General Quadratic-Additive Type Functional Equation and Its Stability

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).