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
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  in 1964. In fact he posed the question “Assume 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?” In 1941 Hyers gave a significant partial solution to this problem in his paper .
Hyers’ result was generalized by Aoki  for additive mappings. In 1978, Rassias and Song  generalized Hyers’ 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, [5–7].
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 . 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  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 , following Cheng and Mordeson , gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type . 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 [15–18].
Definition 1.1. Let be a real vector space. A function is called fuzzy normed on if for all and all (N1) for ,(N2) if and only if for all ,(N3) if ,(N4), (N5) is a nondecreasing function of and ,(N6)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 .
- S. M. Ulam, Problems in Modern Mathematics, Chapter 6, Wiley, New York, NY, USA, 1964.
- D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 1, pp. 222–224, 1941.
- T. Aoki, “On the stability of the linear transformation in banach spaces,” The Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.
- T. M. Rassias and J. H. Song, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, pp. 297–300, 1978.
- 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, “On the stability of functional equations and a problem of ulam,” Acta Applicandae Mathematicae, vol. 62, pp. 123–130, 2000.
- F. Skof, “Local properties and approximations of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.
- A. K. Katsaras, “Fuzzy topological vector spaces 2,” Fuzzy Sets and Systems, vol. 12, no. 2, pp. 143–154, 1984.
- C. Felbin, “Finite dimensional fuzzy normed linear space,” Fuzzy Sets and Systems, vol. 48, no. 2, pp. 239–248, 1992.
- J. Z. Xiao and X. H. Zhu, “Fuzzy normed space of operators and its completeness,” Fuzzy Sets and Systems, vol. 133, no. 3, pp. 389–399, 2003.
- T. Bag and S. K. Samanta, “Finite dimensional fuzzy normed linear spaces,” Journal of Fuzzy Mathematics, vol. 11, no. 3, pp. 687–705, 2003.
- S. C. Cheng and J. N. Mordeson, “Fuzzy linear operators and fuzzy normed linear spaces,” Bulletin of the Calcutta Mathematical Society, vol. 86, pp. 429–436, 1994.
- I. Kramosil and J. Michalek, “Fuzzy metric and statistical metric spaces,” Kybernetika, vol. 11, no. 5, pp. 326–334, 1975.
- A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy versions of Hyers-Ulam-Rassias theorem,” Fuzzy Sets and Systems, vol. 159, no. 6, pp. 720–729, 2008.
- A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, “Fuzzy stability of the jensen functional equation,” Fuzzy Sets and Systems, vol. 159, no. 6, pp. 730–738, 2008.
- A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy almost quadratic functions,” Results in Mathematics, vol. 52, no. 1-2, pp. 161–177, 2008.
- A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy approximately cubic mappings,” Information Sciences, vol. 178, no. 19, pp. 3791–3798, 2008.
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.