Fuzzy Stability of a General Quadratic Functional Equation
We investigate a fuzzy version of stability for the functional equation in the sense of M. Mirmostafaee and M. S. Moslehian.
1. 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  in 1940. In the next year, Hyers  gave a partial solution of Ulam's problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki  for additive mappings, and by Rassias  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  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 , following Cheng and Mordeson , gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type . In 2008, Mirmostafaee and Moslehian  obtained a fuzzy version of stability for the Cauchy functional equation: In the same year, they  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  and Jun and Kim  obtained a stability of the functional equation (3) by taking and composing an additive map and a quadratic map to prove the existence of a general quadratic function which is close to the given function . In their processing, is approximate to the odd part of , and is close to the even part of it, respectively.
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 , starting from a given function , which converges to the desired function 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 , respectively. According to our proposal in this paper, we can take the desired approximate solution at once. Therefore, this idea is a refinement with respect to the simplicity of the proof.
2. Fuzzy Stability of the Functional Equation (3)
We use the definition of a fuzzy normed space given in  to exhibit a reasonable fuzzy version of stability for the general quadratic functional equation in the fuzzy normed linear space.
Definition 1 (see ). Let be a real linear space. A function (the so-called fuzzy subset) is said to be a fuzzy norm on if for all and all , (N1) for , (N2) if and only if for all , (N3) if , (N4), (N5) is a nondecreasing function on , and .
The pair is called a fuzzy normed linear space. Let be a fuzzy normed linear space. Let be a sequence in . Then is said to be convergent if there exists such that , for all . In this case, is called the limit of the sequence and we denote it by . A sequence , in is called Cauchy if for each and each , there exists such that for all and all , we have . 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 be a fuzzy normed space, and let be a fuzzy Banach space. For a given mapping , we use the abbreviation for all . Recall means that is a general quadratic function. For given , the function is called a fuzzy -almost general quadratic function, if for all and . Now we get the general stability result in the fuzzy normed linear setting.
Theorem 1. Let be a positive real number with . And let be a fuzzy -almost general quadratic function from a fuzzy normed space into a fuzzy Banach space . Then there is a unique general quadratic function such that for all and , where .
Proof. We will prove the theorem in three cases, , , and .
Case 1. Let . We define the function by for all . Notice that , , and for all and . Together with (N3), (N4) and (5), this equation implies that if thenfor all and . Let be given. Since , there is such that We observe that for some , the series converges for . It guarantees that, for an arbitrary given , there exists such that for each and . Together with (N5) and (9), this implies that for all . Hence is a Cauchy sequence in the fuzzy Banach space . And so we can define a mapping by for all . Moreover, if we put in (9), we have for all . Next we will show that is a desired general quadratic function. Using (N4), we have for all and . The first six terms on the right hand side of (15) tend to 1 as by the definition of and (N2), and the last term holds for all . By (N3) and (5), we obtain for all and . Since , together with (N5), we can deduce that the last term of (15) also tends to 1 as . It follows from (15) that for all and . By (N2), it leads us to prove that is a general quadratic function.
For an arbitrary fixed and , choose and . Since is the limit of , there is such that . By (14), we haveBecause is arbitrary, we get the inequality (6) in this case.
Finally, to prove the uniqueness of , let be another general quadratic function satisfying (6). Then by (8), we get for all and . Together with (N4) and (6), this implies that for all and . Observe that, for , the last term of the above inequality tends to 1 as by (N5). This implies that , and so we get for all by (N2).
Case 2. Let , and let be a function defined by for all . Then we also have , , and for all and . If , then In the similar argument following (9) of the previous case, we can define the limit of the Cauchy sequence in the Banach fuzzy space . Moreover, putting in the above inequality, we have for each and . To prove that is a general quadratic function, we have enough to show that the last term of (15) in Case 1 tends to 1 as . By (N3) and (5), we get for each and . Observe that all the terms on the right-hand side of the above inequality tend to 1 as , since . Hence, together with the similar argument after (15), we can say that , for all . 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 , let be another general quadratic function satisfying (6). Then, together with (N4), (6), and (20), we have for all and . Since ; in this case, both terms on the right hand side of the above inequality tend to 1 as by (N5). This implies that , and so for all by (N2).
Case 3. Finally, we take and define by for all . Then we have , , and for all and . Moreover if , thenfor all , , and . Similar to the previous cases, it leads us to define the function by . Putting in the above inequality, we have for all and . Notice that for all and . Since , all terms on the right hand side tend to 1 as , which implies that the last term of (15) tends to 1 as . Therefore, we can say that . 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 , let be another general quadratic function satisfying (6). Then by (20), we get for all and . Observe that, for , the last term tends to 1 as by (N5). This implies that and , for all by (N2).
We can use Theorem 1 to get a classical result in the framework of normed spaces. Let be a normed linear space. Then we can define a fuzzy norm on by following where and . Suppose that is a function into a Banach space such that for all , where and . Let be a fuzzy norm on . Then we get for all and . Consider the case . This implies that and so either or or in this case. Hence, for , we have for all and . Therefore, in every case, the inequality holds. It means that is a fuzzy -almost general quadratic function, and by Theorem 1, we get the following stability result.
Corollary 1. Let be a normed linear space, and let be a Banach space. If for all , where and , then there is a unique general quadratic function such that for all .
Remark 1. Consider a function satisfying (5) for all and a real number . Take any . If we choose a real number with , then we have for all . Since , we have . This implies that and so for all and . By (N2), it allows us to get , for all .
S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1968.
M. Adam and S. Czerwik, “On the stability of the quadratic functional equation in topological spaces,” Banach Journal of Mathematical Analysis, vol. 1, no. 2, pp. 245–251, 2007.View at: Google Scholar
G. H. Kim, “On the stability of functional equations with square-symmetric operation,” Mathematical Inequalities & Applications, vol. 4, no. 2, pp. 257–266, 2001.View at: Google Scholar
Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2,” The Journal of Chungcheong Mathematical Society, vol. 22, pp. 201–209, 2009.View at: Google Scholar
T. Bag and S. K. Samanta, “Finite dimensional fuzzy normed linear spaces,” Journal of Fuzzy Mathematics, vol. 11, no. 3, pp. 687–705, 2003.View at: Google Scholar
S. C. Cheng and J. N. Mordeson, “Fuzzy linear operators and fuzzy normed linear spaces,” Bulletin of the Calcutta Mathematical Society, vol. 86, no. 5, pp. 429–436, 1994.View at: Google Scholar
I. Kramosil and J. Michálek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 5, pp. 336–344, 1975.View at: Google Scholar
K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam stability of a general quadratic functional equation,” Journal of Applied Mathematics & Computing, vol. 15, no. 3, pp. 377–392, 2004.View at: Google Scholar