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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2011, Article IDΒ 504802, 16 pages
http://dx.doi.org/10.1155/2011/504802
Research Article

Fuzzy Stability of a Quadratic-Additive Functional Equation

Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea

Received 27 April 2011; Accepted 25 August 2011

Academic Editor: NaseerΒ Shahzad

Copyright Β© 2011 Sun Sook Jin and Yang Hi Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate a fuzzy version of stability for the functional equation .

1. Introduction

A classical question in the theory of functional equations is β€œwhen 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 the unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5–16].

In 1984, Katsaras [17] 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 [18], following Cheng and Mordeson [19], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of the Kramosil and MichΓ‘lek type [20]. In 2008, Mirmostafaee and Moslehian [21] introduced for the first time the notion of fuzzy Hyers-Ulam-Rassias stability. They obtained a fuzzy version of stability for the Cauchy functional equation whose solution is called an additive mapping. In the same year, they [22] proved a fuzzy version of stability for the quadratic functional equation whose solution is called a quadratic mapping. Now we consider the quadratic-additive functional equation whose solution is called a quadratic-additive mapping. In [23], Chang et al. obtained a stability of the quadratic-additive functional equation by taking and composing an additive mapping and a quadratic mapping to prove the existence of a quadratic-additive 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 quadratic-additive functional equation 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), Chang et al. 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.

2. Fuzzy Stability of the Quadratic-Additive Functional Equation

We use the definition of a fuzzy normed space given in [18] to exhibit a reasonable fuzzy version of stability for the quadratic-additive functional equation in the fuzzy normed linear space.

Definition 2.1 (see [18]). Let be a real linear space. A mapping (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 mapping 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 . For given , the mapping is called a fuzzy -almost quadratic-additive mapping if for all and . 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 quadratic-additive mapping from a fuzzy normed space into a fuzzy Banach space . Then there is a unique quadratic-additive mapping such that for all and , where .

Proof. It follows from (2.2) and (N4) that for all . By (N2), we have . We will prove the theorem in three cases: , , and . Case 1. Let and let be a mapping defined by for all . Notice that 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 (2.7), we have 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 (2.7), we have for all . Next we will show that is a desired quadratic-additive mapping. Using (N4), we have for all and . The first eleven terms on the right-hand side of (2.13) 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 (2.13) also tends to 1 as . It follows from (2.13) that for each and . By (N2), this means that for all . 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 (2.12), we have since . Because is arbitrary, we get inequality (2.3) in this case. Finally, to prove the uniqueness of , let be another quadratic-additive mapping satisfying (2.3). Then by (2.6), 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 also have and for all and . If , then we have for all and . In a similar argument following (2.7) 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 quadratic-additive mapping, we have enough to show that the last term of (2.13) 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, with arguments similar to those of (2.13)–(2.15), we can say that for all . Recall that, in Case 1, inequality (2.3) follows from (2.12). By the same reasoning, we get (2.3) from (2.24) in this case.
Now to prove the uniqueness of , let be another quadratic-additive mapping satisfying (2.3). Then, together with (N4), (2.3), and (2.18), 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 and . Notice that for each and . Since , all terms on the right-hand side tend to 1 as , which implies that the last term of (2.13) tends to 1 as . Therefore, we can say that . Moreover, using arguments similar to those of (2.13)–(2.15) in Case 1, we get inequality (2.3) from (2.30) in this case. To prove the uniqueness of , let be another quadratic-additive mapping satisfying (2.3). Then by (2.18), 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).

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 for all . Since , we have . This implies that and so for all and . By (N2), we are allowed to get for all . In other words, is itself a quadratic-additive mapping if is a fuzzy -almost quadratic-additive mapping for the case .

Corollary 2.4. Let be an even mapping satisfying all of the conditions of Theorem 2.2. Then there is a unique quadratic mapping such that for all and , where .

Proof. Let be defined as in Theorem 2.2. Since is an even mapping, we obtain for all . Notice that and for all and . From these, using the similar method in Theorem 2.2, we obtain the quadratic-additive mapping satisfying (2.36). Notice that for all , is even, , and for all . Hence, we get for all . This means that is a quadratic mapping.

Corollary 2.5. Let be an odd mapping satisfying all of the conditions of Theorem 2.2. Then there is a unique additive mapping such that for all and , where .

Proof. Let be defined as in Theorem 2.2. Since is an odd mapping, we obtain for all . Notice that and for all and . From these, using the similar method in Theorem 2.2, we obtain the quadratic-additive mapping satisfying (2.40). Notice that for all is odd, , and for all . Hence, we get for all . This means that is an additive mapping.

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 , see [22]. 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 , , , 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 quadratic-additive mapping, and by Theorem 2.2, we get the following stability result.

Corollary 2.6. Let be a normed linear space and let be a Banach space. If satisfies for all , where and , then there is a unique quadratic-additive mapping such that for all .

Lemma 2.7. If is a mapping such that for all with , then for all .

Proof. We easily know that for all . Using a similar method, we have , , , , , , , , , , , , , and for all , as we desired.

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