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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 270954, 16 pages
http://dx.doi.org/10.1155/2012/270954
Research Article

Hyers-Ulam Stability of Jensen Functional Inequality in p-Banach Spaces

Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Republic of Korea

Received 2 May 2012; Accepted 6 July 2012

Academic Editor: Nicole Brillouet-Belluot

Copyright © 2012 Hark-Mahn Kim 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 prove the Hyers-Ulam stability of the following Jensen functional inequality in -Banach spaces for any fixed nonzero integer .

1. Introduction

The stability problem of equations originated from a question of Ulam [1] concerning the stability of group homomorphisms.

We are given a group and a metric group with metric . Given , does there exist a number such that if satisfies for all , then a homomorphism exists with for all ?

In 1941, Hyers [2] considered the case of approximately additive mappings between Banach spaces and proved the following result.

Suppose that and are Banach spaces and satisfies the following condition: if there is a number such that for all , then the limit exists for all and there exists a unique additive mapping such that Moreover, if is continuous in for each , then the mapping is -linear.

The method which was provided by Hyers, and which produces the additive mapping , is called a direct method. This method is the most important and most powerful tool for studying the stability of various functional equations. Hyers' theorem was generalized by Aoki [3] and Bourgin [4] for additive mappings by considering an unbounded Cauchy difference. In 1978, Rassias [5] also provided a generalization of Hyers' theorem for linear mappings which allows the Cauchy difference to be unbounded. Let and be two Banach spaces and let be a mapping such that is continuous in for each fixed . Assume that there exist and such that Then, there exists a unique -linear mapping such that for all . A generalized result of Rassias' theorem was obtained by Găvruţa in [6] and Jung in [7]. In 1990, Rassias [8] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for . In 1991, Gajda [9], following the same approach as in [5], gave an affirmative solution to this question for . It was shown by Gajda [9], as well as by Rassias and Šemrl [10], that one cannot prove a Rassias' type theorem when . The counterexamples of Gajda [9], as well as of Rassias and Šemrl [10], have stimulated several mathematicians to invent new approximately additive or approximately linear mappings.

We recall some basic facts concerning quasinormed spaces and some preliminary results. Let be a real linear space. A quasinorm is a real-valued function on satisfying the following:(1) for all and if and only if .(2) for all and all .(3)There is a constant such that for all .

The pair is called a quasinormed space if is a quasinorm on [11, 12]. The smallest possible is called the modulus of concavity of . A quasi-Banach space is a complete quasinormed space.

A quasinorm is called a -norm if for all . In this case, a quasi-Banach space is called a -Banach space.

Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem [12], each quasinorm is equivalent to some -norm (see also [11]). Since it is much easier to work with -norms, henceforth, we restrict our attention mainly to -norms. We observe that if are nonnegative real numbers, then where .

In 2009, Moslehian and Najati [13] introduced the Hyers-Ulam stability of the additive functional inequality: and then have investigated the general solution and the Hyers-Ulam stability problem for the functional inequality. The stability problems of several functional equations in quasi-normed spaces and several functional inequalities have been investigated by a number of authors and there are many interesting results concerning the stability of various functional inequalities [1417].

In this paper, we consider a modified and general Jensen functional inequality: for any fixed nonzero integer . First of all, it is easy to see that a function satisfies the inequality (1.8) if and only if is additive. Thus the inequality (1.8) may be called the Jensen functional inequality and the general solution of inequality (1.8) may be called the Jensen function. In the sequel, we investigate the generalized Hyers-Ulam stability of (1.8) in -Banach spaces for any fixed nonzero integer by using the techniques of [14, 15].

2. Generalized Hyers-Ulam Stability

First, we present the general solution of the inequality (1.8).

Lemma 2.1. Let both and be real vector spaces. A function satisfies (1.8) for all if and only if is additive.

Proof. Letting in (1.8), we have . Putting and in (1.8), we get for all . Hence for all . Replacing by in (1.8), we obtain that is, for all . Putting and in (2.3), we get by oddness of , for all . So is additive.
The proof of the converse is trivial.

From now on, assume that is a quasinormed space with quasinorm and that is a -Banach space with -norm . Let be the modulus of concavity of in .

Before taking up the main subject, given a mapping , we define the difference operator by for all and for any fixed nonzero integer .

Theorem 2.2. Suppose that a mapping with satisfies the functional inequality for all and the perturbing function satisfies for all . Then, there exists a unique additive mapping defined by such that for all .

Proof. Replacing by in (2.6), we obtain for all . Letting and in (2.9), we get for all . Putting and in (2.6), we have for all . Replacing by in (2.11), we obtain for all . It follows from (2.10) and (2.12) that for all . If we replace by in (2.13), then we get that
It follows from (2.14) that for all nonnegative integers and with and . Since the right-hand side of (2.15) tends to zero as , by the convergence of the series (2.7), we obtain that the sequence is Cauchy for all . Because of the fact that is complete, it follows that the sequence converges in . Therefore, we can define a mapping as Moreover, letting and taking in (2.15), we get for all .
It follows from (2.6) and (2.7) that for all . So the mapping is additive.
Next, let be another additive mapping satisfying (2.8). Then, we have for all and all . Taking the limit as , we conclude that for all . This completes the proof.

If we put and in the following corollaries, respectively, then we lead to the desired results.

Corollary 2.3. Let for with and . If a mapping with satisfies the following functional inequality for all , then there exists a unique additive mapping such that for all , where .

Corollary 2.4. Let and for . If a mapping with satisfies the following functional inequality for all , then there exists a unique additive mapping such that for all .

Theorem 2.5. Suppose that a mapping satisfies the functional inequality for all , and the perturbing function satisfies for all . Then, there exists a unique additive mapping defined by such that for all .

Proof. We note that since by the convergence of (2.26). Now, if we replace by in (2.14), for all . Then, it follows from the last inequality that for all nonnegative integer and all . The remaining proof is similar to the corresponding part of Theorem 2.2. This completes the proof.

If we put and in the following corollaries, respectively, then we lead to the desired results.

Corollary 2.6. Let for with and . If a mapping satisfies the following functional inequality for all , then there exists a unique additive mapping such that for all , where .

Corollary 2.7. Let and for . If a mapping satisfies the following functional inequality for all , then there exists a unique additive mapping such that for all .

The following is a simple example that the additive functional inequality is not stable for the singular case in Corollaries 2.4 and 2.7.

Example 2.8. Fix and put . Let be defined by and define by which can be found in [9]. It follows from the same argument as in the example of [9] that satisfies the functional inequality for all . In fact, if , then (2.36) is trivially fulfilled. Next, if , then there exists an such that which implies that Thus, we see that for all . As a result, we infer that for all . Finally, if , then one has by use of boundedness of for all . Therefore, satisfies the functional inequality (2.36) and so for all . However, there do not exist an additive function and a constant such that

Remark 2.9. The stability problem on the singular case in Corollaries 2.3 and 2.6 is not easy and it remains with us unsolved for providing a counterexample on the singular case .

3. Alternative Generalized Hyers-Ulam Stability of (1.8)

From now on, we investigate the generalized Hyers-Ulam stability of the functional inequality (1.8) using the contractive property of perturbing term of the inequality (1.8).

Theorem 3.1. Suppose that a mapping with satisfies the functional inequality for all and there exists a constant with for which the perturbing function satisfies for all . Then, there exists a unique additive mapping given by such that for all .

Proof. It follows from (2.15) and (3.2) that for all nonnegative integers and with and . Since the sequence is Cauchy for all , we can define a mapping by Moreover, letting and in the last inequality yields the approximation (3.3).
The remaining proof is similar to the corresponding part of Theorem 2.2. This completes the proof.

Corollary 3.2. Let be a nontrivial function satisfying If with is a mapping satisfying the following functional inequality for all and for some , then there exists a unique additive mapping such that for all .

Proof. Letting and applying Theorem 3.1 with , we obtain the desired result.

Theorem 3.3. Suppose that a mapping satisfies the functional inequality for all and there exists a constant with for which the perturbing function satisfies for all . Then, there exists a unique additive mapping defined by such that for all .

Proof. We observe that because , which follows from the condition . It follows from (2.29) and (3.10) that for all nonnegative integer and all .
The remaining proof is similar to the corresponding part of Theorem 2.2. This completes the proof.

Corollary 3.4. Let be a nontrivial function satisfying If is a mapping satisfying the following functional inequality for all and for some , then there exists a unique additive mapping such that for all .

Proof. Letting and applying Theorem 3.3 with , we lead to the approximation.

Acknowledgment

This study was supported by the Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (No. 2012R1A1A2008139).

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