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Advances in Fuzzy Systems
Volume 2011 (2011), Article ID 216414, 6 pages
http://dx.doi.org/10.1155/2011/216414
Research Article

Some Classes of Difference Sequence Spaces of Fuzzy Real Numbers Defined by Orlicz Function

1Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Garchuk, Guwahati 781035, India
2Department of Mathematics, Indian Institute of Technology Bombay, Powai, Maharashtra, Mumbai 400076, India

Received 12 August 2011; Accepted 10 November 2011

Academic Editor: Ajith Abraham

Copyright © 2011 Binod Chandra Tripathy and Stuti Borgohain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the classes of generalized difference bounded, convergent, and null sequences of fuzzy real numbers defined by an Orlicz function. Some properties of these sequence spaces like solidness, symmetricity, and convergence-free are studied. We obtain some inclusion relations involving these sequence spaces.

1. Introduction

The concept of fuzzy set theory was introduced by Zadeh in 1965. Later on sequences of fuzzy numbers have been discussed by Syau [1], Tripathy and Baruah [2], Tripathy and Borgohain [3], Tripathy and Dutta [4, 5], Tripathy and Sarma [6, 7], and many others.

Kizmaz [8] defined the difference sequence spaces , , and of complex numbers as follows:

The above spaces are Banach spaces, normed by

The idea of Kizmaz [8] was applied to introduce different type of difference sequence spaces and study their different properties by Et et al. [9], Tripathy et al. [10], Tripathy and Baruah [2], Tripathy and Borgohain [3], Tripathy and Esi [11], Tripathy et al. [12], Tripathy and Mahanta [13], and many others.

Tripathy and Esi [11] introduced a new type of difference sequence spaces as follows. Let be fixed, then

The above sequence spaces are Banach spaces, normed by Tripathy et al. [12] further generalized this notion and introduced the following. For and ,

This generalized difference has the following binomial representation:

An Orlicz function is a function , which is continuous, nondecreasing, and convex with , , for and , as (one may refer to Korasnoselkii and Rutitsky [14]).

An Orlicz function is said to satisfy -condition for all values of , if there exists a constant , such that , for all and for .

Remark 1. An Orlicz function satisfies the inequality , for all with .

Throughout the paper represent the classes of all, absolutely summable, and bounded sequences of fuzzy real numbers, respectively.

2. Definitions and Background

Let . Then the space has linear structure induced by the operations and for , and .

The Hausdorff distance between and of is defined as

It is well known that is a complete metric space. A fuzzy real number on is a function associating each real number with its grade of membership .

A fuzzy real number is called convex if , where .

If there exists such that , then the fuzzy real number is called normal.

A fuzzy real number is said to be upper semicontinuous if for each , , for all is open in the usual topology of .

The class of all upper semi-continuous, normal, convex fuzzy real numbers is denoted by . Let , then the -level set , for , is defined by, and is a nonempty compact convex subset of . The 0-level set, that is, , is the closure of strong 0-cut, that is, . The absolute value of , that is, , is defined by

For , is defined as,

The additive identity and multiplicative identity of are denoted by and , respectively, where ) and . The zero sequence of fuzzy real numbers is denoted by .

The linear structure of induces the addition and scalar multiplication , in terms of -level sets, by and , for each , where

Define, for each , and such that , where is the Hausdorff metric. Clearly, with if . Moreover is a complete, separable, and locally compact metric space.

A sequence of fuzzy real numbers is said to converge to the fuzzy number , if for every , there exists such that , for all .

A sequence space is said to be solid if , whenever and , for all .

Let be a sequence, then denotes the set of all permutations of the elements of , that is, . A sequence space is said to be symmetric if for all .

A sequence space is said to be convergence-free if whenever and implies .

A sequence space is said to be monotone if contains the canonical preimages of all its step spaces.

Lemma 2. A class of sequences is solid which implies that is monotone.

Lindenstrauss and Tzafriri [15] used the notion of Orlicz function and introduced the sequence space:

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 .

In the later stage different classes of Orlicz sequence spaces were introduced and investigated by Altin et al. [16], Et et al. [9], Tripathy et al. [10], Tripathy and Borgohain [3], Tripathy and Hazarika [17], Tripathy and Mahanta [13], Tripathy and Sarma [6, 7, 18], and many others.

In this paper we introduce the following difference sequence spaces:

3. Main Results

Theorem 3. The classes of sequences , , are complete metric spaces by the metric for , , , .

Proof. We establish the result for the class of sequences . The proof for the other cases will follow similarly. It can easily be verified that is a metric space by the metric defined above. Next we show that it is a complete metric space.
Let be a Cauchy sequence in such that . Let be given. For a fixed , choose such that . Then there exits a positive integer such that
By the definition of , we have, which implies
Hence , for are Cauchy sequence in and hence are convergent in , since is a complete metric space.
Let
Also, Since is continuous, we get, which implies is a Cauchy sequence in and so is convergent in , since is complete metric space.
Let (say), in , for each .
We have to prove that
For , we have, from (6) and (19),
Proceeding in this way inductively, we get
Also, , for each .
Next taking , keeping fixed, and by the continuity of , we have the following from (20):
Now on taking the infimum of such ’s, we get
Hence from (17) on taking limit as , we get which implies That is, .
Next we show that .
We know that
Since is continuous and nondecreasing, so we get which implies .
Hence is a complete metric space.
The other cases can be established similarly.
This completes the proof of the theorem.

Result 1. The classes of sequences , , , are neither solid nor monotone in general.

Proof. The result follows from the following example.
Example 4. Consider the sequence space . Let and . Let , for all .
Consider the sequence defined by
Then,
Then, we have , for all .
Hence, we have which implies .
Consider the sequence of scalars defined by
For , we have
For , we have which implies
Hence .
Thus, is not solid in general.

Similarly the other cases can be established. The classes of sequences are not monotone followed by Lemma 2.

Result 2. The classes of sequences , , and are not symmetric in general.

Proof. The result follows from the following example.
Example 5. Let and . Let , for all . Consider the sequence defined by
Then,
Then, , for all , which shows .

Let be a rearrangement of such that . Then we get , for all , which implies
Hence, .
Thus the classes of sequences , , and are not symmetric in general.

Note 1. For , the class of sequences and are symmetric. For and , the class of sequences is symmetric.

Proposition 6. The classes of sequences , , are not convergence-free in general.

Proof. The result follows from the following example.
Example 7. Let and . Let , for all . Consider the sequence defined by Then,
Hence we have , which implies .
Consider the sequence defined by so that
Thus , for all , which implies
Thus .
Hence the classes of sequences , , are not convergence-free in general.

Theorem 8. Let , , and be Orlicz functions satisfying -condition. Then, for , , and , (i), (ii).

Proof. (i) Let . Consider and such that .
Then,
Let
Since is continuous and non-decreasing, we get which implies .
This completes the proof.
(ii) Let .
Then,
The proof follows from the equality which implies that .
This completes the proof.

Proposition 9. One has , for , for , , and .

Proof. Let . Then we have,
Now we have
Proceeding in this way, we have , for , for , , and .
This completes the proof.

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