Advances in Difference Equations

, 2010:150873 | Cite as

Stabilities of Cubic Mappings in Fuzzy Normed Spaces

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Research Article
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Abstract

Rassias(2001) introduced the pioneering cubic functional equation in the history of mathematical analysis: Open image in new window 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: Open image in new window to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation: Open image in new window 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.

Keywords

Banach Space Functional Equation Linear Space Cauchy Sequence Ulam Stability 

1. Introduction

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 [1] 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 [4]. Bag and Samanta [5] 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.

The problem of stability of functional equation originated from a question of Ulam [6] concerning the stability of group homomorphism in 1940. Hyers gave a partial affirmative answer to the question of Ulam for Banach spaces in the next year [7]. Let Open image in new window and Open image in new window be Banach spaces. Assume that Open image in new window satisfies Open image in new window for all Open image in new window and some Open image in new window . Then, there exists a unique additive mapping Open image in new window such that Open image in new window for all Open image in new window . Open image in new window theorem was generalized by Aoki [8] for additive mappings. In 1978, a generalized solution for approximately linear mappings was given by Th. M. Rassias [9]. He considered a mapping Open image in new window satisfying the condition

for all Open image in new window , where Open image in new window and Open image in new window . This result was later extended to all Open image in new window .

In 1982, J. M. Rassias [10] gave a further generalization of the result of Hyers and prove the following theorem using weaker conditions controlled by a product of powers of norms. Let Open image in new window be a mapping from a normed vector space Open image in new window into a Banach space Open image in new window subject to the inequality
for all Open image in new window , where Open image in new window and Open image in new window . Then there exists a unique additive mapping Open image in new window which satisfies

for all Open image in new window . The above mentioned stability involving a product of powers of norms is called Ulam–Gavruta–Rassias stability by various authors [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

In 2008, J. M. Rassias [26] 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, 28, 29, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40].

In 1994, Cheng and Mordeson [2] introduced an idea of a fuzzy norm on a linear space whose associated metric is Kramosil and Michálek type [41]. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [42, 43, 44].

In 2001, J. M. Rassias [45] introduced the pioneering cubic functional equation in history of mathematical analysis, as follows:
and solved the famous Ulam stability problem for this inspiring functional equation. Note that this cubic functional equation (*) was the historic transition from the following famous Euler-Lagrange quadratic functional equation:

to the cubic functional equation (*).

The notion of fuzzy stability of the functional equations was initiated by Mirmostafaee and Moslehian in [46]. Later, several various fuzzy versions of stability were investigated [47, 48]. Now, let us introduce the following functional equation:
Since the cubic function Open image in new window satisfies in this equation, so we promise that (1.5) is called a cubic functional equation and every solution will be called a cubic function. The stability problem for the cubic functional equation was proved by Wiwatwanich and Nakmahachalasint [49] for mapping Open image in new window , where Open image in new window and Open image in new window are real Banach spaces. A number of mathematicians worked on the stability of some types of the cubic equation [45, 50, 51, 52, 53, 54]. In [55], Park and Jung introduced a cubic functional equation different from (1.5) as follows:

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, 57, 58, 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 [48], we give the following notion of a fuzzy norm.

Definition 2.1.

Let Open image in new window be a linear space. A fuzzy subset Open image in new window of Open image in new window into Open image in new window is called a fuzzy norm on Open image in new window if for every Open image in new window and Open image in new window

(N1) Open image in new window for Open image in new window ,

(N2) Open image in new window if and only if Open image in new window for all Open image in new window ,

(N3) Open image in new window if Open image in new window ,

(N4) Open image in new window ,

(N5) Open image in new window is a non-decreasing function on Open image in new window and Open image in new window .

The pair Open image in new window will be referred to as a fuzzy normed linear space. One may regard Open image in new window as the truth value of the statement "the norm of Open image in new window is less than or equal to the real number r ". Let Open image in new window be a normed linear space. One can be easily verify that

is a fuzzy norm on Open image in new window . Other examples of fuzzy normed linear spaces are considered in the main text of this paper.

Note that the fuzzy normed linear space Open image in new window is exactly a Menger probabilistic normed linear space Open image in new window where Open image in new window [60].

Definition 2.2.

A sequence Open image in new window in a fuzzy normed space Open image in new window converges to Open image in new window (one denote Open image in new window ) if for every Open image in new window and Open image in new window , there exists a positive integer Open image in new window such that Open image in new window whenever Open image in new window .

Recall that, a sequence Open image in new window in Open image in new window is called Cauchy if for every Open image in new window and Open image in new window , there exists a positive integer Open image in new window such that for all Open image in new window and all Open image in new window , we have Open image in new window . It is known that every convergent sequence in a fuzzy normed space is Cauchy. The fuzzy normed space Open image in new window is said to be fuzzy Banach space if every Cauchy sequence in Open image in new window is convergent to a point in Open image in new window [46].

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 Open image in new window , every Open image in new window -almost cubic mapping Open image in new window can be approximated by a cubic mapping Open image in new window .

Theorem 3.1.

Let Open image in new window . Let Open image in new window be a linear space, and let Open image in new window be a fuzzy normed space. Suppose that an even function Open image in new window satisfies Open image in new window for all Open image in new window and for all Open image in new window . Suppose that Open image in new window is a fuzzy Banach space. If a function Open image in new window satisfies
for all Open image in new window and Open image in new window , then there exists a unique cubic function Open image in new window which satisfies (1.5) and the inequality

holds for all Open image in new window and Open image in new window .

Proof.

We have the following two cases.

Case 1 ( Open image in new window ).

Replacing Open image in new window by Open image in new window in (3.1) and summing the resulting inequality with (3.1), we get
Since (3.1) and (3.3) hold for any Open image in new window , let us fix Open image in new window for convenience. By ( Open image in new window ), we have
It follows from
and last inequality that
In order to prove convergence of the sequence Open image in new window , we replace Open image in new window by Open image in new window to find that for Open image in new window ,
Replacing Open image in new window by Open image in new window in last inequality to get
Replacing Open image in new window by Open image in new window in last inequality, we observe that
This shows that Open image in new window is a Cauchy sequence in the fuzzy Banach space Open image in new window , therefore this sequence converges to some point Open image in new window . Fix Open image in new window and put Open image in new window in (3.13) to obtain
The first two terms on the right hand side of the above inequality tend to 1 as Open image in new window . Therefore we have
for Open image in new window large enough. By last inequality, we have
Now, we show that Open image in new window is cubic. Use inequality (3.1) with Open image in new window replaced by Open image in new window and Open image in new window by Open image in new window to find that
We conclude that Open image in new window fulfills (1.5). It remains to prove the uniqueness assertion. Let Open image in new window be another cubic mapping satisfying (3.17). Fix Open image in new window . Obviously

Therefore Open image in new window for all Open image in new window , whence Open image in new window .

Case 2 ( Open image in new window ).

We can state the proof in the same pattern as we did in the first case. Replace Open image in new window , Open image in new window by Open image in new window and Open image in new window , respectively in (3.4) to get
We replace Open image in new window and Open image in new window by Open image in new window and Open image in new window in last inequality, respectively, we find that
For each Open image in new window , one can deduce
where Open image in new window . It is easy to see that Open image in new window is a Cauchy sequence in ( Open image in new window ). Since Open image in new window is a fuzzy Banach space, this sequence converges to some point Open image in new window , that is,
Moreover, Open image in new window satisfies (1.5) and

The proof for uniqueness of Open image in new window for this case proceeds similarly to that in the previous case, hence it is omitted.

We note that Open image in new window 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 Open image in new window is assumed in Theorem 3.1.

Remark 3.2.

Let Open image in new window . Suppose that the mapping Open image in new window from Open image in new window into Open image in new window is right continuous. Then we get a fuzzy approximation better than (3.17) as follows.

for large enough Open image in new window . It follows that
Tending Open image in new window to zero we infer

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 [26] and called JMRassias stability by several authors [27, 28, 29, 30].

Corollary 3.3.

Let Open image in new window be a Banach space and let Open image in new window be a real number. Suppose that a function Open image in new window satisfies
for all Open image in new window where Open image in new window . Then there exists a unique cubic function Open image in new window which satisfying (1.5) and the inequality

for all Open image in new window . The function Open image in new window is given by Open image in new window for all Open image in new window

Proof.

It is easy to see that Open image in new window is a fuzzy Banach space. Denote by Open image in new window the map sending each Open image in new window to Open image in new window . By assumption,
is a fuzzy norm on Open image in new window . By Theorem 3.1, there exists a unique cubic function Open image in new window satisfies (1.5) and inequality

for all Open image in new window and Open image in new window . Consequently, Open image in new window .

Definition 3.4.

Let Open image in new window be a mapping where Open image in new window and Open image in new window are fuzzy normed spaces. Open image in new window is said to be sequentially fuzzy continuous at Open image in new window if for any Open image in new window satisfying Open image in new window implies Open image in new window . If Open image in new window is sequentially fuzzy continuous at each point of Open image in new window , then Open image in new window is said to be sequentially fuzzy continuous on Open image in new window .

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 Open image in new window is fuzzy sequentially continuous.

Theorem 3.5.

Denote Open image in new window the fuzzy norm obtained as Corollary 3.3 on Open image in new window . Suppose that conditions of Theorem 3.1 hold. If for every Open image in new window the mappings Open image in new window (from Open image in new window into Open image in new window and Open image in new window (from Open image in new window into Open image in new window are sequentially fuzzy continuous, then the mapping Open image in new window is sequentially continuous and Open image in new window for all Open image in new window .

Proof.

We have the following case.

Case 1 ( Open image in new window ).

It follows form (3.17) and (3.38) that
By the sequentially fuzzy continuity of maps Open image in new window and Open image in new window , we can find some Open image in new window such that for any Open image in new window ,
Hence by last inequality and (3.38), we get
On the other hand,
Hence by last inequality and (3.42), we obtain
Therefore it follows from (3.44), (3.40) and (3.39) that for every Open image in new window ,

Therefore for every choice Open image in new window , Open image in new window and Open image in new window , we can find some Open image in new window such that Open image in new window for every Open image in new window . This shows that Open image in new window .

The proof for Open image in new window proceeds similarly to that in the previous case.

It is not hard to see that Open image in new window for every rational number Open image in new window . Since Open image in new window is a fuzzy sequentially continuous map, by the same reasoning as the proof of [46], the cubic function Open image in new window satisfies Open image in new window for every Open image in new window .

The following corollary is the Hyers-Ulam stability [7] of (1.5).

Corollary 3.6.

Let Open image in new window be a Banach space, and let Open image in new window be a real number. Suppose that a function Open image in new window satisfies
for all Open image in new window . Then there exists a unique cubic function Open image in new window which satisfies (1.5) and the inequality

for all Open image in new window . Moreover, if for each fixed Open image in new window the mapping Open image in new window from Open image in new window to Open image in new window is fuzzy sequentially continuous, then Open image in new window for all Open image in new window .

Proof.

Denote Open image in new window and Open image in new window the fuzzy norms obtained as Corollary 3.3 on Open image in new window and Open image in new window , respectively. This time we choose Open image in new window . By Theorem 3.1, there exists a unique cubic function Open image in new window which satisfies the inequality

for all Open image in new window . It follows that Open image in new window . The rest of proof is an immediate consequence of Theorem 3.5.

Notes

Acknowledgments

The second author would like to thank the office of gifted students at the Semnan university for financial support.

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© A. Ghaffari and A. Alinejad. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsSemnan UniversitySemnanIran

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