• Research Article
• Open access
• Published:

# Approximately Quadratic Mappings on Restricted Domains

## Abstract

We introduce a generalized quadratic functional equation , where , are nonzero real numbers with . We show that this functional equation is quadratic if , are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings.

## 1. Introduction

Under what conditions does there exist a group homomorphism near an approximate group homomorphism? This question concerning the stability of group homomorphisms was posed by Ulam [1]. The case of approximately additive mappings was solved by Hyers [2] on Banach spaces. In 1950 Aoki [3] provided a generalization of the Hyers' theorem for additive mappings and in 1978 Th. M. Rassias [4] generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias' theorem has been generalized by GÄƒvruÅ£a [6] who permitted the Cauchy difference to be bounded by a general control function. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations, see [7â€“24]. We also refer the readers to the books in [25â€“29].

It is easy to see that the quadratic function is a solution of each of the following functional equations:

(1.1)
(1.2)

where , are nonzero real numbers with . So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a function between real vector spaces and is quadratic if and only if there exists a unique symmetric biadditive function such that for all (see [13, 25, 27]).

We prove that the functional equations (1.1) and (1.2) are equivalent if , are nonzero rational numbers. The functional equation (1.1) is a spacial case of (1.2). Indeed, for the case in (1.2), we get (1.1).

In 1983 Skof [30] was the first author to solve the Hyers-Ulam problem for additive mappings on a restricted domain (see also [31â€“33]). In 1998 Jung [34] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [35â€“37]). J. M. Rassias [38] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains.

## 2. Solutions of (1.2)

In this section we show that the functional equation (1.2) is equivalent to the quadratic equation (1.1). That is, every solution of (1.2) is a quadratic function. We recall that , are nonzero real numbers with .

Theorem 2.1.

Let and be real vector spaces and be an odd function satisfying (1.2). If is a rational number, then .

Proof.

Since is odd, . Letting (resp., ) in (1.2), we get

(2.1)

for all . Replacing by in (1.2) and adding the obtained functional equation to (1.2), we get

(2.2)

for all . Replacing by in (2.2) and using (2.1), we have

(2.3)

for all . Again if we replace by in (2.3) and use (2.1), we get

(2.4)

for all . Applying (1.2) and using the oddness of , we have

(2.5)

for all , in . So it follows from (2.4) and (2.5) that

(2.6)

for all , in . It easily follows from (2.6) that is additive, that is, for all . So if is a rational number, then for all in . Therefore, it follows from (2.1) that for all in . Since , are nonzero, we infer that .

Theorem 2.2.

Let and be real vector spaces and be an even function satisfying (1.2). Then satisfies (1.1).

Proof.

Letting in (1.2), we get . Replacing by in (1.2), we get

(2.7)

for all . Replacing by in (2.7) and using the evenness of , we get

(2.8)

for all , in . Adding (2.7) to (2.8), we obtain

(2.9)

for all . Replacing by in (2.7), we get

(2.10)

for all , in . Using (2.7) in (2.10), by a simple computation, we get

(2.11)

for all , in . Putting in (2.11), we get that for all . Therefore, it follows from (2.11) that

(2.12)

for all , in . Replacing by in (2.12), we get that for all . So satisfies (1.1).

Theorem 2.3.

Let be a function between real vector spaces and . If is a rational number, then satisfies (1.2) if and only if satisfies (1.1).

Proof.

Let and be the odd and the even parts of . Suppose that satisfies (1.2). It is clear that and satisfy (1.2). By Theorems 2.1 and 2.2, and satisfies (1.1). Since , we conclude that satisfies (1.1).

Conversely, let satisfy (1.1). Then there exists a unique symmetric biadditive function such that for all (see [13]). Therefore

(2.13)

for all . So satisfies (1.2).

Proposition 2.4.

Let be a linear space with the norm . is an inner product space if and only if there exists a real number such that

(2.14)

for all , where .

Proof.

Let be a function defined by . If is an inner product space, then satisfies (2.14) for all . Conversely, let and the (even) function satisfy (2.14). So satisfies (1.2). By Theorem 2.3, the function satisfies (1.1), that is,

(2.15)

for all . Therefore is an inner product space (see [14]).

Proposition 2.5.

Let and be a linear space with the norm . Suppose that

(2.16)

for all , in , where and . Then .

Proof.

Setting in (2.16), we get

(2.17)

for all in . If we take with in (2.17), we get that . Letting in (2.16), we get

(2.18)

for all in . Letting in (2.16), we get

(2.19)

for all in . Since , it follows from (2.17) and (2.19) that

(2.20)

for all . Using (2.18) and (2.20), we get for all . Hence and (2.18) implies that . Finally, follows from (2.19).

Corollary 2.6.

Let be a linear space with the norm . is an inner product space if and only if there exists a real number and such that

(2.21)

for all , where .

## 3. Stability of (1.2) on Restricted Domains

In this section, we investigate the Hyers-Ulam stability of the functional equation (1.2) on a restricted domain. As an application we use the result to the study of an asymptotic behavior of that equation. It should be mentioned that Skof [39] was the first author who treats the Hyers-Ulam stability of the quadratic equation. Czerwik [8] proved a Hyers-Ulam-Rassias stability theorem on the quadratic equation. As a particular case he proved the following theorem.

Theorem 3.1.

Let be fixed. If a mapping satisfies the inequality

(3.1)

for all , then there exists a unique quadratic mapping such that for all . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

We recall that , are nonzero real numbers with .

Theorem 3.2.

Let and be given. Assume that an even mapping satisfies the inequality

(3.2)

for all with . Then there exists such that satisfies

(3.3)

for all with .

Proof.

Let with . Then, since , we get . So it follows from (3.2) that

(3.4)

for all with . So

(3.5)

for all with .

Let with . We have two cases.

Case 1.

. Then .

Case 2.

. Then we have . So

(3.6)

Therefore we get that from Cases 1 and 2. Hence by (3.4) we have

(3.7)

for all with . Set . Then

(3.8)

for all with . From (3.4) and (3.5), we get the following inequalities:

(3.9)

Using (3.7) and the above inequalities, we get

(3.10)

for all with . If with , then . So it follows from (3.10) that

(3.11)

Letting in (3.11), we get

(3.12)

for all with . Letting (and with ) in (3.11), we get . Therefore it follows from (3.11) and (3.12) that

(3.13)

for all with . Since is even, the inequality (3.13) holds for all with . Therefore

(3.14)

for all with . This completes the proof by letting .

Theorem 3.3.

Let and be given. Assume that an even mapping satisfies the inequality (3.2) for all with . Then satisfies

(3.15)

for all .

Proof.

By Theorem 3.2 there exists such that satisfies (3.3) for all with and (see the proof of Theorem 3.2). Using Theorem 2 of [38], we get that

(3.16)

all .

Theorem 3.4.

Let and be given. Assume that an even mapping satisfies the inequality (3.2) for all with . Then there exists a unique quadratic mapping such that and

(3.17)

for all .

Proof.

The result follows from Theorems 3.1 and 3.3.

Skof [39] has proved an asymptotic property of the additive mappings and Jung [34] has proved an asymptotic property of the quadratic mappings (see also [36]). We prove such a property also for the quadratic mappings.

Corollary 3.5.

An even mapping satisfies (1.2) if and only if the asymptotic condition

(3.18)

holds true.

Proof.

By the asymptotic condition (3.18), there exists a sequence monotonically decreasing to 0 such that

(3.19)

for all with . Hence, it follows from (3.19) and Theorem 3.4 that there exists a unique quadratic mapping such that

(3.20)

for all . Since is a monotonically decreasing sequence, the quadratic mapping satisfies (3.20) for all . The uniqueness of implies for all . Hence, by letting in (3.20), we conclude that is quadratic.

Corollary 3.6.

Let be rational. An even mapping is quadratic if and only if the asymptotic condition (3.18) holds true.

## References

1. Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.

2. Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222â€“224. 10.1073/pnas.27.4.222

3. Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64â€“66. 10.2969/jmsj/00210064

4. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978, 72(2):297â€“300. 10.1090/S0002-9939-1978-0507327-1

5. Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223â€“237. 10.1090/S0002-9904-1951-09511-7

6. GÄƒvruÅ£a P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994, 184(3):431â€“436. 10.1006/jmaa.1994.1211

7. Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984, 27(1â€“2):76â€“86.

8. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der UniversitÃ¤t Hamburg 1992, 62: 59â€“64. 10.1007/BF02941618

9. FaÄ­ziev VA, Rassias ThM, Sahoo PK: The space of -additive mappings on semigroups. Transactions of the American Mathematical Society 2002, 354(11):4455â€“4472. 10.1090/S0002-9947-02-03036-2

10. Forti GL: An existence and stability theorem for a class of functional equations. Stochastica 1980, 4(1):23â€“30. 10.1080/17442508008833155

11. Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995, 50(1â€“2):143â€“190. 10.1007/BF01831117

12. Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996, 48(3â€“4):217â€“235.

13. Kannappan P: Quadratic functional equation and inner product spaces. Results in Mathematics 1995, 27(3â€“4):368â€“372.

14. Jordan P, von Neumann J: On inner products in linear, metric spaces. Annals of Mathematics 1935, 36(3):719â€“723. 10.2307/1968653

15. Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992, 44(2â€“3):125â€“153. 10.1007/BF01830975

16. Isac G, Rassias ThM: Stability of -additive mappings: applications to nonlinear analysis. International Journal of Mathematics and Mathematical Sciences 1996, 19(2):219â€“228. 10.1155/S0161171296000324

17. Jun K-W, Lee Y-H: On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality. Mathematical Inequalities & Applications 2001, 4(1):93â€“118.

18. Najati A: Hyers-Ulam stability of an -Apollonius type quadratic mapping. Bulletin of the Belgian Mathematical Society. Simon Stevin 2007, 14(4):755â€“774.

19. Najati A, Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. Journal of Mathematical Analysis and Applications 2007, 335(2):763â€“778. 10.1016/j.jmaa.2007.02.009

20. Najati A, Park C: The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between -algebras. Journal of Difference Equations and Applications 2008, 14(5):459â€“479. 10.1080/10236190701466546

21. Park C-G: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002, 275(2):711â€“720. 10.1016/S0022-247X(02)00386-4

22. Rassias ThM: On a modified Hyers-Ulam sequence. Journal of Mathematical Analysis and Applications 1991, 158(1):106â€“113. 10.1016/0022-247X(91)90270-A

23. Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000, 62(1):23â€“130. 10.1023/A:1006499223572

24. Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000, 251(1):264â€“284. 10.1006/jmaa.2000.7046

25. AczÃ©l J, Dhombres J: Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462.

26. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.

27. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. BirkhÃ¤user, Boston, Mass, USA; 1998:vi+313.

28. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.

29. Rassias ThM (Ed): Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.

30. Skof F: Sull' approssimazione delle applicazioni localmente -additive. Atti della Accademia delle Scienze di Torino 1983, 117: 377â€“389.

31. Hyers DH, Isac G, Rassias ThM: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proceedings of the American Mathematical Society 1998, 126(2):425â€“430. 10.1090/S0002-9939-98-04060-X

32. Jung S-M: Hyers-Ulam-Rassias stability of Jensen's equation and its application. Proceedings of the American Mathematical Society 1998, 126(11):3137â€“3143. 10.1090/S0002-9939-98-04680-2

33. Jung S-M, Moslehian MS, Sahoo PK: Stability of a generalized Jensen equation on restricted domains. Journal of Mathematical Inequalities 2010, 4: 191â€“206.

34. Jung S-M: On the Hyers-Ulam stability of the functional equations that have the quadratic property. Journal of Mathematical Analysis and Applications 1998, 222(1):126â€“137. 10.1006/jmaa.1998.5916

35. Jung S-M: Stability of the quadratic equation of Pexider type. Abhandlungen aus dem Mathematischen Seminar der UniversitÃ¤t Hamburg 2000, 70: 175â€“190. 10.1007/BF02940912

36. Jung S-M, Kim B: On the stability of the quadratic functional equation on bounded domains. Abhandlungen aus dem Mathematischen Seminar der UniversitÃ¤t Hamburg 1999, 69: 293â€“308. 10.1007/BF02940881

37. Jung S-M, Sahoo PK: Hyers-Ulam stability of the quadratic equation of Pexider type. Journal of the Korean Mathematical Society 2001, 38(3):645â€“656.

38. Rassias JM: On the Ulam stability of mixed type mappings on restricted domains. Journal of Mathematical Analysis and Applications 2002, 276(2):747â€“762. 10.1016/S0022-247X(02)00439-0

39. Skof F: Proprieta' locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53(1):113â€“129. 10.1007/BF02924890

## Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2010-0007143).

## Author information

Authors

### Corresponding author

Correspondence to Soon-Mo Jung.

## Rights and permissions

Reprints and permissions

Najati, A., Jung, SM. Approximately Quadratic Mappings on Restricted Domains. J Inequal Appl 2010, 503458 (2010). https://doi.org/10.1155/2010/503458