Advances in Fuzzy Systems

Volume 2012 (2012), Article ID 150815, 9 pages

doi:10.1155/2012/150815

Research Article

Fuzzy Stability of an -Dimensional Quadratic and Additive Functional Equation

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 15 December 2011; Accepted 6 April 2012

Academic Editor: Uzay Kaymak

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 will investigate a fuzzy version of stability for the functional equation in the sense of Mirmostafaee and Moslehian.

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 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 (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, we investigate a general stability of the -dimensional quadratic and additive functional equation

in the fuzzy normed space. We will call each solution of (3) a quadratic-additive function.

In 2006, Jun and Kim [10] proved the stability of (3) by constructing an additive function and a quadratic function separately and by approximating the given function with a quadratic-additive function . In their approach, and approximate the odd part and the even part of , respectively. However, their method is not efficient in comparison with our method which approximates the function with the quadratic-additive function simultaneously. Indeed, we introduce a Cauchy sequence by making use of the given function , and the sequence converges to a quadratic-additive function which approximates the function in the fuzzy sense. Our idea seems to be a refinement of previous studies.

2. Preliminaries

In 1984, Katsaras [18] defined a fuzzy norm on a linear space to construct a fuzzy structure on the space. Since then, mathematicians have introduced several types of fuzzy norm in different points of view. In particular, adhering to the point of view of Cheng and Mordeson, Bag and Samanta suggested an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type (see [19–21]).

We introduce the definition of a fuzzy normed space to establish a reasonable fuzzy version of stability for the -dimensional quadratic and additive functional equation (3) in the fuzzy normed space (cf. [19]).

Definition 1. Let be a real linear space. A function is said to be a fuzzy norm on if the following conditions are true for all , and all , :

(); ()  if and only if   for all ;() if ;();() is a nondecreasing function on and .

The pair is called a fuzzy normed space. Let be a fuzzy normed space. A sequence in is said to be convergent if there exists an such that for all . In this case, is called the limit of the sequence and we write . A sequence in is called Cauchy if for each and each there exists an such that for all and all . It is known that every convergent sequence in a fuzzy normed space is Cauchy. If every Cauchy sequence in converges in , then the fuzzy norm is said to be complete, and the fuzzy normed space is called a fuzzy Banach space.

In 2008, Mirmostafaee and Moslehian first dealt with the fuzzy stability problems for the additive Cauchy equation, Jensen’s functional equation, and the quadratic functional equation (see [22–24]).

3. Fuzzy Stability of (3)

Let and be a fuzzy normed space and a fuzzy Banach space, respectively. Assume that is a fixed integer greater than 1. For a given function , we use the abbreviation

for all . For a given , the function is called a fuzzy -almost quadratic-additive function if

for all and all .

The following theorem gives a fuzzy version of the stability of the -dimensional quadratic and additive functional equation .

Theorem 2. Let be a positive real number with . If a function is a fuzzy -almost quadratic-additive function from a fuzzy normed space into a fuzzy Banach space , then there exists a unique quadratic-additive function such that

for each and , where .

Proof. It follows from (), (), and (5) that

for all . Thus, it follows from () that .

We will split the proof into three parts according to the value of , namely, , , or .

Case 1. Assume that and let be a function defined by

for all and . Then, and by a long manipulation, we get

for all and for any . Together with (), (), and (5), this equation implies that if then

for all and for all .

Let be given. Since , there exists an such that

We observe that for some , the series converges for . Hence, for an arbitrarily given , there exists an such that

for each and . By () and (10), we have

for all and for each . Hence, is a Cauchy sequence in the fuzzy Banach space , and, thus, we can define a function by

for all . Moreover, if we put and replace with in (10), then we have

for all and for any .

Next, we will show that is a quadratic-additive function. Using () and (4), we have

for all , , and . In view of the definition of and (), the first four terms on the right hand side of (16) tend to 1 as . It follows from (4) and (8) that

for all and . Hence, by () and (), we have

for all and .

By () and (5), we obtain

for all , , and . Since , it follows from () that the last term in (16) also goes to 1 as . In view of (16), we conclude that

for all and . By (), this implies that

for all .

We will now prove the first inequality in (6). Let , let , and let be given. Since is the limit of the sequence , there exists an such that

By () and (15), we have

Because is arbitrary, we get the first inequality in (6) for the case of .

Finally, it remains to prove the uniqueness of . Let be another quadratic-additive function satisfying the first inequality in (6). Then, by (9) and (21), we get

for all and . Together with (), (), (), (6), and (8), this implies that

for all and . This implies that for all and . Thus, it follows from () that

for all .

Case 2. Assume that . Let us define a function by

for all and . Then, we have , and it follows from (4) and (27) that

for all and . If , then it follows from (), (), and (5) that

for all and .

By a similar argument of the preceding part, we can define the limit of the Cauchy sequence in the fuzzy Banach space . Moreover, putting and replacing with in the last inequality yield

for each and for each .

To prove that is a quadratic-additive function, it suffices to show that the last term of (16) tends to 1 as . By (4) and (27), we obtain

for all and . Hence, it follows from (), (), (), and (5) that

for all and , since . Hence, as we did for the preceding case of , it follows from (16) that for all .

By a similar way as the first inequality in (6) follows from (15), we see that the second inequality of (6) follows from (30).

We will prove the uniqueness of . Let be another quadratic-additive function satisfying the second inequality in (6). Notice that (24) also holds true for the case of . Then, by (), (), (), (6), (24), and (27), we have

for all and , since in this case. This implies that for all and , and hence for all by .

Case 3. Finally, we consider the case of . Let us define a function by

for all and . Then, we have , and a somewhat tedious manipulation yields

for all and .

If and are nonnegative integers with , then it follows from , , and (5) that

for all and .

Similarly to the preceding cases, let us define a function by . Putting and replacing with in the last inequality yield

for all and . Notice that

for all . Hence, it follows from (), () and (5) that

for all and , since . Thus, all terms on the right hand side of (16) tend to 1 as . Therefore, it follows from () and (16) that for all . Moreover, using the same argument as in the first case, the third inequality in (6) follows from (37) for the case of .

We will prove the uniqueness of . Let be another quadratic-additive function satisfying the third inequality in (6). Then, (), (), (), (6), (24), and by (34), we get

for all and . This implies that for all and , and hence for all by .

4. Applications

Corollary 3. If an even function satisfies all of the conditions of Theorem 2, then there exists a unique quadratic function such that

for all and , where .

Proof. Let be defined as in Theorem 2. Since is an even function, it follows from (8), (27), and (34) that

for all . Notice that and

for all and . From these equalities, using the similar method as presented in Theorem 2, we obtain the quadratic-additive function satisfying (41). Notice that for all , is even, and for all . Hence, we get

for all , . This implies that is a quadratic function.

Corollary 4. If an odd function satisfies all of the conditions of Theorem 2, then there exists a unique additive function such that

for all and , where and .

Proof. Let be defined as in Theorem 2. Since is an odd function, we obtain

for all . Notice that and

for all and . From these equalities, by using a similar method in Theorem 2, we obtain the quadratic-additive function satisfying (45). Notice that for all , is odd, and for all . Hence, we get

for all , . This implies that is an additive function.

We can use Theorem 2 to prove a classical result in the framework of normed spaces. Let be a normed space and let be a Banach space. Then we can define fuzzy norms by

where , and (see [22]). Suppose a function satisfies

for all , where and . Let be a fuzzy norm on . Then we get

for all and .

Consider the case

This implies that

Hence, there exists an such that . Thus, with , we have

for all and . Therefore, the inequality

holds true. That is, is a fuzzy -almost quadratic-additive function, and by Theorem 2, we obtain the following stability result.

Corollary 5. Let be a normed linear space and let be a Banach space. Assume that is an integer greater than 1 and is a positive real number with . If a function satisfies

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

for all .

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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