Journal of Applied MathematicsVolume 2012 (2012), Article ID 691981, 12 pagesdoi:10.1155/2012/691981

Research Article

Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors

Jae-Young Chung,¹ Dohan Kim,² and John Michael Rassias³

¹Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea²Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea³Section of Mathematics and Informatics, Pedagogical Department E. E., National and Kapodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, 15342 Athens, Greece

Received 20 June 2011; Accepted 5 September 2011

Academic Editor: Junjie Wei

Copyright © 2012 Jae-Young Chung et al. 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 consider the Hyers-Ulam stability problems for the Jensen-type functional equations in general restricted domains. The main purpose of this paper is to find the restricted domains for which the functional inequality satisfied in those domains extends to the inequality for whole domain. As consequences of the results we obtain asymptotic behavior of the equations.

1. Introduction

The Hyers-Ulam stability problems of functional equations was originated by Ulam in 1960 when he proposed the following question [1].

Let be a mapping from a group to a metric group with metric such that Then does there exist a group homomorphism and such that for all ?

One of the first assertions to be obtained is the following result, essentially due to Hyers [2], that gives an answer for the question of Ulam.

Theorem 1.1. Suppose that is an additive semigroup, is a Banach space, , and satisfies the inequality for all . Then there exists a unique function satisfying for which for all .

We call the functions satisfying (1.4) additive functions. Perhaps Aoki in 1950 was the first author treating the generalized version of Hyers’ theorem [3]. Generalizing Hyers’ result he proved that if a mapping between two Banach spaces satisfies with , then there exists a unique additive function such that for all . In 1951 Bourgin [4, 5] stated that if is symmetric in and with for each , then there exists a unique additive function such that for all . Unfortunately, there was no use of these results until 1978 when Rassias [6] dealt with the inequality of Aoki [3]. Following Rassias’ result, a great number of papers on the subject have been published concerning numerous functional equations in various directions [6–15]. Among the results, stability problem in a restricted domain was investigated by Skof, who proved the stability problem of inequality (1.3) in a restricted domain [16, 17]. Developing this result, Jung, Rassias, and M. J. Rassias considered the stability problems in restricted domains for some functional equations including the Jensen functional equation [9] and Jensen-type functional equations [13]. We also refer the reader to [18–27] for some related results on Hyers-Ulam stabilities in restricted conditions. The results can be summarized as follows. Let and be a real normed space and a real Banach space, respectively. For fixed , if satisfies the functional inequalities (such as that of Cauchy, quadratic, Jensen, and Jensen type) for all with (which is the case where the inequalities are given by two indeterminate variables and ), the inequalities hold for all . Following the approach in [28] we consider the Jensen-type equation in various restricted domains in an Abelian group. As applications, we obtain the stability problems for the above equations in more general restricted domains than that of the form , which generalizes and refines the stability theorems in [13]. As an application we obtain asymptotic behaviors of the equations.

2. Stability of Jensen-Type Functional Equations

Throughout this section, we denote by , , and , an Abelian group, a real normed space, and a Banach space, respectively. In this section we consider the Hyers-Ulam stability of the Jensen and Jensen-type functional inequalities for the functions in restricted domains .

Inequalities (2.1) and (2.2) were previously treated by J. M. Rassias and M. J. Rassias [13], who proved the Hyers-Ulam stability of the inequalities in the restricted domain , for the functions :

Theorem 2.1. Let and be fixed. Suppose that satisfies the inequality for all , with . Then there exists a unique additive function such that for all .

Theorem 2.2. Let and be fixed. Suppose that satisfies the inequality for all , with and for all , with . Then there exists a unique additive function such that for all .

We use the following usual notations. We denote by the product group; that is, for , we define . For a subset of and , we define .

For given we denote by the subsets of points of the forms (not necessarily distinct) in , respectively, The set can be viewed as the vertices of rectangles in , and can be viewed as a subset of the vertices of rectangles in .

Definition 2.3. Let . One introduces the following conditions and on . For any , there exists a such that respectively.

The sets can be understood as the translations of and by and , respectively.

There are many interesting examples of the sets satisfying some of the conditions and . We start with some trivial examples.

Example 2.4. Let be a real normed space. For , let Then satisfies if , if and satisfies if , if .

Example 2.5. Let be a real inner product space. For Then satisfies if , if .

Example 2.6. Let be the group of nonsingular square matrices with the operation of matrix multiplication. For , let Then both and satisfy if , if .

In the following one can see that if and are replaced by arbitrary subsets of four points (not necessarily distinct) in , respectively, then the conditions become stronger; that is, there are subsets and which satisfy the conditions and , respectively, but and fail to fulfill the following conditions (2.13) and (2.14), respectively. For any subset of points (not necessarily distinct) in , there exists a such that respectively.

Now we give examples of and which satisfy and , respectively, but not (2.13) and (2.14), respectively.

Example 2.7. Let be the group of integers. Enumerate such that and let . Then it is easy to see that satisfies the condition . Now let with . Then is not contained in for all . Indeed, for any choices of , we have for all . Thus, if for some , then for some . Thus, it follows from the condition that the line segment joining the points of intersects the line in , which contradicts the condition . Similarly, let . Then it is easy to see that satisfies the condition but not (2.14).

Theorem 2.8. Let satisfy the condition and . Suppose that satisfies (2.1) for all . Then there exists an additive function such that for all .

Proof. For given , choose a such that . Replacing by , by ; by , by ; by , by ; by 0, by in (2.1), respectively, we have From (2.18), using the triangle inequality and dividing the result by 2, we have for all . From (2.19), using Theorem 1.1, we get the result.

Let , and let . Then satisfies the condition . Thus, as a direct consequence of Theorem 2.8, we obtain the following (cf. Theorem 2.1).

Corollary 2.9. Let . Suppose that satisfies inequality (2.1) for all , with . Then there exists a unique additive function such that for all .

Theorem 2.10. Let satisfy the condition and . Suppose that satisfies (2.2) for all . Then there exists a unique additive function such that for all .

Proof. For given , choose a such that . Replacing by , by ; by , by ; by , by in (2.2), respectively, we have From (2.22), using the triangle inequality and dividing the result by 2, we have Now by Theorem 1.1, we get the result.

Let , and let . Then satisfies the condition (J2). Thus, as a direct consequence of Theorem 2.10, we generalize and refine Theorem 2.2 as follows.

Corollary 2.11. Let. Suppose that satisfies inequality (2.2) for all , with . Then there exists a unique additive function such that for all .

Remark 2.12. Corollary 2.11 refines Theorem 2.2 in both the bounds and the condition (2.6).

Now we discuss other possible restricted domains. We assume that is a 2-divisible Abelian group. For given , we denote by , One can see that and consist of the vertices of parallelograms in , respectively.

Definition 2.13. Let . One introduces the following conditions on . For any , there exists a such that respectively.

Example 2.14. Let be a real normed space. For , let Then satisfies and if , and satisfies and if .

Example 2.15. Let be a real inner product space. For , Then satisfies if .

Example 2.16. Let be the group of nonsingular square matrices with the operation of matrix multiplication. For , let Then and satisfy both and ( if .

From now on, we assume that is a 2-divisible Abelian group.

Theorem 2.17. Let satisfy the condition and . Suppose that satisfies (2.1) for all . Then there exists a unique additive function such that for all .

Proof. For given , choose a such that . Replacing by , by ; by , by ; by , by ; by , by in (2.1), respectively, we have From (2.31), using the triangle inequality, we have for all . Replacing by , by in (2.32), we have for all . From (2.33), using Theorem 1.1, we get the result.

Let with , and let . Then satisfies the conditions and . Thus, as a direct consequence of Theorem 2.17 we generalize Theorem 2.1 as follows.

Corollary 2.18. Let with . Suppose that satisfies the inequality (2.1) for all , with . Then there exists a unique additive function such that for all .

Theorem 2.19. Let satisfy the condition and . Suppose that satisfies (2.2) for all . Then there exists a unique additive function such that for all .

Proof. For given , choose a such that . Replacing by , by ; by , by ; by , by ; by , by in (2.2), respectively, we have From (2.36), using the triangle inequality, we have for all . Replacing by , by in (2.37) and using Theorem 1.1, we get the result.

As a direct consequence of Theorem 2.19, we have the following.

Corollary 2.20. Let with . Suppose that satisfies the inequality (2.2) for all , with . Then there exists a unique additive function such that for all .

3. Asymptotic Behavior of the Equations

In this section we discuss asymptotic behaviors of the equations which gives refined versions of the results in [13].

Using Theorems 2.8 and 2.17, we have the following (cf. [13]).

Theorem 3.1. Let satisfy or . Suppose that satisfies the asymptotic condition as . Then there exists a unique additive function such that for all .

Proof. By the condition (3.1), for each , there exists such that for all with . Let . Then satisfies both the conditions and . By Theorems 2.8 and 2.17, there exists a unique additive function such that for all . Putting in (3.4) and using the triangle inequality, we have for all . Using the additivity of , we have for all . Letting in (3.4), we get the result.

Corollary 3.2. Let satisfy one of the conditions: . Suppose that satisfies the condition as . Then there exists a unique additive function such that for all .

Using Theorems 2.10 and 2.19, we have the following (cf. [13]).

Theorem 3.3. Let satisfy or . Suppose that satisfies the condition as . Then is an additive function.

Corollary 3.4. Let satisfy one of the conditions: . Suppose that satisfies the condition as . Then is an additive function.

Acknowledgments

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (no. 2011-0003898), and the second author was partially supported by the Research Institute of Mathematics, Seoul National University.

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