Research Article | Open Access

Ehsan Movahednia, "Fuzzy Stability of Quadratic Functional Equations in General Cases", *International Scholarly Research Notices*, vol. 2011, Article ID 503164, 8 pages, 2011. https://doi.org/10.5402/2011/503164

# Fuzzy Stability of Quadratic Functional Equations in General Cases

**Academic Editor:**W. Sun

#### Abstract

The aim of this paper is to investigate fuzzy Hyers-Ulam-Rassias stability of the general case of quadratic functional equation where and fixed integers with . These functional equations are equivalent. This has been proven by Ulam, 1964.

#### 1. Introduction and Preliminaries

The stability problem of functional equations was raised by Ulam [1] in 1964. In fact he posed the question βAssume that a function satisfies a functional equation approximately according to some convention. Is it then possible to find near this function a function satisfying the equation accurately?β In 1941 Hyers gave a significant partial solution to this problem in his paper [2].

Hyersβ result was generalized by Aoki [3] for additive mappings. In 1978, Rassias and Song [4] generalized Hyersβ result, a fact which rekindled interest in the field. Such type of stability is now called the Ulam-Hyers-Rassias stability of functional equations.We refer the curious readers for further information on such problems to, for example, [5β7].

The functional equation is said to be a simple quadratic functional equation. The first person that investigated the stability of the simple quadratic equation was Skof [8]. He proved that, if is a mapping from a normed space into a Banach space satisfying then there is a unique simple quadratic function such that In 1984, Katsaras [9] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a linear space from various points of view [10, 11]. In particular, in 2003, Bag and Samanta [12], following Cheng and Mordeson [13], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [14]. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces.

Recently, considerable attention has been increasing to the problem of fuzzy stability of functional equations. Several fuzzy stability results concerning Cauchy, Jensen, simple quadratic, and cubic functional equations have been investigated [15β18].

*Definition 1.1. *Let be a real vector space. A function is called* fuzzy normed *on if for all and all (N_{1}) for ,(N_{2}) if and only if for all ,(N_{3}) if ,(N_{4}),
(N_{5}) is a nondecreasing function of and ,(N_{6})for is continuous on ,the pair () is called a *fuzzy normed vector space*.

*Example 1.2. *Let be a normed linear space. One can easily verify that, for each *, *
defines a fuzzy norm on .

*Definition 1.3. *Let be a fuzzy normed vector space. A sequence in is said to be convergent or converges if there exists an such that for all *.* In this case*, * is called the limit of the sequence , and one denotes it by

*Definition 1.4. *Let be a fuzzy normed vector space. A sequence in is said to be Cauchy if for each and each there exists an such that

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

#### 2. Main Results

Let be a linear space and () a fuzzy normed space. Let be a fuzzy Banach space and a function satisfying where and for all , such that is a function, and for some number with . Then, there exists a unique quadratic functional equation such that

*Proof. *Putting and in (2.1), we have that
Therefore
and
Now, replacing in (2.6),
and then by the assumption that and property of Definition 1.1 we obtain that
By comparing (2.6) and (2.8) and using property we obtain that
Again, by replacing , in (2.9),
Thus
By comparing (2.6), and (2.11) we obtain that
With following this process we obtain that
If , , then . Replacing by in (2.13) gives
By replacing in (2.14) we obtain that
Thus
It follows that
Let , and let be given. Since , there is some such that
Fix some . The convergence of series guarantees that there exists some such that, for each , the inequality holds. It follows that
Hence is a Cauchy sequence in fuzzy Banach space , and thus this sequence converges to some . It means that
Furthermore by putting in (2.17),
Next we will show that is quadratic. Let , and then we have that
The first six terms on the right-hand side of the above inequality tend to 1 as , and the seventh term, by (2.1), is greater than or equal to
which tends to 1 as . Therefore
for each and . So by property , we have that
Therefore is quadratic function. For every and , by (2.22), for large enough , we have that
Let be another quadratic function from to which satisfies (2.3). Since, for each ,
We have that
for each . Due to ,
for each and . Therefore .

#### References

- S. M. Ulam,
*Problems in Modern Mathematics*, Chapter 6, Wiley, New York, NY, USA, 1964. View at: MathSciNet - 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, no. 1, 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
*The Journal of the Mathematical Society of Japan*, vol. 2, pp. 64β66, 1950. View at: Google Scholar - T. M. Rassias and J. H. Song, βOn the stability of the linear mapping in Banach spaces,β
*Proceedings of the American Mathematical Society*, vol. 72, pp. 297β300, 1978. View at: Google Scholar - S. Czerwik, Ed.,
*Stability of Functional Equations of Ulam-Hyers-Rassias Type*, Hadronic Press, Palm Harbor, Fla, USA, 2003. - S. M. Jung,
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*, Hadronic Press, Palm Harbor, Fla, USA, 2001. - T. M. Rassias, βOn the stability of functional equations and a problem of ulam,β
*Acta Applicandae Mathematicae*, vol. 62, pp. 123β130, 2000. View at: Google Scholar - F. Skof, βLocal properties and approximations of operators,β
*Rendiconti del Seminario Matematico e Fisico di Milano*, vol. 53, pp. 113β129, 1983. View at: Google Scholar - A. K. Katsaras, βFuzzy topological vector spaces 2,β
*Fuzzy Sets and Systems*, vol. 12, no. 2, pp. 143β154, 1984. View at: Google Scholar - C. Felbin, βFinite dimensional fuzzy normed linear space,β
*Fuzzy Sets and Systems*, vol. 48, no. 2, pp. 239β248, 1992. View at: Google Scholar - J. Z. Xiao and X. H. Zhu, βFuzzy normed space of operators and its completeness,β
*Fuzzy Sets and Systems*, vol. 133, no. 3, pp. 389β399, 2003. 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, pp. 429β436, 1994. View at: Google Scholar - I. Kramosil and J. Michalek, βFuzzy metric and statistical metric spaces,β
*Kybernetika*, vol. 11, no. 5, pp. 326β334, 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, 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 - 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 - A. K. Mirmostafaee and M. S. Moslehian, βFuzzy approximately cubic mappings,β
*Information Sciences*, vol. 178, no. 19, pp. 3791β3798, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2011 Ehsan Movahednia. 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.