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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 869274, 9 pages
http://dx.doi.org/10.1155/2013/869274
Research Article

Approximate Euler-Lagrange Quadratic Mappings in Fuzzy Banach Spaces

Department of Mathematics, Chungnam National University, Daejeon 305-764, Republic of Korea

Received 18 June 2013; Accepted 9 August 2013

Academic Editor: Bing Xu

Copyright © 2013 Hark-Mahn Kim and Juri 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.

Abstract

We consider general solution and the generalized Hyers-Ulam stability of an Euler-Lagrange quadratic functional equation in fuzzy Banach spaces, where , are nonzero rational numbers with , .

1. Introduction

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 additive mappings on Banach spaces. Hyers's theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gǎvruta [5] by replacing the unbounded Cauchy difference by a general control function.

The functional equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. Cholewa [6] noticed that the theorem of F. Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [7] proved the Hyers-Ulam stability of the quadratic functional equation. In particular, Rassias investigated the Hyers-Ulam stability for the relative Euler-Lagrange functional equation in [810]. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [1114]).

The theory of fuzzy space has much progressed as the theory of randomness has developed. Some mathematicians have defined fuzzy norms on a vector space from various points of view [1519]. Following Cheng and Mordeson [20] and Bag and Samanta [15] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [21] and investigated some properties of fuzzy normed spaces [22].

We use the definition of fuzzy normed spaces given [15, 18, 23].

Definition 1 (see [15, 18, 23]). Let be a real vector space. A function is said to be a fuzzy norm on if, for all and all ,   for ;   if and only if for all ;   for ;  ;   is a nondecreasing function on 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 [18, 24].

Definition 2 (see [15, 18, 23]). Let be a fuzzy normed vector space. A sequence in is said to be convergent or converges to if there exists an such that for all . In this case, is called the limit of the sequence , and one denotes it by -.

Definition 3 (see [15, 18, 23]). 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 , one has .

It is well known that every convergent sequence in a fuzzy normed space is a Cauchy sequence. 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.

It is said that a mapping between fuzzy normed spaces and is continuous at if, for each sequence converging to , the sequence converges to . If is continuous at each , then is said to be continuous on (see [22]).

We recall the fixed point theorem from [25], which is needed in Section 4.

Theorem 4 (see [25, 26]). Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either for all nonnegative integers or there exists a positive integer such that(1), for all ;(2)the sequence converges to a fixed point of ;(3) is the unique fixed point of in the set ;(4), for all .

In 1996, Isac and Rassias [27] were the first to provide new application of fixed point theorems to the proof of stability theory of functional equations. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [2830] and references therein).

Recently, Kim et al. [31] investigated the solution and the stability of the Euler-Lagrange quadratic functional equation where , are non-zero rational numbers with .

Najati and Jung [32] have observed the Hyers-Ulam stability of the generalized quadratic functional equation where , are non-zero rational numbers with .

In this paper, we generalize the above quadratic functional equation (5) to investigate the generalized Hyers-Ulam stability of an Euler-Lagrange quadratic functional equation in fuzzy Banach spaces, where are non-zero rational numbers with , . In particular, if in the functional equation (6), then is trivial and so (6) reduces to (5).

2. General Solution of (6)

Lemma 5 (see [31]). A mapping between linear spaces satisfies the functional equation where , are non-zero rational numbers with if and only if is quadratic.

Lemma 6. Let and be vector spaces and an odd function satisfying (6). Then .

Proof. Putting (resp., ) in (6), we get for all . Replacing by in (6) and adding the obtained functional equation to (6), we get for all . Replacing by in (9) and using (8), we get for all . Again if we replace by in (10) and use (8), we get for all . Exchanging for in (6) and using the oddness of , we have for all . Replacing by in (12) and adding the obtained functional equation to (12), we get for all . So it follows from (11) and (13) that for all . It easily follows from (14) that is additive; that is, for all . Since is a rational number, for all . Therefore, it follows from (8) that for all . Since , are nonzero, we infer that if .
If , then , and thus we see easily that by the similar argument above.

Lemma 7. Let and be vector spaces and an even function satisfying (6). Then is quadratic.

Proof. Putting in (6), we get since . Replacing by in (6), we obtain for all . Replacing by in (15) and using the evenness of , we get for all . Adding (15) and (16), we get for all . Thus (17) can be rewritten by where , for all . Therefore, it follows from Lemma 5 that is quadratic.

Theorem 8. Let be a function between vector spaces and . Then satisfies (6) if and only if is quadratic.

Proof. Let and be the odd and the even parts of . Suppose that satisfies (6). It is clear that and satisfy (6). By Lemmas 6 and 7, and is quadratic. Since , we conclude that is quadratic.
Conversely, if a mapping is quadratic, then it is easy to see that satisfies (6).

3. Stability of (6) by Direct Method

Throughout this paper, we assume that is a linear space, is a fuzzy Banach space, and is a fuzzy normed space.

For notational convenience, given a mapping , we define a difference operator of (6) by for all .

Theorem 9. Assume that a mapping with satisfies the inequality and is a mapping for which there is a constant satisfying such that for all and all . Then one can find a unique Euler-Lagrange quadratic mapping satisfying the equation and the inequality for all .

Proof. We observe from (21) that for all . Putting in (20), we obtain for all . Therefore it follows from (23), (24) that for all and any integer . So which yields for all and any integers , . Hence one obtains for all and any integers , , . Since is convergent series, we see by taking the limit in the last inequality that a sequence is Cauchy in the fuzzy Banach space and so it converges in . Therefore a mapping defined by is well defined for all . It means that , , for all . In addition, we see from (26) that and so, for any , for sufficiently large and for all and all . Since is arbitrary and is left continuous, we obtain for all , which yields the approximation (22).
In addition, it is clear from (20) and that the following relation holds for all and all . Therefore, we obtain by use of that which implies by . Thus we find that is an Euler-Lagrange quadratic mapping satisfying (6) and (22) near the approximate quadratic mapping .
To prove the aforementioned uniqueness, we assume now that there is another quadratic mapping which satisfies (22). Then one establishes by using the equality and (22) that which tends to 1 as by . Therefore one obtains for all , completing the proof of uniqueness.

We remark that, if in Theorem 9, then as , and so for all . Hence for all and is itself a quadratic mapping.

Theorem 10. Assume that a mapping with satisfies the inequality and is a mapping for which there is a constant satisfying such that for all and all . Then one can find a unique Euler-Lagrange quadratic mapping satisfying the equation and the inequality for all .

Proof. It follows from (24) and (38) that for all . Therefore it follows that for all and any integer . Thus we see from the last inequality that
The remaining assertion goes through by the similar way to the corresponding part of Theorem 9.

We also observe that, if in Theorem 10, then as , and so for all . Hence and is itself a quadratic mapping.

Corollary 11. Let be a normed space and () a fuzzy normed space. Assume that there exist real numbers and is real number such that either or . If a mapping with satisfies the inequality for all and all . Then one can find a unique Euler-Lagrange quadratic mapping satisfying the equation and the inequality for all and all .

Proof. Taking and applying Theorems 9 and 10, we obtain the desired approximation, respectively.

Corollary 12. Assume that, for , there exists a real number such that a mapping with satisfies the inequality for all and all . Then one can find a unique Euler-Lagrange quadratic mapping satisfying the equation and the inequality for all and all .

We remark that, if , then , and so . Thus we get that is itself a quadratic mapping.

4. Stability of (6) by Fixed Point Method

Now, in the next theorem, we are going to consider a stability problem concerning the stability of (6) by using a fixed point theorem of the alternative for contraction mappings on generalized complete metric spaces due to Margolis and Diaz [25].

Theorem 13. Assume that there exists constant with and satisfying such that a mapping with satisfies the inequality for all ,   , and is a mapping satisfying for all and all . Then there exists a unique Euler-Lagrange quadratic mapping satisfying the equation and the inequality for all and all .

Proof. We consider the set of functions and define a generalized metric on as follows: Then one can easily see that is a complete generalized metric space [33, 34].
Now, we define an operator as for all , .
We first prove that is strictly contractive on . For any , let be any constant with . Then we deduce from the use of (48) and the definition of that Since is arbitrary constant with , we see that, for any , which implies is strictly contractive with constant on .
We now want to show that . If we put ,    in (47), then we arrive at which yields and so for all .
Using the fixed point theorem of the alternative for contractions on generalized complete metric spaces due to Margolis and Diaz [25], we see the following (i), (ii), and (iii).
(i) There is a mapping with such that and is a fixed point of the operator ; that is, for all . Thus we can get for all and all .
(ii) Consider as . Thus we obtain for all and all , that is; the mapping given by is welldefined for all . In addition, it follows from conditions (47), (48), and that for all . Therefore we obtain by use of , (59), and (60) which implies by , and so the mapping is quadratic satisfying (6).
(iii) The mapping is a unique fixed point of the operator in the set . Thus if we assume that there exists another Euler-Lagrange type quadratic mapping satisfying (49), then and so is a fixed point of the operator and . By the uniqueness of the fixed point of in , we find that , which proves the uniqueness of satisfying (49). This ends the proof of the theorem.

Theorem 14. Assume that there exists constant with and satisfying such that a mapping with satisfies the inequality for all ,   , and is a mapping satisfying for all . Then there exists a unique Euler-Lagrange quadratic mapping satisfying the equation and the inequality for all .

Proof. The proof of this theorem is similar to that of Theorem 13.

Remark 15. In a real space with a fuzzy norm , the stability result obtained by the direct method is somewhat different from the stability result obtained by the fixed point method as follows. Let be a normed space and a Banach space. Let a mapping with satisfy the inequality for all and if . Assume that there exist real numbers and such that either ,   , , resp.) or ,   , , resp.). Then there exists a unique quadratic function which satisfies the inequality: for all and if , which is verified by using the direct method together with the following inequality for all .
On the other hand, assume that there exist real numbers and such that either , (, , resp.) or , (, , resp.). Then there exists a unique quadratic function which satisfies the inequality for all and if , which is established by using the fixed point method together with Therefore, we observe that the corresponding subsequential four stability results by the direct method are sharper than the corresponding subsequential four stability results obtained by the fixed point method.

Acknowledgment

This work was supported by research fund of Chungnam National University.

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