Journal of Applied Mathematics

Volume 2012 (2012), Article ID 547865, 15 pages

doi:10.1155/2012/547865

Research Article

Stability of an -Dimensional Mixed-Type Additive and Quadratic Functional Equation in Random Normed Spaces

Yang-Hi Lee¹ and Soon-Mo Jung²

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 5 August 2011; Accepted 28 November 2011

Academic Editor: Xianhua Tang

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 the -dimensional mixed-type additive and quadratic functional equation in random normed spaces by applying 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 that

for 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 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] addressed the Hyers’s stability theorem and attempted to weaken the condition for the bound of the norm of Cauchy difference and generalized the 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 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 [6–17]).

In this paper, applying the fixed point method, we prove the Hyers-Ulam-Rassias stability of the -dimensional mixed-type additive and quadratic functional equation

in random normed spaces. Every solution of (1.3) is called a quadratic-additive function.

Throughout this paper, let be an integer larger than 1.

2. Preliminaries

We introduce some terminologies, notations, and conventions usually used in the theory of random normed spaces (see [18, 19]). The set of all probability distribution functions is denoted by

Let us define . The set is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all . The maximal element for in this order is the distribution function given by

Definition 2.1 (See [18]). A function is called a continuous triangular norm (briefly, continuous -norm) if satisfies the following conditions:

is commutative and associative;τ is continuous; for all ; for all with and .

Typical examples of continuous -norms are , , and .

Definition 2.2 (See [19]). Let be a vector space, a continuous -norm, and let be a function satisfying the following conditions:

for all if and only if ; for all , , and for all ; for all and all .

A triple is called a random normed space (briefly, RN-space).

If is a normed space, we can define a function by

for all and . Then is a random normed space, which is called the induced random normed space.

Definition 2.3. Let be an -space.

A sequence in is said to be convergent to a point if, for every and , there exists a positive integer such that whenever .A sequence in is called a Cauchy sequence if, for every and , there exists a positive integer such that whenever .An RN-space is called complete if and only if every Cauchy sequence in converges to a point in .

Definition 2.4. Let be a nonempty set. A function is called a generalized metric on if and only if satisfies

if and only if ; for all ; for all .

We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to [20] or [21].

Theorem 2.5 (See [20, 21]). 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:

the sequence converges to a fixed point of ; is the unique fixed point of in ;if , then

In 2003, Radu [22] noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative (Theorem 2.5). 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 also [23–26]). Furthermore, Miheţ and Radu [27] applied the fixed point method to prove the stability theorems of the additive Cauchy equation in random normed spaces.

In 2009, Towanlong and Nakmahachalasint [28] established the general solution and the stability of the -dimensional mixed-type additive and quadratic functional equation (1.3) by using the direct method. According to [28], a function is a quadratic-additive function, where and are vector spaces, if and only if there exist an additive function and a quadratic function such that for all .

3. Hyers-Ulam-Rassias Stability

Throughout this paper, let be a real vector space and let be a complete RN-space. For a given function , we use the following abbreviation:

for all .

We will now prove the stability of the functional equation (1.3) in random normed spaces by using fixed point method.

Theorem 3.1. Let be a real vector space, an RN-space, a complete RN-space, and let be a function. Assume that satisfies one of the following conditions:

for some ; for some

for all and . If a function satisfies and

for all and , then there exists a unique function such that

for all and

for all and , where , and .

Proof. We will first treat the case where satisfies the condition . Let be the set of all functions with , and let us define a generalized metric on by

It is not difficult to show that is a complete generalized metric space (see [29] or [30, 31]).

Consider the operator defined by

Then we can apply induction on to prove

for all and .

Let and let be an arbitrary constant with . For some satisfying the condition , it follows from the definition of , , , and that

for all and , which implies that

That is, is a strict contraction with the Lipschitz constant .

Moreover, by , , and (3.2), we see that

for all and . Hence, it follows from the definition of that

Now, in view of Theorem 2.5, the sequence converges to the unique “fixed point” of in the set and is represented by

for all .

By Theorem 2.5, (3.11), and the definition of , we have

that is, the first inequality in (3.4) holds true.

We will now show that is a quadratic-additive function. It follows from and the definition of that

for all , , and . Due to the definition of , the first four terms on the right-hand side of the above inequality tend to 1 as .

By a somewhat tedious manipulation, we have

Hence, it follows from , , definition of , (3.2), and that

which tends to 1 as for all and . Therefore, (3.14) implies that

for any and . By , this implies that for all , which ends the proof of the first part.

Now, assume that satisfies the condition . Let be the same as given in the first part. We now consider the operator defined by

for all and . Notice that

for all and .

Let and let be an arbitrary constant with . From , , the definition of , and , we have

for all , , and for some satisfying , which implies that

That is, is a strict contraction with the Lipschitz constant .

Moreover, by , (3.2), and , we see that

for all and . This implies that by the definition of . Therefore, according to Theorem 2.5, the sequence converges to the unique “fixed point” of in the set and is represented by

for all . Since

the second inequality in (3.4) holds true.

Next, we will show that is a quadratic-additive function. As we did in the first part, we obtain the inequality (3.14). In view of the definition of , the first four terms on the right-hand side of the inequality (3.14) tend to 1 as . Furthermore, a long manipulation yields

Thus, it follows from , , definition of , (3.2), and that

which tends to 1 as for all and . Therefore, it follows from (3.14) that

for any and . By , this implies that

for all , which ends the proof.

By a similar way presented in the proof of Theorem 3.1, we can also prove the preceding theorem if the domains of relevant functions include 0.

Theorem 3.2. Let be a real vector space, an RN-space, a complete RN-space, and let be a function. Assume that satisfies one of the conditions and in Theorem 3.1 for all and . If a function satisfies and (3.2) for all and , then there exists a unique quadratic-additive function satisfying (3.4) for all and .

Now, we obtain general Hyers-Ulam stability results of (1.3) in normed spaces. If is a normed space, then is an induced random normed space. We get the following result.

Corollary 3.3. Let be a real vector space, a complete normed space, and let be a function. Assume that satisfies one of the following conditions:

for some ; for some

for all . If a function satisfies and

for all , then there exists a unique function such that

for all and

for all .

Proof. Let us put

for all , , and . If satisfies the condition for all and for some , then

for all and , that is, satisfies the condition . In a similar way, we can show that if satisfies , then it satisfies the condition .

Moreover, we get

for all and , that is, satisfies the inequality (3.2) for all .

According to Theorem 3.1, there exists a unique function such that

for all and

for all and , which ends the proof.

We now prove the Hyers-Ulam-Rassias stability of (1.3) in the framework of normed spaces.

Corollary 3.4. Let be a real normed space, , and let be a complete normed space. If a function satisfies and

for all and for some , then there exists a unique quadratic-additive function such that

for all .

Proof. If we put

then the induced random normed space satisfies the conditions stated in Theorem 3.2 with .

Corollary 3.5. Let be a real normed space, , and let be a complete normed space. If a function satisfies and

for all and for some , then there exists a unique quadratic-additive function satisfying

Proof. If we put , , and define

for all and , then we have

that is, satisfies condition given in Theorem 3.1 for all and . We moreover get

that is, satisfies the inequality (3.2) for all and .

According to Theorem 3.2, there exists a unique quadratic-additive function satisfying

for all , or equivalently

for all , which ends the proof.

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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