Journal of Applied Mathematics

Volume 2014 (2014), Article ID 514192, 11 pages

http://dx.doi.org/10.1155/2014/514192

Research Article

Fuzzy Stability of the Generalized Version of Drygas Functional Equation

Chang Il Kim,¹ Giljun Han,¹ and Seong-A Shim²

¹Department of Mathematics Education, Dankook University, 152 Jukjeon-ro, Suji-gu, Yongin-si, Gyeonggi-do 448-701, Republic of Korea

²Department of Mathematics, Sungshin Women's University, 249-1 Dongseon-Dong 3-Ga, Seongbuk-gu, Seoul 136-742, Republic of Korea

Received 4 February 2014; Accepted 5 April 2014; Published 27 April 2014

Academic Editor: Roberto Natalini

Copyright © 2014 Chang Il Kim et al. 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 consider the functional equation for a fixed rational number with and a fixed real number . We study the solution of the equation between linear spaces and prove the generalized Hyers-Ulam stability for it when the target space is a fuzzy normed space.

1. Introduction and Preliminaries

In 1940, Ulam proposed the following stability problem (cf. [1]):

“Let be a group and a metric group with the metric . Given a constant , does there exist a constant such that if a mapping satisfies for all , then there exists a unique homomorphism with for all ?”

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] and Moslehian and Rassias [4] for additive mappings, and by Rassias [5] 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 [6–10]).

Recently, the stability problems in the fuzzy spaces have been extensively studied (see [11–13]). The concept of fuzzy norm on a linear space was introduced by Katsaras [14] in 1984. Later, Cheng and Mordeson [15] gave a new definition of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [16]. In 2008, for the first time, Mirmostafaee and Moslehian [12, 13] used the definition of a fuzzy norm in [17] to obtain a fuzzy version of stability for the Cauchy functional equation

and the quadratic functional equation

We call a solution of (1) an additive mapping and a solution of (2) is called a quadratic mapping. Also, the equation

is called Drygas functional equation (see [18, 19] for details).

Najati and Moghimi [20] investigated the generalized Hyers-Ulam stability for functional equation derived from additive and quadratic functions on quasi-Banach spaces.

In this paper, we introduce the following functional equation for a fixed rational number with and a fixed real number :

with . It is easy to see that the function is a solution of the functional equation (4), so we can expect that a solution of (4) is additive-quadratic type. We note that the left-hand side of (4) is essentially the sum of two Drygas functionals and . In Section 2, a complete characterization of the solution of (4) is given. In Section 3, we prove the stability for (4) in fuzzy Banach spaces. One can find some kinds of gaps for finding in Theorems 13 and 14. In Theorem 15, we resolve these gaps for special and practical case of . Also, we give an example related to Theorem 15. We list some definitions related to fuzzy normed spaces.

Definition 1. Let be a real vector space. A function is called 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 of and ;(N6)for any , is continuous on .

In this case, the pair is called a fuzzy normed space.

Definition 2. Let be a fuzzy normed space. A sequence in is said to be convergent if there exists an such that for all . In this case, is called the limit of the sequence    in X and one denotes it by .

Definition 3. Let be a fuzzy normed space. A sequence in is said to be Cauchy if for any , , there is an such that for any and any positive integer , .

It is well known that every convergent sequence in a fuzzy normed space is Cauchy. A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and the complete fuzzy normed space is called a fuzzy Banach space.

2. Solution of (4)

In this section, we investigate solutions of (4) between linear spaces and by separating cases into odd functions and even functions. And then, in Theorem 8, it can be concluded that any solution of (4) is additive-quadratic type. We start with the odd function case.

Lemma 4. Let be an odd mapping with satisfying (4). Suppose that . Then is an additive mapping.

Proof. Since is an odd mapping, the functional equation (4) can be written by

for all .

If , then it is easy to check that is additive. Suppose that . Letting in (5), we have

for all . Replacing by in (5), we have

for all ; letting in (7), we have

for all . Replacing and by and in (5), respectively, by (6), we have

for all . Letting in (9), we have

for all , and letting in (5), we have

for all . By (5), (7), (8), (9), (10), and (11), we have

for all . If , then by (12), we have

for all and is additive-cubic ([20]). Since for all , is additive.

Lemma 5. Let be an odd mapping with satisfying (4). Suppose that . Then is an additive mapping.

Proof. Since is an odd mapping, the functional equation (4) can be written by

for all . Replacing by in (14), we have

for all , and interchanging and in (15), we have

for all . Replacing and by and in (5), respectively, we have

for all . By (16) and (17), we have

for all .

Replacing and by and in (5), respectively, we have

for all . Letting in (19), we have

for all . Replacing by in (14), we have

for all and by (20) and (21), we have

for all . Replacing by in (18), we have

for all and by (22) and (23), we have

for all . Since , we have

for all and letting in (25), we have

for all . By (22) and (23), we have

for all . Letting in (5), we have

for all , and by (27) and (28), we have

for all . Since , we have

for all , and hence is additive.

Combining Lemmas 4 and 5, we can get the following theorem.

Theorem 6. Let be an odd mapping with satisfying (4). Then is an additive mapping.

Now if we assume that is an even function, (4) turns into the following equation with :

And in [21], the authors proved the following theorem.

Theorem 7 (see [21]). Let be a mapping with . Then is quadratic if and only if satisfies (31) for all , a fixed nonzero rational number , and fixed real numbers , with .

By Theorems 6 and 7, we have the following theorem which is the conclusion of this section.

Theorem 8. Let be a mapping with . Then satisfies (4) if and only if is an additive-quadratic mapping.

3. The Generalized Hyers-Ulam Stability for (4)

In this section, we prove the generalized Hyers-Ulam stability of functional equation (4) in fuzzy normed spaces. Throughout this section, we assume that is a linear space, is a fuzzy Banach space, and is a fuzzy normed space.

For any mapping , we define the difference operator by

for all .

Theorem 9. Let be a function and let be a real number such that and

for all and all . Let be an odd mapping such that and

for all and all . Then there exists a unique additive mapping such that the inequality

holds for all and all .

Proof. Since is an odd mapping, the inequality (34) is equivalent to the following inequality:

for all and all . By (33) and (N3), we have

for all and all , and so by (37), we have

for all and all . Letting in (36), by (N3), we have

for all and all . By (38), (39), and (N3), we have

for all , all , and all positive integers . Hence by (40) and (N4), for any , we have

for all , all , and all positive integers . So for any , we have

for all , all , all nonnegative integers , and all positive integers . Thus, by (42), for any , we have

for all , all , all nonnegative integers , and all positive integers . Since is convergent, , and so by the usual argument is a Cauchy sequence in . Since is a fuzzy Banach space, there is a mapping defined by

for all . Moreover by (41), we have

for all , all , and all positive integers . Let be a real number with . Then, by (43), (44), (N4), and (N5), we have

for sufficiently large positive integer , all , and all or . Since is continuous on for all from (N2) and (N6), by taking , we get

for all and all , and so we have (35).

By (33), (34), and (N3), we have

for all and all . Since and

for all and all , by (44), (48), and (N4), we have

for sufficiently large , all , and all or . Since , for all , and so, by (N2), for all . By Theorem 8, is additive.

To prove the uniqueness of , let be another additive mapping satisfying (35). Then for any and a positive integer , , and so by (45),

holds for all , all positive integers , and all . Since , , and so for all .

Now we deal with the even function case.

Theorem 10. Let be a function and let be a real number such that and

for all and all . Let be an even mapping satisfying and (34). Then there exists a unique quadratic mapping such that the inequality

holds for all and all .

Proof. Since is an even mapping, the inequality (34) is equivalent to the following inequality:

for all and all . By (52) and (N3), we have

for all and all , and so by (55), we have

for all and all . Letting in (54), by (N3), we have

for all and all . By (52), (56), (57), and (N3), we have

for all , all , and all positive integers . Hence by (58) and (N4), for any , we have

for all , all , and all positive integers . So for any , we have

for all , all , all nonnegative integers , and all positive integers . Thus, by (60) and (N3), for any , we have

for all , all , all nonnegative integers , and all positive integers . Since is convergent, , and so by the usual argument again, is a Cauchy sequence in . Since is a fuzzy Banach space, there is a mapping defined by

for all . Moreover by (59), we have

for all , all , and all positive integers . Let be a real number with . Then, by (62), (63), and (N4), we have

for sufficiently large positive integer , all , and all or . Since is continuous on for all from (N2) and (N6), we get

for all and all , and so we have (53). By (34) and (N5), we have

for all and all . Since

for all and all , and by (62), (66), and (N4), we have

for sufficiently large , all , and all or .

Since , for all , and so by (N2), for all . By Theorem 8, is quadratic.

To prove the uniqueness of , let be another quadratic mapping satisfying (53). Then for any and a positive integer , , and so by (63),

holds for all , all positive integers , and all . Since , , and so for all .

Now we consider the next two theorems which are similar to Theorems 9 and 10. The proofs are straightforward and similar to those of Theorems 9 and 10.

Theorem 11. Let be a function and let be a real number such that such that

for all and all . Let be an odd mapping with satisfying (34). Then there exists a unique additive mapping such that the inequality

holds for all and all .

Theorem 12. Let be a function and let be a real number such that such that

for all and all . Let be an even mapping with satisfying (34). Then there exists a unique additive mapping such that the inequality

holds for all and all .

By combining Theorems 9 and 10, we can have the following theorem which is the main theorem of the paper.

Theorem 13. Let be a function and let be a real number such that such that

for all and all . Let be a mapping with satisfying (34). Then there exists a unique additive-quadratic mapping such that the inequality

holds for all and all , where

Proof. By (34), we have

for all and . By Theorem 9, there is a unique additive mapping such that

holds for all and all .

By (34), we have

for all and . By Theorem 10, there is a unique quadratic mapping such that

holds for all and all . Let . Then by (78) and (80), we have (75).

The uniqueness of satisfying (75) is trivial.

Also, if we combine Theorems 11 and 12, we have the following theorem.

Theorem 14. Let be a function and let be a real number such that such that

for all and all . Let be a mapping with satisfying (34). Then there exists a unique additive-quadratic mapping such that the inequality

holds for all and all , where

Among the examples of the function , there are lots of meaningful ones satisfying for all real numbers and for some real number . The following theorem says that a strong and useful result can be obtained in such cases.

Theorem 15. Let be a function and let be a real number such that such that

for all and all . Let be a mapping with satisfying (34). Then there exists a unique additive-quadratic mapping such that the inequality

holds for all and all , where

We can use Theorem 15 to get a classical result in the framework of normed spaces. For example, it is well known that for any normed space , mappings , defined by

are fuzzy norms on X. In [12, 13, 22], some examples are provided for the fuzzy norms , and other fuzzy norms. Here especially using the fuzzy norm and taking , we have the following example.

Example 16. Let be a mapping such that and

for all , a fixed rational number , and a fixed positive number such that . Then there exists a unique additive-quadratic mapping such that the inequality

holds for all .

The condition in Example 16 is indispensable. The following example shows that the inequality (88) is not stable for , especially in the case of . We will give the proof when , and the proof when is similar.

Example 17. Let be a real number with . Define mappings by

and a mapping by

We will show that satisfies the following inequality:

for all . Here,

But there do not exist an additive-quadratic mapping and a nonnegative constant such that

for all .

Proof. Note that , , and for all . First, suppose that . Then . Now suppose that . Then there is a positive integer such that

and so

Hence, we have

and so for any ,

for all . Thus,

Note that for all , , and . First, suppose that . Then for all . Now suppose that . Then there is a positive integer such that

and so

Hence, we have

and so for any ,

for all . Thus,

Hence, we have

for all , and so we have (92).

Suppose that there exist an additive mapping , a quadratic mapping , and a nonnegative constant such that satisfies (94). Since , by (94), we have

for all and all positive integers , and so for all . Since is additive,

for all . Take a positive integer such that , and pick with . Then

which is in contradiction with (107).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, NY, USA, 1960.

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, pp. 222–224, 1941.

T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

M. S. Moslehian and T. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007.

Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.

P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.

K. Cieplinski, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations—a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151–164, 2012.

St. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59–64, 1992.

P. Găvruta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.

M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361–376, 2006.

H.-M. Kim, J. M. Rassias, and J. Lee, “Fuzzy approximation of Euler-Lagrange quadratic mappings,” Journal of Inequalities and Applications, vol. 2013, article 358, pp. 1–15, 2013.

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 versions of Hyers-Ulam-Rassias theorem,” Fuzzy Sets and Systems, vol. 159, no. 6, pp. 720–729, 2008.

A. K. Katsaras, “Fuzzy topological vector spaces. II,” Fuzzy Sets and Systems, vol. 12, no. 2, pp. 143–154, 1984.

S. C. Cheng and J. N. Mordeson, “Fuzzy linear operators and fuzzy normed linear spaces,” Bulletin of the Calcutta Mathematical Society, vol. 86, no. 5, pp. 429–436, 1994.

I. Kramosil and J. Michálek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 5, pp. 336–344, 1975.

T. Bag and S. K. Samanta, “Finite dimensional fuzzy normed linear spaces,” Journal of Fuzzy Mathematics, vol. 11, no. 3, pp. 687–705, 2003.

H. Drygas, “Quasi-inner products and their applications,” in Advances in Multivariate Statistical Analysis, A. K. Gupta, Ed., pp. 13–30, Reidel, Dordrecht, The Netherlands, 1987.

V. A. Faiziev and P. K. Sahoo, “On the stability of Drygas functional equation on groups,” Banach Journal of Mathematical Analysis, vol. 1, no. 1, pp. 43–55, 2007.

A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399–415, 2008.

C. I. Kim, G. Han, and S.-A. Shim, “Hyers-Ulam stability for a class of quadratic functional equations via a typical form,” Abstract and Applied Analysis, vol. 2013, Article ID 283173, 8 pages, 2013.

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.