Research Article | Open Access

# Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type

**Academic Editor:**Pierpaolo D'Urso

#### Abstract

We study different properties of convergent, null, and bounded double sequence spaces of fuzzy real numbers like completeness, solidness, sequence algebra, symmetricity, convergence-free, and so forth. We prove some inclusion results too.

#### 1. Introduction

Throughout the paper, a double sequence is denoted by , a double infinite array of elements , where each is a fuzzy real number.

The initial work on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Móricz [3], Tripathy [4], Basarir and Sonalcan [5], and many others. Hardy [2] introduced the notion of regular convergence for double sequences.

The concept of paranormed sequences was studied by Nakano [6] and Simons [7] at the initial stage. Later on, it was studied by many others.

After the introduction of fuzzy real numbers, different classes of sequences of fuzzy real numbers were introduced and studied by Tripathy and Nanda [8], Choudhary and Tripathy [9], Tripathy et al. [10–13], Tripathy and Dutta [14–16], Tripathy and Borgogain [17], Tripathy and Das [18], and many others.

Let denote the set of all closed and bounded intervals on , the real line. For , , we define where and . It is known that is a complete metric space.

A *fuzzy real number * is a fuzzy set on , that is, a mapping associating each real number with its grade of membership .

The *-level* set of the fuzzy real number , for , is defined as .

The set of all upper semicontinuous, normal, and convex fuzzy real numbers is denoted by , and throughout the paper, by a fuzzy real number, we mean that the number belongs to .

Let , and let the -level sets be ; the product of and is defined by

#### 2. Definitions and Preliminaries

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 set of all real numbers can be embedded in . For , is defined by

The* absolute *value, of , is defined by (see, e.g., [19])

A fuzzy real number is called *nonnegative* if , for all . The set of all nonnegative fuzzy real numbers is denoted by .

Let be defined by

Then defines a metric on .

The additive identity and multiplicative identity in are denoted by , respectively.

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

A sequence of fuzzy numbers converges to a fuzzy number if both and hold for every [20].

A sequence of generalized fuzzy numbers converges weakly to a generalized fuzzy number (and we write ) if distribution functions converge weakly to and converge weakly to [21].

A double sequence of fuzzy real numbers is said to be* convergent in Pringsheim’s sense *to the fuzzy real number if, for every , there exists , such that , for all , .

A double sequence of fuzzy real numbers is said to be* regularly convergent *if it converges in Pringsheim’s sense, and the following limits exist:

A fuzzy real number sequence is said to be *bounded *if , for some .

For and , we define

Throughout the paper , , , , , and denote the classes of *all*, *bounded*, *convergent in Pringsheim*’*s sense*,* null in Pringsheim’s sense*,* regularly convergent*, and *regularly null* fuzzy real number sequences, respectively.

A double sequence space is said to be *solid* (or* normal*) if , whenever , for all , , for some .

Let , and let be a double sequence space. A *K-step space* of is a sequence space

A *canonical preimage* of a sequence is a sequence defined as follows:

A *canonical preimage of a step space * is a set of canonical preimages of all elements in .

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

From the above definitions, we have the following remark.

*Remark 1. *A sequence space is solid is monotone.

A double sequence space is said to be *symmetric* if , whenever , where is a permutation of .

A double sequence space is said to be *sequence algebra* if , whenever , .

A double sequence space is said to be *convergence-free* if , whenever ,* and * implies that .

Sequences of fuzzy real numbers relative to the paranormed sequence spaces were studied by Choudhary and Tripathy [9].

In this paper, we introduce the following sequence spaces of fuzzy real numbers.

Let be a sequence of positive real numbers

For , we get the class .

Also a fuzzy sequence if , and the following limits exist:

For the class of sequences , .

We define , .

#### 3. Main Results

Theorem 2. *Let be bounded. Then, the classes of sequences , , , , and are complete metric spaces with respect to the metric defined by
*

*Proof. *We prove the result for *.* Let be a Cauchy sequence in . Then, for a given , there exists such that

Since is complete, there exist fuzzy numbers such that , for each , *. *

Taking in (13), we have

Using the triangular inequality
we have . Hence, is complete.

*Property 1. *The space is symmetric, but the spaces , , , , , and are not symmetric.

*Proof. *Obviously the space is symmetric. For the other spaces, consider the following example.*Example* 3. Consider the sequence space . Let , for all and , otherwise. Let the sequence be defined by
and for ,

Let be a rearrangement of defined by
and for ,

Then, , but . Hence, is not symmetric. Similarly, it can be established that the other spaces are also not symmetric.

Theorem 4. *The spaces *, , , *and ** are solid. *

*Proof. * Consider the sequence space . Let , and let be such that .

The result follows from the inequality

Hence, the space is solid. Similarly, the other spaces are also solid.

*Property 2. *The spaces , , and are not monotone and hence are not solid.

*Proof. *The result follows from the following example.*Example* 5. Consider the sequence space . Let for even and , otherwise. Let . Let be defined by the following:

for all , ,
Then, . Let be the canonical preimage of for the subsequence of . Then,
Then, . Thus, is not monotone. Similarly, the other spaces are also not monotone. Hence, the spaces , , and are not solid.

*Property 3. *The spaces , , , , , , and are not convergence-free.

The result follows from the following example.

*Example 6. * Consider the sequence space . Let , for all , , otherwise. Consider the sequence defined by
and for other values,

Let the sequence be defined by
and for other values,
Then, , but . Hence, the space is not convergence-free. Similarly, the other spaces are also not convergence-free.

Theorem 7. *, for , , , . The inclusions are strict.*

*Proof. *Since convergent sequences are bounded, the proof is clear.

Theorem 8. * Let , for all . Then, for , , , , , .*

*Proof. *Consider the sequence spaces and . Let .

Then, , for all .

The result follows from the inequality .

Theorem 9. *The spaces *, , , , , , *and * *are sequence algebras. *

*Proof. *Consider the sequence space . Let , . Then, the result follows immediately from the inequality

#### Acknowledgment

The author’s work is supported by UGC Project no. F. 5-294/2009-10 (MRP/NERO).

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#### Copyright

Copyright © 2013 Bipul Sarma. 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.