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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 564923, 8 pages
http://dx.doi.org/10.1155/2013/564923
Research Article

Generalized Hyers-Ulam Stability of Quadratic Functional Inequality

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

Received 17 June 2013; Revised 7 October 2013; Accepted 30 October 2013

Academic Editor: Krzysztof Ciepliński

Copyright © 2013 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 establish the general solution of the functional inequality and then investigate the generalized Hyers-Ulam stability of this inequality in Banach spaces and in non-Archimedean Banach spaces.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, 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 ?

Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces by using, so called, direct method. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias’ theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Rassias’ approach.

Let and be vector spaces. A mapping is called quadratic if and only if it is a solution of the quadratic functional equation: for all . It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all , where the mapping is given by ([6, 7]). The Hyers-Ulam stability of the quadratic functional equation (1) was first proved by Skof [8] for functions , where is a normed space and is a Banach space. Cholewa [9] noticed that Skof’s theorem is also valid if is replaced by an abelian group. In 1992, Czerwik [10] proved the generalized Hyers-Ulam stability of quadratic functional equation (1) in the spirit of Rassias approach by using direct method [11].

The generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by Cădariu and Radu [12, 13] by using fixed point method (see also [14, 15]).

Gilányi [16] and Rätz [17] proved that for a function mapping from an abelian group divisible by 2 into an inner product space , the functional inequality implies the Jordan-von Neumann functional equation

Fechner [18] and Gilányi [19] have proved the generalized Hyers-Ulam stability of the functional inequality (2). Park et al. [20] have investigated the generalized Hyers-Ulam stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, and Kim et al. [21] have proved the generalized Hyers-Ulam stability of Jensen functional inequality in -Banach spaces. The stability problems of several functional equations and inequalities have been extensively investigated by a number of authors and there are many interesting results concerning the stability of various functional equations and inequalities [6, 22].

In 2001, Bae and Kim [23] investigated the Hyers-Ulam stability of the quadratic functional equation which is equivalent to the original quadratic functional equation (1). Now, let us turn our attention to investigate the generalized stability problem of the following quadratic functional inequality:

In this paper, we make an attempt to establish the generalized Hyers-Ulam stability of a new quadratic functional inequality (5) by using fixed point method and direct method. In Section 2, we establish the general solution of the functional inequality (5). And then we prove the generalized Hyers-Ulam stability of the functional inequality (5) in Banach spaces by using fixed point method. In Section 3, we verify the generalized Hyers-Ulam stability of the functional inequality (5) in Banach spaces by using direct method. In Section 4, we investigate the generalized Hyers-Ulam stability of the functional inequality (5) in non-Archimedean Banach spaces by using fixed point method. In Section 5, we prove the generalized Hyers-Ulam stability of the functional inequality (5) in non-Archimedean Banach spaces by using direct method.

2. Stability of (5) by Fixed Point Method

In [24], Rassias introduced the following equality: for a fixed integer . Let , be real vector space. It has been proved that if a mapping satisfies for all with , then the mapping is realized as the sum of an additive mapping and a quadratic mapping [25]. Park [25] and Jang et al. [26] have proved the generalized Hyers-Ulam stability of the functional equation (7). In particular, if and even function satisfies (7), then it is easy to see that satisfies the equation for all . Thus, we first consider the general solution of functional equation (8) to verify the general solution of functional inequality (5).

Lemma 1. Let both and be vector spaces. A function satisfies (8) if and only if is quadratic.

Proof. If we put in (8), then we have . Letting in (8), we have for all . Similarly, we can easily show that and for all . Replacing by in (8), we get for all . Substituting for in (8), we obtain for all . Switching with in (9) yields for all . Adding (9) to (10) and using (8), we arrive at for all . Letting in (8), we obtain for all . Switching with in (12) and using the evenness of , one gets for all . Adding (12) to (13) and using (8), we have for all . Putting and in (8), we get for all . From (14) and (15), it follows that for all . By (11) and (16), we obtain for all . Letting and in (17), we have for all . Interchanging for in (18) and using the evenness of , one gets for all . Substituting for in (18), we get by virtue of (19) which yields for all . Thus, the mapping is quadratic.
The proof of the converse is trivial.

Now, we present the general solution of the functional inequality (5) by using Lemma 1.

Lemma 2. Let both and be vector spaces. A mapping satisfies the functional inequality (5) for all if and only if is quadratic.

Proof. Let satisfy the functional inequality (5). If we replace in (5) by , then we have . Replacing by in (5), we obtain for all . Letting and in (5), we get for all . Putting in (22) yields for all . From (23) and (24), we have for all . Thus, it follows from (22) that for all . So is quadratic by Lemma 1.
The proof of the converse is trivial.

Let be a set. A function is called a generalized metric on if satisfies the following:(1) if and only if ;(2) for all ;(3) for all .

Before taking up the main subject, we recall the fixed point theorem from [14].

Theorem 3 (see [14]). Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either for all nonnegative integers or there exists a positive integer such that(1), for all ;(2) the sequence converges to a fixed point of ;(3) is the unique fixed point of in the set ;(4) for all .

In this part, let be a normed space and a Banach space.

Theorem 4. Suppose that a mapping with satisfies the functional inequality for all and that there exists a constant with for which the perturbing function satisfies for all . Then, there exists a unique quadratic mapping given by (, resp.) such that for all .

Proof. Replacing by in (27), we obtain for all . Letting and in (27), we get for all . Putting in (30) yields for all . From (31) and (32), it follows that and so for all .
Consider the set of mappings and introduce the generalized metric on : where, as usual, . It is easy to show that is a complete generalized metric space (see the proof of Theorem 3.1 of [27]).
Now we consider a linear mapping such that for all . Then it is well-known that is a strictly contractive mapping with Lipschitz constant , and it follows from (34) that . By Theorem 3, there exists a mapping satisfying the following.(1) is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set . This implies that is a unique fixed point of such that there exists a satisfying for all .(2) as . This implies the equality for all .(3), which implies that the inequality (29) holds.
Now, we show that the mapping is quadratic. It follows from (27) and (28) that for all . Thus, it follows from Lemma 2 that the mapping is quadratic, as desired.

We obtain the following corollary concerning the stability for approximate mappings controlled by a sum of powers of norms.

Corollary 5. Let be a real number and a positive real number with . If a mapping with satisfies the inequality for all , then we can find a unique quadratic mapping satisfying the inequality for all .

3. Stability of (5) by Direct Method

We now investigate stability problem of the quadratic functional inequality (5) with perturbed control function . In this section, let be a normed space and a Banach space.

Theorem 6. Suppose that a mapping with satisfies the functional inequality (27) and that the perturbing function satisfies for all . Then, there exists a unique quadratic mapping defined by (, resp.) such that for all .

Proof. It follows from (33) that for all . Therefore, we prove from inequality (44) that for any integers and with for all . Since the right-hand side of (45) tends to zero as , 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 (45), we get the desired inequality (43).
It follows from (27) and (42) that for all . So the mapping is quadratic.
Next, let be another quadratic mapping satisfying (43). Then, we have for all and all . Taking the limit as , we conclude that for all . This completes the proof.

We obtain the following corollary concerning the stability for approximate mappings controlled by a sum of powers of norms.

Corollary 7. Let be a real number and a positive real number with or for all . If a mapping with satisfies the inequality for all , then we can find a unique quadratic mapping satisfying the inequality for all .

4. Stability of (5) in Non-Archimedean Spaces by Fixed Point Method

A non-Archimedean valuation in a field is a function : with the following:(i) if and only if ;(ii) for all ;(iii) for all .

Any field endowed with a non-Archimedean valuation is said to be a non-Archimedean field; in any such field we have and for all .

Definition 8. Let X be a linear space over a field with a non-Archimedean nontrivial valuation . A function is said to be a non-Archimedean norm if it satisfies the following conditions:(i) if and only if ;(ii), for all and ;(iii), for all .Then is called a non-Archimedean normed space.

Definition 9. Let be a sequence in a non-Archimedean normed space .(1) A sequence in a non-Archimedean space is a Cauchy sequence if the sequence converges to zero.(2) The sequence is said to be convergent if, for any , there are a positive integer and such that Then the point is called the limit of the sequence , which is denoted by .(3) If every Cauchy sequence in converges, then the non-Archimedean normed space is called a non-Archimedean Banach space.

In 2007, Moslehian and Rassias [28] proved the generalized Hyers-Ulam stability of the Cauchy and quadratic functional equations in non-Archimedean normed spaces. Some papers [29, 30] on the stability of various functional equations in non-Archimedean normed spaces have been published after their stability results.

In this section, assume that is a non-Archimedean normed space and that is a non-Archimedean Banach space. Now, we are going to investigate the stability of the functional inequality (5) in non-Archimedean Banach space by using fixed point method.

Theorem 10. Suppose that a mapping with satisfies the functional inequality (27) and that there exists a constant with for which the perturbing function satisfies for all . Then, there exists a unique quadratic mapping given by (, resp.) such that for all , where for all .

Proof. From (31) and (32), we get by using the non-Archimedean norm
and so for all . Applying the similar argument to the corresponding proof of Theorem 4 on the complete generalized metric space , we get the desired result.

We obtain the following corollary concerning the stability for approximate mappings controlled by a sum of powers of norms.

Corollary 11. Let be a real number and a positive real number with . If a mapping with satisfies the inequality for all , then we can find a unique quadratic mapping satisfying the inequality for all .

5. Stability of (5) in Non-Archimedean Spaces by Direct Method

Now, we are going to investigate the stability of the functional inequality (5) in non-Archimedean Banach space by direct method. In this section, assume that is a non-Archimedean normed space and that is a non-Archimedean Banach space.

Theorem 12. Suppose that a mapping with satisfies the functional inequality (27) and that is a function such that for all and exists for all , where for all . Then there exists a quadratic mapping defined by , (, resp.) such that for all . Moreover, if for all , then is a unique quadratic mapping satisfying (61).

Proof. Replacing by and dividing by in (55), we have for all . It follows from (58) and (63) that the sequence is Cauchy for all , and the sequence converges in the non-Archimedean Banach space . Therefore, we can define a mapping as Applying the similar argument to the corresponding proof of Theorem 6, we get the required result.

Corollary 13. Let   be a function satisfying (i)   for all and (ii) .
Suppose that with satisfies the inequality for all and for some . Then there exists a unique quadratic mapping such that for all .

Proof. Letting , we obtain for all . It follows from (60) that By direct calculation, exists and holds for all . Applying Theorem 12, we conclude that for all .

Corollary 14. Let be a function satisfying (i) for all and (ii) .
Suppose that a mapping with satisfies the inequality for all and for some . Then there exists a unique quadratic mapping such that for all .

Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2012R1A1A2008139).

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