 one obtains f (2x) =8f (x) and functional Equation (1.2) reduces to the functional equation studied by Sahoo. Thus the solutions of (1.2) can be obtained immediately from the solutions the functional equation studied by Sahoo [33].

In [18], Jun et al. extended the cubic functional Equation (1.2) to the following cubic functional equation

f ( x + ky ) + f ( x - ky ) + f ( kx ) = k 2 f ( x + y ) + k 2 f ( x - y ) + ( k 3 - 2 k 2 + 2 ) f ( x )(1.3) where k ≥ 2 is an integer. They considered the general solution of the functional Equation (1.3), and then proved the generalized Hyers-Ulam stability of the Equation (1.3) in Banach spaces. Furthermore, they proved the stability of the functional Equation (1.3) by using fixed point method. Recently considerable attention has been given to the problem of the stability of functional equations in Felbin’s type fuzzy normed linear spaces. Some stability results concerning concerning Cauchy, quadratic, cubic and generalized radical reciprocal quadratic functional equations have been investigated [8, 31] in Felbin’s type spaces.

The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the cubic functional Equation (1.3) in Felbin’s type fuzzy normed linear spaces using the direct and fixed point methods. We also present some applications of our main results for the stability of the cubic functional Equation (1.3) from a real normed space to a Banach space.

     Following [8, 31], we present some definitions and preliminary results from the theory of fuzzy real numbers, which will help to investigate Hyers-Ulam stability in Felbin’s type fuzzy normed linear spaces.

In [11], Grantner et al. take the fuzzy real number as a decreasing mapping from the real line to the unit interval or lattice in general. Lowen [23] treated the fuzzy real numbers as non-decreasing, left continuous mapping from the real line to the unit interval so that its supremum over ℝ is 1. Also fuzzy arithmetic operations on L-fuzzy real line were studied by Rodabaugh [30], where he showed that the binary addition is the only extension of addition to ℝ ( L ) .

Hoehle [13] especially emphasized the role of fuzzy real numbers as modeling a fuzzy threshold softening the notion of Dedekind cut. In this paper a fuzzy number is taken as a fuzzy normal and convex mapping from the real line to the unit interval. The concept of the fuzzy metric space has been studied by Kaleva [19, 20] by using fuzzy number as a fuzzy set on the real axis. Kaleva [21] also has recently showed that a fuzzy metric space can be embedded in a complete fuzzy metric space.

Felbin [9] introduced the concept of fuzzy normed linear space, and Xiao and Zhu [37] studied its linear topological structures and some basic properties of a fuzzy normed linear space. It is known that theories of classical normed space and Menger probabilistic normed spaces are special cases of fuzzy normed linear spaces.

Let η be a fuzzy subset on ℝ , i.e., a mapping η : ℝ → [ 0 , 1 ] associating with each real number t its grade of membership η (t).

Definition 2.1. (cf. [9]) A fuzzy subset η on ℝ is called a fuzzy real number, whose α-level set is denoted by [η]  _α , i.e., [η]  _α  = {t : η (t) ≥ α}, if it satisfies two axioms:

(N1) There exists t 0 ∈ ℝ such that η (t₀) =1;

(N2) For each α ∈ ( 0 , 1 ] , [ η ] α = [ η α - , η α + ] where - ∞ < η α - ≤ η α + < + ∞ .

The set of all fuzzy real numbers denoted by F ( ℝ ) . If η ∈ F ( ℝ ) and η (t) =0 whenever t < 0, then η is called a non-negative fuzzy real number and F ∗ ( ℝ ) denotes the set of all non-negative fuzzy real numbers. The interested reader can also find further details in [2, 32].

The number 0 ¯ stands for the fuzzy real number as: 0 ¯ ( t ) = { 1 , t = 0 , 0 , t ≠ 0 . Clearly, 0 ¯ ∈ F ∗ ( ℝ ) . Also the set of all real numbers can be embedded in F ( ℝ ) because if r ∈ (- ∞ , + ∞), then r ¯ ∈ F ( ℝ ) satisfies r ¯ ( t ) = 0 ¯ ( t - r ) .

Definition 2.2. (cf. [9]) Define a partial ordering ⪯ in F ( ℝ ) by η ⪯ δ if and only if η α - ≤ δ α - and η α + ≤ δ α + for all α ∈ (0, 1]. The strict inequality in F ( ℝ ) is defined by η ≺ δ if and only if η α - < δ α - and η α + < δ α + for all α ∈ (0, 1].

Definition 2.3. (cf. [37]) Let X be a real linear space, L and R (respectively, left norm and right norm) be symmetric and non-decreasing mappings from [0, 1] × [0, 1] into [0, 1] satisfying L (0, 0) =0, R (1, 1) =1. Then ∥· ∥ is called a fuzzy norm and (X, ∥ · ∥ , L, R) is a fuzzy normed linear space (abbreviated to FNLS) if the mapping ∥· ∥ from X into F ∗ ( ℝ ) satisfies the following axioms, where [ ∥ x ∥ ] α = [ ∥ x ∥ ] α - , ∥ x ∥ ] α + ] for x ∈ X and α ∈ (0, 1]:

(A1) ∥ x ∥ = 0 ¯ if and only if x = 0;

(A2) ∥rx∥ = |r| ⊙ ∥ x ∥ for all x ∈ X and r ∈ (- ∞ , + ∞);

(A3) For all x, y ∈ X:

(A3L) if s ≤ ∥ x ∥ 1 - , t ≤ ∥ y ∥ 1 - and s + t ≤ ∥ x + y ∥ 1 - , then ∥x + y ∥ (s + t) ≥ L (∥ x ∥ (s) , ∥ y ∥ (t));

(A3R) if s ≥ ∥ x ∥ 1 - , t ≥ ∥ y ∥ 1 - and s + t ≥ ∥ x + y ∥ 1 - , then ∥x + y ∥ (s + t) ≤ L (∥ x ∥ (s) , ∥ y ∥ (t)).

Lemma 2.1. (cf. [38]). Let (X, ∥ · ∥ , L, R) be an FNLS and suppose that (R1) R (a, b) ≤ max(a, b); (R2) ∀α ∈ (0, 1] , ∃ β ∈ (0, α] such that R (β, y) ≤ α for all y ∈ (0, α); (R3) lim a → 0 + R ( a , a ) = 0 . Then (R1) ⇒ (R2) ⇒ (R3), but not conversely.

Lemma 2.2. (cf. [31]). Let (X, ∥ · ∥ , L, R) be an FNLS and suppose that (L1) L (a, b) ≥ min(a, b); (L2) ∀α ∈ (0, 1] , ∃ β ∈ [α, 1] such that L (β, γ) ≥ α for all γ ∈ [α, 1]; (L3) lim a → 1 - L ( a , a ) = 1 . Then (L1) ⇒ (L2) ⇒ (L3), but not conversely.

Definition 2.4. (cf. [37]) Let (X, ∥ · ∥ , L, R) be an FNLS and lim a → 0 + R ( a , a ) = 0 . A sequence { x n } n = 1 ∞ ⊆ X is said to converge to x ∈ X, denoted by lim n → ∞ x n = x , if lim n → ∞ ∥ x n - x ∥ α + = 0 for every α ∈ (0, 1], and is called a Cauchy sequence if lim m , n → ∞ ∥ x m - x n ∥ α + = 0 for every α ∈ (0, 1]. A subset A ⊆ X is said to be complete if every Cauchy sequence in A, converges in A. The fuzzy normed space (X, ∥ · ∥ , L, R) is said to be a fuzzy Banach space if it is complete.

Theorem 2.1. (cf. [38]) Let (X, ∥ · ∥ , L, R) be an FNLS satisfying (R2). Then: (1) For each α ∈ (0, 1], ∥ · ∥ α + is a continuous mapping from X into ℝ . (2) For any n ∈ ℤ + and { x i } i = 1 n we have ∀ α ∈ ( 0 , 1 ] , ∃ β ∈ ( 0 , α ] ; ∥ ∑ i = 1 n x i ∥ α + ≤ ∑ i = 1 n ∥ x i ∥ β + .

3 Stability of Equation (1.3): Direct method

    In this section, we prove the generalized Hyers-Ulam stability of the functional Equation (1.3) in Felbin’s type fuzzy normed linear spaces by using the direct method. For notational simplicity, given mapping f : X → Y, we define the difference operator D _k f : X → Y of the functional Equation (1.2) by D k f ( x , y ) : = f ( x + ky ) + f ( x - ky ) + f ( kx ) - k 2 f ( x + y ) - k 2 f ( x - y ) - ( k 3 - 2 k 2 + 2 ) f ( x ) for all x, y ∈ X, where k ≥ 2 is an integer.

Theorem 3.1. Let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R2). Suppose that f : X → Y be a mapping for which there is a function φ : X 2 → F ∗ ( ℝ ) such that φ ˜ α ( x , y ) : = ∑ i = 0 ∞ 1 k 3 i ( φ ( k i x , k i y ) ) α + < ∞(3.1) and ∥ D k f ( x , y ) ∥ ⪯ φ ( x , y )(3.2) for all x, y ∈ X and all α ∈ (0, 1]. Then there exists a unique cubic mapping C : X → Y such that ∀ α ∈ ( 0 , 1 ] , ∃ β ∈ ( 0 , α ] , s . t . ∥ f ( x ) - C ( x ) ∥ α + ≤ 1 k 3 φ ˜ β ( x , 0 )(3.3) for all x ∈ X.

Proof. Letting y = 0 in (3.2), we have

∥ f ( kx ) - k 3 f ( x ) ∥ ⪯ φ ( x , 0 )(3.4) for all x ∈ X. Replacing x by k ⁿ x and multiply the both sides of (3.4) by 1 k 3 ( n + 1 ) in the fuzzy scalar multiplication sense, we obtain

∥ 1 k 3 ( n + 1 ) f ( k n + 1 x ) - 1 k 3 n f ( k n x ) ∥ ⪯ 1 k 3 ( n + 1 ) ⊙ φ ( k n x , 0 )(3.5) for all x ∈ X and all non-negative integers n ∈ ℕ . By Theorem 2.1 and the inequality (3.5), we conclude that for all α ∈ (0, 1] there exists β ∈ (0, α] such that

∥ 1 k 3 ( n + 1 ) f ( k n + 1 x ) - 1 k 3 m f ( k m x ) ∥ α + ≤ ∑ i = m n ∥ 1 k 3 ( i + 1 ) f ( k i + 1 x ) - 1 k 3 i f ( k i x ) ∥ β + ≤ 1 k 3 ∑ i = m n 1 k 3 i ( φ ( k i x , 0 ) ) β +(3.6) for all x ∈ X and all non-negative integers m and n with n ≥ m. Now (3.1) and (3.6) imply that { 1 k 3 n f ( k n x ) } is a fuzzy Cauchy sequence in Y for each x ∈ X. Since Y is a fuzzy Banach space, the sequence { 1 k 3 n f ( k n x ) } converges for each x ∈ X. So one can define the mapping C : X → Y by

C ( x ) : = lim n → ∞ 1 k 3 n f ( k n x )(3.7) for all x ∈ X. Letting m = 0 and passing the limit n→ ∞ in (3.6), by the continuity of ∥ · ∥ α + , we get

∥ f ( x ) - C ( x ) ∥ α + ≤ 1 k 3 ∑ i = 0 ∞ 1 k 3 i ( φ ( k i x , 0 ) ) β +(3.8) for all x ∈ X. Therefore, we obtain (3.3). Now we show that C is cubic and unique. Applying (3.1), (3.2) and the continuity of ∥ · ∥ α + , we have

∥ D k C ( x , y ) ∥ α + = lim n → ∞ ∥ 1 k 3 n D k f ( k n x , k n y ) ∥ α + = lim n → ∞ 1 k 3 n ∥ D k f ( k n x , k n y ) ∥ α + ≤ lim n → ∞ 1 k 3 n ( φ ( k n x , k n y ) ) α + = 0(3.9) for all x, y ∈ X. Hence the mapping C satisfies the functional Equation (1.3). Therefore by [18], we get that the mapping C : X → Y is cubic.

To prove the uniqueness of C, let C′ : X → Y be another cubic mapping satisfying (1.3) and (3.3). By Theorem 2.1, since

∥ C ( x ) - C ′ ( x ) ∥ α + = lim n → ∞ 1 k 3 n ∥ f ( k n x ) - C ′ ( k n x ) ∥ α + ≤ lim n → ∞ 1 k 3 ( n + 1 ) ∑ i = 0 ∞ 1 k 3 i ( φ ( k i x , 0 ) ) β + ≤ 1 k 3 lim n → ∞ ∑ i = n ∞ 1 k 3 i ( φ ( k i x , 0 ) ) β + = 0

for all x ∈ X. Then, we conclude that C (x) = C′ (x) for all x ∈ X, as desired. This completes the proof of the theorem.□

Remark 1. The above Theorem 3.1 remains true if ∥ · ∥ α + is replaced by ∥ · ∥ α - in (3.3) and the fuzzy Banach space Y satisfies (L2) and (R2).

The following Theorem 3.2 is an alternative result of Theorem 3.1.

Theorem 3.2. Let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space such that R (a, b) ≤ max(a, b) and L (a, b) ≥ min(a, b). Let f : X → Y be a mapping for which there is a function φ : X 2 → F ∗ ( ℝ ) satisfying (3.1) and (3.2) for all x, y ∈ X and all α ∈ (0, 1]. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ ⪯ φ ¯ ( x , 0 )(3.10) for all x ∈ X, where φ ¯ ( x , 0 ) is a fuzzy real number generated by the families of nested bounded closed intervals [a _α , b _α ] such that a α = 1 k 3 ∑ i = 0 ∞ 1 k 3 i ( φ ( k i x , 0 ) ) α - , b α = 1 k 3 ∑ i = 0 ∞ 1 k 3 i ( φ ( k i x , 0 ) ) α + for all x ∈ X.

Theorem 3.3. Let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R2). Suppose that f : X → Y be a mapping for which there is a function φ : X 2 → F ∗ ( ℝ ) such that Φ ˜ α ( x , y ) : = ∑ i = 1 ∞ k 3 i ( φ ( x k i , y k i ) ) α + < ∞(3.11) and ∥ D k f ( x , y ) ∥ ⪯ φ ( x , y )(3.12) for all x, y ∈ X and all α ∈ (0, 1]. Then there exists a unique cubic mapping C : X → Y such that ∀ α ∈ ( 0 , 1 ] , ∃ β ∈ ( 0 , α ] , s . t . ∥ f ( x ) - C ( x ) ∥ α + ≤ 1 k 3 Φ ˜ β ( x , 0 )(3.13) for all x ∈ X.

Proof. Letting y = 0 in (3.12), we have

∥ f ( kx ) - k 3 f ( x ) ∥ ⪯ φ ( x , 0 )(3.14) for all x ∈ X. If we replace x by x k n + 1 and multiply the both sides of (3.14) by k³ⁿ in the fuzzy scalar multiplication sense, we obtain

∥ k 3 n f ( x k n ) - k 3 ( n + 1 ) f ( x k n + 1 ) ∥ ⪯ k 3 n ⊙ φ ( x k n + 1 , 0 )(3.15) for all x ∈ X and all non-negative integers n ∈ ℕ . By Theorem 2.1 and the inequality (3.15), we conclude that for all α ∈ (0, 1] there exists β ∈ (0, α] such that

∥ k 3 ( n + 1 ) f ( x k n + 1 ) - k 3 m f ( x k m ) ∥ α + ≤ ∑ i = m n ∥ k 3 ( i + 1 ) f ( x k i + 1 ) - k 3 i f ( x k i ) ∥ β + ≤ 1 k 3 ∑ i = m n k 3 ( i + 1 ) ( φ ( x k i + 1 , 0 ) ) β +(3.16) for all x ∈ X and all non-negative integers m and n with n ≥ m. Now (3.11) and (3.16) imply that { k 3 n f ( x k n ) } is a fuzzy Cauchy sequence in Y for each x ∈ X. Since Y is a fuzzy Banach space, the sequence { k 3 n f ( x k n ) } converges for each x ∈ X. The rest of this proof is similar to the proof of Theorem 3.□

The same discussion in Remark 1 holds for Theorem 3.3. Also the following theorem is an alternative result of Theorem 3.3.

Theorem 3.4. Let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space such that R (a, b) ≤ max(a, b) and L (a, b) ≥ min(a, b). Let f : X → Y be a mapping for which there is a function φ : X 2 → F ∗ ( ℝ ) satisfying (3.11) and (3.12) for all x, y ∈ X and all α ∈ (0, 1]. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ ⪯ φ ¯ ( x , 0 )(3.17) for all x ∈ X, where φ ¯ ( x , 0 ) is a fuzzy real number generated by the families of nested bounded closed intervals [a _α , b _α ] such that a α = 1 k 3 ∑ i = 1 ∞ k 3 i ( φ ( x k i , 0 ) ) α - , b α = 1 k 3 ∑ i = 1 ∞ k 3 i ( φ ( x k i , 0 ) ) α + for all x ∈ X.

Corollary 3.1. Let μ be non-negative fuzzy real number, and let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R2). Suppose that a mapping f : X → Y satisfies the inequality ∥ D k f ( x , y ) ∥ ⪯ μ(3.18) for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∀ α ∈ ( 0 , 1 ] , ∃ β ∈ ( 0 , α ] , s . t . ∥ f ( x ) - C ( x ) ∥ α + ≤ 1 k 3 - 1 μ β +(3.19) for all x ∈ X.

Proof. The asserted result in Corollary 3.1 can be easily derived by considering φ (x, y) : = μ for all x, y ∈ X in Theorem 3.1. □

Corollary 3.2. Let μ be non-negative fuzzy real number, and let r, s be non-negative real numbers such that r, s > 3 or 0 < r, s < 3. Let X be a fuzzy normed linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R2). Suppose that a mapping f : X → Y satisfies the inequality ∥ D k f ( x , y ) ∥ ⪯ μ ⊗ ( ∥ x ∥ X r ⊕ ∥ y ∥ X s )(3.20) for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∀ α ∈ ( 0 , 1 ] , ∃ β ∈ ( 0 , α ] , s . t . ∥ f ( x ) - C ( x ) ∥ α + ≤ μ β + | k r - k 3 | ( ∥ x ∥ X r ) β +(3.21) for all x ∈ X.

Proof. The proof follows immediately by taking φ ( x , y ) : = μ ⊗ ( ∥ x ∥ X r ⊕ ∥ y ∥ X s ) for all x, y ∈ X in Theorem 3.1 and Theorem 3.3.□

    In this section, we will prove the generalized Hyers-Ulam stability of the functional Equation (1.3) using fixed point method. Before proceeding to the proof of the main results, we begin with the next Lemma 4.1 is due to Diaz and Margolis [7], which is extensively applied to the stability theory of functional equations.

Lemma 4.1. ([7]). Let (E, d) be a complete generalized metric space and J : E → E be a strictly contractive mapping with Lipschitz constant L < 1. Then for each fixed element x ∈ E, either d ( J n x , J n + 1 x ) = ∞ for all nonnegative integers n or there exists a positive integer n₀ such that (i) d (J ⁿ x, Jⁿ⁺¹x) < ∞ ,     ∀ n ≥ n₀; (ii) the sequence {J ⁿ x} is convergent to a fixed point y^* of J; (iii) y^* is the unique fixed point of J in the set E^∗ : = {y ∈ E ∣ d (J ^{n ₀} x, y) < + ∞}; (iv) d ( y , y * ) ≤ 1 1 - L d ( y , Jy ) , ∀ y ∈ E ∗ .

Theorem 4.1. Let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R1). Suppose that f : X → Y be a mapping for which there is a function φ : X 2 → F ∗ ( ℝ ) such that lim n → ∞ 1 k 3 n ( φ ( k n x , k n y ) ) α + = 0(4.1) and ∥ D k f ( x , y ) ∥ ⪯ φ ( x , y )(4.2) for all x, y ∈ X and all α ∈ (0, 1]. If there exists a 0 < L < l such that φ ( x , x ) ⪯ k 3 L ⊙ φ ( x k , x k ) for all x ∈ X, then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ α + ≤ 1 k 3 ( 1 - L ) φ ( x , 0 ) α +(4.3) for all x ∈ X and all α ∈ (0, 1].

Proof. Let S : ={ g| g : X → Y }, and introduce a generalized metric d on S as follows: d ( g , q ) : = inf { ρ ∈ ℝ + | ∥ g ( x ) - q ( x ) ∥ α + ≤ ρ φ ( x , 0 ) α + , ∀ x ∈ X , ∀ α ∈ ( 0 , 1 ] } . It is easy to prove that (S, d) is a complete generalized metric space [5, 24]. Now, we define the mapping J : S → S given by

J g ( x ) : = 1 k 3 g ( kx ) , for all g ∈ S and x ∈ X .(4.4)

Let g, q ∈ S and let ρ ∈ ℝ + be an arbitrary constant with d (g, q) ≤ ρ. From the definition of d, we get ∥ g ( x ) - q ( x ) ∥ α + ≤ ρ φ ( x , 0 ) α + for all x ∈ X and all α ∈ (0, 1]. Therefore, we get

∥ J g ( x ) - J q ( x ) ∥ α + = ∥ 1 k 3 g ( kx ) - 1 k 3 q ( kx ) ∥ α + ≤ ρ k 3 φ ( kx , 0 ) α + ≤ L ρ φ ( x , 0 ) α +(4.5) for some L < 1 and for all x ∈ X, and all α ∈ (0, 1]. Hence, it holds that d ( J g , J q ) ≤ L ρ , that is, d ( J g , J q ) ≤ Ld ( g , q ) for all g, q ∈ S.

Letting y = 0 in (4.2), we get

∥ f ( kx ) - k 3 f ( x ) ∥ ⪯ φ ( x , 0 )(4.6) for all x ∈ X. So ∥ f ( x ) - 1 k 3 f ( kx ) ∥ ⪯ 1 k 3 ⊙ φ ( x , 0 ) . Therefore we can conclude that

∥ f ( x ) - 1 k 3 f ( kx ) ∥ α + ≤ 1 k 3 φ ( x , 0 ) α +(4.7) for all x ∈ X and all α ∈ (0, 1]. Hence, by (4.7), we obtain the inequality d ( f , J f ) ≤ 1 k 3 . Therefore according to Lemma 4.1, the sequence J n f converges to a fixed point C of J , that is, C : X → Y , lim n → ∞ 1 k 3 n f ( k n x ) = C ( x ) for all x ∈ X, and

C ( kx ) = k 3 C ( x )(4.8) for all x ∈ X. Also C is the unique fixed point of J in the set S^∗ = {g ∈ S : d (f, g) < ∞}. This implies that C is a unique mapping satisfying (4.8) such that there exists a ρ ∈ ℝ + such that ∥ f ( x ) - C ( x ) ∥ α + ≤ ρ φ ( x , 0 ) α + for all x ∈ X and all α ∈ (0, 1]. Also, d ( f , C ) ≤ 1 1 - L d ( f , J f ) ≤ 1 k 3 ( 1 - L ) . This means that the inequality (4.3) holds. It follows from (4.1) and (4.2) and continuity of ∥ · ∥ α + that ∥ D k C ( x , y ) ∥ α + = lim n → ∞ ∥ 1 k 3 n D k f ( k n x , k n y ) ∥ α + = lim n → ∞ 1 k 3 n ∥ D k f ( k n x , k n y ) ∥ α + ≤ lim n → ∞ 1 k 3 n ( φ ( k n x , k n y ) ) α + = 0 for all x, y ∈ X and all α ∈ (0, 1]. Hence the mapping C satisfies the functional Equation (1.3). Therefore by [18], C : X → Y is a cubic mapping, as desired.□

Corollary 4.1. Let μ be non-negative fuzzy real number, and let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R1). Suppose that the mapping f : X → Y satisfies (3.18) for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ α + ≤ 1 k 3 - 1 μ α + for all x ∈ X and all α ∈ (0, 1].

Proof. The asserted result in Corollary 4.1 can be easily derived by considering φ (x, y) : = μ for all x, y ∈ X and choosing L = 1 k 3 in Theorem 4.1. □

Corollary 4.2. Let μ be non-negative fuzzy real number, and let r, s be non-negative real numbers such that 0 < r, s < 3. Let X be a fuzzy normed linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R1). Suppose that a mapping f : X → Y satisfies (3.20) for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ α + ≤ μ α + k 3 - k r ( ∥ x ∥ X r ) α +(4.9) for all x ∈ X.

Proof. The proof follows from Theorem 4.1 by taking φ ( x , y ) : = μ ⊗ ( ∥ x ∥ X r ⊕ ∥ y ∥ X s ) for all x, y ∈ X. Then we can choose L = k^r-3 and we get the desired result.□

Theorem 4.2. Let X be a linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R1). Let φ : X 2 → F ∗ ( ℝ ) ba a function such that there exists an 0 < L < 1 such that φ ( x , x ) ⪯ L k 3 ⊙ φ ( kx , kx ) and lim n → ∞ k 3 n ( φ ( x k n , y k n ) ) α + = 0(4.10) for all x, y ∈ X and all α ∈ (0, 1]. Suppose that f : X → Y be a mapping satisfies (4.2) for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ α + ≤ L k 3 ( 1 - L ) φ ( x , 0 ) α +(4.11) for all x ∈ X and all α ∈ (0, 1].

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 4.1. Now, we consider the mapping J : S → S defined by

J g ( x ) : = k 3 g ( x k ) , for all g ∈ S and x ∈ X .(4.12)

It follows from (4.7) that

∥ f ( x ) - k 3 f ( x k ) ∥ α + ≤ L k 3 φ ( x , 0 ) α +(4.13) for all x ∈ X and all α ∈ (0, 1]. Thus d ( f , J f ) ≤ L k 3 . So d ( f , C ) ≤ 1 1 - L d ( f , J f ) ≤ L k 3 ( 1 - L ) .

The rest of the proof is similar to the proof of Theorem 4.1. □

Corollary 4.3. Let μ be non-negative fuzzy real number, and let r, s be non-negative real numbers such that 0 < r, s < 3. Let X be a fuzzy normed linear space and (Y, ∥ · ∥ , L, R) be a fuzzy Banach space satisfying (R1). Suppose that a mapping f : X → Y satisfies (3.20) for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ α + ≤ μ α + k r - k 3 ( ∥ x ∥ X r ) α +(4.14) for all x ∈ X.

Proof. The proof follows from Theorem 4.2 by taking φ ( x , y ) : = μ ⊗ ( ∥ x ∥ X r ⊕ ∥ y ∥ X s ) for all x, y ∈ X. Then by choosing L = k^3-r we get the desired result.   □

     Throughout this section, let X be a real normed space with a norm ∥ · ∥  _X , and Y a real Banach space with a norm ∥ · ∥  _Y . Now, we will prove the generalized Hyers-Ulam stability of the functional Equation (1.3) in Banach spaces by using Theorem 3.1 and Theorem 3.3.

Theorem 5.1. ([18]). Suppose that f : X → Y be a mapping for which there exists a function ψ : X² → [0, ∞) such that ψ ˜ ( x , y ) : = ∑ i = 0 ∞ 1 k 3 i ( ψ ( k i x , k i y ) < ∞(5.1)and ∥ D k f ( x , y ) ∥ ≤ ψ ( x , y )(5.2)for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ ≤ 1 k 3 ψ ˜ ( x , 0 )(5.3)for all x ∈ X.

Proof. Letting ∥ y ∥ ( t ) = 0 ¯ ( t - ∥ y ∥ Y ) , R (a, b) ≤ max(a, b) and L (a, b) ≥ min(a, b). It is easy to see that (Y, ∥ · ∥ , L, R) is a fuzzy Banach space. By (5.1) and (5.2), we obtain ψ α ˜ ( x , y ) : = ∑ i = 0 ∞ 1 k 3 i ( ψ ¯ ( k i x , k i y ) ) α + < ∞ and ∥ D k f ( x , y ) ∥ ⪯ ψ ¯ ( x , y ) for all x, y ∈ X. The proof follows from Theorem 3.1 by taking φ ( x , y ) = ψ ¯ ( x , y ) for all x, y ∈ X. Then we get the desired result.□

Theorem 5.2. ([18]). Suppose that f : X → Y be a mapping for which there exists a function ψ : X² → [0, ∞) such that Ψ ˜ ( x , y ) : = ∑ i = 1 ∞ k 3 i ( ψ ( x k i , y k i ) ) < ∞(5.4)and ∥ D k f ( x , y ) ∥ ≤ ψ ( x , y )(5.5)for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ ≤ 1 k 3 Ψ ˜ ( x , 0 )(5.6)for all x ∈ X.

Proof. The proof is similar to the proof of Theorem 5.1.□

Corollary 5.1. ([18]). Let μ, r, s be non-negative real numbers such that r, s > 3 or 0 < r, s < 3, . Suppose that a mapping f : X → Y satisfies the inequality ∥ D k f ( x , y ) ∥ ≤ { μ , μ ( ∥ x ∥ X r + ∥ y ∥ X s )(5.7) for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ f ( x ) - C ( x ) ∥ ≤ { μ k 3 - 1 , μ | k r - k 3 | ∥ x ∥ X r(5.8)for all x ∈ X.

Proof. The results follows from Theorem 5.1 and Theorem 5.2. □

We use the direct and fixed point methods to establish the Hyers-Ulam stability of a more general cubic functional equation of the form f ( x + ky ) + f ( x - ky ) + f ( kx ) = k 2 f ( x + y ) + k 2 f ( x - y ) + ( k 3 - 2 k 2 + 2 ) f ( x ) in the framework of Felbin’s type fuzzy normed linear spaces. We therefore provide a link two various discipline: Felbin’s fuzzy type spaces and functional equations. We have show that the Hyers-Ulam stability for functional equations in Banach space can be deduced from the stability of functional equations in Felbin’s type fuzzy normed linear spaces. These circumstances can be applied to other significant functional equations.