Generalized Ulam-Hyers Stability Results of a Quadratic Functional Equation in Felbin’s Type Fuzzy Normed Linear Spaces

This paper presents the generalized Ulam-Hyers stability of the following quadratic functional equation f ( x + y 2 − z ) + f ( y + z 2 − x ) + f ( z + x 2 − y ) = 3 4 (f (z − x) + f (z − y) + f (x − y)) in Felbin’s type fuzzy normed linear spaces (f-NLS) using direct and fixed point methods.


Introduction
In 1940, S.M. Ulam [37] posed the stability problem for approximate homomorphisms. In 1941, D.H. Hyers [12] provided a partial solution to Ulam's problem for mappings between Banach spaces.
In 1950, T. Aoki [2] generalized Hyers' Theorem for additive mappings. In 1978, Th.M.Rassias [28] proved a further generalzation of Hyers' Theorem by introducing the concept of the unbounded Cauchy difference for the sum of powers of two p−norms. During the last three decades the stability theorem of Th.M. Rassias [28] provided a lot of influence for the development of stability theory of a large variety of functional equations. This new concept is known today with the term Hyers-Ulam-Rassias stability for functional equations. Following the innovative approach of Th.M. Rassias similar theorems were formulated and proved by a number of mathematicians. For example four years later in 1982, J.M. Rassias [27] proved a similar theorem for the case "the unbounded Cauchy difference" is the "product of two p−norms".
In 1994, the above stability results were further extended by P. Gavruta [9] who considered a more general control function in the real variables x, y for the unbounded Cauchy difference in the spirit of Th.M. Rassias' stability approach. In 2008, a special case of Gavruta's theorem for the unbounded Cauchy difference was obtained in [31] by considereing the summation of both the sum and product of two p−norms.
The functional equation is said to be quadratic functional equation because the quadratic function f (x) = ax 2 is a solution of the functional equation (1.1).
This paper presents the generalized Ulam-Hyers stability of the following quadratic functional equa- in Felbin's fuzzy normed linear spaces (f-NLS) using direct and fixed point methods

Some preliminaries on fuzzy real number
This section, some preliminaries in the theory of fuzzy real numbers are given. Furthermore, we give some definition which help to investigate the stability in Felbin's type normed linear spaces.
In [10] Grantner takes the fuzzy real number as a decreasing mapping from the real line to the unit interval or lattice in general. Lowen [23] applies 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 [34], 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 real 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 also has recently showed that a fuzzy metric space can be embedded in a complete fuzzy metric space [21].
In [8], Felbin introduced the concept of fuzzy normed linear space (f-NLS); Xiao and Zhu [38] 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 ηt.
Definition 2.1. [8] A fuzzy subset η on R is called a fuzzy real number, whose α−level set is denoted if it satisfies two axioms: (N1) There exists t 0 ∈ R such that η(t 0 ) = 1.
The set of all fuzzy real number is 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 number0 stands for the fuzzy real number as: Clearly,0 ∈ F * (R). Also the set of all real numbers can be embedded in F (R) because if r ∈ (−∞, ∞), The additive and multiplicative identities in F (R) are0 and1, respectively. Let η be defined as Definition 2.3. [8] For k, 0 ∈ R, fuzzy scalar multiplication k η is defined as (k η)(t) = η(t/k) and 0 η is defined to be 0.
[38] Let X be a real linear space, L and R (respectively, left norm and right norm) be symmetric and non-decreasing mappings from Then || · || is called a fuzzy norm and (X, || · ||, L, R) is a fuzzy normed linear space (abbreviated to Let (X, || · ||, L, R) be an f-NLS, and suppose that Then (R1) ⇒ (R2) ⇒ (R3) but not conversely.
Theorem 2.1. [35] Let (X, || · ||, L, R) be an f-NLS and lim Hausdorff topological vector space, whose neighborhood base of origin is Definition 2.8. Let (X, || · ||, L, R) be an f-NLS. 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 (f-BS) if it is complete.

Felbin's stability results: Direct method
The generalized Ulam-Hyers stability of the quadratic functional equation is discussed in this section.
Through out this section, let U be a normed space and V be a Banach space respectively. Define a mapping F : for all x, y , z ∈ U. for all x, y , z ∈ U and let F : U 3 → V be an even function satisfying the inequality for all x, y , z ∈ U. Then there exists a unique quadratic function Q : for all x ∈ U. The mapping Q(x) is defined by for all x ∈ U.
Replacing (x, y , z) by (x, x, −x) and dividing by 2 2 in (1.2), we get for all x ∈ U. Replacing x by 2x in (3.5) and divided by 2 2 , we get for all x ∈ U. Combining (3.5) and (3.6), we obtain for all x ∈ U. Using induction on a positive integer n, we obtain that for all x ∈ U. In order to prove the convergence of the sequence f (2 n x) 2 2n , replace x by 2 m x and divided by 2 2m in (3.8), for any m, n > 0 , we arrive for all x ∈ U. Since the right hand side of the inequality (3.9) tends to 0 as m → ∞, the sequence f (2 n x) 2 2n is Cauchy sequence. Since Y is complete, there exists a mapping Q : U → Y such that Letting n → ∞ in (3.8), we see that (3.3) holds for all x ∈ U. Now we need to prove Q satisfies (1.2), replacing (x, y , z) by (2 n x, 2 n y , 2 n z) and divided by 2 2n in (3.2), we arrive 1 2 2n Df (2 n x, 2 n y , 2 n z) + α ≺ 1 2 2n ξ (2 n x, 2 n y , 2 n z) + α for all x, y , z ∈ U. Hence we arrive Q (2 n x, 2 n y , 2 n z) + α = 0.
Hence Q satisfies (1.2) for all x, y , z ∈ U. In order to prove Q is unique, let Q (x) be another quadratic mapping satisfying (3.3) and (1.2). Then for all x ∈ U. Hence Q is unique.
For j = −1, we can prove the similar stability result. Hence it completes the proof.

Felbin's stability results: Fixed point method This section deals with the generalized Ulam -Hyers stability of the functional equation (1.2) in
Banach space using fixed point method.
Now we will recall the fundamental results in fixed point theory. (ii)The fixed point for each given element x * is globally attractive, that is (A2) li m n→∞ T n x = x * , for any starting point x ∈ X; (iii) One has the following estimation inequalities: (i ) d(T n x, T n+1 x) < ∞ for all n ≥ n 0 ; (ii )The sequence (T n x) is convergent to a fixed point y * of T (ii i ) y * is the unique fixed point of T in the set Y = {y ∈ X : d(T n 0 x, y ) < ∞}; (iv ) d(y * , y ) ≤ 1 1−L d(y , T y ) for all y ∈ Y. where µ i = 2 if i = 0 and µ i = 1 2 if i = 1 such that the functional inequality with for all x, y , z ∈ U. If there exists L = L(i ) < 1 such that the function has the property for all x ∈ U, then there exists a unique quadratic mapping Q : U → Y satisfying the functional equation (1. 2) and for all x ∈ U.
Proof. Consider the set and introduce the generalized metric on Ω, It is easy to see that (Ω, d) is complete.
for all x ∈ U. Now p, q ∈ Ω, we have This implies d(T p, T q) ≺ Ld(p, q), for all p, q ∈ Ω . i.e., T is a strictly contractive mapping on Ω with Lipschitz constant L.
Replacing (x, y , z) by (x, x, −x) in (1.2), we get, Hence from the above inequality, we have for all x ∈ U. Using (4.3) and (4.4) for the case i = 0, it reduces to Again replacing x = x 2 in (4.6), we get, for all x ∈ U. Using (4.3) and (4.4) for the case i = 1 it reduces to In both cases, we arrive Therefore (A1) holds.
By (A2), it follows that there exists a fixed point Q of T in Ω such that for all x ∈ U.
To prove Q : U → Y is quadratic. Replacing (x, y , z) by for all x ∈ U and K > 0. Finally by (A4), we obtain This completes the proof of the theorem.
The following corollary is an immediate consequence of Theorem 4.3 concerning the stability of (1.2).
Hence it completes the proof.

Conclusion
This article has proved the stability results of the quadratic functional equation in Felbin's fuzzy normed linear spaces (f-NLS) using both the direct and fixed point methods.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.