A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

Abstract

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

1. Introduction

Let and be abelian groups. For any mapping , we define for all . A mapping is called an additive mapping (or a quadratic mapping) if satisfies the functional equation (or ) for all . We notice that the mappings given by and are solutions of and , respectively.

A mapping is called a quadratic-additive mapping if and only if is represented by the sum of an additive mapping and a quadratic mapping. A functional equation is called a quadratic-additive type functional equation if and only if each of its solutions is a quadratic-additive mapping. The mapping given by is a solution of the quadratic-additive type functional equation. In the study of the stability problems of quadratic-additive type functional equations, the uniqueness problem frequently occurs under various conditions. However, no author has succeeded in proving the relevant uniqueness (see [1–3]).

In this paper, we prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. In Section 4, we apply our uniqueness theorem to complement stability theorems of the papers [4, 5] where the uniqueness has not been proved. Indeed, this uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for various quadratic-additive type functional equations.

2. Main Result

Let be a real vector space and a real normed space. For any mapping , and denote the even part and the odd part of , respectively.

In the following theorem, we prove that if, for any given mapping , there exists a mapping (near ) with some properties possessed by additive or quadratic or quadratic-additive mappings, then the mapping is uniquely determined.

Theorem 1. Let be a real constant, let be a function satisfying one of the following conditions, for all , and let be an arbitrarily given mapping. If there exists a mapping such that for all and for all , then is determined by for all . In other words, is the unique mapping satisfying (5) and (6).

Proof. Assume that is a mapping satisfying (5) and (6) for a given mapping .
First, we consider the case when satisfies condition (2) or (3). It then follows from (2), (3), (5), and (6) that for all . That is, it holds that for all .
If satisfies condition (2), then we use (2), (5), and (6) to get for all . Thus, it holds that for all when satisfies (2). Therefore, we get for all when satisfies (2).
On the other hand, if satisfies condition (3), then we have for all . Therefore, we have for all when satisfies (3). Hence, we get for all provided satisfies (3).
Now, we deal with the case when satisfies condition (4). Then, we have for all . Thus, it follows that for all provided satisfies (4). Similarly, if satisfies condition (4), then we have for all . It thus holds that for all when satisfies (4). Therefore, if satisfies (4), then we get for all . Altogether, is uniquely determined for each case.

3. Applications

In general, it is not easy to apply Theorem 1 in practical applications. Hence, we introduce two corollaries which are easily applicable to investigating the uniqueness problems in the generalized Hyers-Ulam stability of functional equations. For the exact definition of the generalized Hyers-Ulam stability, we refer the reader to [6].

Corollary 2. Let be a real constant, let be a function satisfying one of the following conditions, for all , and let be an arbitrarily given mapping. If there exists a mapping satisfying (5) for all and (6) for all , then is a unique mapping satisfying (5) and (6).

Proof. If satisfies (16), then we have that is, satisfies condition (2) for all .
For the case of (17), it holds that that is, satisfies condition (4) for all . Hence, our assertion is true in view of Theorem 1.

Corollary 3. Let be a real constant, let be functions satisfying each of the following conditions, for all , and let be an arbitrarily given mapping. If there exists a mapping satisfying the inequality for all and the conditions in (6) for all , then is a unique mapping satisfying (6) and (21).

Proof. If we set , then it follows from (20) that for all . We make change of the summation indices in the preceding equality with and to get for any . Hence, we get for all . On the other hand, we use the above equality to get for all . From the above two equalities, we conclude that for all .
Similarly, we have for all . If we make change of the summation indices in the last equality with and , then we get for any . Thus, we obtain for each . Thus, it holds that for each . Theorem 1 implies that our conclusion for this corollary is true.

In the following corollary, we prove that if for any given mapping there exists an additive or a quadratic or a quadratic-additive mapping near , then the mapping is uniquely determined.

The proofs of the following two corollaries immediately follow from Corollaries 2 and 3, respectively, because each of additive, quadratic, and quadratic-additive mappings satisfies both conditions in (6) for any given rational number .

Corollary 4. Let be a rational number, let be a function satisfying condition (16) or (17) for all , and let be an arbitrarily given mapping. If there exists an additive or a quadratic or a quadratic-additive mapping satisfying inequality (5), then is uniquely determined.

Corollary 5. Let be a rational number, let be functions satisfying each of the conditions in (20) for all , and let be an arbitrarily given mapping. If there exists an additive or a quadratic or a quadratic-additive mapping satisfying (21), then is uniquely determined.

If then satisfies (2), if then satisfies (3), or if then satisfies (4). Hence, by Theorem 1, we have the following corollaries concerning the Hyers-Ulam-Rassias stability. For the detailed concept of the Hyers-Ulam-Rassias stability, we refer to [7–9].

Corollary 6. Let and be real constants, let , be real normed spaces, and let be an arbitrarily given mapping. If there exists a mapping satisfying the inequality for all and (6) for all , then is a unique mapping satisfying (6) and (31).

Since each of additive, quadratic, and quadratic-additive mappings satisfies the conditions in (6), using Corollary 6, we can easily prove the following corollary.

Corollary 7. Let and be real constants, let , be real normed spaces, and let be an arbitrarily given mapping. If there exists an additive or a quadratic or a quadratic-additive mapping satisfying the inequality for all , then is uniquely determined.

4. Discussions

We now define Nakmahachalasint [5, Theorem  3.1] investigated the generalized Hyers-Ulam stability of the functional equation as we see in the following theorem.

Theorem 8. Let be an integer, let be a real vector space, let be a Banach space, and let be an even function. Define for all . If or for all , and a mapping satisfies and for all , then there exists a unique mapping satisfying and if (34) holds, or if (35) holds, for all .

We remark that neither (37) nor (38) assures the uniqueness of near because and may be very large even though is very small.

Using Corollary 2, we improve Theorem 8 by giving proof for the uniqueness of .

Theorem 9. Let be an integer, let be a real vector space, let be a Banach space, and let be an even function. Define for all . If one of (34) and (35) holds for all and a mapping satisfies and (36) for all , then there exists a unique mapping satisfying and if (34) holds, or if (35) holds, for all .

Proof. In Theorem 8 or [5, Theorem  3.1], Nakmahachalasint showed that there exists a quadratic-additive mapping such that if (34) holds, then or if condition (35) holds, then for all . Hence, there exists a quadratic-additive mapping such that for all . Moreover, Nakmahachalasint [5, Theorem  2.1] proved that the even part of each solution of is a quadratic mapping, while its odd part is an additive mapping. Hence, every solution of satisfies the conditions in (6).
Therefore, by (43), Corollary 2 implies that is uniquely determined.

Now we define for a positive real constant . The mapping defined by is a solution of the functional equation , where are real constants. For the case , Jin and Lee [4] investigated the stability of the equation in Fuzzy spaces.

Assume that is a solution of the functional equation . Then we have and . Moreover, we know that for all . Therefore, we get for all ; that is, is a quadratic mapping and is an additive mapping. Therefore, if a mapping satisfies the functional equation for all , then is a quadratic-additive mapping such that for all . But we do not know that all quadratic-additive mappings satisfy the equalities in (47) if is not a rational number.

Theorem 10. Let be an arbitrary real constant. If a mapping satisfies for all with a positive real constant , then there exists a unique quadratic-additive mapping such that for all .

Proof. Notice that . Since , if , then it follows from (48) that for all . So, it is easy to show that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define a mapping by for all .
Moreover, if we put and let in (50), we obtain the inequality for all . From the definition of , we get for all .
If , then it follows from (48) that for all . So, it is easy to show that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define a mapping by for all .
Moreover, if we put and let in (54), we obtain inequality (52) for all . From the definition of , we get for all .
Since , if , then it follows from (48) that for all . So, it is easy to show that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define a mapping by for all .
Moreover, if we put and let in (57), we obtain the inequality for all . From the definition of , we get for all .
If , then it follows from (48) that for all . So, it is easy to show that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define a mapping by for all .
Moreover, if we put and let in (61), we obtain inequality (59) for all . From the definition of , we get for all .
For any case, we have that for all and that is, is a quadratic-additive mapping satisfying (49). By Corollary 6 and (47), is a unique quadratic-additive mapping satisfying (49).

Conflict of Interests

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

Acknowledgment

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. 2012R1A1A4A01002971).