Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type

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.

Academic Editor:Pierpaolo D'Urso

1, Department of Mathematics, Madhab Choudhury College-Gauhati University, Barpeta, Assam 781301, India

Received 7 November 2012; Revised 3 January 2013; Accepted 16 January 2013

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.

1. Introduction

Throughout the paper, a double sequence is denoted by [figure omitted; refer to PDF] , a double infinite array of elements [figure omitted; refer to PDF] , where each [figure omitted; refer to PDF] is a fuzzy real number.

The initial work on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Moricz [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 [figure omitted; refer to PDF] denote the set of all closed and bounded intervals [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , the real line. For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . It is known that [figure omitted; refer to PDF] is a complete metric space.

A fuzzy real number [figure omitted; refer to PDF] is a fuzzy set on [figure omitted; refer to PDF] , that is, a mapping [figure omitted; refer to PDF] associating each real number [figure omitted; refer to PDF] with its grade of membership [figure omitted; refer to PDF] .

The [figure omitted; refer to PDF] -level set [figure omitted; refer to PDF] of the fuzzy real number [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , is defined as [figure omitted; refer to PDF] .

The set of all upper semicontinuous, normal, and convex fuzzy real numbers is denoted by [figure omitted; refer to PDF] , and throughout the paper, by a fuzzy real number, we mean that the number belongs to [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] , and let the [figure omitted; refer to PDF] -level sets be [figure omitted; refer to PDF] ; the product of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF]

2. Definitions and Preliminaries

A fuzzy real number [figure omitted; refer to PDF] is called convex if [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

If there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , then the fuzzy real number [figure omitted; refer to PDF] is called normal.

A fuzzy real number [figure omitted; refer to PDF] is said to be upper semicontinuous if, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , is open in the usual topology of [figure omitted; refer to PDF] .

The set [figure omitted; refer to PDF] of all real numbers can be embedded in [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF]

The absolute value, [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , is defined by (see, e.g., [19]) [figure omitted; refer to PDF]

A fuzzy real number [figure omitted; refer to PDF] is called nonnegative if [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] . The set of all nonnegative fuzzy real numbers is denoted by [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF] defines a metric on [figure omitted; refer to PDF] .

The additive identity and multiplicative identity in [figure omitted; refer to PDF] are denoted by [figure omitted; refer to PDF] , respectively.

A sequence [figure omitted; refer to PDF] of fuzzy real numbers is said to be convergent to the fuzzy real number [figure omitted; refer to PDF] if, for every [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] .

A sequence of fuzzy numbers [figure omitted; refer to PDF] converges to a fuzzy number [figure omitted; refer to PDF] if both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] hold for every [figure omitted; refer to PDF] [20].

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

A double sequence [figure omitted; refer to PDF] of fuzzy real numbers is said to be convergent in Pringsheim's sense to the fuzzy real number [figure omitted; refer to PDF] if, for every [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

A double sequence [figure omitted; refer to PDF] of fuzzy real numbers is said to be regularly convergent if it converges in Pringsheim's sense, and the following limits exist: [figure omitted; refer to PDF]

A fuzzy real number sequence [figure omitted; refer to PDF] is said to be bounded if [figure omitted; refer to PDF] , for some [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF]

Throughout the paper [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] is said to be solid (or normal ) if [figure omitted; refer to PDF] , whenever [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , for some [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] be a double sequence space. A K-step space of [figure omitted; refer to PDF] is a sequence space [figure omitted; refer to PDF]

A canonical preimage of a sequence [figure omitted; refer to PDF] is a sequence [figure omitted; refer to PDF] defined as follows: [figure omitted; refer to PDF]

A canonical preimage of a step space [figure omitted; refer to PDF] is a set of canonical preimages of all elements in [figure omitted; refer to PDF] .

A double sequence space [figure omitted; refer to PDF] is said to be monotone if [figure omitted; refer to PDF] contains the canonical preimage of all its step spaces.

From the above definitions, we have the following remark.

Remark 1.

A sequence space [figure omitted; refer to PDF] is solid [figure omitted; refer to PDF] [figure omitted; refer to PDF] is monotone.

A double sequence space [figure omitted; refer to PDF] is said to be symmetric if [figure omitted; refer to PDF] , whenever [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a permutation of [figure omitted; refer to PDF] .

A double sequence space [figure omitted; refer to PDF] is said to be sequence algebra if [figure omitted; refer to PDF] , whenever [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

A double sequence space [figure omitted; refer to PDF] is said to be convergence-free if [figure omitted; refer to PDF] , whenever [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] .

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 [figure omitted; refer to PDF] be a sequence of positive real numbers [figure omitted; refer to PDF]

For [figure omitted; refer to PDF] , we get the class [figure omitted; refer to PDF] .

Also a fuzzy sequence [figure omitted; refer to PDF] if [figure omitted; refer to PDF] , and the following limits exist: [figure omitted; refer to PDF]

For the class of sequences [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

We define [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

3. Main Results

Theorem 2.

Let [figure omitted; refer to PDF] be bounded. Then, the classes of sequences [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are complete metric spaces with respect to the metric defined by [figure omitted; refer to PDF]

Proof.

We prove the result for [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be a Cauchy sequence in [figure omitted; refer to PDF] . Then, for a given [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] is complete, there exist fuzzy numbers [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Taking [figure omitted; refer to PDF] in (13), we have [figure omitted; refer to PDF]

Using the triangular inequality [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] . Hence, [figure omitted; refer to PDF] is complete.

Property 1.

The space [figure omitted; refer to PDF] is symmetric, but the spaces [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are not symmetric.

Proof.

Obviously the space [figure omitted; refer to PDF] is symmetric. For the other spaces, consider the following example.

Example 3. Consider the sequence space [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , otherwise. Let the sequence [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF] and for [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be a rearrangement of [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] and for [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Then, [figure omitted; refer to PDF] , but [figure omitted; refer to PDF] . Hence, [figure omitted; refer to PDF] is not symmetric. Similarly, it can be established that the other spaces are also not symmetric.

Theorem 4.

The spaces [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are solid.

Proof.

Consider the sequence space [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] be such that [figure omitted; refer to PDF] .

The result follows from the inequality [figure omitted; refer to PDF]

Hence, the space [figure omitted; refer to PDF] is solid. Similarly, the other spaces are also solid.

Property 2.

The spaces [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are not monotone and hence are not solid.

Proof.

The result follows from the following example.

Example 5. Consider the sequence space [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] for [figure omitted; refer to PDF] even and [figure omitted; refer to PDF] , otherwise. Let [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be defined by the following:

for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the canonical preimage of [figure omitted; refer to PDF] for the subsequence [figure omitted; refer to PDF] of [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] is not monotone. Similarly, the other spaces are also not monotone. Hence, the spaces [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are not solid.

Property 3.

The spaces [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are not convergence-free.

The result follows from the following example.

Example 6.

Consider the sequence space [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , otherwise. Consider the sequence [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] and for other values, [figure omitted; refer to PDF]

Let the sequence [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF] and for other values, [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] , but [figure omitted; refer to PDF] . Hence, the space [figure omitted; refer to PDF] is not convergence-free. Similarly, the other spaces are also not convergence-free.

Theorem 7.

[figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The inclusions are strict.

Proof.

Since convergent sequences are bounded, the proof is clear.

Theorem 8.

Let [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Proof.

Consider the sequence spaces [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] .

Then, [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] .

The result follows from the inequality [figure omitted; refer to PDF] .

Theorem 9.

The spaces [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are sequence algebras.

Proof.

Consider the sequence space [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, the result follows immediately from the inequality [figure omitted; refer to PDF]

Acknowledgment

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

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