Ulam Type Stability ofA-Quadratic Mappings in Fuzzy Modular ∗-Algebras

In this paper, we find the solution of the following quadratic functional equation n ∑1≤i<j≤n Q ( xi − xj ) = ∑ i=1 Q ( ∑j 6=i xj − (n− 1)xi ) , which is derived from the gravity of the n distinct vectors x1, · · · , xn in an inner product space, and prove that the stability results of the A-quadratic mappings in μ-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and β-homogeneous property.

defined between probabilistic modular spaces in the probabilistic sense. After then, Shen and Chen [15] following the idea of probabilistic modular spaces and the definition of fuzzy metric spaces based on George and Veeramani's sense [16], applied fuzzy concept to the classical notions of modular spaces. Using Khamsi's fixed point theorem in modular spaces [17], Wongkum and Kumam [18] proved the stability of sextic functional equations in fuzzy modular spaces equipped necessarily with lower semicontinuity and β-homogeneous property.
In the present paper, concerning the stability problem for the following functional equation which is derived from the gravity of the n-distinct vectors in an inner product space, we investigate the stability problem for A-quadratic mappings in µ-complete convex fuzzy modular * -algebras of the following functional equation without using lower semicontinuity and β-homogeneous property.
, the gravity of the n distinct vectors, then we get the following identity which is equivalent to the equation for any distinct vectors X 1 , X 2 , · · · , X n .
Employing the above equality (1), we introduce the new functional equation: for a mapping Q : U → V and for all vectors x 1 , · · · , x n ∈ U, where U and V are linear spaces and n ≥ 3 is a positive integer. From now on, we introduce some basic definitions of fuzzy modular * -algebras. (1) • is commutative, associative; (2) a • 1 = a; Three common examples of the t-norm are (1) a • M b = min{a, b}; (2) a • p b = a · b; (3) a • L b = max{a + b − 1, 0}. For more example, we refer to [19]. Throughout this paper, we denote that
(1). {x n } is said to be µ-convergent to a point x ∈ X if for any t > 0, (2). {x n } is called µ-Cauchy if for each ε > 0 and each t > 0, there exists n 1 such that, for all n ≥ n 1 and all (3). If each Cauchy sequence is convergent, then the fuzzy modular space is said to be complete.

Fuzzy Modular Stability for A-Quadratic Mappings
First of all, we find out the general solution of (1.3) in the class of mappings between vector spaces.
Proof. Let Q satisfy Equation (2). One finds that Q(0) = 0 and Q(ax) = a 2 Q(x) by changing (x, y) to (0, 0) and (x, 0) in (3), respectively, where a := n − 1 is a positive integer with a ≥ 2. Putting x 1 := x, x 2 := y and x i := 0 for all i = 3, · · · , n in (2), we get for all x, y ∈ U. Using [23] [Theorem 1], we obtain that Q is a generalized polynomial map of degree at most 4. Therefore, for all x ∈ U, where A k : U k → V is a k-additive symmetric map (k = 1, · · · , 4) and A 0 ∈ V. Since a is an integer, we get Let A be a complex * -algebra with unit and let M be a left A-module. We call a mapping Q : M → A an A-quadratic mapping if both relations Q(ax) = aQ(x)a * and Q(x + y) + Q(x − y) = 2Q(x) + 2Q(y) are fulfilled for all a ∈ A, x, y ∈ M [24]. For the sake of convenience, we define the following: In addition, let • be defined by minimum t-norm and A M be the set of all mapping from M to A, Q A (M, A) be the set of all A-quadratic mappings from M to A. Now, we present a stability of the A-quadratic mapping concerning Equation (2) in µ-complete convex fuzzy modular * -algebras without using β-homogeneous properties. Theorem 2. Let (A, µ, •) be µ-complete convex fuzzy modular * -algebra with norm · and M be a left A-module, (X, µ , •) fuzzy modular space, U (A) the unitary group of A. Assume that there exist two mappings f ∈ A M and ε ∈ X M n such that , and either f is measurable or f (tx) is continuous in t ∈ R for each fixed x ∈ M. Then, there exists a unique mapping Q ∈ Q A (M, A) that satisfies Equation (2) and the inequality for all x ∈ M and t > 0, where (n − 1) 2 nt 6(n 2 − (i + 1)n + 1) .
Proof. Define a mapping g : for all x ∈ M. Then, for each x ∈ M, the following equation is obtained: for all i = 1, · · · , n − 1 and for all j = 1, · · · , n, where For each fixed (i, j) ∈ J , one obtains from ∑ m k=1 1+2( (n − 1) 2 nt 6(n 2 − (i + 1)n + 1) for all t > 0 and x ∈ M, m ∈ N. Then, it follows from the above inequality that for all x ∈ M and t > 0. Therefore, we prove from this relation that, for any integers m, p, for all t > 0, x ∈ M. Since the right-hand side of the above inequality tends to 1 as m → ∞, the sequence { g((n−1) m x) (n−1) 2m } is µ-Cauchy and thus converges in A. Hence, we may define a mapping Q : M → A as for all x ∈ M and t > 0. In addition, we claim that the mapping Q satisfies (2). For this purpose, we calculate the following inequality: for all x ∈ M, u ∈ U (A), m ∈ N, t > 0, where L := n 3 − n 2 + 2n + 2 2 . This means that D u Q(x 1 , · · · , x n ) = 0 for all x 1 , · · · x n ∈ M, u ∈ U (A). Hence, the mapping Q satisfies (2) and so Q((n − 1)x) = (n − 1) 2 Q(x) for all x ∈ M. It follows that To prove the uniqueness, let Q be another mapping satisfying (2) and for all x ∈ M. Thus, we have for all x ∈ M, t > 0. Taking the limit as m → ∞, then we conclude that Q(x) = Q (x) for all x ∈ M. Under the assumption that either f is measurable or f (tx) is continuous in t ∈ R for each fixed x ∈ M, the quadratic mapping Q satisfies Q(tx) = t 2 Q(x) for all x ∈ M and for all t ∈ R by the same reasoning as the proof of [25]. That is, Q is R-quadratic. Let P := n 4 −2n 3 +3n 2 −3n+14 4 . Putting x 1 := −(n − 1) k x and x i := 0 for all i = 2, · · · , n in (4) and dividing the resulting inequality by (n − 1) 2k , we have Pt n(n − 1) •µ D u f (−(n − 1) k x, 0, · · · , 0), (n − 1) 2k Pt •µ f (0), 4(n − 1) 2k Pt (n − 2)(n − 1)n(n + 1) for all x ∈ M, u ∈ U (A), t > 0. Taking k → ∞ and using the evenness of Q, we obtain that Q(ux) = uQ(x)u * for all x ∈ M and for each u ∈ U (A). The last relation is also true for u = 0. Now, let a be a nonzero element in A and K a positive integer greater than 4 a . Then, we have a K < 1 4 < 1 − 2 3 . By [26] [Theorem 1], there exist three elements u 1 , u 2 , u 3 ∈ U (A) such that 3 a K = u 1 + u 2 + u 3 . Thus, we calculate in conjunction with [27] [Lemma 2.1] that for all a ∈ A(a = 0) and for all x ∈ M. Thus, the unique R-quadratic mapping Q is also A-quadratic, as desired. This completes the proof.
for all t > 0, which implies f (0) = 0. From Equation (6), we get the following equality for all x ∈ M and all t > 0. Using ∆ n−1 -condition and convexity of µ, we find the following inequality