Generalized lacunary -statistically convergent sequences of fuzzy numbers using a modulus function
© Srivastava and Mohanta; licensee Springer. 2013
Received: 12 May 2013
Accepted: 16 October 2013
Published: 25 November 2013
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.
The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh , 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 , Nanda , Nuray and Savaş , Mursaleen and Basarır , Malkowsky et al. , Tripathy and Chandra , Tripathy and Borgogain  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  introduced the idea of statistical convergence of fuzzy numbers and Nuray  introduced the related concept of convergence with the help of a lacunary sequence. Using their ideas, many authors such as Kwon and Shim , Bilgin , Altin et al. , Esi , 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.
For each , the set is a closed, bounded and nonempty interval of ℝ.
for any and .
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 .
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 .
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 )
Lemma 2.2 (Maddox )
where , and λ is any complex number.
Lemma 2.3 (Maddox )
where , .
3 Main results
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.
In this case, we write . Thus the class denotes the set of all lacunary strongly -summable sequences of fuzzy numbers.
Let , and for all , the sequence space represents the space studied by Kwon and Shim .
Let , and for all , the sequence space denotes the space investigated by Bilgin .
Let , the sequence space denotes the space studied by Altin et al. .
Let and for all , the sequence space reduces to the space investigated by Esi .
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. □
Next we show that the limit point , for which the proof is as follows.
which implies as , i.e., and hence the sequence space is a complete metric space. □
If f is a bounded modulus function and , then .
Proof Easy, so omitted. □
if and only if .
if and only if ,
Proof (i) Assume that . Then there exists such that for sufficiently large r.
Since . So both the terms and converge to 0 as and hence converges to 0 as , i.e., and hence .
Conversely, let imply but .
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 .
which implies as and it follows that .
(Necessity) Suppose that implies but .
which implies that .
It proves that and hence the result follows immediately. □
if and only if .
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 .
then implies .
which implies 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 .
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. □
then if and only if .
Since as , from equation (3.14) we have as , which implies that condition (ii) is satisfied.
Then, for any , if for any , and if for some r, as . Hence .
But for , which implies . □
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