STABILITY OF A CUBIC FUNCTIONAL EQUATION IN FUZZY NORMED SPACE

In this paper, the authors investigate the general solution of a new cubic functional equation 3f(x + 3y) − f(3x + y) = 12[f(x + y) + f(x − y)] + 80f(y) − 48f(x) and discuss its generalized Hyers Ulam Rassias stability in Banach spaces and stability in fuzzy normed spaces.


Introduction
In 1940, S.M.Ulam [28] raised the following question. Under what conditions does there exist an additive mapping near an approximately addition mapping?. The case of approximately additive functions was solved by D.H.Hyers [11] under the assumption that for ǫ > 0 and f : E 1 → E 2 be such that f (x + y) − f (x) − f (y) ≤ ε for all x, y ∈ E 1 then there exist a unique additive mapping T : In 1978, a generalized version of the theorem of Hyers for approximately linear mapping was given by Th.M.Rassias [22]. He proved that for a mapping f : E 1 → E 2 be such that f (tx) is continuous in t ∈ R and for each fixed x ∈ E 1 assume that there exist constant ε > 0 and p ∈ [0, 1) with A number of mathematicians were attracted by the result of Th.M.Rassias. The stability concept that was introduced and investigated by Rassias is called the Hyers-Ulam-Rassias stability.
During the last decades, the stability problems of several functional equations have been extensively investigated by a number of authors [1,5,10,13,23,24,25].
In 1982-1989, J.M.Rassias [20,21] replaced the sum appeared in right hand side of the equation (1.1) by the product of powers of norms. Infact, he proved the following theorem. Theorem 1.1. Let f : E 1 → E 2 be a mapping from a normed vector space E 1 into Banach space E 2 subject to the inequality for all x, y ∈ E 1 , where ε and p are constants with ε > 0 and 0 ≤ p < 1 2 . Then the limit exist for all x ∈ E 1 and L : E 1 → E 2 is the unique additive mapping which satisfies for all x ∈ E 1 . If p > 1 2 the inequality (1.3) holds for x, y ∈ E 1 and the limit exist for all x ∈ E 1 and A : E 1 → E 2 is the unique additive mapping which satisfies for all x ∈ E 1 .
The functional equation is called a quadratic functional equation. Infact, every solution of the quadratic equation (1.8) is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation (1.8) was discussed by Skof [27], Chowlewa [7], Czerwik [8] in different settings. Jun and Kim [12], Park and Jung [19] introduced the following functional equations and investigated its general solution and the generalized Hyers-Ulam-Rassias stability respectively. The functional equations (1.9) and (1.10) are called cubic functional equation because the function f (x) = cx 3 is a solution of the above functional equation (1.9) and (1.10).
In modeling applied problems only partial information may be known (or) there may be a degree of uncertainty in the parameters used in the model or some measurments may be imprecise. Due to such features, we are tempted to consider the study of functional equations in the fuzzy setting.
For the last 40 years, fuzzy theory has become very active area of research and a lot of development has been made in the theory of fuzzy sets to find the fuzzy analogues of the classical set theory. This branch finds a wide range of applications in the field of science and engineering.
A.K.Katsaras [14] introduced an idea of fuzzy norm on a linear space in 1984, in the same year Cpmgxin Wu and Jinxuan Fang [29] introduced a notion of fuzzy normed space to give a generalization of the Kolmogoroff normalized theorem for fuzzy topological linear spaces. In 1991, R.Biswas [4] defined and studied fuzzy inner product spaces in linear space. In 1992, C.Felbin [9] introduced an alternative definition of a fuzzy norm on a linear topological structures of a fuzzy normed linear spaces. In 1994, S.C.Cheng and J.N.Mordeson [6] introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of I.Kramosil and J.Michalek [15]. In 2003, T.Bag and S.K.Samanta [2] modified the definition of S.C.Cheng and J.N.Mordeson [6] by removing a regular condition. Recently many various result have been investigated by numerous authors one can refer to [3,16,17,26].
Before we proceed to the main theorems, we will introduce a definition and an example to illustrate the idea of fuzzy norm. Definition. Let X be a real linear space. A function N : X × R → [0, 1] is said to be fuzzy norm on X if for all x, y ∈ X and all a, b ∈ R : The pair (X, N ) is called a fuzzy normed linear space. One may regard N (x, a) as the truth value of the statement the norm of x is less than or equal to the real number a . Example 1.1. Let (X, · ) be a normed linear space. Then is a fuzzy norm on X . In the following we will suppose that N (x, ·) is left continuous for every x. A fuzzy normed linear space is a pair (X, N ), where X is a real linear space and N is a fuzzy norm on X. Let (X, N ) be a fuzzy normed linear space. A sequence {x n } in X is said to be convergent if there exist x ∈ X such that lim n→∞ N (x n − x, t) = 1(t > 0).
In that case, x is called the limit of the sequence {x n } and we write N − lim n→∞ x n = x.
A sequence {x n } in fuzzy normed space (X, N ) is called cauchy if for each ǫ > 0 and δ > 0, there exist n 0 ∈ N such that If each cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.
In this paper, we introduce a new cubic functional equation of the form and discuss its general solution in Section 2. In Section 3, we investigate the generalized Hyers -Ulam -Rassias stability for the functional equation (1.11). We have also given the counter example to illustrate the non-stability of the functional equation (1.11) for some cases. In Section 4, we obtain the fuzzy stability for the functional equation (1.11). Now we proceed to find the general solution of the functional equation (1.11).

The General Solution of the Functional Equation (1.11)
In this section, we obtain the general solution of the functional equation (1.11). Through out this section, let X and Y be real vector spaces.
for all x, y ∈ X. Proof. Putting x = y = 0 in (1.11), we get f (0) = 0. Let y = 0 in (1.11), we obtain for all x ∈ X. Setting x = 0 in equation (1.11) and using (2.2), we get for all y ∈ X. Replacing y by x (1.11), we obtain for all x ∈ X. Again replacing x by 3y in (1.11) and using equation (2.4), we get for all y ∈ X. Substituting y by 3x in equation (1.11) and using (2.4) and (2.5), we obtain for all x ∈ X. Replacing y by y − x in (1.11) and using oddness of f , we arrive for all x, y ∈ X. Interchanging x and y in (2.7), we get for all x, y ∈ X. Replacing x by x − y in (1.11), we obtain for all x, y ∈ X. Multiplying the above equation by 3 and using (2.8), we get for all x, y ∈ X. Replacing y by −y in (2.10), we obtain for all x, y ∈ X. Using (2.10) in (2.11), we arrive for all x, y ∈ X. Multiplying equation (2.13) by 2 and adding with (2.12), we arrive (2.1) for all x, y ∈ X.

Generalized Hyers -Ulam -Rassias Stability of (1.11)
In this section, we consider X to be a real vector space and Y to be a Banach space, we present the Hyers -Ulam -Rassias stability of the functional equation (1.11) involving sum of powers of norms. Let us denote for all x, y ∈ X. for all x, y ∈ X. If a function f : X → Y satisfies for all x, y ∈ X, then there exist a unique cubic function C : X → Y which satisfies (1.11) and The function C is given by Proof. Setting y = 0 in (3.2) and dividing by 27, we get for all x ∈ X. Replacing x by 3x and dividing by 27 in equation (3.5), we obtain for all x ∈ X. Adding (3.5) and (3.6), we get for all x ∈ X. Generalizing, we get for all x ∈ X. Now we have to prove that the sequence {f (3 n x)/27 n } is a cauchy sequence for all x ∈ X. For every positive integer n, m and for all x ∈ X, consider for all x ∈ X. By condition (3.1), the right-hand side approaches 0 as n → ∞. Thus, the sequence is a cauchy sequence. Due to the completeness of the Banach space Y is well-defined. We can see that (3.4) holds.
Taking the limit as n → ∞, using (3.1) and (3.4) the above equation becomes for all x, y ∈ X. Therefore C satisfies (1.11).
To prove the uniqueness of C, suppose that there exist another cubic function S : X → Y such that S satisfies (1.11) and (3.3), we have By condition (3.1), the right-hand side approaches 0 as n → ∞, and it follows that C(x) = S(x) for all x ∈ X. Hence, C is unique. This completes the proof of the theorem. for all x, y ∈ X. If a function C : X → Y satisfies for all x, y ∈ X, then there exist a unique cubic function C : X → Y which satisfies (1.11) and The function C is given by Proof. The proof begins in the same manner as that of Theorem 3.1. The only difference starts with the replacement of inequality (3.5) by replacing x by x 3 . The following Corollary is the immediate consequences of Theorem 3.1 and 3.2 which gives the Hyers-Ulam and generalized Hyers-Ulam stability of the functional equation (1.11).
Corollary 3.1. Let Y be a Banach space and let ε ≥ 0 be a real number. If a function f : X → Y satisfies the functional equation for all x, y ∈ X, then there exist a unique cubic function C : X → Y defined by 27 n which satisfies the equation (1.11) and the inequality Moreover, if for each fixed x ∈ X the mapping t → f (tx) from R to X is continuous, then C (rx) = r 3 C(x) for all r ∈ R.
Corollary 3.2. Let X and Y be a real normed space and a Banach space respectively. If a function f : X → Y satisfies the functional equation with 0 ≤ p < 3 or p > 3 for some ε ≥ 0 and for all x, y ∈ X, then there exist a unique cubic function C : X → Y such that Now we will provide an example to illustrate that the functional equation (1.11) is not stable for p = 3 in Corollary 3.4.
Example 3.1. Let φ : R → R be a function defined by where a > 0 is a constant and a function f : R → R by Then f satisfies the functional inequality for all x, y ∈ R. Then there do not exist a cubic mapping C : R → R and a constant β > 0 such that .
Therefore we see that f is bounded. We are going to prove that f satisfies (3.14).
We claim that the cubic functional equation (1.11) is not stable for p = 3 in Corollary 3.4. Suppose on the contrary, there exist a cubic mapping C : R → R and a constant β > 0 satisfying (3.15). Since f is bounded and continuous for all x ∈ R, C is bounded on any open interval containing the origin and continuous at the origin. In view of Corollary 3.4, C(x) must have the form C(x) = kx 3 for any x in R. Thus we obtain that |f (x)| ≤ (β + |k|) |x| 3 . (3.17) But we can choose a positive integer m with ma > β + |k|. If x ∈ 0, 1 3 m−1 , then 3 n x ∈ (0, 1) for all n = 0, 1, . . . , m − 1. For this x, we get which contradicts (3.17). Therefore the cubic functional equation (1.11) is not stable in sense of Ulam, Hyers and Rassias if p = 3, assumed in the inequality (3.13).
We obtain the following Corollary for Theorem 3.1 and 3.2, which gives J.M. Rassias stability of the functional equation (1.11) Corollary 3.3. Let X and Y be a real normed space and a Banach space respectively. If a function f : X → Y satisfies the functional equation with 0 < p < 1 or p > 1 for some ε > 0 and for all x, y ∈ X, then there exist a unique cubic function C : X → Y such that

Fuzzy Stability of the Functional Equation (1.11)
Throughout this section, assume that X , (Z, N ′ ) and (Y, N ) are linear space, fuzzy normed space and fuzzy Banach space respectively. We now investigate the fuzzy stability of the functional equation (1.11). Theorem 4.1. Let β ∈ {1, −1} be fixed and let φ 1 : X × X → Z be a mapping such that for some α > 0 with α for all x ∈ X and all a > 0 , and for all x, y ∈ X and all a > 0. Suppose an odd mapping f : X → Y with f (0) = 0 satisfies the inequality for all a > 0 and all x, y ∈ X. Then the limit exist for all x ∈ X and the mapping C : X → Y is the unique cubic mapping satisfying for all x ∈ X and all a > 0. Proof. Let β = 1. Letting y = 0 in (4.2), we get for all x ∈ X and all a > 0. Replacing x by 3 n x in (4.4), we obtain for all x ∈ X and all a > 0. Using (4.1), we get for all x ∈ X and all a > 0. Replacing a by α n a in (4.6), we get for all x ∈ X and all a > 0. It follows from for all x ∈ X and all a > 0. Replacing x by 3 m x in (4.8),we get and so for all x ∈ X, a > 0 and all m, n ≥ 0 . Replacing a by for all x ∈ X, a > 0 and all m, n ≥ 0. Since 0 < α < 27 and  for all x ∈ X and all a > 0. Taking the limit as n → ∞ and using (N 6 ), we get for all x ∈ X and all a > 0. Now we claim that C is cubic. Replacing x, y by 3 n x, 3 n y in (4.2) respectively, we get N 1 27 n D f (3 n x, 3 n y), a ≥ N ′ (φ 1 (3 n x, 3 n y), 27 n a) for all x, y ∈ X and all a > 0 . Since lim n→∞ N ′ (φ 1 (3 n x, 3 n y), 27 n a) = 1 C satisfies the functional equation (1.11). Hence C : X → Y is Cubic. To prove the uniqueness of C, let C ′ : X → Y be another cubic mapping satisfying (4.3). Fix x ∈ X, clearly C(3 n x) = 27 n C(x) and C ′ (3 n x) = 27 n C ′ (x) for all x ∈ Xand all n ∈ N . It follows from (4. Thus N (C(x) − C ′ (x), a) = 1 for all x ∈ X and all a > 0 , and so C(x) = C ′ (x). For β = −1 , we can prove the result by similar method.