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Stability results in ℒ-fuzzy normed spaces for a cubic functional equation
Journal of Inequalities and Applications volume 2012, Article number: 249 (2012)
Abstract
We establish some stability results concerning the functional equation
where is a fixed integer, in the setting of ℒ-fuzzy normed spaces that in turn generalize a Hyers-Ulam stability result in the framework of classical normed spaces.
MSC:39B22, 39B52, 39B72, 46S40, 47S40.
1 Introduction and preliminaries
The theory of fuzzy sets was introduced by Zadeh [1] in 1965. After the pioneering work of Zadeh, there has been a great effort to obtain fuzzy analogues of classical theories. Among other fields, a progressive development is made in the field of fuzzy topology [2–9]. One of problems in ℒ-fuzzy topology is to obtain an appropriate concept of ℒ-fuzzy metric spaces and ℒ-fuzzy normed spaces. In 1984, Katsaras [10] defined a fuzzy norm on a linear space, and at the same year Wu and Fang [11] also introduced a fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for a fuzzy topological linear space. Some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [2, 3, 9, 12–14]. In 1994, Cheng and Mordeson introduced the definition of a fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [15]. In 2003, Bag and Samanta [16] modified the definition of Cheng and Mordeson [17] by removing a regular condition. In 2004, Park [18] introduced and studied the notion of intuitionistic fuzzy metric spaces. In 2006, Saadati and Park introduced and studied the notion of intuitionistic fuzzy normed spaces.
On the other hand, the study of stability problems for a functional equation is related to a question of Ulam [19], concerning the stability of group homomorphisms, affirmatively answered for Banach spaces by Hyers [20]. Subsequently, the result of Hyers was generalized by Aoki [21] for additive mappings and by Rassias [22] for linear mappings by considering an unbounded Cauchy difference. We refer the interested readers for more information on such problems to the papers [19, 23–30].
Let X and Y be real linear spaces and let be a mapping. If , the cubic function , where c is a real constant, clearly satisfies the functional equation
Hence, the above equation is called the cubic functional equation. Recently, Cho, Saadati and Wang [31] introduced the functional equation
which has () as a solution for .
In this paper, we investigate the Hyers-Ulam stability of the functional equation as follows:
where is a fixed integer.
We recall some definitions and results for our main result in this paper.
A triangular norm (shorter t-norm) is a binary operation on the unit interval , i.e., a function satisfying the following four axioms: for all ,
-
(i)
(commutativity);
-
(ii)
(associativity);
-
(iii)
(boundary condition);
-
(iv)
whenever (monotonicity).
Basic examples are the Łukasiewicz t-norm and the t-norms , and , where , , and
for all .
For all and all t-norms T, let . For all and all t-norms T, define by the recursion equation for all . A t-norm T is said to be of Hadžić type (we denote it by ) if the family is equicontinuous at (see [12]).
Other important triangular norms are as follows (see [32]):
-
The Sugeno-Weber family is defined by , and
if .
-
The Domby family is defined by , if , , if and
if .
-
The Aczel-Alsina family is defined by , if , , if and
if .
A t-norm T can be extended (by associativity) in a unique way to an n-array operation by taking, for any , the value defined by
A t-norm T can also be extended to a countable operation by taking, for any sequence in , the value
The limit on the right-hand side of (1.2) exists since the sequence is non-increasing and bounded from below.
Proposition 1.1 [32]
-
(1)
For , the following equivalence holds:
-
(2)
If T is of Hadžić type, then
for all sequence in such that .
-
(3)
If , then
-
(4)
If , then
We give some definitions and related lemmas for our main result.
Definition 1.2 [17]
Let be a complete lattice and U a non-empty set called universe. An ℒ-fuzzy set on U is defined by a mapping . For any , represents the degree (in L) to which u satisfies .
Lemma 1.3 [16]
Consider the set and the operation defined by
for all . Then is a complete lattice.
Definition 1.4 [33]
An intuitionistic fuzzy set on a universe U is an object , where and for all are called the membership degree and the non-membership degree, respectively, of u in and, further, they satisfy .
We presented the classical definition of t-norms which can be straightforwardly extended to any lattice . Define first and .
Definition 1.5 A triangular norm (t-norm) on ℒ is a mapping satisfying the following conditions:
-
(i)
() () (boundary condition);
-
(ii)
() () (commutativity);
-
(iii)
() () (associativity);
-
(iv)
() () (monotonicity).
A t-norm can also be defined recursively an -array operation for each by and
for all and .
The t-norm ℳ defined by
is a continuous t-norm.
Definition 1.6 A t-norm on is said to be t-representable if there exist a t-norm T and a t-conorm S on such that
for all , .
Definition 1.7
-
(1)
A negator on ℒ is any decreasing mapping satisfying and .
-
(2)
If a negator on ℒ satisfies for all , then is called an involution negator.
-
(3)
The negator on defined as for all is called the standard negator on .
Definition 1.8 The 3-tuple is said to be an ℒ-fuzzy normed space if V is a vector space, is a continuous t-norm on ℒ and is an ℒ-fuzzy set on satisfying the following conditions: for all and ,
-
(i)
;
-
(ii)
;
-
(iii)
for all ;
-
(iv)
;
-
(v)
is continuous;
-
(vi)
and .
In this case, is called an ℒ-fuzzy norm. If is an intuitionistic fuzzy set and the t-norm is t-representable, then the 3-tuple is said to be an intuitionistic fuzzy normed space.
Definition 1.9 (see [15])
Let be an ℒ-fuzzy normed space.
-
(1)
A sequence in is called a Cauchy sequence if, for any and for any , there exists a positive integer such that
for all and , where is a negator on ℒ.
-
(2)
A sequence in is said to be convergent to a point in the ℒ-fuzzy normed space (denoted by ) if wherever for all .
-
(3)
If every Cauchy sequence in is convergent in V, then the ℒ-fuzzy normed space is said to be complete and the ℒ-fuzzy normed space is called an ℒ-fuzzy Banach space.
Lemma 1.10 Let be an ℒ-fuzzy norm on V. Then we have the following.
-
(1)
is non-decreasing with respect to for all x in V.
-
(2)
for all x, y in V and all .
Definition 1.11 Let be an ℒ-fuzzy normed space. For any , we define the open ball with center and radius as
A subset is called open if, for all , there exist and such that .
Let denote the family of all open subsets of V. Then is called the topology induced by the ℒ-fuzzy norm .
2 Main results
In this section, we study the stability of the functional equation (1.1) in ℒ-fuzzy normed spaces.
Theorem 2.1 Let X be a linear space and a complete ℒ-fuzzy normed space. Let be a mapping with and Q be an ℒ-fuzzy set on satisfying
for all and all . If
and
for all and , then there exists a unique cubic mapping such that
for all and .
Proof Putting in (2.1), we have
for all and all . Therefore, it follows that
which implies that
that is,
for all , and all . Since , we get for all . By the triangle inequality, it follows that
for all , and all .
In order to prove the convergence of the sequence , we replace x with in (2.3) to find that for all ,
for all , and all . Since the right-hand side of the inequality tend to as m tends to infinity, the sequence is a Cauchy sequence. Thus, we may define
for all .
Now we show that C is a cubic mapping. Replacing x, y with and , respectively, in (2.1), it follows that
for all , and all . Taking the limit as , we find that C satisfies (1.1) for all .
To prove (2.2), taking the limit as in (2.3), we have (2.2).
Finally, to prove the uniqueness of the cubic mapping C subject to (2.2), let us assume that there exists another cubic mapping which satisfies (2.2). Obviously, we have
for all and all . Hence, it follows from (2.2) that
for all and all . This proves the uniqueness of C and completes the proof. □
Corollary 2.2 Let be an ℒ-fuzzy normed space and be a complete ℒ-fuzzy normed space. If is a mapping such that
for all and all . If
and
for all and all , then there exists a unique cubic mapping such that
for all and all .
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Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number 2012003499).
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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
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Lee, SB., Park, WG. & Bae, JH. Stability results in ℒ-fuzzy normed spaces for a cubic functional equation. J Inequal Appl 2012, 249 (2012). https://doi.org/10.1186/1029-242X-2012-249
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DOI: https://doi.org/10.1186/1029-242X-2012-249