Abstract and Applied AnalysisVolume 2012 (2012), Article ID 926390, 23 pagesdoi:10.1155/2012/926390

Research Article

On the Hyers-Ulam Stability of a General Mixed Additive and Cubic Functional Equation in n-Banach Spaces

Tian Zhou Xu¹ and John Michael Rassias²

¹School of Mathematics, Beijing Institute of Technology, Beijing 100081, China²Pedagogical Department E.E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Athens 15342, Greece

Received 7 January 2012; Accepted 15 February 2012

Academic Editor: Krzysztof Cieplinski

Copyright © 2012 Tian Zhou Xu and John Michael Rassias. 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.

The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.

1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation?

If the problem accepts a unique solution, we say the equation is stable (see [1]). The study of stability problems for functional equations is related to a question of Ulam [2] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [3]. The result of Hyers was generalized by Aoki [4] for approximate additive mappings and by Rassias [5] for approximate linear mappings by allowing the Cauchy difference operator to be controlled by . In 1994, a generalization of Rassias’ theorem was obtained by Gvruţa [6], who replaced by a general control function . On the other hand, several further interesting discussions, modifications, extensions, and generalizations of the original problem of Ulam have been proposed (see, e.g. [7–12] and the references therein).

Recently, Park [9] investigated the approximate additive mappings, approximate Jensen mappings, and approximate quadratic mappings in 2-Banach spaces and proved the generalized Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation, and the quadratic functional equation in 2-Banach spaces. This is the first result for the stability problem of functional equations in 2-Banach spaces.

In [11, 12], we introduced the following mixed additive-cubic functional equation for fixed integers with , : with , and investigated the generalized Hyers-Ulam stability of (1.1) in quasi-Banach spaces and non-Archimedean fuzzy normed spaces, respectively.

In this paper, we investigate, approximate mixed additive-cubic mappings in -Banach spaces. That is, we prove the generalized Hyers-Ulam stability of a general mixed additive-cubic equation (1.1) in -Banach spaces by the direct method.

The concept of 2-normed spaces was initially developed by Gähler [13, 14] in the middle of 1960s, while that of -normed spaces can be found in [15, 16]. Since then, many others have studied this concept and obtained various results; see for instance [15, 17–19].

We recall some basic facts concerning -normed spaces and some preliminary results.

Definition 1.1. Let , and let be a real linear space with dim and a function satisfying the following properties:(N1) if and only if are linearly dependent,(N2) is invariant under permutation,(N3),(N4) for all and . Then the function is called an -norm on and the pair is called an -normed space.

Example 1.2. For , the Euclidean -norm is defined by where for each .

Example 1.3. The standard -norm on , a real inner product space of dimension dim , is as follows: where denotes the inner product on . If , then this -norm is exactly the same as the Euclidean -norm mentioned earlier. For , this -norm is the usual norm .

Definition 1.4. A sequence in an -normed space is said to converge to some in the -norm if for every .

Definition 1.5. A sequence in an -normed space is said to be a Cauchy sequence with respect to the -norm if for every . If every Cauchy sequence in converges to some , then is said to be complete with respect to the -norm. Any complete -normed space is said to be an -Banach space.

Now we state the following results as lemma (see [9] for the details).

Lemma 1.6. Let be an -normed space. Then, (1)For and , a real number, for all ,(2) for all ,(3)if for all , then ,(4)for a convergent sequence in , for all .

2. Approximate Mixed Additive-Cubic Mappings

In this section, we investigate the generalized Hyers-Ulam stability of the generalized mixed additive-cubic functional equation in -Banach spaces. Let be a linear space and an -Banach space. For convenience, we use the following abbreviation for a given mapping : for all .

Theorem 2.1. Let be a linear space and an -Banach space. Let be a mapping with for which there is a function such that for all . Then, there is a unique additive mapping such that for all , where

Proof. Letting in (2.3), we get for all . Putting in (2.3), we have for all . Thus for all . Letting in (2.3), we get for all . Letting in (2.3), we have for all . Letting in (2.3), we have for all . Replacing and by and in (2.3), respectively, we get for all . Replacing and by and in (2.3), respectively, we get for all . Replacing and by and in (2.3), respectively, we have for all . Setting in (2.3), we have for all . Letting in (2.3), we have for all . Letting in (2.3), we have for all . By (2.6), (2.7), (2.13), (2.15), and (2.16), we get for all . By (2.6), (2.10), and (2.11), we have for all . It follows from (2.12) and (2.19) that for all . By (2.14) and (2.20), we have for all . By (2.6), (2.15), (2.16), and (2.17), we have for all . It follows from (2.6), (2.8), (2.9), and (2.22) that for all . Hence, for all . By (2.9), we have for all . From (2.23) and (2.25), we have for all . Also, from (2.18) and (2.26), we get for all .On the other hand, it follows from (2.21) and (2.27) that for all . Therefore by (2.24) and (2.28), we get for all .Now, let be the mapping defined by for all . Then, (2.29) means that for all . Also, we get for all . Replacing by in (2.31) and dividing both sides of (2.31) by , we get for all and all integers . For all integers with , we have for all . So, we get for all . This shows that the sequence is a Cauchy sequence in . Since is an -Banach space, the sequence converges. So, we can define a mapping by for all . Putting , then passing the limit in (2.33), and using Lemma 1.6(4), we get for all .Now we show that is additive. By Lemma 1.6, (2.2), (2.32), and (2.35), we have for all . By Lemma 1.6(3), for all . Also, by Lemma 1.6(4), (2.2), (2.3), and (2.35), we get for all . By Lemma 1.6(3), for all . Hence, the mapping satisfies (1.1). By [11, Lemma 2.3], the mapping is additive. Therefore, implies that the mapping is additive.To prove the uniqueness of , let be another additive mapping satisfying (2.4). Fix . Clearly, and for all . It follows from (2.4) that for all , and . By (2.2), we see that the right-hand side of the above inequality tends to 0 as . Therefore, for all . By Lemma 1.6, we can conclude that for all . So, . This proves the uniqueness of .

Theorem 2.2. Let be a linear space and an -Banach space. Let be a mapping with for which there is a function such that for all . Then, there is a unique additive mapping such that for all , where is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.1.

Corollary 2.3. Let be a normed space and an -Banach space. Let such that , and let be a mapping with such that for all . Then, there exists a unique additive mapping such that for all , where

Proof. Define for all , and apply Theorems 2.1 and 2.2.

The following example shows that the assumption cannot be omitted in Corollary 2.3.

Example 2.4. Let be a linear space over . Define by , where , , ( is the imaginary unit). Then, is a 2-normed linear space.Let defined byConsider the function defined by for all , where . Then, satisfies the functional inequality for all , but there do not exist an additive mapping and a constant such that for all .It is clear that for all . If or for all , then the inequality (2.47) holds. Now suppose that . Then, there exists an integer such that Hence, for all . From the definition of and (2.48), we obtain that Therefore, satisfies (2.47). Now, we claim that the functional equation (1.1) is not stable for in Corollary 2.3. Suppose on the contrary that there exist an additive mapping and a constant such that for all . Then, there exists a constant such that for all rational numbers . So, we obtain that for all rational numbers and all . Let with . If is a rational number in and (), then for all , and we get which contradicts (2.50).

Theorem 2.5. Let be a linear space and an -Banach space. Let be a mapping with for which there is a function such that for all . Then, there is a unique cubic mapping such that for all , where is defined as in Theorem 2.1.

Proof. As in the proof of Theorem 2.1, we have for all , where is defined as in Theorem 2.1.Now, let be the mapping defined by . By (2.55), we have for all . Replacing by in (2.56) and dividing both sides of (2.56) by , we get for all and all integers . For all integers with , we have for all . So, we get for all . This shows that the sequence is a Cauchy sequence in . Since is an -Banach space, the sequence converges. So, we can define a mapping by for all . Putting , then passing the limit in (2.58), and using Lemma 1.6(4), we get for all . Now we show that is cubic. By Lemma 1.6, (2.52), (2.58), and (2.60), we have for all . By Lemma 1.6(3), for all . Also, by Lemma 1.6(4), (2.52), (2.53), and (2.60), we get for all . By Lemma 1.6(3), for all . Hence the mapping satisfies (1.1). By [11, Lemma 2.3], the mapping is cubic. Therefore, implies that the mapping is cubic.To prove the uniqueness of , let be another cubic mapping satisfying (2.54). Fix . Clearly, and for all . It follows from (2.54) that for all , and . By (2.52), we see that the right-hand side of the above inequality tends to 0 as . Therefore, for all . By Lemma 1.6, we can conclude that for all . So . This proves the uniqueness of .

Theorem 2.6. Let be a linear space and an -Banach space. Let be a mapping with for which there is a function such that for all . Then, there is a unique cubic mapping such that for all , where is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.5.

Corollary 2.7. Let be a normed space and an -Banach space. Let such that , and let be a mapping with such that for all . Then, there exists a unique cubic mapping such that for all , where is defined as in Corollary 2.3.

Proof. Define for all , and apply Theorems 2.5 and 2.6.

The following example shows that the the generalized Hyers-Ulam stability problem for the case of was excluded in Corollary 2.7.

Example 2.8. Let be a linear space over , and let be defined as in Example 2.4. Then, is a 2-normed linear space.Let be defined byConsider the function defined by for all , where . Then, satisfies the functional inequality for all , but there do not exist a cubic mapping and a constant such that for all .It is clear that for all . If or for all , then inequality (2.71) holds. Now suppose that . Then, there exists an integer such that Hence, for all . From the definition of and (2.72), we obtain that Therefore, satisfies (2.71). Now, we claim that the functional equation (1.1) is not stable for in Corollary 2.7. Suppose on the contrary that there exist a cubic mapping and a constant such that for all . Then, there exists a constant such that for all rational numbers . So, we obtain that for all rational numbers and all . Let with . If is a rational number in and (), then for all , and we get which contradicts (2.74).

Theorem 2.9. Let be a linear space and an -Banach space. Let be a mapping with for which there is a function such that for all . Then, there exist a unique additive mapping and a unique cubic mapping such that for all , where is defined as in Theorem 2.1.

Proof. By Theorems 2.1 and 2.5, there exist an additive mapping and a cubic mapping such that for all . Hence, for all . So, we obtain (2.78) by letting and for all .To prove the uniqueness of and , let be another additive and cubic mapping satisfying (2.78). Fix . Let and . So, for all . Then (2.76) implies that for all . Thus, . So, it follows from (2.81) that for all . Therefore, .

Similarly to Theorem 2.9, one can prove the following result.

Theorem 2.10. Let be a linear space and an -Banach space. Let be a mapping with for which there is a function such that for all . Then, there exist a unique additive mapping and a unique cubic mapping such that for all , where is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems 2.2 and 2.6.

Theorem 2.11. Let be a linear space and an -Banach space. Let be a mapping with for which there is a function such that for all . Then, there exist a unique additive mapping and a unique cubic mapping such that for all , where is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems 2.2 and 2.5.

Corollary 2.12. Let be a normed space and an -Banach space. Let , , and let be a mapping with such that for all . Then, there exist a unique additive mapping and a unique cubic mapping such that for all , where is defined as in Corollary 2.3.

Proof. Define for all , and apply Theorems 2.9–2.11.

Remark 2.13. The generalized Hyers-Ulam stability problem for the cases of and was excluded in Corollary 2.12 (see Examples 2.4 and 2.8).

Acknowledgments

The authors would like to thank the Editor Professor Krzysztof Ciepliński and anonymous referees for their valuable comments and suggestions. The first author was supported by the National Natural Science Foundation of China (NNSFC) (grant No. 11171022).

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