Abstract and Applied Analysis
Volume 2009 (2009), Article ID 535678, 17 pages
doi:10.1155/2009/535678
Research Article

Fuzzy Stability of Jensen-Type Quadratic Functional Equations

Sun-Young Jang,1 Jung Rye Lee,2 Choonkil Park,3 and Dong Yun Shin4

1Department of Mathematics, University of Ulsan, Ulsan 680-749, South Korea
2Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea
3Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
4Department of Mathematics, University of Seoul, Seoul 130-743, South Korea

Received 29 December 2008; Revised 26 March 2009; Accepted 10 April 2009

Academic Editor: John Rassias

Copyright © 2009 Sun-Young Jang 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 prove the generalized Hyers-Ulam stability of the following quadratic functional equations 2 𝑓 ( ( 𝑥 + 𝑦 ) / 2 ) + 2 𝑓 ( ( 𝑥 − 𝑦 ) / 2 ) = 𝑓 ( 𝑥 ) + 𝑓 ( 𝑦 ) and 𝑓 ( 𝑎 𝑥 + 𝑎 𝑦 ) + ( 𝑎 𝑥 − 𝑎 𝑦 ) = 2 𝑎 2 𝑓 ( 𝑥 ) + 2 𝑎 2 𝑓 ( 𝑦 ) in fuzzy Banach spaces for a nonzero real number 𝑎 with 𝑎 ≠ ± 1 / 2 .

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 ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝 by ‖ 𝑥 ‖ 𝑝 ⋅ ‖ 𝑦 ‖ 𝑞 for 𝑝 , 𝑞 ∈ ℝ with 𝑝 + 𝑞 ≠ 1 (see also [7, 8] for a number of other new results). The papers of J. M. Rassias [6–8] introduced the Ulam- Găvruţa-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 𝑓 ∶ 𝑋 → 𝑌 , where 𝑋 is a normed space and 𝑌 is a Banach space. Cholewa [13] noticed that the theorem of Skof is still true if the relevant domain 𝑋 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 Michálek 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 𝑋 be a real vector space. A function 𝑁 ∶ 𝑋 × ℝ → [ 0 , 1 ] is called a fuzzy norm on 𝑋 if for all 𝑥 , 𝑦 ∈ 𝑋 and all 𝑠 , 𝑡 ∈ ℝ ,
( 𝑁 1 ) 𝑁 ( 𝑥 , 𝑡 ) = 0 for 𝑡 ≤ 0 ;
( 𝑁 2 ) 𝑥 = 0 if and only if 𝑁 ( 𝑥 , 𝑡 ) = 1 for all 𝑡 > 0 ;
( 𝑁 3 ) 𝑁 ( 𝑐 𝑥 , 𝑡 ) = 𝑁 ( 𝑥 , 𝑡 / | 𝑐 | ) if 𝑐 ≠ 0 ;
( 𝑁 4 ) 𝑁 ( 𝑥 + 𝑦 , 𝑠 + 𝑡 ) ≥ m i n { 𝑁 ( 𝑥 , 𝑠 ) , 𝑁 ( 𝑦 , 𝑡 ) } ;
( 𝑁 5 ) 𝑁 ( 𝑥 , ⋅ ) is a non-decreasing function of ℝ and l i m 𝑡 → ∞ 𝑁 ( 𝑥 , 𝑡 ) = 1 ;
( 𝑁 6 ) for 𝑥 ≠ 0 , 𝑁 ( 𝑥 , ⋅ ) is continuous on ℝ .
The pair ( 𝑋 , 𝑁 ) 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 ( 𝑋 , 𝑁 ) be a fuzzy normed vector space. A sequence { 𝑥 𝑛 } in 𝑋 is said to be convergent or converge if there exists an 𝑥 ∈ 𝑋 such that l i m 𝑛 → ∞ 𝑁 ( 𝑥 𝑛 − 𝑥 , 𝑡 ) = 1 for all 𝑡 > 0 . In this case, 𝑥 is called the limit of the sequence { 𝑥 𝑛 } and we denote it by N- l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑥 .

Definition 1.3 (see [31, 35–38]). Let ( 𝑋 , 𝑁 ) be a fuzzy normed vector space. A sequence { 𝑥 𝑛 } in 𝑋 is called Cauchy if for each 𝜀 > 0 and each 𝑡 > 0 there exists an 𝑛 0 ∈ ℕ such that for all 𝑛 ≥ 𝑛 0 and all 𝑝 > 0 , we have 𝑁 ( 𝑥 𝑛 + 𝑝 − 𝑥 𝑛 , 𝑡 ) > 1 − 𝜀 .
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 𝑓 ∶ 𝑋 → 𝑌 between fuzzy normed vector spaces 𝑋 and 𝑌 is continuous at a point 𝑥 0 ∈ 𝑋 if for each sequence { 𝑥 𝑛 } converging to 𝑥 0 in 𝑋 , then the sequence { 𝑓 ( 𝑥 𝑛 ) } converges to 𝑓 ( 𝑥 0 ) . If 𝑓 ∶ 𝑋 → 𝑌 is continuous at each 𝑥 ∈ 𝑋 , then 𝑓 ∶ 𝑋 → 𝑌 is said to be continuous on 𝑋 (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 𝑋 is a vector space and that ( 𝑌 , 𝑁 ) is a fuzzy Banach space. Let 𝑎 be a nonzero real number with 𝑎 ≠ ( ± 1 / 2 ).

2. Fuzzy Stability of Quadratic Functional Equations

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

Theorem 2.1. Let 𝑓 ∶ 𝑋 → 𝑌 be an even mapping with 𝑓 ( 0 ) = 0 . Suppose that 𝜑 is a mapping from 𝑋 to a fuzzy normed space ( 𝑍 , 𝑁 ) such that for all 𝑥 , 𝑦 ∈ 𝑋 ⧵ { 0 } and all positive real numbers 𝑡 , 𝑠 . If 𝜑 ( 3 𝑥 ) = 𝛼 𝜑 ( 𝑥 ) for some positive real number 𝛼 with 𝛼 < 9 , then there is a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 such that 𝑄 ( 𝑥 ) = 𝑁 - l i m 𝑛 → ∞ 𝑓 ( 3 𝑛 𝑥 ) / 9 𝑛 and where

Proof. Putting 𝑦 = 3 𝑥 and 𝑠 = 𝑡 in (2.1), we get for all 𝑥 ∈ 𝑋 and all 𝑡 > 0 . Replacing 𝑥 by 2 𝑥 , 𝑦 by 0 , and 𝑠 by 𝑡 in (2.1), we obtain Thus and so Then by the assumption, Replacing 𝑥 by 3 𝑛 𝑥 in (2.7) and applying (2.8), we get Thus for each 𝑛 > 𝑚 we have Let 𝜀 > 0 and 𝛿 > 0 be given. Since l i m 𝑡 → ∞ 𝑀 ( 𝑥 , 𝑡 ) = 1 , there is some 𝑡 0 > 0 such that 𝑀 ( 𝑥 , 𝑡 0 ) > 1 − 𝜀 . Since ∑ ∞ 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 𝛼 𝑘 𝑡 0 / 9 𝑘 < ∞ , there is some 𝑛 0 ∈ ℕ such that ∑ 𝑛 − 1 𝑘 = 𝑚 0 𝑥 0 2 0 0 𝑑 𝛼 𝑘 𝑡 0 / 9 𝑘 < 𝛿 for 𝑛 > 𝑚 ≥ 𝑛 0 . It follows that for all 𝑡 ≥ 𝑡 0 . This shows that the sequence { 𝑓 ( 3 𝑛 𝑥 ) / 9 𝑛 } is Cauchy in ( 𝑌 , 𝑁 ) . Since ( 𝑌 , 𝑁 ) is complete, { 𝑓 ( 3 𝑛 𝑥 ) / 9 𝑛 } converges to some 𝑄 ( 𝑥 ) ∈ 𝑌 . Thus we can define a mapping 𝑄 ∶ 𝑋 → 𝑌 by 𝑄 ( 𝑥 ) ∶ = 𝑁 − l i m 𝑡 → ∞ 𝑓 ( 3 𝑛 𝑥 ) / 9 𝑛 . Moreover, if we put 𝑚 = 0 in (2.10), then we observe that Thus Next we show that 𝑄 is quadratic. Let 𝑥 , 𝑦 ∈ 𝑋 . Then we have The first four terms on the right-hand side of the above inequality tend to 1 as 𝑛 → ∞ and the fifth term, by (2.1), is greater than or equal to which tends to 1 as 𝑛 → ∞ . Hence for all 𝑥 , 𝑦 ∈ 𝑋 and all 𝑡 > 0 . This means that 𝑄 satisfies the Jensen quadratic functional equation and so it is quadratic.
Next, we approximate the difference between 𝑓 and 𝑄 in a fuzzy sense. For every 𝑥 ∈ 𝑋 and 𝑡 > 0 , by (2.13), for large enough 𝑛 , we have The uniqueness assertion can be proved by a standard fashion; cf. [36]: Let 𝑄 ′ be another quadratic mapping from 𝑋 into 𝑌 , which satisfies the required inequality. Then for each 𝑥 ∈ 𝑋 and 𝑡 > 0 , Since 𝑄 and 𝑄 are quadratic, for all 𝑥 ∈ 𝑋 , all 𝑡 > 0 and all 𝑛 ∈ ℕ .
Since 0 < 𝛼 < 9 , l i m 𝑛 → ∞ ( 9 / 𝛼 ) 𝑛 = ∞ . Hence the right-hand side of the above inequality tends to 1 as 𝑛 → ∞ . It follows that 𝑄 ( 𝑥 ) = 𝑄 ( 𝑥 ) for all 𝑥 ∈ 𝑋 .

Theorem 2.2. Let 𝑓 ∶ 𝑋 → 𝑌 be an even mapping with 𝑓 ( 0 ) = 0 . Suppose that 𝜑 is a mapping from 𝑋 to a fuzzy normed space ( 𝑍 , 𝑁 ) satisfying (2.1). If 𝜑 ( 3 𝑥 ) = 𝛼 𝜑 ( 𝑥 ) for some real number 𝛼 with 𝛼 > 9 , then there is a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 such that 𝑄 ( 𝑥 ) = 𝑁 - l i m 𝑛 → ∞ 9 𝑛 𝑓 ( 𝑥 / 3 𝑛 ) and where

Proof. It follows from (2.7) that Then by the assumption, Replacing 𝑥 by 𝑥 / 3 𝑛 in (2.22) and applying (2.23), we get Thus for each 𝑛 > 𝑚 we have
Let 𝜀 > 0 and 𝛿 > 0 be given. Since l i m 𝑡 → ∞ 𝑀 ( 𝑥 , 𝑡 ) = 1 , there is some 𝑡 0 > 0 such that 𝑀 ( 𝑥 , 𝑡 0 ) > 1 − 𝜀 . Since ∑ ∞ 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 9 𝑘 𝑡 0 / 𝛼 𝑘 < ∞ , there is some 𝑛 0 ∈ ℕ such that ∑ 𝑛 − 1 𝑘 = 𝑚 0 𝑥 0 2 0 0 𝑑 9 𝑘 𝑡 0 / 𝛼 𝑘 < 𝛿 for 𝑛 > 𝑚 ≥ 𝑛 0 . It follows that for all 𝑡 ≥ 𝑡 0 . This shows that the sequence { 9 𝑛 𝑓 ( 𝑥 / 3 𝑛 ) } is Cauchy in ( 𝑌 , 𝑁 ) . Since ( 𝑌 , 𝑁 ) is complete, { 9 𝑛 𝑓 ( 𝑥 / 3 𝑛 ) } converges to some 𝑄 ( 𝑥 ) ∈ 𝑌 . Thus we can define a mapping 𝑄 ∶ 𝑋 → 𝑌 by 𝑄 ( 𝑥 ) ∶ = 𝑁 - l i m 𝑡 → ∞ 9 𝑛 𝑓 ( 𝑥 / 3 𝑛 ) . Moreover, if we put 𝑚 = 0 in ( 2 . 8 ) , then we observe that Thus The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.3. Let 𝑓 ∶ 𝑋 → 𝑌 be a mapping with 𝑓 ( 0 ) = 0 . Suppose that 𝜑 is a mapping from 𝑋 to a fuzzy normed space ( 𝑍 , 𝑁 ) satisfying (2.1). If 𝜑 ( 2 𝑥 ) = 𝛼 𝜑 ( 𝑥 ) for some positive real number 𝛼 with 𝛼 < 4 , then there is a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 such that 𝑄 ( 𝑥 ) = 𝑁 - l i m 𝑛 → ∞ 𝑓 ( 2 𝑛 𝑥 ) / 4 𝑛 and where 𝑀 ( 𝑥 , 𝑡 ) ∶ = m i n { 𝑁 ( 𝜑 ( 2 𝑥 ) , 2 𝑡 ) , 𝑁 ( 𝜑 ( 0 ) , 2 𝑡 ) } .

Proof. Letting 𝑦 = 0 and replacing 𝑥 by 2 𝑥 and 𝑠 by 𝑡 in (2.1), we obtain Thus Then by the assumption, Replacing 𝑥 by 2 𝑛 𝑥 in (2.31) and applying (2.32), we get Thus for each 𝑛 > 𝑚 we have
Let 𝜀 > 0 and 𝛿 > 0 be given. Since l i m 𝑡 → ∞ 𝑀 ( 𝑥 , 𝑡 ) = 1 , there is some 𝑡 0 > 0 such that 𝑀 ( 𝑥 , 𝑡 0 ) > 1 − 𝜀 . Since ∑ ∞ 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 𝛼 𝑘 𝑡 0 / 4 𝑘 < ∞ , there is some 𝑛 0 ∈ ℕ such that ∑ 𝑛 − 1 𝑘 = 𝑚 0 𝑥 0 2 0 0 𝑑 𝛼 𝑘 𝑡 0 / 4 𝑘 < 𝛿 for 𝑛 > 𝑚 ≥ 𝑛 0 . It follows that for all 𝑡 ≥ 𝑡 0 . This shows that the sequence { 𝑓 ( 2 𝑛 𝑥 ) / 4 𝑛 } is Cauchy in ( 𝑌 , 𝑁 ) . Since ( 𝑌 , 𝑁 ) is complete, { 𝑓 ( 2 𝑛 𝑥 ) / 4 𝑛 } converges to some 𝑄 ( 𝑥 ) ∈ 𝑌 . Thus we can define a mapping 𝑄 ∶ 𝑋 → 𝑌 by 𝑄 ( 𝑥 ) ∶ = 𝑁 - l i m 𝑡 → ∞ 𝑓 ( 2 𝑛 𝑥 ) / 4 𝑛 . Moreover, if we put 𝑚 = 0 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 𝑓 ∶ 𝑋 → 𝑌 be a mapping with 𝑓 ( 0 ) = 0 . Suppose that 𝜑 is a mapping from 𝑋 to a fuzzy normed space ( 𝑍 , 𝑁 ) satisfying (2.1). If 𝜑 ( 2 𝑥 ) = 𝛼 𝜑 ( 𝑥 ) for some real number 𝛼 with 𝛼 > 4 , then there is a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 such that 𝑄 ( 𝑥 ) = 𝑁 - l i m 𝑛 → ∞ 4 𝑛 𝑓 ( 𝑥 / 2 𝑛 ) and where 𝑀 ( 𝑥 , 𝑡 ) ∶ = m i n { 𝑁 ( 𝜑 ( 𝑥 ) , 𝑡 / 2 ) , 𝑁 ( 𝜑 ( 0 ) , 𝑡 / 2 ) } .

Proof. It follows from (2.31) that Then by the assumption, Replacing 𝑥 by 𝑥 / 2 𝑛 in (2.39) and applying (2.40), we get Thus for each 𝑛 > 𝑚 we have
Let 𝜀 > 0 and 𝛿 > 0 be given. Since l i m 𝑡 → ∞ 𝑀 ( 𝑥 , 𝑡 ) = 1 , there is some 𝑡 0 > 0 such that 𝑀 ( 𝑥 , 𝑡 0 ) > 1 − 𝜀 . Since ∑ ∞ 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 4 𝑘 𝑡 0 / 𝛼 𝑘 < ∞ , there is some 𝑛 0 ∈ ℕ such that ∑ 𝑛 − 1 𝑘 = 𝑚 0 𝑥 0 2 0 0 𝑑 4 𝑘 𝑡 0 / 𝛼 𝑘 < 𝛿 for 𝑛 > 𝑚 ≥ 𝑛 0 . It follows that for all 𝑡 ≥ 𝑡 0 . This shows that the sequence { 4 𝑛 𝑓 ( 𝑥 / 2 𝑛 ) } is Cauchy in ( 𝑌 , 𝑁 ) . Since ( 𝑌 , 𝑁 ) is complete, { 4 𝑛 𝑓 ( 𝑥 / 2 𝑛 ) } converges to some 𝑄 ( 𝑥 ) ∈ 𝑌 . Thus we can define a mapping 𝑄 ∶ 𝑋 → 𝑌 by 𝑄 ( 𝑥 ) ∶ = 𝑁 - l i m 𝑡 → ∞ 4 𝑛 𝑓 ( 𝑥 / 2 𝑛 ) . Moreover, if we put 𝑚 = 0 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 𝑎 ≠ ( ± 1 / 2 ) .

Theorem 2.5. Let | 2 𝑎 | > 1 and 𝑓 ∶ 𝑋 → 𝑌 a mapping with 𝑓 ( 0 ) = 0 . Suppose that 𝜑 is a mapping from 𝑋 to a fuzzy normed space ( 𝑍 , 𝑁 ) such that for all 𝑥 , 𝑦 ∈ 𝑋 ⧵ { 0 } and all positive real numbers 𝑡 , 𝑠 . If 𝜑 ( 2 𝑎 𝑥 ) = 𝛼 𝜑 ( 𝑥 ) for some positive real number 𝛼 with 0 < 𝛼 < 4 𝑎 2 , then there is a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 such that 𝑄 ( 𝑥 ) = 𝑁 - l i m 𝑛 → ∞ 𝑓 ( ( 2 𝑎 ) 𝑛 𝑥 ) / ( 2 𝑎 ) 2 𝑛 and for all 𝑥 ∈ 𝑋 and all 𝑡 > 0 .

Proof. Putting 𝑦 = 𝑥 and 𝑠 = 𝑡 in (2.46), we get for all 𝑥 ∈ 𝑋 and all 𝑡 > 0 . Thus and so Replacing 𝑥 by ( 2 𝑎 ) 𝑛 𝑥 in (2.50), we get Thus for each 𝑛 > 𝑚 we have
Let 𝜀 > 0 and 𝛿 > 0 be given. Since l i m 𝑡 → ∞ 𝑁 ( 𝜑 ( 𝑥 ) , 2 𝑎 2 𝑡 ) = 1 , there is some 𝑡 0 > 0 such that 𝑁 ( 𝜑 ( 𝑥 ) , 2 𝑎 2 𝑡 0 ) > 1 − 𝜀 . Since ∑ ∞ 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 𝛼 𝑘 𝑡 0 / ( 2 𝑎 ) 2 𝑘 < ∞ , there is some 𝑛 0 ∈ ℕ such that ∑ 𝑛 − 1 𝑘 = 𝑚 0 𝑥 0 2 0 0 𝑑 𝛼 𝑘 𝑡 0 / ( 2 𝑎 ) 2 𝑘 < 𝛿 for 𝑛 > 𝑚 ≥ 𝑛 0 . It follows that for all 𝑡 ≥ 𝑡 0 . This shows that the sequence { 𝑓 ( ( 2 𝑎 ) 𝑛 𝑥 ) / ( 2 𝑎 ) 2 𝑛 } is Cauchy in ( 𝑌 , 𝑁 ) . Since ( 𝑌 , 𝑁 ) is complete, { 𝑓 ( ( 2 𝑎 ) 𝑛 𝑥 ) / ( 2 𝑎 ) 2 𝑛 } converges to some 𝑄 ( 𝑥 ) ∈ 𝑌 . Thus we can define a mapping 𝑄 ∶ 𝑋 → 𝑌 by 𝑄 ( 𝑥 ) ∶ = 𝑁 - l i m 𝑡 → ∞ 𝑓 ( ( 2 𝑎 ) 𝑛 𝑥 ) / ( 2 𝑎 ) 2 𝑛 . Moreover, if we put 𝑚 = 0 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. Let | 2 𝑎 | < 1 and 𝑓 ∶ 𝑋 → 𝑌 a mapping with 𝑓 ( 0 ) = 0 . Suppose that 𝜑 is a mapping from 𝑋 to a fuzzy normed space ( 𝑍 , 𝑁 ) satisfying (2.46). If 𝜑 ( 2 𝑎 𝑥 ) = 𝛼 𝜑 ( 𝑥 ) for some real number 𝛼 with 𝛼 > 4 𝑎 2 , then there is a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 such that 𝑄 ( 𝑥 ) = 𝑁 - l i m 𝑛 → ∞ ( 2 𝑎 ) 2 𝑛 𝑓 ( 𝑥 / ( 2 𝑎 ) 𝑛 ) and for all 𝑥 ∈ 𝑋 and all 𝑡 > 0 .

Proof. It follows from (2.50) that for all 𝑥 ∈ 𝑋 and all 𝑡 > 0 . Thus Replacing 𝑥 by 𝑥 / ( 2 𝑎 ) 𝑛 in (2.58), we get Thus for each 𝑛 > 𝑚 we have Let 𝜀 > 0 and 𝛿 > 0 be given. Since l i m 𝑡 → ∞ 𝑁 ( 𝜑 ( 𝑥 ) , ( 𝛼 / 2 ) 𝑡 ) = 1 , there is some 𝑡 0 > 0 such that 𝑁 ( 𝜑 ( 𝑥 ) , ( 𝛼 / 2 ) 𝑡 0 ) > 1 − 𝜀 . Since ∑ ∞ 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 ( 2 𝑎 ) 2 𝑘 𝑡 0 / 𝛼 𝑘 < ∞ , there is some 𝑛 0 ∈ ℕ such that ∑ 𝑛 − 1 𝑘 = 𝑚 0 𝑥 0 2 0 0 𝑑 ( 2 𝑎 ) 2 𝑘 𝑡 0 / 𝛼 𝑘 < 𝛿 for 𝑛 > 𝑚 ≥ 𝑛 0 . It follows that for all 𝑡 ≥ 𝑡 0 . This shows that the sequence { ( 2 𝑎 ) 2 𝑛 𝑓 ( 𝑥 / ( 2 𝑎 ) 𝑛 ) } is Cauchy in ( 𝑌 , 𝑁 ) . Since ( 𝑌 , 𝑁 ) is complete, { ( 2 𝑎 ) 2 𝑛 𝑓 ( 𝑥 / ( 2 𝑎 ) 𝑛 ) } converges to some 𝑄 ( 𝑥 ) ∈ 𝑌 . Thus we can define a mapping 𝑄 ∶ 𝑋 → 𝑌 by 𝑄 ( 𝑥 ) ∶ = 𝑁 - l i m 𝑡 → ∞ ( 2 𝑎 ) 2 𝑛 𝑓 ( 𝑥 / ( 2 𝑎 ) 𝑛 ) . Moreover, if we put 𝑚 = 0 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).

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