Journal of Function Spaces

Volume 2014 (2014), Article ID 349281, 9 pages

http://dx.doi.org/10.1155/2014/349281

Research Article

On the Stability of Cubic and Quadratic Mapping in Random Normed Spaces under Arbitrary -Norms

J. Vahidi,¹ S. J. Lee,² F. Fallah,³ and R. Ahmadi⁴

¹Department of Mathematics, Iran University of Science and Technology, Tehran, Iran

²Department of Mathematics, Daejin University, Kyeonggi 487-711, Republic of Korea

³Department of Mathematics, Mazandaran University of Science and Technology, Behshahr, Iran

⁴Marand Technical College, University of Tabriz, Tabriz, Iran

Received 15 November 2013; Revised 7 March 2014; Accepted 20 March 2014; Published 30 April 2014

Academic Editor: Zbigniew Leśniak

Copyright © 2014 J. Vahidi 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 stability of some functional equations in the random normed spaces under arbitrary -norms.

1. Introduction and Preliminaries

The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper [4] of Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. We refer the interested readers for more information on such problems to the papers [5–9]. In addition, some authors investigated the stability in the settings of fuzzy, probabilistic, and random normed spaces, and for more details see [10–22].

The functional equation

is said to be the cubic functional equation since the function is its solution. Every solution of the cubic functional equation is said to be a cubic mapping. The stability problem for the cubic functional equation was solved by Skof [23] for mappings , where is a normed space and is a Banach space.

The functional equation

is said to be the quadratic functional equation since the function is its solution. Every solution of the quadratic functional equation is said to be a quadratic mapping (see [8, 9]).

Throughout this paper, the space of all probability distribution functions is denoted by and the subset is the set , where denotes the left limit of the function at the point . The space is partially ordered by the usual pointwise ordering of functions; that is, if and only if for all in . The maximal element for in this order is the distribution function given by

Definition 1 (see [24]). A mapping is a continuous triangular norm (briefly, a -norm) if satisfies the following conditions:

(i)is commutative and associative;(ii) is continuous;(iii) for all ;(iv) whenever and     ().

Recall that, if is a -norm and is a given sequence of numbers in , is defined recursively by and for .

Definition 2 (see [25]). A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into such that, the following conditions hold:

 (RN1) for all if and only if ; (RN2) for all , ; (RN3) for all and .

Definition 3. Let be an RN-space. Consider the following.

(1)A sequence in is said to be convergent to in if, for every and , there exists a positive integer such that whenever .(2)A sequence in is called Cauchy if, for every and , there exists a positive integer such that whenever .(3)An RN-space is said to be complete if every Cauchy sequence in is convergent to a point in .

Theorem 4 (see [24]). If is an RN-space and is a sequence such that , then almost everywhere.

2. On the Stability of Cubic Mappings in RN-Spaces

Theorem 5. Let be a normed space, let be an RN-space, and let be a mapping such that for some ,

and let be a complete RN-space. If is an odd mapping such that and

then there exists a unique cubic mapping such that

Proof. Letting in (6) we get

thus

since is an odd mapping and then we have

Replacing by in (6)

we get

from Definition 2 we have

Therefore by (10) and (12), we get

Replacing by and letting in (14) we get

Then

Replacing by in (16) we obtain

then

Whence

and using (4) we have

therefore

and from (21) we have

also, from above inequality we have

Then

By replacing by in (25) we observe that

then

Using (4) and Definition 2 we have

when , tend to . Then is a Cauchy sequence in . Since is a complete RN-space this sequence converges to some point . Fix and put in (28). Then

and from (29) for every , ,  , we have

We know when tend , then , and . Therefore taking the and using (30), and also by using Definition 1(iii) and Definition 2 and we get

and since was arbitrary, by taking in (31), we have

which implies that the Inequality (7) holds. Replacing and by and in (6), respectively, we get

that by division by the we have

then

for all , and for all . Since , we conclude that fulfills (1). To prove the uniqueness of the cubic function , let us assume that there exists a quadratic function which satisfies (7). Obviously we have and for all and . It follows from (7) and (32) that for each ,

then , for all . And so . This completes the proof.

3. On the Stability of Quadratic Mappings in RN-Spaces

Theorem 6. Let be a normed space, let be an RN-space, and let be a mapping such that for some ,

and let be a complete RN-space. If is an even mapping such that and

then there exists a unique quadratic mapping such that

Proof. Letting in (38) we get

Since is an even mapping and then we have

Replacing by in (38) we get

and by (41) and (42), we get

Therefore

Replacing by and letting in (44), we get

Then

that is,

Replacing by in (46) we obtain

then

By using (2) and (37) we have

so

and from (51) we have

and also, from the above inequality we have,

Hence

By replacing by in (54) we obtain

then

Using (37) and Definition 2 we observe that

when , tend to . Then is a Cauchy sequence in . Since is a complete RN-space this sequence converges to some point . Fix and put in (57). Thus

for every , , , and from (58) we have

we know when tend , then , and . Therefore taking the and using (59), also by using Definitions 1 and 2 we get

since was arbitrary, and by taking in (60), we get

which implies that the Inequality (39) holds. Replacing and by and in (38), respectively, we get 

that by division by the we have

for all and for all , then fulfills (1). To prove the uniqueness of the quadratic function , let us assume that there exists a quadratic function which satisfies (39). Obviously we have and for all . From (39) and (61) we have

Since , we get , and , then , for all . And so . This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referee for giving useful comments and suggestions for the improvement of this paper.

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