Abstract

We prove the generalized Hyers-Ulam stability of the following quadratic functional equations and in fuzzy Banach spaces for a nonzero real number with .

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The work of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

J. M. Rassias [6] proved a similar stability theorem in which he replaced the factor by for with (see also [7, 8] for a number of other new results). The papers of J. M. Rassias [6–8] introduced the Ulam- GΔƒvruΕ£a-Rassias stability of functional equations. See also [9–11].

The functional equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [12] for mappings , where is a normed space and is a Banach space. Cholewa [13] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. In [14], Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation.

J. M. Rassias [15] introduced and solved the stability problem of Ulam for the Euler-Lagrange-type quadratic functional equation

motivated from the following pertinent algebraic equation

The solution of the functional equation (1.2) is called a Euler-Lagrange-type quadratic mapping. J. M. Rassias [16, 17] introduced and investigated the relative functional equations. In addition, J. M. Rassias [18] generalized the algebraic equation (1.3) to the following equation and introduced and investigated the general pertinent Euler-Lagrange quadratic mappings. Analogous quadratic mappings were introduced and investigated in [19, 20].

These Euler-Lagrange mappings are named Euler-Lagrange-Rassias mappings and the corresponding Euler-Lagrange equations are called Euler-Lagrange-Rassias equations. Before 1992, these mappings and equations were not known at all in functional equations and inequalities. However, a completely different kind of Euler-Lagrange partial differential equations are known in calculus of variations. Therefore, we think that J. M. Rassias' introduction of Euler-Lagrange mappings and equations in functional equations and inequalities provides an interesting cornerstone in analysis. Already some mathematicians have employed these Euler-Lagrange mappings.

Recently, Jun and Kim [21] solved the stability problem of Ulam for another Euler-Lagrange-Rassias-type quadratic functional equation. Jun and Kim [22] introduced and investigated the following quadratic functional equation of Euler-Lagrange-Rassias type:

whose solution is said to be a generalized quadratic mapping of Euler-Lagrange-Rassias type.

During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [9, 23–26]).

Katsaras [27] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [28–30]. In particular, Bag and Samanta [31], following Cheng and Mordeson [32], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and MichΓ‘lek type [33]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [34].

We use the definition of fuzzy normed spaces given in [31] and [35–38] to investigate a fuzzy version of the generalized Hyers-Ulam stability for the quadratic functional equations

in the fuzzy normed vector space setting.

Definition 1.1 (see [31, 35–38]). Let be a real vector space. A function is called a fuzzy norm on if for all and all ,
() for ;
() if and only if for all ;
() if ;
() ;
() is a non-decreasing function of and ;
() for , is continuous on .
The pair is called a fuzzy normed vector space.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [35–38].

Definition 1.2 (see [31, 35–38]). Let be a fuzzy normed vector space. A sequence in is said to be convergent or converge if there exists an such that for all In this case, is called the limit of the sequence and we denote it by N-

Definition 1.3 (see [31, 35–38]). Let be a fuzzy normed vector space. A sequence in is called Cauchy if for each and each there exists an such that for all and all we have
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces and is continuous at a point if for each sequence converging to in , then the sequence converges to . If is continuous at each , then is said to be continuous on (see [34]).
In this paper, we prove the generalized Hyers-Ulam stability of the quadratic functional equations (1.6) and (1.7) in fuzzy Banach spaces.
Throughout this paper, assume that is a vector space and that is a fuzzy Banach space. Let be a nonzero real number with ).

2. Fuzzy Stability of Quadratic Functional Equations

We prove the fuzzy stability of the quadratic functional equation (1.6).

Theorem 2.1. Letbe an even mapping with. Suppose that is a mapping from to a fuzzy normed spacesuch thatfor alland all positive real numbers. Iffor some positive real numberwith, then there is a unique quadratic mappingsuch that- and where

Proof. Putting and in (2.1), we get for all and all Replacing by by , and by in (2.1), we obtain Thus and so Then by the assumption, Replacing by in (2.7) and applying (2.8), we get Thus for each we have Let and be given. Since , there is some such that . Since , there is some such that for . It follows that for all . This shows that the sequence is Cauchy in . Since is complete, converges to some . Thus we can define a mapping by . Moreover, if we put in (2.10), then we observe that Thus Next we show that is quadratic. Let . Then we have The first four terms on the right-hand side of the above inequality tend to 1 as and the fifth term, by (2.1), is greater than or equal to which tends to 1 as . Hence for all and all . This means that satisfies the Jensen quadratic functional equation and so it is quadratic.
Next, we approximate the difference between and in a fuzzy sense. For every and , by (2.13), for large enough , we have The uniqueness assertion can be proved by a standard fashion; cf. [36]: Let be another quadratic mapping from into , which satisfies the required inequality. Then for each and , Since and are quadratic, for all , all and all .
Since , . Hence the right-hand side of the above inequality tends to 1 as . It follows that for all .

Theorem 2.2. Letbe an even mapping with. Suppose thatis a mapping fromto a fuzzy normed space satisfying (2.1). Iffor some real numberwith , then there is a unique quadratic mappingsuch that- and where

Proof. It follows from (2.7) that Then by the assumption, Replacing by in (2.22) and applying (2.23), we get Thus for each we have
Let and be given. Since , there is some such that . Since , there is some such that for . It follows that for all . This shows that the sequence is Cauchy in . Since is complete, converges to some . Thus we can define a mapping by -. Moreover, if we put in , then we observe that Thus The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.3. Letbe a mapping with. Suppose thatis a mapping fromto a fuzzy normed space satisfying (2.1). Iffor some positive real number with, then there is a unique quadratic mappingsuch that-and where .

Proof. Letting and replacing by and by in (2.1), we obtain Thus Then by the assumption, Replacing by in (2.31) and applying (2.32), we get Thus for each we have
Let and be given. Since , there is some such that . Since , there is some such that for . It follows that for all . This shows that the sequence is Cauchy in . Since is complete, converges to some . Thus we can define a mapping by -. Moreover, if we put in (2.34), then we observe that Thus The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.4. Let be a mapping with. Suppose thatis a mapping fromto a fuzzy normed space satisfying (2.1). Iffor some real numberwith, then there is a unique quadratic mappingsuch that-and where

Proof. It follows from (2.31) that Then by the assumption, Replacing by in (2.39) and applying (2.40), we get Thus for each we have
Let and be given. Since , there is some such that . Since , there is some such that for . It follows that for all . This shows that the sequence is Cauchy in . Since is complete, converges to some . Thus we can define a mapping by -. Moreover, if we put in (2.42), then we observe that Thus The rest of the proof is similar to the proof of Theorem 2.1.

Now we prove the fuzzy stability of the quadratic functional equation (1.7) for the case .

Theorem 2.5. Letanda mapping with. Suppose thatis a mapping fromto a fuzzy normed spacesuch thatfor alland all positive real numbers. Iffor some positive real numberwith, then there is a unique quadratic mappingsuch that-and for all and all.

Proof. Putting and in (2.46), we get for all and all Thus and so Replacing by in (2.50), we get Thus for each we have
Let and be given. Since , there is some such that . Since , there is some such that for . It follows that for all . This shows that the sequence is Cauchy in . Since is complete, converges to some . Thus we can define a mapping by -. Moreover, if we put in (2.52), then we observe that Thus The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.6. Letanda mapping with. Suppose thatis a mapping fromto a fuzzy normed spacesatisfying (2.46). Iffor some real numberwith, then there is a unique quadratic mappingsuch that-andfor alland all.

Proof. It follows from (2.50) that for all and all Thus Replacing by in (2.58), we get Thus for each we have Let and be given. Since , there is some such that . Since , there is some such that for . It follows that for all . This shows that the sequence is Cauchy in . Since is complete, converges to some . Thus we can define a mapping by -. Moreover, if we put in (2.60), then we observe that Thus The rest of the proof is similar to the proof of Theorem 2.1.

Acknowledgment

Dr. Sun-Young Jang was supported by the Research Fund of University ofUlsan in 2008, and Dr. Choonkil Park was supported by National ResearchFoundation of Korea (NRF-2009-0070788).