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 andthen 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 havefor 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.