Research Article | Open Access

# Fuzzy Stability of a General Quadratic Functional Equation

**Academic Editor:**Rustom M. Mamlook

#### Abstract

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 βwhen is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?β. 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 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 [5β15].

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 MichΓ‘lek 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) *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 [22] and Jun and Kim [23] 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 [17] to exhibit a reasonable fuzzy version of stability for the general quadratic functional equation in the fuzzy normed linear space.

*Definition 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 , (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 [21]. 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 .

#### References

- S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience, New York, NY, USA, 1968. - D. H. Hyers, βOn the stability of the linear functional equation,β
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222β224, 1941. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Aoki, βOn the stability of the linear transformation in Banach spaces,β
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64β66, 1950. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Th. M. Rassias, βOn the stability of the linear mapping in Banach spaces,β
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297β300, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Adam and S. Czerwik, βOn the stability of the quadratic functional equation in topological spaces,β
*Banach Journal of Mathematical Analysis*, vol. 1, no. 2, pp. 245β251, 2007. View at: Google Scholar - St. Czerwik, βOn the stability of the quadratic mapping in normed spaces,β
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*, vol. 62, pp. 59β64, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Găvruţa, βA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,β
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431β436, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. H. Kim, βOn the stability of functional equations with square-symmetric operation,β
*Mathematical Inequalities & Applications*, vol. 4, no. 2, pp. 257β266, 2001. View at: Google Scholar - Y.-H. Lee, βOn the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2,β
*The Journal of Chungcheong Mathematical Society*, vol. 22, pp. 201β209, 2009. View at: Google Scholar - Y.-H. Lee, βOn the stability of the monomial functional equation,β
*Bulletin of the Korean Mathematical Society*, vol. 45, no. 2, pp. 397β403, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y.-H. Lee and K.-W. Jun, βA generalization of the Hyers-Ulam-Rassias stability of Jensen's equation,β
*Journal of Mathematical Analysis and Applications*, vol. 238, no. 1, pp. 305β315, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y.-H. Lee and K.-W. Jun, βA generalization of the Hyers-Ulam-Rassias stability of the Pexider equation,β
*Journal of Mathematical Analysis and Applications*, vol. 246, no. 2, pp. 627β638, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y.-H. Lee and K.-W. Jun, βOn the stability of approximately additive mappings,β
*Proceedings of the American Mathematical Society*, vol. 128, no. 5, pp. 1361β1369, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, βFuzzy stability of the Jensen functional equation,β
*Fuzzy Sets and Systems*, vol. 159, no. 6, pp. 730β738, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Skof, βLocal properties and approximation of operators,β
*Rendiconti del Seminario Matematico e Fisico di Milano*, vol. 53, pp. 113β129, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. K. Katsaras, βFuzzy topological vector spaces. II,β
*Fuzzy Sets and Systems*, vol. 12, no. 2, pp. 143β154, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Bag and S. K. Samanta, βFinite dimensional fuzzy normed linear spaces,β
*Journal of Fuzzy Mathematics*, vol. 11, no. 3, pp. 687β705, 2003. View at: Google Scholar - S. C. Cheng and J. N. Mordeson, βFuzzy linear operators and fuzzy normed linear spaces,β
*Bulletin of the Calcutta Mathematical Society*, vol. 86, no. 5, pp. 429β436, 1994. View at: Google Scholar - I. Kramosil and J. Michálek, βFuzzy metrics and statistical metric spaces,β
*Kybernetika*, vol. 11, no. 5, pp. 336β344, 1975. View at: Google Scholar - A. K. Mirmostafaee and M. S. Moslehian, βFuzzy versions of Hyers-Ulam-Rassias theorem,β
*Fuzzy Sets and Systems*, vol. 159, no. 6, pp. 720β729, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. K. Mirmostafaee and M. S. Moslehian, βFuzzy almost quadratic functions,β
*Results in Mathematics*, vol. 52, no. 1-2, pp. 161β177, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-M. Kim, βHyers-Ulam stability of a general quadratic functional equation,β
*Publications de l'Institut Mathématique*, vol. 73(87), pp. 129β137, 2003. View at: Publisher Site | Google Scholar - K.-W. Jun and H.-M. Kim, βThe generalized Hyers-Ulam stability of a general quadratic functional equation,β
*Journal of Applied Mathematics & Computing*, vol. 15, no. 3, pp. 377β392, 2004. View at: Google Scholar

#### Copyright

Copyright © 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.