Fuzzy Stability of the Generalized Version of Drygas Functional Equation

In the next year, Hyers [2] gave a partial solution ofUlam’s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki [3] and Moslehian and Rassias [4] for additive mappings, and by Rassias [5] for linear mappings, to consider the stability problemwith unboundedCauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians (see [6–10]). Recently, the stability problems in the fuzzy spaces have been extensively studied (see [11–13]). The concept of fuzzy norm on a linear space was introduced by Katsaras [14] in 1984. Later, Cheng and Mordeson [15] gave a new definition of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [16]. In 2008, for the first time, Mirmostafaee and Moslehian [12, 13] used the definition of a fuzzy norm in [17] to obtain a fuzzy version of stability for the Cauchy functional equation


Introduction and Preliminaries
In 1940, Ulam proposed the following stability problem (cf.[1]): "Let  1 be a group and  2 a metric group with the metric .Given a constant  > 0, does there exist a constant  > 0 such that if a mapping  :  1 →  2 satisfies ((), ()()) <  for all ,  ∈  1 , then there exists a unique homomorphism ℎ :  1 →  2 with ((), ℎ()) <  for all  ∈  1 ?" 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] and Moslehian and Rassias [4] for additive mappings, and by Rassias [5] 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 [6][7][8][9][10]).
Najati and Moghimi [20] investigated the generalized Hyers-Ulam stability for functional equation derived from additive and quadratic functions on quasi-Banach spaces.with (0) = 0.It is easy to see that the function () =  2 +  is a solution of the functional equation ( 4), so we can expect that a solution of ( 4) is additive-quadratic type.We note that the left-hand side of ( 4) is essentially the sum of two Drygas functionals   (, ) and   (, ).In Section 2, a complete characterization of the solution of ( 4) is given.In Section 3, we prove the stability for (4) in fuzzy Banach spaces.One can find some kinds of gaps for finding  in Theorems 13 and 14.In Theorem 15, we resolve these gaps for special and practical case of (, ).Also, we give an example related to Theorem 15.We list some definitions related to fuzzy normed spaces.(N6) for any  ̸ = 0, (, ⋅) is continuous on R.
In this case, the pair (, ) is called a fuzzy normed space.
Definition 2. Let (, ) be a fuzzy normed space.A sequence {  } in  is said to be convergent if there exists an  ∈  such that lim  → ∞ (  − , ) = 1 for all  > 0. In this case,  is called the limit of the sequence {  } in X and one denotes it by  − lim  → ∞   = .
Definition 3. Let (, ) be a fuzzy normed space.A sequence {  } in  is said to be Cauchy if for any  > 0,  > 0, there is an  ∈  such that for any  ≥  and any positive integer , ( + −   , ) > 1 − .
It is well known that every convergent sequence in a fuzzy normed space is Cauchy.A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and the complete fuzzy normed space is called a fuzzy Banach space.(4) In this section, we investigate solutions of (4) between linear spaces  and  by separating cases into odd functions and even functions.And then, in Theorem 8, it can be concluded that any solution of (4) is additive-quadratic type.We start with the odd function case.Lemma 4. Let  :  →  be an odd mapping with (0) = 0 satisfying (4).Suppose that  ̸ = − .Then  is an additive mapping.

Solution of
Proof.Since  is an odd mapping, the functional equation ( 4) can be written by for all ,  ∈ .
Proof.Since  is an odd mapping, the functional equation ( 4) can be written by for all ,  ∈ .Replacing  by  +  in ( 14), we have for all ,  ∈ , and interchanging  and  in ( 15), we have for all ,  ∈ .Replacing  and  by  +  and  in ( 5), respectively, we have for all ,  ∈ .By ( 16) and ( 17), we have for all ,  ∈ .
Combining Lemmas 4 and 5, we can get the following theorem.Theorem 6.Let  :  →  be an odd mapping with (0) = 0 satisfying (4).Then  is an additive mapping.Now if we assume that  is an even function, (4) turns into the following equation with  = 1: And in [21], the authors proved the following theorem.
By Theorems 6 and 7, we have the following theorem which is the conclusion of this section.Theorem 8. Let  :  →  be a mapping with (0) = 0. Then  satisfies (4) if and only if  is an additive-quadratic mapping.

The Generalized Hyers-Ulam
Stability for (4) In this section, we prove the generalized Hyers-Ulam stability of functional equation ( 4) in fuzzy normed spaces.Throughout this section, we assume that  is a linear space, (, ) is a fuzzy Banach space, and (,   ) is a fuzzy normed space.
For any mapping  :  → , we define the difference operator  : holds for all  ∈  and all  > 0.
Proof.Since  is an odd mapping, the inequality (34) is equivalent to the following inequality: for all ,  ∈  and all  > 0. By (33) and (N3), we have for all ,  ∈  and all  > 0, and so by (37), we have for all ,  ∈  and all  > 0. Letting  = 0 in (36), by (N3), we have for all  ∈  and all  > 0. By (38), (39), and (N3), we have for all  ∈ , all  > 0, and all positive integers .Hence by ( 40) and (N4), for any  ∈ , we have for all  ∈ , all  > 0, and all positive integers .So for any  ∈ , we have for all  ∈ , all  > 0, all nonnegative integers , and all positive integers .Thus, by (42), for any  ∈ , we have for all  ∈ , all  > 0, all nonnegative integers , and all positive integers .
Now we deal with the even function case.