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Generalized lacunary -statistically convergent sequences of fuzzy numbers using a modulus function
Journal of Inequalities and Applications volume 2013, Article number: 559 (2013)
Abstract
In this paper, we introduce the space of lacunary strongly -summable sequences of fuzzy numbers and discuss relations between -statistically convergent sequences and lacunary -statistically convergent sequences of fuzzy numbers. We also study inclusion relations using different arbitrary lacunary sequences.
MSC:40A05, 40C05, 46A45.
1 Introduction
The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [1], and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as topological spaces, similarity relations and fuzzy orderings, fuzzy mathematical programming etc. Later on, various types of sequence spaces of fuzzy numbers have been constructed by several authors such as Matloka [2], Nanda [3], Nuray and Savaş [4], Mursaleen and Basarır [5], Malkowsky et al. [6], Tripathy and Chandra [7], Tripathy and Borgogain [8] and so on. Later on, the fuzzy sequence space got momentum after the introduction of new convergence methods and theories in the process as well as the requirement. Some of them are statistical convergence, lacunary statistical convergence etc.
Nuray and Savaş [4] introduced the idea of statistical convergence of fuzzy numbers and Nuray [9] introduced the related concept of convergence with the help of a lacunary sequence. Using their ideas, many authors such as Kwon and Shim [10], Bilgin [11], Altin et al. [12], Esi [13], Tripathy and Baruah [14, 15], Tripathy and Dutta [16, 17] and others constructed different types of sequence spaces.
2 Definitions and preliminaries
Definition 2.1 A fuzzy number X is a mapping associating each real number t with its grade of membership .
Definition 2.2 If there exists such that , then the fuzzy number X is called normal.
Definition 2.3 A fuzzy number X is said to be convex if , where .
Definition 2.4 A fuzzy number X is said to be upper semi-continuous if for each , for all is open in the usual topology of ℝ.
denotes the set of all upper semi-continuous, normal, convex fuzzy numbers such that is compact, where denotes the closure of the set in the usual topology of ℝ.
Definition 2.5 The set forms a linear space under addition and scalar multiplication in terms of α-level sets as defined below:
where is given as
For each , the set is a closed, bounded and nonempty interval of ℝ.
Let D denote the set of all closed and bounded intervals on the real line ℝ. For , is a complete metric space where the metric d is defined as
for any and .
It is easy to verify that , defined by
is a metric on .
Definition 2.6 Let be a sequence of fuzzy numbers. Then the sequence is said to be -convergent to the fuzzy number , denoted as , if for every , there exists a positive integer such that for all .
Definition 2.7 The sequence of fuzzy numbers is said to be -statistically convergent to the fuzzy number ξ if for every ,
In this case, we write and denotes the set of all -statistically convergent sequences of fuzzy numbers.
By a lacunary sequence we mean an increasing sequence of integers such that and as . Throughout this paper, the intervals determined by θ will be denoted by and .
Definition 2.8 Let θ be a lacunary sequence. A lacunary refinement of is a lacunary sequence satisfying .
Definition 2.9 The sequence of fuzzy numbers is said to be lacunary -statistically convergent to the fuzzy number ξ if for every ,
In this case, we write , and denotes the set of all lacunary -statistically convergent sequences of fuzzy numbers.
Definition 2.10 A metric on is said to be translation invariant if for all .
Lemma 2.1 (Mursaleen and Basarır [5])
If is a translation invariant metric on , then
-
(i)
,
-
(ii)
, .
Lemma 2.2 (Maddox [18])
Let , for all k be sequences of complex numbers, and let be a bounded sequence of positive real numbers, then
and
where , and λ is any complex number.
Lemma 2.3 (Maddox [18])
Let , for all k be sequences of complex numbers and , then
where , .
3 Main results
Now, we introduce the space as follows:
where is a lacunary sequence, f is any modulus function and is any sequence of strictly positive real numbers.
Now we define a lacunary strongly -summable sequence of fuzzy numbers as follows.
The sequence is said to be lacunary strongly -summable to the fuzzy number if for every ,
In this case, we write . Thus the class denotes the set of all lacunary strongly -summable sequences of fuzzy numbers.
For suitable choices of m, f and , some of the known sequence spaces, which are obtained from the space , are as follows.
-
(i)
Let , and for all , the sequence space represents the space studied by Kwon and Shim [10].
-
(ii)
Let , and for all , the sequence space denotes the space investigated by Bilgin [11].
-
(iii)
Let , the sequence space denotes the space studied by Altin et al. [12].
-
(iv)
Let and for all , the sequence space reduces to the space investigated by Esi [13].
Thus, the study of the sequence space gives a unified approach to many of the earlier known spaces.
Theorem 3.1 Let be a bounded sequence of positive real numbers. Then the class is a linear space over ℝ.
Proof Using Lemma 2.1, Lemma 2.2, the subadditivity property of a modulus function f and the result , it is easy to show that is a linear space over the real field ℝ. □
Theorem 3.2 Let be a bounded sequence of positive real numbers such that . Then the sequence space is a complete metric space with respect to the metric
where .
Proof It is easy to verify that h gives a metric on . To show completeness, let us assume that is a Cauchy sequence in , where for each . So, for given , there exists such that
i.e.,
This means that
as well as
Since f is a modulus function, so equation (3.1) gives for all , for some such that and for each , i.e.,
Similarly, as f is a modulus function, so by choosing suitable , equation (3.2) gives for all and for each k, i.e.,
Using equation (3.3) and equation (3.4), it can be easily shown that is a Cauchy sequence in . But is complete, so the sequence converges to in as .
Keeping u fixed and letting in equation (3.1) and equation (3.2), we get
and
i.e.,
Next we show that the limit point , for which the proof is as follows.
Since for , so for each u, there exist and such that
Similarly, for each v, there exist and such that
Now let and . Then from equation (3.2), equation (3.6), equation (3.7) and using Lemma 2.2, we have
Now using the fact that the modulus function is monotone and for a suitable choice of , we get
i.e., is a Cauchy sequence in . Since is complete, so there exists such that as . Keeping u fixed and letting in equation (3.8), we get
Hence, from equation (3.5), equation (3.6) and equation (3.9), we have
which implies as , i.e., and hence the sequence space is a complete metric space. □
Theorem 3.3 Let be a lacunary sequence and be a sequence of fuzzy numbers. Then:
-
(i)
implies .
-
(ii)
If f is a bounded modulus function and , then .
Proof Easy, so omitted. □
Theorem 3.4 Let be a lacunary sequence, and let be a sequence of fuzzy numbers. Then:
-
(i)
if and only if .
-
(ii)
if and only if ,
where
and
Proof (i) Assume that . Then there exists such that for sufficiently large r.
Let . Then
Consider
Since , we have
and
Since . So both the terms and converge to 0 as and hence converges to 0 as , i.e., and hence .
Conversely, let imply but .
Since θ is a lacunary sequence, we can select a subsequence of θ satisfying
Define
Now we prove that the sequence X for which , which is defined as above, is strongly -summable but not lacunary strongly -summable. Since
and
which implies , but if n is sufficiently large integer and j is the unique integer with , then, since implies , we have
i.e., . This shows that , which leads to a contradiction. Therefore it proves that .
(ii) (Sufficiency) Suppose . Then there exists a constant such that for all . Let . So .
Let . Then we can find and such that
and
Then if n is any integer with , where , then we can write
which implies as and it follows that .
(Necessity) Suppose that implies but .
Since θ is any lacunary sequence, we can select a subsequence of θ satisfying and define a bounded difference sequence by
Let be given. Then we have if , then
and if , then
which implies that .
For the above sequence and for ,
and for ,
It proves that and hence the result follows immediately. □
Theorem 3.5 Let be a lacunary sequence and be a sequence of fuzzy numbers. Then:
-
(i)
if and only if .
-
(ii)
if and only if .
Proof The proof can be established following the technique used in Theorem 3.4. □
Theorem 3.6 If is a lacunary refinement of and , then .
Theorem 3.7 Let be a lacunary refinement of the lacunary sequence . Let and , , be the corresponding intervals of θ and β, respectively, and and , respectively. If there exists such that
then implies .
Proof Let be given, and for every , we can find such that , then we have
which implies that . □
Theorem 3.8 Let and be two lacunary sequences. Let and , , be the corresponding intervals of θ and β, respectively, and , , and and . If there exists such that
then implies .
Proof Let . Then α is a lacunary refinement of both the lacunary sequences β and θ. The interval sequence of α is .
Let be given, , and for every , we can find such that and . Then we have
which implies that .
Since α is a lacunary refinement of β, so by Theorem 3.6 it follows that implies and hence the result follows. □
Theorem 3.9 Let and be two lacunary sequences. Let , , , and , . If there exists such that
then if and only if .
Definition 3.1 Let and be two lacunary sequences. Let , , . There exist a sequence X and a fuzzy number L such that and if and only if there exist and which satisfy the following conditions:
-
(i)
;
-
(ii)
;
-
(iii)
, .
Proof Let there exist a sequence and a fuzzy number ξ such that and . Then we can find a subsequence , and such that
For each , there exist a positive integer and a whole number such that
Then we can write
Since , so for sufficiently large , we have
It can be seen that . Hence, by using equation (3.11) in equation (3.10), we get
This implies that , which implies that at least one of the following two inequalities holds:
or
Suppose that equation (3.12) holds. Since , so using equation (3.12), we conclude that , which proves (iii).
For such , chosen in the proof of (iii), from equation (3.12), we have
Since as , from equation (3.14) we have as , which implies that condition (ii) is satisfied.
Conversely, let for the two lacunary sequences and there exist sequences and which satisfy the above three conditions. Define
Then, for any , if for any , and if for some r, as . Hence .
But for , which implies . □
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Authors’ contributions
PDS in this paper introduced the space of lacunary strongly -summable sequences of fuzzy numbers by using a modulus function, defined a suitable metric for the completeness property and gave the idea to introduce the relation between two arbitrary lacunary sequences. SM proved the relation between the spaces of lacunary strongly -summable sequences using a modulus function and -summable sequences using a modulus function of fuzzy numbers and does some inclusion relations between any two arbitrary lacunary sequences. All authors read and approved the final manuscript.
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Srivastava, P.D., Mohanta, S. Generalized lacunary -statistically convergent sequences of fuzzy numbers using a modulus function. J Inequal Appl 2013, 559 (2013). https://doi.org/10.1186/1029-242X-2013-559
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DOI: https://doi.org/10.1186/1029-242X-2013-559