Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 534120 | 15 pages | https://doi.org/10.1155/2011/534120

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

Academic Editor: Yoshikazu Giga
Received14 Mar 2011
Accepted22 Jun 2011
Published18 Aug 2011

Abstract

We investigate a fuzzy version of stability for the functional equation in the sense of Mirmostafaee and 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 [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 [515].

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 Michalek 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 (1.1) 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 and a quadratic mapping to prove the existence of a general quadratic mapping which is close to the given mapping . 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 functional equation deriving from quadratic and additive mappings (1.3) in the fuzzy normed linear space. To do it, we introduce a Cauchy sequence , starting from a given mapping , which converges to the desired mapping 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 , 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 (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 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 , for , if and only if for all , if , , is a non-decreasing 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 a fuzzy Banach space. For a given mapping , we use the abbreviation for all . Recall means that is a general quadratic mapping. For given , the mapping is called a fuzzy -almost general quadratic mapping if for all and all . Now, we get the general stability result in the fuzzy normed linear setting.

Theorem 2.2. Let be a positive real number with . And let be a fuzzy q-almost general quadratic mapping from a fuzzy normed space into a fuzzy Banach space . Then, there is a unique general quadratic mapping such that for each and , where .

Proof. We will prove the theorem in three cases, , , and .
Case 1. Let . We define a mapping by for all . Then, , , and for all and . Together with (N3), (N4), and (2.2), this equation implies that if , then for 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 . By (N5) and (2.6) we have for all . Recall for all . Thus, becomes a Cauchy sequence for all . Since is complete, we can define a mapping by for all . Moreover, if we put in (2.6), we have for all . Next, we will show that is a general quadratic mapping. Using (N4), we have for all and . The first six terms on the right hand side of (2.12) tend to 1 as by the definition of and (N2), and the last term holds for all . By (N3) and (2.2), we obtain for all and . Since , together with (N5), we can deduce that the last term of (2.12) also tends to 1 as . It follows from (2.12) that for all and . Since , and for all , this means that for all by (N2).
Now, we approximate the difference between and in a fuzzy sense. For an arbitrary fixed and , choose and . Since is the limit of , there is such that . By (2.11), we have Because is arbitrary and , we get (2.3) in this case.
Finally, to prove the uniqueness of , let be another general quadratic mapping satisfying (2.3). Then, by (2.5), we get for all and . Together with (N4) and (2.3), 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 mapping defined by for all . Then, we have , , and for all and . If , then we have for all and . In the similar argument following (2.6) 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 all and . To prove that is a general quadratic mapping, we have enough to show that the last term of (2.12) in Case 1 tends to 1 as . By (N3) and (2.2), we get for all 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 (2.12), we can say that for all . 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 , let be another general quadratic mapping satisfying (2.3). Then, together with (N4), (2.3), and (2.17), 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 which implies that if , then for all and . Similar to the previous cases, it leads us to define the mapping by . Putting in the above inequality, we have for all . Notice that for all and . Since , both terms on the right-hand side tend to 1 as , which implies that the last term of (2.12) tends to 1 as . Therefore, we can say that . Moreover, using the similar argument after (2.12) in Case 1, we get (2.3) from (2.29) in this case. To prove the uniqueness of , let be another general quadratic mapping satisfying (2.3). Then, by (2.17), 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). This completes the proof.

Remark 2.3. Consider a mapping satisfying (2.2) 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 . Since , , and for all , this means that for all by (N2). In other words, is itself a general quadratic mapping if is a fuzzy -almost general quadratic mapping for the case .

We can use Theorem 2.2 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 where and [21]. Suppose that is a mapping 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 in this case. Hence, for , we have for all and . Therefore, in every case, holds. It means that is a fuzzy -almost general quadratic mapping, and by Theorem 2.2, we get the following stability result.

Corollary 2.4. Let be a normed linear space, and let be a Banach space. If satisfies