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Stability results in non-Archimedean -fuzzy normed spaces for a cubic functional equation
Journal of Inequalities and Applications volume 2012, Article number: 193 (2012)
Abstract
We establish some stability results concerning the functional equation
where is a fixed integer in the setting of non-Archimedean -fuzzy normed spaces.
MSC:39B52, 46S10, 46S40, 47S10, 47S40.
1 Introduction
The theory of fuzzy sets was introduced by Zadeh [34] 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 has been made in the field of fuzzy topology [1, 8, 9, 12–14, 18, 22, 30]. One of the problems in -fuzzy topology is to obtain an appropriate concept of -fuzzy metric spaces and -fuzzy normed spaces. In 2004, Park [23] introduced and studied the notion of intuitionistic fuzzy metric spaces. In 2006, Saadati and Park [28] 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 the question of Ulam [33] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [19]. Subsequently, the result of Hyers was generalized by Aoki [2] for additive mappings and by Rassias [24] for linear mappings by considering an unbounded Cauchy difference. We refer the interested readers for more information on such problems to the papers [4, 6, 11, 20, 21, 25, 29, 32, 33].
Let X and Y be real linear spaces and 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 [5] 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.
2 Preliminaries
In this section, 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 Hadžić type (we denote it by ) if the family is equicontinuous at (see [15]).
Other important triangular norms are as follows (see [16]):
-
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 r-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 side of (2.1) exists since the sequence is non-increasing and bounded below.
Proposition 2.1 [16]
-
(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
3 -fuzzy normed spaces
In this section, we give some definitions and related lemmas for our main result.
Definition 3.1 [10]
Let be a complete lattice and U a non-empty set called a universe. An -fuzzy set on U is defined by a mapping . For any , represents the degree (in L) to which u satisfies .
Lemma 3.2 [7]
Consider the set and the operation defined by
for all . Then is a complete lattice.
Definition 3.3 [3]
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 furthermore, they satisfy .
In Section 2, we presented the classical definition of t-norms, which can be straight-forwardly extended to any lattice . Define first and .
Definition 3.4 A triangular norm (t-norm) on is a mapping satisfying the following conditions:
-
(i)
for all (boundary condition);
-
(ii)
for all (commutativity);
-
(iii)
for all (associativity);
-
(iv)
and for all (monotonicity).
A t-norm can also be defined recursively as an -array operation for each by and
for all and .
The t-norm defined by
is a continuous t-norm.
Definition 3.5 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 3.6
-
(1)
A negator on is any decreasing mapping satisfying and .
-
(2)
If a negator on satisfies for all , then is called an involution negator.
-
(3)
The negator on defined as for all is called the standard negator on .
Definition 3.7 The 3-tuple is said to be an -fuzzy normed space if V is a vector space, is a continuous t-norm on and is an -fuzzy set on satisfying the following conditions: for all and ,
-
(a)
;
-
(b)
;
-
(c)
for all ;
-
(d)
;
-
(e)
is continuous;
-
(f)
and .
In this case, is called an -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 3.8 (see [27])
Let be an -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 .
-
(2)
A sequence in is said to be convergent to a point in the -fuzzy normed space (denoted by\hspace*{-6pt} \hspace*{-6pt}) if wherever for all .
-
(3)
If every Cauchy sequence in is convergent in V, then the -fuzzy normed space is said to be complete and the -fuzzy normed space is called an -fuzzy Banach space.
Lemma 3.9 [26]
Let be an -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 3.10 Let be an -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 -fuzzy norm .
4 -fuzzy Hyers-Ulam stability for cubic functional equations in non-Archimedean -fuzzy normed spaces
In 1897, Hensel [17] introduced the field with valuation in which does not have the Archimedean property.
Definition 4.1 Let be a field. A non-Archimedean absolute value on is a function such that, for any ,
-
(1)
and equality holds if and only if ,
-
(2)
,
-
(3)
(the strict triangle inequality).
Note that for each integer n. We always assume, in addition, that is non-trivial, i.e., there is such that .
Definition 4.2 A non-Archimedean -fuzzy normed space is a triple , where V is a vector space, is a continuous t-norm on and is an -fuzzy set on satisfying the following conditions: for all and ,
-
(a)
;
-
(b)
;
-
(c)
for all ;
-
(d)
;
-
(e)
is continuous;
-
(f)
and .
From now on, let be a non-Archimedean field, X a vector space over and a non-Archimedean -fuzzy Banach space over . We investigate the Hyers-Ulam stability of the cubic functional equation (1.1).
Next, we define an -fuzzy approximately cubic mapping. Let Ψ be an -fuzzy set on such that is non-decreasing,
and
for all , all and all .
Definition 4.3 A mapping is said to be Ψ-approximately cubic if
for all and all .
The following is the main result in this paper.
Theorem 4.4 Let be a Ψ-approximately cubic mapping. If and there are and an integer k () with such that
and
for all and all , then there exists a unique cubic mapping such that
for all and all , where
for all and all .
Proof Let be the identity mapping . First, we show, by induction on j, that
for all , all and all . Putting in (4.1), we obtain
for all and all . This proves (4.4) for . Let (4.4) hold for some . Replacing y by 0 and x by in (4.1), we get
for all and all . Since , it follows that
for all and all . Thus (4.4) holds for all . In particular, we have
for all and all . Replacing x by in the above inequality and using the inequality (4.2), we obtain
for all , all and all , and so
for all , all and all . Hence it follows that
for all , all and all . Since for all and all , is a Cauchy sequence in the non-Archimedean -fuzzy Banach space . Hence we can define a mapping such that
for all and all . Next, for all , all and all , we have
for all and all , and so
for all and all . Taking the limit as in the above inequality, we obtain
for all and all , which proves (4.3).
Since is continuous, from the well-known result in an -fuzzy (probabilistic) normed space (see [31], Chapter 12), it follows that
for all and all . On the other hand, replacing x, y by , in (4.1) and (4.2), we get
for all and all . Since , we infer that C is a cubic mapping.
For the uniqueness of C, let be another cubic mapping such that
for all and all . Then we have, for all and all ,
Therefore, from (4.5), we conclude that . This completes the proof. □
Corollary 4.5 Let and let be a Ψ-approximately cubic mapping. If and there are and an integer k () with such that
and
for all and all , then there exists a unique cubic mapping such that
for all and all , where
for all and all .
Proof Since
for all and all and is of type, it follows from Proposition 2.1 that
for all and all . Therefore, if we apply Theorem 4.4, then we get the conclusion. This completes the proof. □
Example 4.6 Let be a non-Archimedean Banach space, a non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) in which
for all and all and a complete non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space). Define
for all and all . It is easy to see that (4.2) holds for . Also, since
for all and all , we have
for all and all . Let be a Ψ-approximately cubic mapping. A straightforward computation shows that
for all and all . Therefore, all the conditions of Theorem 4.4 hold, and so there exists a unique cubic mapping such that
for all and all .
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Acknowledgements
This research was supported by 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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Bae, JH., Lee, SB. & Park, WG. Stability results in non-Archimedean -fuzzy normed spaces for a cubic functional equation. J Inequal Appl 2012, 193 (2012). https://doi.org/10.1186/1029-242X-2012-193
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DOI: https://doi.org/10.1186/1029-242X-2012-193