Stabilities of Cubic Mappings in Fuzzy Normed Spaces
© A. Ghaffari and A. Alinejad. 2010
Received: 15 January 2010
Accepted: 11 May 2010
Published: 9 June 2010
Rassias(2001) introduced the pioneering cubic functional equation in the history of mathematical analysis: and solved the pertinent famous Ulam stability problem for this inspiring equation. This Rassias cubic functional equation was the historic transition from the following famous Euler-Lagrange-Rassias quadratic functional equation: to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation: in fuzzy normed linear spaces. We use the definition of fuzzy normed linear spaces to establish a fuzzy version of a generalized Hyers-Ulam-Rassias stability for above equation in the fuzzy normed linear space setting. The fuzzy sequentially continuity of the cubic mappings is discussed.
Studies on fuzzy normed linear spaces are relatively recent in the field of fuzzy functional analysis. The notion of fuzzyness has a wide application in many areas of science. In 1984, Katsaras  first introduced a definition of fuzzy norm on a linear space. Later, several notions of fuzzy norm have been introduced and discussed from different points of view [2, 3]. Concepts of sectional fuzzy continuous mappings and strong uniformly convex fuzzy normed linear spaces have been introduced by Bag and Samanta . Bag and Samanta  introduced a notion of boundedness of a linear operator between fuzzy normed spaces, and studied the relation between fuzzy continuity and fuzzy boundedness. They studied boundedness of linear operators over fuzzy normed linear spaces such as fuzzy continuity, sequential fuzzy continuity, weakly fuzzy continuity and strongly fuzzy continuity.
for all , where and . This result was later extended to all .
In 2008, J. M. Rassias  generalized even further the above two stabilities via a new stability involving a mixed product-sum of powers of norms, called JMRassias stability by several authors [27–30].
In the last two decades, several form of mixed type functional equation and its Ulam–Hyers stability are dealt in various spaces like Fuzzy normed spaces, Random normed spaces, Quasi–Banach spaces, Quasinormed linear spaces and Banach algebra by various authors like [31–40].
In 1994, Cheng and Mordeson  introduced an idea of a fuzzy norm on a linear space whose associated metric is Kramosil and Michálek type . Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [42–44].
to the cubic functional equation (*).
and investigated the generalized Hyers-Ulam-Rassias stability for this equation on abelian groups. They also obtained results in sense of Hyers-Ulam stability and Hyers-Ulam-Rassias stability. A number of results concerning the stability of different functional equations can be found in [23, 56–59].
In this paper, we prove the Hyers-Ulam-Rassias stability of the cubic functional equation (1.5) in fuzzy normed spaces. Later, we will show that there exists a close relationship between the fuzzy sequentially continuity behavior of a cubic function, control function and the unique cubic mapping which approximates the cubic map.
2. Notation and Preliminary Results
In this section some definitions and preliminary results are given which will be used in this paper. Following , we give the following notion of a fuzzy norm.
Let be a linear space. A fuzzy subset of into is called a fuzzy norm on if for every and
(N1) for ,
(N2) if and only if for all ,
(N3) if ,
(N5) is a non-decreasing function on and .
is a fuzzy norm on . Other examples of fuzzy normed linear spaces are considered in the main text of this paper.
Note that the fuzzy normed linear space is exactly a Menger probabilistic normed linear space where .
A sequence in a fuzzy normed space converges to (one denote ) if for every and , there exists a positive integer such that whenever .
Recall that, a sequence in is called Cauchy if for every and , there exists a positive integer such that for all and all , we have . It is known that every convergent sequence in a fuzzy normed space is Cauchy. The fuzzy normed space is said to be fuzzy Banach space if every Cauchy sequence in is convergent to a point in .
3. Main Results
We will investigate the generalized Hyers-Ulam type theorem of the functional equation (1.5) in fuzzy normed spaces. In the following theorem, we will show that under special circumstances on the control function , every -almost cubic mapping can be approximated by a cubic mapping .
holds for all and .
We have the following two cases.
Case 1 ( ).
Therefore for all , whence .
Case 2 ( ).
The proof for uniqueness of for this case proceeds similarly to that in the previous case, hence it is omitted.
We note that need not be equal to 27. But we do not guarantee whether the cubic equation is stable in the sense of Hyers, Ulam and Rassias if is assumed in Theorem 3.1.
Let . Suppose that the mapping from into is right continuous. Then we get a fuzzy approximation better than (3.17) as follows.
From Theorem 3.1, we obtain the following corollary concerning the stability of (1.5) in the sense of the JMRassias stability of functional equations controlled by the mixed product-sum of powers of norms introduced by J. M. Rassias  and called JMRassias stability by several authors [27–30].
for all . The function is given by for all
for all and . Consequently, .
Let be a mapping where and are fuzzy normed spaces. is said to be sequentially fuzzy continuous at if for any satisfying implies . If is sequentially fuzzy continuous at each point of , then is said to be sequentially fuzzy continuous on .
For the various definitions of continuity and also defining a topology on a fuzzy normed space we refer the interested reader to [61, 62]. Now we examine some conditions under which the cubic mapping found in Theorem 3.1 to be continuous. In the following theorem, we investigate fuzzy sequentially continuity of cubic mappings in fuzzy normed spaces. Indeed, we will show that under some extra conditions on Theorem 3.1, the cubic mapping is fuzzy sequentially continuous.
Denote the fuzzy norm obtained as Corollary 3.3 on . Suppose that conditions of Theorem 3.1 hold. If for every the mappings (from into and (from into are sequentially fuzzy continuous, then the mapping is sequentially continuous and for all .
We have the following case.
Case 1 ( ).
Therefore for every choice , and , we can find some such that for every . This shows that .
The proof for proceeds similarly to that in the previous case.
It is not hard to see that for every rational number . Since is a fuzzy sequentially continuous map, by the same reasoning as the proof of , the cubic function satisfies for every .
The following corollary is the Hyers-Ulam stability  of (1.5).
for all . Moreover, if for each fixed the mapping from to is fuzzy sequentially continuous, then for all .
for all . It follows that . The rest of proof is an immediate consequence of Theorem 3.5.
The second author would like to thank the office of gifted students at the Semnan university for financial support.
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