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Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 494781, 7 pages
Approximate Quadratic-Additive Mappings in Fuzzy Normed Spaces
1Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Republic of Korea
2Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea
Received 11 March 2014; Revised 9 May 2014; Accepted 16 May 2014; Published 26 May 2014
Academic Editor: Dorian Popa
Copyright © 2014 Ick-Soon Chang and Yang-Hi Lee. 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.
We examine the generalized Hyers-Ulam stability of the following functional equation: in the fuzzy normed spaces with the fixed point method.
The problem of stability for functional equations originated from questions of Ulam  concerning the stability of group homomorphisms. Hyers  had answered affirmatively the question of Ulam for Banach spaces. The theorem of Hyers was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. Thereafter, many interesting results of the generalized Hyers-Ulam stability to a number of functional equations and mappings have been investigated. Especially, Cădariu and Radu  observed that the existence of the solution for a functional equation and the estimation of the difference with the given mapping can be obtained from the fixed point alternative. This method is called a fixed point method. Also, they [6, 7] applied this method to prove the stability theorems of the additive functional equation.
Katsaras  defined a fuzzy norm on a linear space to construct a fuzzy structure on the space. Since then, some mathematicians have introduced several types of fuzzy norm in different points of view. In particular, Bag and Samanta, following Cheng and Mordeson, gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [9–11]. In 2008, Mirmostafaee and Moslehian [12, 13] obtained a fuzzy stability for the additive functional equation and for the quadratic functional equation.
On the other hand, there are some papers where several results of stability for different functional equations are proved in probabilistic metric and random normed spaces (see, e.g., [14–17]); after that the results were established in fuzzy normed spaces or in non-Archimedean fuzzy normed spaces [18–21]. In these papers except , the fixed point method is used. Moreover, in some of them another type of metric is used (see, e.g., ).
In this paper, we take into account the generalized Hyers-Ulam stability of the following quadratic-additive type functional equation: in the fuzzy normed spaces via the fixed point method. First of all, it is known that if a mapping satisfies the functional equation (1), then is quadratic-additive mapping in . Thus the functional equation (1) may be called the quadratic-additive type functional equation and the general solution of functional equation (1) may be called the quadratic-additive mapping. However, the stability problem for the functional equation (1.1) in  is not investigated and so in this paper we deal with the stability of this equation.
Theorem 1 (the fixed point alternative). Assume that is a complete generalized metric space and is a strict contraction with the Lipschitz constant . If there exists a nonnegative integer such that for some , then the following statements are true. (F1)The sequence converges to a fixed point of .(F2) is the unique fixed point of in .(F3)If , then
Definition 2. Let be a real linear space. A function is said to be a fuzzy norm on if the following conditions are true: (N1) for all and ;(N2) if and only if for all ;(N3) for all and with ;(N4) for all and ;(N5) is a nondecreasing function on and for all .
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 this paper, we note that the triangular norm is used (see the definition of the fuzzy norm, in the axiom ()), while in some recent papers properties of generalized Hyers-Ulam stability by taking other triangular norms have been discussed (e.g., of Hadžić type, in  ).
3. Generalized Hyers-Ulam Stability of (1)
Let and be a fuzzy normed space and a fuzzy Banach space, respectively. For a given mapping , we use the abbreviation for all .
In the following theorem, we investigate the stability problems of the functional equation (1) between fuzzy normed spaces.
Theorem 3. Let and be fuzzy normed spaces and let be a fuzzy Banach space. Assume that a mapping satisfies one of the following conditions: (i) for some ;(ii) for some ;(iii) for some for all and . If a mapping with satisfies for all and , then there exists a unique quadratic-additive mapping such that for all and , where Moreover, if is continuous in under the condition , then the mapping is a quadratic-additive mapping.
Proof. We will take into account three different cases for the assumption of .
Case 1. Assume that satisfies the condition (i). We consider the set of functions and introduce a generalized metric on by
We first prove that is a generalized metric on . If , that is, then we see that for all and all , which means that for all and all . It follows that So we get for all .
Conversely, if for all , then we have for all and . So we know that
Of course, it is easily checked that for all .
Let such that and . Then for all and all . Thus we find that This implies that . Hence we yield that . Therefore is a generalized metric on .
Now if we define a function by for all , then we have for all and all .
For any , let be an arbitrary constant with . The definition of provides that, for , for all , which implies that . Thus is a strictly contractive self-mapping of with the Lipschitz constant .
Moreover, by (4), we see that for all . The above inequality and the definition of show that .
According to Theorem 1, the sequence converges to a unique fixed point of in the set , which is represented by for all . We observe that This guarantees that inequality (5) holds.
Next, we are in the position to prove that is quadratic-additive mapping. Now, we figure out the relation for all and all . The first fifteen terms on the right-hand side of the above inequality tend to as by the definition of . Moreover, we find that which tends to as , since . Therefore inequality (23) gives that for all and . So we deduce that for all .
In order to show the uniqueness of , we assume that is another quadratic-additive mapping satisfying (5), and then we yield that for all . That is, is another fixed point of . Since is a unique fixed point of in the set , we conclude that .
Case 2. Assume that satisfies the condition (ii). The proof of this case can be carried out similarly as the proof of Case 1. In particular, assume that is continuous in . If , are any fixed nonzero integers, then we have for all and . Since is arbitrary, we have for all and . From these and the following equality: we get the inequality for all . Due to the previous inequality and the fact that , we obtain that .
Case 3. Assume that satisfies the condition . Let the set be as in the proof of Case 1. Now we take into account the function defined by for all and . Note that and for all . Let and let be an arbitrary constant with . From the definition of , we have for all , which means that . Hence is a strictly contractive self-mapping of with the Lipschitz constant .
Moreover, by (4), we see that for all . It implies that by the definition of . Therefore, according to Theorem 1, the sequence converges to a unique fixed point of in the set , which is represented by for all . Since inequality (5) holds.
Next, we will show that is quadratic-additive mapping. As in the previous case, we have inequality (23) for all and all . The first terms on the right-hand side of inequality (23) tend to as by the definition of . Now consider that which tends to as for all . Therefore it follows from (23) that for all and . That is, for all .
In particular, instead of the assumption of Theorem 3 that is a fuzzy normed space, it is enough to consider that is a linear space. Moreover, we can use Theorem 3 to get a classical result in the framework of normed spaces. Let be a normed linear space. Then we can define a fuzzy norm on by following where and , and see .
Theorem 4. Let and be a normed space and a complete normed space, respectively. If a mapping satisfies for all and a fixed , then there exists a unique quadratic-additive mapping such that for all .
Proof. Let be a fuzzy norm on . Then we get for all and all and for all and all . Therefore, it follows from (40) that If we define a mapping by then we see that and satisfy the conditions of Theorem 3 with for and for . So we feel that (41) holds for all .
Theorem 5. Let and be a normed space and a complete normed space, respectively. Assume that is a mapping defined by for all and a fixed . Suppose that is a mapping such that for all . Then the mapping is a quadratic-additive mapping.
Proof. If we define a mapping by then and are fulfilled in the conditions of Theorem 3 with . Based on the fact that is continuous in under the condition (ii), we arrive at the desired conclusion.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for giving useful suggestions and for the improvement of this paper. The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2A10004419).
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