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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 482936, 14 pages
http://dx.doi.org/10.1155/2012/482936
Research Article

A Fixed Point Approach to the Stability of an -Dimensional Mixed-Type Additive and Quadratic Functional Equation

1Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea
2Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of Korea

Received 8 October 2011; Accepted 3 December 2011

Academic Editor: Alberto d'Onofrio

Copyright © 2012 Yang-Hi Lee and Soon-Mo Jung. 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 investigate the stability problems for a functional equation by using the fixed point method.

1. Introduction

In 1940, Ulam [1] gave a wide-ranging talk before a Mathematical Colloquium at the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms.

Let be a group and let be a metric group with a metric . Given , does there exist a such that if a function satisfies the inequality for all , then there is a homomorphism with for all ?

If the answer is affirmative, we say that the functional equation for homomorphisms is stable. Hyers [2] was the first mathematician to present the result concerning the stability of functional equations. He answered the question of Ulam for the case where and are assumed to be Banach spaces. This result of Hyers is stated as follows.

Let be a function between Banach spaces such thatfor some and for all . Then the limit exists for each , and is the unique additive function such that for every . Moreover, if is continuous in for each fixed , then the function is linear.

We remark that the additive function is directly constructed from the given function and this method is called the direct method. The direct method is a very powerful method for studying the stability problems of various functional equations. Taking this famous result into consideration, the additive Cauchy equation is said to have the Hyers-Ulam stability on if, for every function satisfying the inequality (1.1) for some and for all , there exists an additive function such that is bounded on .

In 1950, Aoki [3] generalized the theorem of Hyers for additive functions, and in the following year, Bourgin [4] extended the theorem without proof. Unfortunately, it seems that their results failed to receive attention from mathematicians at that time. No one has made use of these results for a long time.

In 1978, Rassias [5] actually rediscovered the result of Aoki; he proved the following theorem of Hyers for linear functions.

Let be a function between Banach spaces. If satisfies the functional inequality for some , with and for all , then there exists a unique additive function such that for each . If, in addition, is continuous in for each fixed , then the function is linear.

This result of Rassias attracted a number of mathematicians who began to be stimulated to investigate the stability problems of functional equations. By regarding a large influence of Ulam, Hyers, and Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Aoki and Rassias is called the Hyers-Ulam-Rassias stability. For the last thirty years many results concerning the Hyers-Ulam-Rassias stability of various functional equations have been obtained (see [616]). For some discussion of possible definitions of stability for functional equations, see [17].

The following functional equation is called an -dimensional mixed-type additive and quadratic functional equation, and each solution of (1.3) is called a quadratic-additive function. Recently, Towanlong and Nakmahachalasint [18] solved the -dimensional mixed-type additive and quadratic functional equation.

Let and be vector spaces. A function satisfies (1.3) for all if and only if there exist an additive function and a quadratic function such that for all .

Moreover, they also investigated the Hyers-Ulam-Rassias stability of (1.3) by using the direct method (see [18]). Indeed, they tried to approximate the even and odd parts of each solution of a perturbed inequality by the even and odd parts of an “exact” solution of (1.3), respectively.

In Theorems 3.1 and 3.3 of this paper, we will apply the fixed point method and prove the Hyers-Ulam-Rassias stability of the -dimensional mixed-type additive and quadratic functional equation. The advantage of this paper, in comparison with [18], is to approximate each solution of a perturbed inequality by an “exact” solution of (1.3), and we obtain sharper estimations in consequence of this advantage.

Throughout this paper, let be a (real or complex) vector space, a Banach space, and an integer larger than 1.

2. Preliminaries

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

We remark that the only difference between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.

We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to [19].

Theorem 2.1. Let be a complete generalized metric space. Assume that is a strict contraction with the Lipschitz constant . If there exists a nonnegative integer such that for some , then the following statements are true. (i)The sequence converges to a fixed point of . (ii) is the unique fixed point of in . (iii)If , then

In 1991, Baker applied the fixed point method to prove the Hyers-Ulam stability of a nonlinear functional equation (see [20]). Thereafter, Radu noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative (Theorem 2.1). Indeed, he applied the fixed point method to prove the existence of a solution of the inequality (1.1) and investigated the Hyers-Ulam stability of the additive Cauchy equation (see [21] and also [2226]). For a somewhat different fixed point approach to stability of functional equations, see [27, 28].

3. Hyers-Ulam-Rassias Stability

Let be a (real or complex) vector space and let be a Banach space. For a given function , we use the following abbreviation: for all .

In the following theorem, we prove the stability of the functional equation (1.3) by using the fixed point method.

Theorem 3.1. Let be a given function. Assume that a function satisfies and the inequality for all . If there exists a constant such that has the property for all , then there exists a unique function such that for all and for all , where the function is defined by for all . In particular, is represented by for all .

Proof. Let be the set of all functions with . We introduce a generalized metric on by where is given in (3.5). It is not difficult to show that is a complete generalized metric space (see [29] or [30, 31]).
Now we consider the operator defined by for all . We can apply induction on to prove for all and .
Let and let be an arbitrary constant with . By the definition of , (3.3), and (3.5), we have for all , which implies that for any . That is, is a strict contraction with the Lipschitz constant .
Moreover, by (3.2), we see that for all , that is, (see the definition of ). Therefore, according to Theorem 2.1, the sequence converges to the unique “fixed point” of in the set and is represented by (3.6) for all . Notice that . By Theorem 2.1, we have which implies the validity of (3.4).
By a somewhat tedious manipulation, it follows from (3.2), (3.3), and (3.6) that for all .

We can prove the following corollary by a way similar to that presented in the proof of the preceding theorem.

Corollary 3.2. Let be a given function such that for all . Assume that a function satisfies and the inequality (3.2) for all . If there exists a constant such that has the property (3.3) for all , then there exists a unique quadratic-additive function such that for all .

Proof. Notice that for all (see the proof of Theorem 3.1). By a similar method used in the proof of Theorem 3.1, we can show that there exists a unique quadratic-additive function satisfying (3.15).

In the following theorem, we prove the Hyers-Ulam-Rassias stability of (1.3) under the condition (3.17) instead of (3.3).

Theorem 3.3. Given , assume that a function satisfies the condition and the inequality (3.2) for all . If there exists a constant such that for all , then there exists a unique function such that for all and for all , where is defined by (3.5). In particular, is represented by for all .

Proof. Let and be defined as in the proof of Theorem 3.1. Then is a complete generalized metric space. We now consider the operator defined by for all and .
Notice that for all and .
Let and let be an arbitrary constant with . By the definition of , together with (3.5) and (3.17), we have for all . Thus, we get for any . That is, is a strict contraction with the Lipschitz constant .
Moreover, it follows from (3.2), (3.5), and (3.17) that for all , which implies that . Therefore, according to Theorem 2.1, the sequence converges to the unique “fixed point” of in the set and is represented by (3.19). In view of Theorem 2.1, we have Hence, the inequality (3.18) is true.
In a similar way presented in the proof of Theorem 3.1, it follows from (3.2), (3.17), and (3.19) that for all .

We can prove the following corollary by a similar way as we did in the proof of Corollary 3.2. We omit the proof.

Corollary 3.4. Let be a given function such that for all . Assume that a function satisfies and the inequality (3.2) for all . If there exists a constant such that has the property (3.17) for all , then there exists a unique quadratic-additive function such that for all .

4. Applications

For a given function , we will use the following abbreviation: for all .

Corollary 4.1. Let be functions for which and there exist functions such that for all and . If there exists a constant such that and for all and , then there exist additive functions such that for all , where are defined by for all and . In particular, the functions and are represented by for all .

Proof. By a long manipulation, we obtain for all and . If we put for all and , then it follows from (4.2) and (4.10) that for all and . By (4.3) and (4.4), we know that and satisfy (3.3) and (3.17), respectively. Furthermore, for all and .
Therefore, according to Corollary 3.2, there exists a unique quadratic-additive function satisfying (4.5) and is represented by (3.6). By (4.2) and (4.3), we see for all . From these and (3.6), we get (4.8).
Moreover, we have for all . Taking the limit as in the aforementioned inequality, we get for all .
Now, according to Corollary 3.4, there exists a unique function satisfying (4.6) and is represented by (3.19). Due to (4.2) and (4.4), we get as well as for all . From these and (3.19), we get (4.9).
Moreover, we have for all . Taking the limit as in the aforementioned inequality, we get for all .

Corollary 4.2. Let be a normed space and . If a function satisfies and the inequality for all and for some , then there exists a unique function such that for all and for all .

Proof. If we put for all and then our assertions follow from Theorems 3.1 and 3.3.

Corollary 4.3. Let be a normed space. If a function satisfies and the inequality for all and for some and , then there exists a unique quadratic-additive function such that for all . In particular, if , then is a quadratic-additive function.

Proof. This corollary follows from Corollaries 3.2 and 3.4 by putting for all with defined in (4.23).

Corollary 4.4. Let be a normed space and let be real constants with . If a function satisfies and the inequality for all , then there exists a unique function such that for all and for all , where .

Proof. If we put for all and is defined in (4.23), then our assertion follows from Theorems 3.1 and 3.3.

Acknowledgment

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).

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