International Journal of Mathematics and Mathematical Sciences

Volume 2016 (2016), Article ID 1793065, 10 pages

http://dx.doi.org/10.1155/2016/1793065

## General Quadratic-Additive Type Functional Equation and Its Stability

^{1}Department of Mathematics Education, Gongju National University of Education, Gongju 32553, Republic of Korea^{2}Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Republic of Korea

Received 9 December 2015; Accepted 28 February 2016

Academic Editor: Vladimir V. Mityushev

Copyright © 2016 Yang-Hi Lee and Soon-Mo Jung. 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 investigate the general functional equation of the form − − , whose solutions are quadratic-additive mappings in connection with stability problems.

#### 1. Introduction

In 1940, Ulam [1] posed an important problem concerning the stability of group homomorphisms. In the following year, Hyers [2] solved the problem for the case of Cauchy additive functional equation. After a period longer than two decades, Rassias [3] generalized Hyers’ result and then Găvruta [4] extended Rassias’ result by allowing unbounded control functions. The concept of stability introduced by Rassias and Găvruta is known today with the term “generalized Hyers-Ulam stability” of functional equations.

A solution to the functional equationis called an additive mapping and a solution to the functional equationis called a quadratic mapping. If a mapping can be expressed by the sum of an additive mapping and a quadratic mapping, then we call the mapping a quadratic-additive mapping. Now, we consider the general quadratic-additive type functional equationwith nonzero real constants , , and . The mapping is a solution to this functional equation, where , are real constants. For the case , the stability of the functional equation (3) was investigated by some mathematicians (see [5] for ).

In this paper, we will prove that if , , and are nonzero real constants, then every solution to the functional equation (3) is a quadratic-additive mapping and, conversely, we will also prove that every quadratic-additive mapping is a solution to (3) provided , , and are rational constants. Moreover, we will prove the generalized Hyers-Ulam stability of the functional equation (3).

We remark here that (3) is a special form of the general linear equation, whose stability was investigated by Bahyrycz and Olko [6] via a different method from that we apply in this paper. In particular, the main result of this paper is a generalization of [6].

#### 2. Preliminaries

Throughout this paper, let and be vector spaces over , let be a real normed space, let be a real Banach space, and let , , and be positive real constants.

For a given mapping , we use the following abbreviations: for all .

The following lemmas were proved in [7].

Lemma 1. *Let , be real numbers with . If a mapping satisfies the functional equation for all , then is quadratic and is additive.*

Lemma 2. *Let , be rational constants with . A mapping satisfies for all if and only if is quadratic and is additive.*

Since the equality holds for all , the next couple of lemmas are direct consequences of Lemmas 1 and 2.

Lemma 3. *Let , be real numbers with . If a mapping satisfies the functional equation for all , then is quadratic and is additive.*

Lemma 4. *Let , be rational constants with . A mapping satisfies for all if and only if is quadratic and is additive.*

Lemma 5. *Let , be rational constants. If is a quadratic-additive mapping, then is a solution to the functional equation for all .*

*Proof. *Let be a quadratic-additive mapping. Then, is a quadratic mapping and is an additive mapping. Hence, we have , , and , where is an arbitrary rational number.

First, we will prove that the mapping satisfies for arbitrary rational numbers and . If or , then the equality holds for all . When , the equality follows from the equality for all . If and are nonzero rational constants with , then there exists an integer satisfying . According to Lemma 4, the equality implies that satisfies for all . Altogether, the mapping satisfies for arbitrary rational numbers and .

On the other hand, the equality implies that satisfies for arbitrary rational numbers and . Since the equality holds for all , we conclude that the mapping satisfies for all .

Lemma 6. *If real numbers , , and satisfy , then either , or , or .*

*Proof. *First, assume that and , , and are real constants satisfying , , , and . We then obtain the equalities , , and , which contradict the fact .

Second, assume that , , and are nonzero real constants satisfying conditions , , , and . We then obtain the equalities , , and , which contradict the fact .

We will now prove that is a quadratic-additive mapping provided is a solution to the functional equation for all .

*Remark 7. *We remark that if and are permutations, then we have

Theorem 8. *Let , , and be nonzero real numbers. If a mapping satisfies the functional equation (with , when ), then is a quadratic-additive mapping.*

*Proof. *In view of Remark 7, it is enough to check the following three cases: (1), (2), and (3). Notice that , when .

(1) If , , and are real constants with , then we can assume that without loss of generality by Remark 7. In view of Lemma 1, the equality implies that is a quadratic-additive mapping. In particular, we know that every solution to is a quadratic-additive mapping.

(2) If , , and are real constants with , then we can assume that without loss of generality due to Remark 7. We can easily show the validity of the following equalities:for all . Since every solution to is a quadratic-additive mapping, the equality implies that is a quadratic-additive mapping.

(3) By Lemma 6, we conclude that if , , and are real constants with , then either , or , or . By Remark 7, we can assume that without loss of generality. On account of Lemma 3, the equality implies that is a quadratic-additive mapping.

Theorem 9. *Let , , and be rational numbers. If is a quadratic-additive mapping, then satisfies the functional equation .*

*Proof. *Let be a quadratic-additive mapping. Then we easily show that is a quadratic mapping and is an additive mapping. Hence, we have , , and , where is an arbitrary rational number. In view of Lemma 5, we see that the mappings and satisfy the equalities and . Therefore, the equalities imply that the equalities and hold for all . From the equality for all , we get the equality , as desired.

The next theorem is a direct consequence of Theorems 8 and 9.

Theorem 10. *Let , , and be nonzero rational constants. A mapping satisfies for all if and only if is a quadratic-additive mapping.*

#### 3. Main Results

In the following theorems, Theorems of [8] can be slightly modified for the case when and without altering their proofs.

Theorem 11. *Given a real number with , let be a function satisfying the conditionfor all and let be a function satisfying the conditionfor all . If a mapping satisfies andfor all and if satisfiesfor all , then there exists a unique mapping such thatholds for all ,hold for all , and holds for all .*

Theorem 12. *Given a real constant with , let be a function satisfying the conditionfor all and let be a function satisfying the conditionfor all . If a mapping satisfies and (16) for all and if satisfies (17) for all , then there exists a unique mapping satisfying (18) for all and (19) for all and satisfyingfor all .*

Theorem 13. *Given a real constant with , let be a function satisfying the conditionsfor all and let be a function satisfying the conditionsfor all . If a mapping satisfies and (16) for all and if satisfies (17) for all , then there exists a unique mapping satisfying (18) for all and the equalities in (19) for all , such thatholds for all .*

It is to be noted that we can replace with in Theorems 11–13.

Lemma 14. *Let , , and be real constants. If a mapping satisfies and for all , then*(i)*,*(ii)*,*(iii)*,*(iv)*,*(v)*,*(vi)*,*(vii)*,*(viii)*,*(ix)*,*(x)*,*(xi)*,*(xii)* for all .*

*Proof. *Using the equalities for all and by Remark 7, we can easily prove the assertions.

Since every solution to the functional equation (3) is a quadratic-additive mapping according to Theorem 8 provided , , and are nonzero real constants, we can prove the following set of theorems by using Theorems 11–13.

Theorem 15. *For nonzero real constants , , and with , let be a function satisfying the conditionfor all . If a mapping satisfies andfor all , then there exists a unique quadratic-additive mapping such that (18) holds for all andholds for all .*

*Proof. *It follows from (29) that for all . If we put and , then satisfies condition (14) and inequality (16) for all . In view of Theorem 11, there exists a unique mapping satisfying equality (18) for all and the equalities in (19) for all , such that inequality (30) holds for all .

In view of Lemma 14, the equalities in (19) follow from equality (18). Hence, there exists a unique mapping satisfying equality (18) for all and inequality (30) for all .

By the same way as in the proof of Theorem 15, we can prove the following couple of theorems. Hence, we omit their proofs.

Theorem 16. *Given nonzero real constants , , and with , let be a function satisfying the conditionfor all . If a mapping satisfies and inequality (29) for all , then there exists a unique quadratic-additive mapping satisfying (18) for all andfor all .*

Theorem 17. *Given nonzero real constants , , and satisfying , let be a function satisfying the conditionsfor all . If a mapping satisfies and equality (29) for all , then there exists a unique quadratic-additive mapping satisfying equality (18) for all andfor all .*

By the same way as in the proofs of Theorems 18–23, we can use Lemma 14 to prove Theorems 15–17.

Theorem 18. *For nonzero real constants , , and with , let be a function satisfying the conditionfor all . If a mapping satisfies and (29) for all , then there exists a unique quadratic-additive mapping satisfying (18) for all andfor all .*

*Proof. *If we put and , then it follows from (29) thatfor all . For this case, conditions (14)–(17) in Theorem 11 are satisfied. Hence, there exists a unique mapping satisfying equality (18) for all and the equalities in (19) for all and satisfying inequality (37) for all . Since equality (18) implies equalities (19) in Lemma 14, there exists a unique mapping satisfying equality (18) for all as well as inequality (37) for all .

Theorem 19. *Given nonzero real constants , , and with , assume that is a function satisfying the conditionfor all . If a mapping satisfies and equality (29) for all , then there exists a unique quadratic-additive mapping satisfying (18) for all andfor all .*

Theorem 20. *Let , , and be nonzero real constants with . Assume that is a function satisfying the conditionsfor all . If a mapping satisfies and equality (29) for all , then there exists a unique quadratic-additive mapping satisfying (18) for all andfor all .*

Since every quadratic-additive mapping is a solution to the functional equation (3) and every solution to the functional equation (3) is a quadratic-additive mapping provided , , and are nonzero rational numbers by Theorem 9, the following set of theorems are direct consequences of Theorems 15–20.

Theorem 21. *For nonzero rational constants , , and , let be a function satisfying condition (28) for all . If a mapping satisfies and (29) for all , then there exists a unique quadratic-additive mapping satisfying (30) for all .*

Theorem 22. *For nonzero rational constants , , and , let be a function satisfying condition (32) for all . If a mapping satisfies and inequality (29) for all , then there exists a unique quadratic-additive mapping satisfying (33) for all .*

Theorem 23. *Given nonzero rational constants , , and , assume that is a function satisfying the conditions in (34) for all . If a mapping satisfies and equality (29) for all , then there exists a unique quadratic-additive mapping satisfying (35) for all .*

Theorem 24. *Let , , and be nonzero rational constants and let be a function satisfying condition (36) for all . If a mapping satisfies and (29) for all , then there exists a unique quadratic-additive mapping satisfying (37) for all .*

Theorem 25. *For nonzero rational constants , , and , let be a function satisfying condition (39) for all . If a mapping satisfies and equality (29) for all , then there exists a unique quadratic-additive mapping satisfying (40) for all .*

Theorem 26. *Given nonzero rational constants , , and , suppose is a function satisfying the conditions in (41) for all . If a mapping satisfies and equality (29) for all , then there exists a unique quadratic-additive mapping satisfying (42) for all .*

We remember that is a real normed space and is a real Banach space.

Corollary 27. *Assume that , , and are nonzero real constants with , is a real number with , and is a positive real number. If a mapping satisfies and the inequalityfor all , then there exists a unique quadratic-additive mapping such that holds for all andholds for all .*

*Proof. *If we put for all , then satisfies either the first inequality in (28) when and , or the second inequality in (28) when and , or the first inequality in (32) when and , or the second inequality in (32) when and , or the first inequality in (34) when and , or the second inequality in (34) when and for all . Therefore, by Theorems 15–17, we obtain the desired inequality (44).

The following corollary follows from Theorems 18–20.

Corollary 28. *Assume that , , and are nonzero real constants with , is a real number with , and is a positive real number. If a mapping satisfies and inequality (43) for all , then there exists a unique quadratic-additive mapping such that holds for all andholds for all .*

*Proof. *If we put for all , then satisfies either the first inequality in (36) when and , or the second inequality in (36) when and , or the first inequality in (39) when and , or the second inequality in (39) when and , or the first inequality in (41) when and , or the second inequality in (41) when and for all . Therefore, by Theorems 18–20, we obtain the desired inequality (45).

The following corollary follows from Theorems 21–23.

Corollary 29. *Let , , and be nonzero rational constants with and let be nonnegative real numbers with . If a mapping satisfies and inequality (43) for all , then there exists a unique quadratic-additive mapping such that inequality (44) holds for all .*

The following corollary follows from Theorems 24–26.

Corollary 30. *Let , , and be nonzero rational constants with and let be nonnegative real numbers with . If a mapping satisfies and inequality (43) for all , then there exists a unique quadratic-additive mapping such that inequality (45) holds for all .*

#### Competing Interests

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

#### Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.

#### Acknowledgments

Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2015R1D1A1A02061826).

#### References

- S. M. Ulam,
*Problems in Modern Mathematics*, John Wiley & Sons, New York, NY, USA, 1964. View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - T. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y.-H. Lee and S.-M. Jung, “Generalized Hyers-Ulam stability of a 3-dimensional quadratic-additive type functional equation,”
*International Journal of Mathematical Analysis*, vol. 9, no. 9–12, pp. 527–540, 2015. View at Publisher · View at Google Scholar · View at Scopus - A. Bahyrycz and J. Olko, “On stability of the general linear equation,”
*Aequationes Mathematicae*, vol. 89, no. 6, pp. 1461–1474, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y.-H. Lee and S.-M. Jung, “Stability of some 2-dimensional functional equations,”
*International Journal of Mathematical Analysis*, vol. 10, no. 4, pp. 171–190, 2016. View at Google Scholar - Y.-H. Lee and S.-M. Jung, “A general stability theorem for a class of functional equations including quadratic-additive functional equations,”
*Journal of Computational Analysis and Applications*, In press.