Fuzzy Stability of a Functional Equation Deriving from Quadratic and Additive Mappings

and Applied Analysis 3 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 Y,N ′ a fuzzy Banach space. For a given mapping f : X → Y , we use the abbreviation Df ( x, y ) : f ( 2x y ) f ( 2x − y 2f x − fx y − fx − y − 2f 2x , 2.1 for all x, y ∈ X. Recall Df ≡ 0 means that f is a general quadratic mapping. For given q > 0, the mapping f is called a fuzzy q-almost general quadratic mapping if N ′ ( Df ( x, y ) , t s ) ≥ minN x, s ,Ny, tq, 2.2 for all x, y ∈ X \ {0} and all s, t ∈ 0,∞ . Now, we get the general stability result in the fuzzy normed linear setting. Theorem 2.2. Let q be a positive real number with q / 1/2, 1. And let f be a fuzzy q-almost general quadratic mapping from a fuzzy normed space X,N into a fuzzy Banach space Y,N ′ . Then, there is a unique general quadratic mapping F : X → Y such that N ′ ( F x − f x , t ≥ sup 0<t′<t N ( x, t′q 7 2p 3p 4p / |4−2p|3p 5 2 · 2p 3p / 2|2−2p| q ) ,


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 considering 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  In the same year, they 21 proved a fuzzy version of stability for the quadratic functional equation We call a solution of 1.1 an additive mapping and a solution of 1.2 is called a quadratic mapping.Now, we consider the functional equation which is called a functional equation deriving from quadratic and additive mappings.We call a solution of 1.3 a general quadratic mapping.In 2008, Najati and Moghimi 22 obtained a stability of the functional equation 1.3 by taking and composing an additive mapping A and a quadratic mapping Q to prove the existence of a general quadratic mapping F which is close to the given mapping 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 f x f −x /2 − f 0 of it, respectively.In this paper, we get a general stability result of the functional equation deriving from quadratic and additive mappings 1.3 in the fuzzy normed linear space.To do it, we introduce a Cauchy sequence {J n f x }, starting from a given mapping f, which converges to the desired mapping F in the fuzzy sense.As we mentioned before, in previous studies of stability problem of 1.3 , 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 1.3
We use the definition of a fuzzy normed space given in 17 to exhibit a reasonable fuzzy version of stability for the functional equation deriving from quadratic and additive mappings in the fuzzy normed linear space.Definition 2.1 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 N − lim n → ∞ x n x.A sequence {x n } 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 Y, N a fuzzy Banach space.For a given mapping f : X → Y , we use the abbreviation for all x, y ∈ X. Recall Df ≡ 0 means that f is a general quadratic mapping.For given q > 0, the mapping f is called a fuzzy q-almost general quadratic mapping if N Df x, y , t s ≥ min N x, s q , N y, t q , 2.2 for all x, y ∈ X \ {0} and all s, t ∈ 0, ∞ .Now, we get the general stability result in the fuzzy normed linear setting.
Theorem 2.2.Let q be a positive real number with q / 1/2, 1.And let f be a fuzzy q-almost general quadratic mapping from a fuzzy normed space X, N into a fuzzy Banach space Y, N .Then, there is a unique general quadratic mapping F : X → Y such that for each x ∈ X and t > 0, where p 1/q.
Case 1.Let q > 1.We define a mapping J n f : X → Y by for all x ∈ X.Then, J 0 f x f x , J j f 0 f 0 , and for all x ∈ X \ {0} and j ≥ 0. Together with N3 , N4 , and 2.2 , this equation implies that if n m > m ≥ 0, then We observe that for some t > t 0 , the series ∞ j 0 7 2 p 3 p 4 p / 4 j 1 • 3 p 5 2 • 2 p 3 p / 2 j 2 2 jp t p converges for p 1/q < 1.It guarantees that for an arbitrary given c > 0, there exists n 0 ≥ 0 such that for each m ≥ n 0 and n > 0. By N5 and 2.6 we have for all x ∈ X \ {0}.Recall J n f 0 f 0 for all n > 0. Thus, {J n f x } becomes a Cauchy sequence for all x ∈ X.Since Y, N is complete, we can define a mapping F : X → Y by for all x ∈ X.Moreover, if we put m 0 in 2.6 , we have

2.11
for all x ∈ X. Next, we will show that F is a general quadratic mapping.Using N4 , we have 2.12 for all x, y ∈ X \ {0} and n ∈ N. The first six terms on the right hand side of 2.12 tend to 1 as n → ∞ by the definition of F and N2 , and the last term holds for all x, y ∈ X \ {0}.By N3 and 2.2 , we obtain 2.14 for all x, y ∈ X \ {0} and n ∈ N. Since q > 1, together with N5 , we can deduce that the last term of 2.12 also tends to 1 as n → ∞.It follows from 2.12 that N DF x, y , t 1, 2.15 for all x, y ∈ X \ {0} and t > 0. Since DF 0, 0 0, DF x, 0 0 and DF 0, y 0 for all x, y ∈ X \ {0}, this means that DF x, y 0 for all x, y ∈ X by N2 .
Abstract and Applied Analysis 7 Now, we approximate the difference between f and F in a fuzzy sense.For an arbitrary fixed x ∈ X and t > 0, choose 0 < ε < 1 and 0 < t < t.Since F is the limit of

2.16
Because 0 < ε < 1 is arbitrary and F 0 f 0 , we get 2.3 in this case.Finally, to prove the uniqueness of F, let F : X → Y be another general quadratic mapping satisfying 2.3 .Then, by 2.5 , we get 2.17 for all x ∈ X and n ∈ N. Together with N4 and 2.3 , this implies that Abstract and Applied Analysis for all x ∈ X and n ∈ N. Observe that for q 1/p, 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 mapping defined by for all x ∈ X.Then, we have J 0 f x f x , J j f 0 f 0 , and Df −2 j x/3, −2 j x/3 for all x ∈ X and j ≥ 0. If n m > m ≥ 0, then we have min N Df 2 j x/3, 2 j x/3 for all x ∈ X and t > 0. In the similar argument following 2.6 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

2.23
for all x ∈ X and t > 0. To prove that F is a general quadratic mapping, we have enough to show that the last term of 2.12 in Case 1 tends to 1 as n → ∞.By N3 and 2.2 , we get ≥ min N x, 2 2q−1 n−4q t q , N y, 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 , 2.24 for all x, y ∈ X \ {0} 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 2.12 , we can say that DF x, y 0 for all x, y ∈ X. Recall that in Case 1, 2.3 follows from 2.11 .By the same reasoning, we get 2.3 from 2.23 in this case.Now, to prove the uniqueness of F, let F be another general quadratic mapping satisfying 2.3 .Then, together with N4 , 2.3 , and 2.17 , we have ∞ 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 2.28 for all x ∈ X \ {0} and t > 0. Similar to the previous cases, it leads us to define the mapping F : X → Y by F x : N − lim n → ∞ J n f x .Putting m 0 in the above inequality, we have ≥ min N x, 2 1−2q n−3q t q , N y, 2 1−2q n−3q t q , 2.30 for all x, y ∈ X \ {0} and t > 0. Since 0 < q < 1/2, both terms on the right-hand side tend to 1 as n → ∞, which implies that the last term of 2.12 tends to 1 as n → ∞.Therefore, we can say that DF ≡ 0.Moreover, using the similar argument after 2.12 in Case 1, we get Abstract and Applied Analysis 13 2.3 from 2.29 in this case.To prove the uniqueness of F, let F : X → Y be another general quadratic mapping satisfying 2.3 .Then, by 2.17 , we get 2.31 for all x ∈ X and n ∈ N. Observe that for 0 < q < 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 .This completes the proof.
Remark 2.3.Consider a mapping f : X → Y satisfying 2.2 for all x, y ∈ X \ {0} and a real number q < 0. Take any t > 0. If we choose a real number s with 0 < 2s < t, then we have N Df x, y , t ≥ N Df x, y , 2s ≥ min N x, s q , N y, s q , 2.32 for all x, y ∈ X \ {0}.Since q < 0, we have lim s → 0 s q ∞.This implies that lim s → 0 N x, s q lim s → 0 N y, s q 1, 2.33 and so N Df x, y , t 1, 2.34 for all t > 0 and x, y ∈ X \ {0}.Since DF 0, 0 0, DF x, 0 0, and DF 0, y 0 for all x, y ∈ X \ {0}, this means that DF x, y 0 for all x, y ∈ X by N2 .In other words, f is itself a general quadratic mapping if f is a fuzzy q-almost general quadratic mapping for the case q < 0.
We can use Theorem 2.2 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 and so, either x p ≥ t or y p ≥ s in this case.Hence, for q 1/p, we have min N X x, s q , N X y, t q 0, 2.39 for all x, y ∈ X and s, t > 0. Therefore, in every case, N Y Df x, y , t s ≥ min N X x, s q , N X y, t q 2.40 holds.It means that f is a fuzzy q-almost general quadratic mapping, and by Theorem 2.
and Michalek type 19 .In 2008, Mirmostafaee and Moslehian 20 obtained a fuzzy version of stability for the Cauchy functional equation f x y − f x − f y 0. 1.1 X and t ∈ R 21 .Suppose that f : X → Y is a mapping into a Banach space Y, | • | such that ∈ X, where p > 0 and p / 1, 2. Let N Y be a fuzzy norm on Y .Then, we get for all x, y ∈ X and s, t ∈ R. Consider the case N Y Df x, y , t s 0. This implies that 2, we get the following stability result.Let X, • be a normed linear space, and let Y, | • | be a Banach space.If f : X → Y satisfies for all x ∈ X.