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Some Orlicz extended ℐ-convergent A-summable classes of sequences of fuzzy numbers
Journal of Inequalities and Applicationsvolume 2013, Article number: 479 (2013)
The article introduces some classes of sequences of fuzzy numbers extended by Orlicz functions by using the notions of ℐ-convergence and matrix transformation and investigates the classes for relationship between them as well as establishes some relevant properties. Further, the Hukuhara difference property is employed to derive a new kind of spaces and prove that such spaces can be equipped with a linear topological structure.
MSC:40A05, 40D25, 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 fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming. Matloka  introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties. Later on, the sequences of fuzzy numbers were discussed by Diamond and Kloeden , Nanda , Esi , Dutta [6–8] and many others.
A fuzzy number is a fuzzy set on the real axis, i.e., a mapping which satisfies the following four conditions:
u is normal, i.e., there exists such that .
u is fuzzy convex, i.e., for all and for all .
u is upper semi-continuous.
The set is compact, where denotes the closure of the set in the usual topology of R.
We denote the set of all fuzzy numbers on R by and call it the space of fuzzy numbers. λ-level set of is defined by
The set is a closed, bounded and non-empty interval for each which is defined by . R can be embedded in since each can be regarded as a fuzzy number
Let W be the set of all closed bounded intervals A of real numbers such that . Define the relation s on W as follows:
Lemma 1 (Talo and Basar )
Let and . Then
is a complete metric space.
Lemma 2 (Talo and Basar )
The following statements hold:
for all .
If as , then as .
The notion of ℐ-convergence was initially introduced by Kostyrko et al. . Later on, it was further investigated from the sequence space point of view and linked with the summability theory by Salat et al. [12, 13], Tripathy and Hazarika [14–16] and Kumar and Kumar  and many others. For some other related works, one may refer to Altinok et al. , Altin et al. [19–22], Çolak et al. , Güngör  and many others.
Let X be a non-empty set, then a family of sets (the class of all subsets of X) is called an ideal if and only if for each , we have and for each and each , we have . A non-empty family of sets is a filter on X if and only if , for each , we have and for each and each , we have . An ideal ℐ is called non-trivial ideal if and . Clearly, is a non-trivial ideal if and only if is a filter on X. A non-trivial ideal is called admissible if and only if . A non-trivial ideal ℐ is maximal if there cannot exist any non-trivial ideal containing ℐ as a subset. Further details on ideals of can be found in Kostyrko et al. .
Lemma 3 (Kostyrko et al. [, Lemma 5.1])
If is a maximal ideal, then for each , we have either or .
Example 1 If we take , then is a non-trivial admissible ideal of N and the corresponding convergence coincides with the usual convergence.
Example 2 If we take , where denotes the asymptotic density of the set A, then is a non-trivial admissible ideal of N and the corresponding convergence coincides with the statistical convergence.
Recall in  that an Orlicz function M is a continuous, convex, nondecreasing function defined for such that and . If the convexity of an Orlicz function is replaced by , then this function is called the modulus function and characterized by Ruckle . The Orlicz function M is said to satisfy -condition for all values of u if there exists such that , .
Lemma 4 Let M be an Orlicz function which satisfies -condition, and let . Then, for each , we have for some constant .
Two Orlicz functions and are said to be equivalent if there exist positive constants α, β and such that
for all x with .
Lindenstrauss and Tzafriri  studied some Orlicz-type sequence spaces defined as follows:
The space with the norm
becomes a Banach space which is called an Orlicz sequence space. The space is closely related to the space which is an Orlicz sequence space with for .
Throughout the article, N and R denote the set of positive integers and the set of real numbers, respectively. The zero sequence is denoted by θ.
Let be an infinite matrix of real numbers. We write if converges for each i.
Throughout the paper, denotes the set of all sequences of fuzzy numbers.
Definition 1 A set is said to be solid if whenever for all and .
The following well-known inequality will be used throughout the article. Let be any sequence of positive real numbers with , , then
for all and . Also, for all .
2 Some new sequence spaces
Let ℐ be an admissible ideal of the non-empty set S, and let be a bounded sequence of positive real numbers. Let be a sequence of Orlicz functions, let be an infinite matrix of real numbers, and let be a sequence of fuzzy numbers. Then we introduce the following sequence spaces:
3 Main results
In this section we investigate the main results of this paper.
Theorem 1 The spaces , , and are linear over the field of reals.
Proof We give the proof for the space only, and the others will follow similarly. Let and be two elements in . Then there exist and such that
Let α, β be two reals. By the continuity of the Orlicz functions ’s, we have the following inequality:
Hence we have the following inclusion:
This completes the proof. □
It is not possible in general to find some fuzzy number such that (called the Hukuhara difference when it exists). Since every real number is a fuzzy number, we can assume that is such a set of sequences of fuzzy numbers with the Hukuhara difference property.
For the next result, we consider to be the space of sequences of fuzzy numbers with the Hukuhara difference property.
Theorem 2 The space is a paranormed space (not totally paranormed) with the paranorm g defined by
Proof Clearly, and . Let and be two elements in . Now, for , we put
Let us take . Then, using the convexity of Orlicz functions ’s, we obtain
which in turn gives us
Let , where , and let as . To prove that as , we put
By the continuity of the sequence , we observe that
From the above inequality it follows that
Note that for all . Hence, by our assumption, the right-hand side of relation (3.1) tends to 0 as and the result follows. This completes the proof. □
Theorem 3 Let and be two sequences of Orlicz functions. Then the following statements hold:
provided is such that .
Proof (i) Let be given. Choose such that . Choose such that implies that for each . Let be any element. Put
Then, by the definition of ideal, we have . If , we have
Using the continuity of the sequence from relation (3.2), we have
Consequently, we get
This implies that
This completes the proof.
Let . Then the result follows from the following inequality:
Taking in the proof of the above theorem, we have the following corollary.
Corollary 1 Let and be two sequences of Orlicz functions. Then the following statements hold:
provided is such that .
The proofs of the following two theorems are easy and so they are omitted.
Theorem 4 Let and be bounded, then
Theorem 5 For any two sequences of positive real numbers and , the following statement holds:
Proposition 1 The sequence spaces are solid for and .
Proof We give the proof of the proposition for only. Let and be such that for all . Then, for given , we have
Again the set
Hence and so . Thus the space is solid. □
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The authors declare that they have no competing interests.
AE proposed the research content. HD formulated the structure of the paper. ABK collected necessary literature for review and study. AE and ABK constructed the new spaces. HD equipped the spaces with a linear topological structure. The proofs of the results were completed through combined efforts of all the authors. All authors read and approved the final manuscript.