Fuzzy Stability of a General Quadratic Functional Equation

A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?”. Such a problem, called a stability problem of the functional equation, was formulated by Ulam [1] in 1940. In the next year, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki [3] for additive mappings, and by Rassias [4] for linear mappings, to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5–15]. In 1984, Katsaras [16] defined a fuzzy norm on a linear space to construct a fuzzy structure on the space. Since then, some mathematicians have introduced several types of fuzzy norm in different points of view. In particular, Bag and Samanta [17], following Cheng and Mordeson [18], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [19]. In 2008, Mirmostafaee and Moslehian [20] obtained a fuzzy version of stability for the Cauchy functional equation:


Introduction and Preliminaries
A classical question in the theory of functional equations is "when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?".Such a problem, called a stability problem of the functional equation, was formulated by Ulam [1] in 1940.In the next year, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings.Subsequently, his result was generalized by Aoki [3] for additive mappings, and by Rassias [4] for linear mappings, to consider the stability problem with unbounded Cauchy differences.During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5][6][7][8][9][10][11][12][13][14][15].
In 1984, Katsaras [16] defined a fuzzy norm on a linear space to construct a fuzzy structure on the space.Since then, some mathematicians have introduced several types of fuzzy norm in different points of view.In particular, Bag and Samanta [17], following Cheng and Mordeson [18], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [19].In 2008, Mirmostafaee and Moslehian [20] obtained a fuzzy version of stability for the Cauchy functional equation: In the same year, they [21] proved a fuzzy version of stability for the quadratic functional equation: We call a solution of (1) an additive map, and a solution of (2) is called a quadratic map.Now we consider the functional equation: which is called a general quadratic functional equation.We call a solution of (3) a general quadratic function.Recently, Kim [22] and Jun and Kim [23] obtained a stability of the functional equation ( 3) by taking and composing an additive map A and a quadratic map Q to prove the existence of a general quadratic function F which is close to the given function f .In their processing, A is approximate to the odd part ( f (x) − f (−x))/2 of f , and Q is close to the even part In this paper, we get a general stability result of the general quadratic functional equation (3) in the fuzzy normed linear space.To do it, we introduce a Cauchy sequence {J n f (x)}, starting from a given function f , which converges to the desired function F in the fuzzy sense.As we mentioned before, in previous studies of stability problem of (3), they attempted to get stability theorems by handling the odd and even part of f , respectively.According to our proposal in this paper, we can take the desired approximate solution F at once.Therefore, this idea is a refinement with respect to the simplicity of the proof.

Fuzzy Stability of the Functional Equation (3)
We use the definition of a fuzzy normed space given in [17] to exhibit a reasonable fuzzy version of stability for the general quadratic functional equation in the fuzzy normed linear space.
Definition 1 (see [17]).Let X be a real linear space.A function N : X × R → [0, 1] (the so-called fuzzy subset) is said to be a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, The pair (X, N) is called a fuzzy normed linear space.Let (X, N) be a fuzzy normed linear space.Let {x n } be a sequence in X.Then {x n } is said to be convergent if there exists x ∈ X such that lim n → ∞ N(x n − x, t) = 1, for all t > 0. In this case, x is called the limit of the sequence {x n } and we denote it by in X is called Cauchy if for each ε > 0 and each t > 0, there exists n 0 such that for all n ≥ n 0 and all p > 0, we have N(x n+p − x n , t) > 1 − ε.It is known that every convergent sequence in a fuzzy normed space is Cauchy.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.
Let (X, N) be a fuzzy normed space, and let (Y , N ) be a fuzzy Banach space.For a given mapping f : X → Y , we use the abbreviation for all x, y, z ∈ X. Recall D f ≡ 0 means that f is a general quadratic function.For given q > 0, the function f is called a fuzzy q-almost general quadratic function, if for all x, y, z ∈ X and r, s, t ∈ [0, ∞).Now we get the general stability result in the fuzzy normed linear setting.
Theorem 1.Let q be a positive real number with q / = 1/2, 1.And let f be a fuzzy q-almost general quadratic function froma fuzzy normed space (X, N) into a fuzzy Banach space (Y , N ).Then there is a unique general quadratic function F : for all x ∈ X and t > 0, where p = 1/q.
Case 1.Let q > 1.We define the function J n f : X → Y by for all x ∈ X.Notice that for all x ∈ X and j ≥ 0. Together with (N3), (N4) and ( 5), this equation implies that if n + m > m ≥ 0 then for all x ∈ X and t > 0. Let ε > 0 be given.Since lim t → ∞ N(x, t) = 1, there is t 0 > 0 such that We observe that for some t > t 0 , the series ∞ j=0 ((5 + 2 p )/4 • 2 p (2 p /4) j +((2 p /2)+(7/4))(2 p /2) j ) t p converges for p = 1/q < 1.It guarantees that, for an arbitrary given c > 0, there exists for each m ≥ n 0 and n > 0. Together with (N5) and ( 9), this implies that for all x ∈ X. Hence {J n f (x)} is a Cauchy sequence in the fuzzy Banach space (Y , N ).And so we can define a mapping for all x ∈ X.Moreover, if we put m = 0 in (9), we have for all x ∈ X.Next we will show that F is a desired general quadratic function.Using (N4), we have for all x, y, z ∈ X and n ∈ N. The first six terms on the right hand side of (15) tend to 1 as n → ∞ by the definition of F and (N2), and the last term holds , , , for all x, y, z ∈ X.By (N3) and ( 5), we obtain , for all x, y, z ∈ X and n ∈ N. Since q > 1, together with (N5), we can deduce that the last term of ( 15) also tends to 1 as n → ∞.It follows from (15) that for all x, y, z ∈ X and t > 0. By (N2), it leads us to prove that F is a general quadratic function.
For an arbitrary fixed x ∈ X and t > 0, choose 0 < ε < 1 and 0 < t < t.Since F is the limit of Because 0 < ε < 1 is arbitrary, we get the inequality (6) in this case.Finally, to prove the uniqueness of F, let F : X → Y be another general quadratic function satisfying (6).Then by (8), we get for all x ∈ X and n ∈ N. Together with (N4) and ( 6), this implies that for all x ∈ X and n ∈ N. Observe that, for q = 1/ p > 1, the last term of the above inequality tends to 1 as n → ∞ by (N5).This implies that N (F(x)−F (x), t) = 1, and so we get for all x ∈ X by (N2).
Case 2. Let 1/2 < q < 1, and let J n f : X → Y be a function defined by for all x ∈ X.Then we also have J 0 f (x) = f (x), J j f (0) = f (0), and for all x ∈ X and j ≥ 0. If n + m > m ≥ 0, then In the similar argument following ( 9) of the previous case, we can define the limit F(x) := N − lim n → ∞ J n f (x) of the Cauchy sequence {J n f (x)} in the Banach fuzzy space Y .Moreover, putting m = 0 in the above inequality, we have for each x ∈ X and t > 0. To prove that F is a general quadratic function, we have enough to show that the last term of (15) in Case 1 tends to 1 as n → ∞.By (N3) and ( 5), we get , , , ≥ min N x, 2 (2q−1)n−4q t q , N y, 2 (2q−1)n−4q t q , N z, 2 (2q−1)n−4q t q , N x, 2 (1−q)n−4q t q , N y, 2 (1−q)n−4q t q , N z, 2 (1−q)n−4q t q , (27 for each x, y, z ∈ X and t > 0. Observe that all the terms on the right-hand side of the above inequality tend to 1 as n → ∞, since 1/2 < q < 1.Hence, together with the similar argument after (15), we can say that DF(x, y, z) = 0, for all x, y, z ∈ X. Recall, in Case 1, the inequality (6) follows from (14).By the same reasoning, we get ( 6) from (26) in this case.Now to prove the uniqueness of F, let F be another general quadratic function satisfying (6).Then, together with (N4), (6), and (20), we have Advances in Fuzzy Systems −2q = ∞; in this case, both terms on the right hand side of the above inequality tend to 1 as n → ∞ by (N5).This implies that N (F(x) − F (x), t) = 1, and so F(x) = F (x) for all x ∈ X by (N2).
Case 3. Finally, we take 0 < q < 1/2 and define J n f : X → Y by for all x ∈ X.Then we have J 0 f (x) = f (x), J j f (0) = f (0), and for all x ∈ X and j ≥ 0. Moreover if n + m > m ≥ 0, then for all x ∈ X, j ≥ 0, and t > 0. Similar to the previous cases, it leads us to define the function F : X → Y by F(x) := N − lim n → ∞ J n f (x).Putting m = 0 in the above inequality, we have for all x ∈ X and t > 0. Notice that , , for all x, y, z ∈ X and t > 0. Since 0 < q < 1/2, all terms on the right hand side tend to 1 as n → ∞, which implies that the last term of (15) tends to 1 as n → ∞.Therefore, we can say that DF ≡ 0.Moreover, using the similar argument after (15) in Case 1, we get the inequality (6) from (32) in this case.To prove the uniqueness of F, let F : X → Y be another general quadratic function satisfying (6).Then by (20), we get for all x ∈ X and n ∈ N. Observe that, for 0 < q = 1/ p < 1/2, the last term tends to 1 as n → ∞ by (N5).This implies that N (F(x) − F (x), t) = 1 and F(x) = F (x), for all x ∈ X by (N2).
We can use Theorem 1 to get a classical result in the framework of normed spaces.Let (X, • ) be a normed linear space.Then we can define a fuzzy norm N X on X by following where x ∈ X and t ∈ R [21].Suppose that f : for all x, y, z ∈ X, where p > 0 and p / = 1, 2. Let N Y be a fuzzy norm on Y .Then we get Advances in Fuzzy Systems 9 for all x, y, z ∈ X and r, s, t ∈ R. Consider the case N Y (D f (x, y, z), r + s + t) = 0.This implies that x p + y p + z p ≥ D f x, y, z ≥ r + s + t, (38) and so either x p ≥ r or y p ≥ s or z p ≥ t in this case.Hence, for q = 1/ p, we have min N X (x, r q ), N X y, s q , N X (z, t q ) = 0, (39) for all x, y, z ∈ X and r, s, t > 0. Therefore, in every case, the inequality N Y D f x, y, z , r + s + t ≥ min N X (x, r q ), N X y, s q , N X (z, t q ) , ( for all x ∈ X. Remark 1.Consider a function f : X → Y satisfying (5) for all x, y, z ∈ X \ {0} and a real number q < 0. Take any t > 0.
holds.It means that f is a fuzzy q-almost general quadratic function, and by Theorem 1, we get the following stability result.1,2,then there is a unique general quadratic function F :X → Y such that F(x) − f (x) ≤ 5 + 2 p 2 p |4 − 2 p | + 2 • 2 p + 7 2|2 − 2 p | x p ,(42) Corollary 1.Let (X, • ) be a normed linear space, and let(Y , | • | ) be a Banach space.If D f x, y, z ≤ x p + y p + z p (41)for all x, y, z ∈ X, where p > 0 and p / =