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.

1. Introduction

Throughout the paper, a double sequence is denoted by ⟨𝑋_𝑛𝑘⟩, a double infinite array of elements 𝑋_𝑛𝑘, where each 𝑋_𝑛𝑘 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 𝐷 denote the set of all closed and bounded intervals 𝑋=[𝑎₁,𝑎₂] on 𝑅, the real line. For 𝑋, 𝑌∈𝐷, we define 𝑑(𝑋,𝑌)=max𝑎₁−𝑏₁,𝑎₂−𝑏₂, (1) where 𝑋=[𝑎₁,𝑎₂] and 𝑌=[𝑏₁,𝑏₂]. It is known that (𝐷,𝑑)  is a complete metric space.

A fuzzy real number 𝑋 is a fuzzy set on 𝑅, that is, a mapping 𝑋∶𝑅→𝐼(=[0,1]) associating each real number 𝑡 with its grade of membership 𝑋(𝑡).

The 𝛼-level set [𝑋]^𝛼 of the fuzzy real number 𝑋, for 0<𝛼≤1, is defined as [𝑋]^𝛼={𝑡∈𝑅∶𝑋(𝑡)≥𝛼}.

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

Let 𝑋,𝑌∈𝑅(𝐼), and let the 𝛼-level sets be [𝑋]_𝛼=[𝑎^𝛼₁,𝑎^𝛼₂],[𝑌]_𝛼=[𝑏^𝛼₁,𝑏^𝛼₂],and𝛼∈[0,1]; the product of 𝑋 and 𝑌 is defined by [𝑋⊗𝑌]_𝛼=min_{{}𝑖,𝑗∈1,2}𝑎^𝛼_𝑖⋅𝑏^𝛼_𝑗,max_{{}𝑖,𝑗∈1,2}𝑎^𝛼_𝑖⋅𝑏^𝛼_𝑗. (2)

2. Definitions and Preliminaries

A fuzzy real number 𝑋 is called convex if 𝑋(𝑡)≥𝑋(𝑠)∧𝑋(𝑟)=min(𝑋(𝑠),𝑋(𝑡)), where 𝑠<𝑡<𝑟.

If there exists 𝑡₀∈𝑅 such that 𝑋(𝑡₀)=1, then the fuzzy real number 𝑋 is called normal.

A fuzzy real number 𝑋 is said to be upper semicontinuous if, for each 𝜀>0, 𝑋⁻¹([0,𝑎+𝜀)), for all 𝑎∈𝐼, is open in the usual topology of 𝑅.

The set 𝑅 of all real numbers can be embedded in 𝑅(𝐼). For 𝑟∈𝑅, 𝑟∈𝑅(𝐼) is defined by 𝑟(𝑡)=1,for𝑡=𝑟,0,for𝑡≠𝑟. (3)

The absolute value, |𝑋| of 𝑋∈𝑅(𝐼), is defined by (see, e.g., [19]) |𝑋𝑡){𝑋(𝑡)(|𝑋𝑡)|(=max,𝑋−𝑡)}if𝑡≥0,|(=0if𝑡<0. (4)

A fuzzy real number 𝑋 is called nonnegative if 𝑋(𝑡)=0, for all 𝑡<0. The set of all nonnegative fuzzy real numbers is denoted by 𝑅^∗(𝐼).

Let 𝑑∶𝑅(𝐼)×𝑅(𝐼)→𝑅 be defined by 𝑑(𝑋,𝑌)=sup_0≤𝛼≤1𝑑[𝑋]^𝛼,[𝑌]^𝛼. (5)

Then 𝑑 defines a metric on 𝑅(𝐼).

The additive identity and multiplicative identity in 𝑅(𝐼) are denoted by 0and1, respectively.

A sequence (𝑋_𝑘) of fuzzy real numbers is said to be convergent to the fuzzy real number 𝐿 if, for every 𝜀>0, there exists 𝑛₀∈𝑁 such that 𝑑(𝑋_𝑘,𝐿)<𝜀, for all 𝑘≥𝑛₀.

A sequence of fuzzy numbers (𝑓_𝑛) converges to a fuzzy number 𝑓 if both lim_𝑛→∞[𝑓_𝑛]⁻_𝛼=[𝑓]⁻_𝛼 and lim_𝑛→∞[𝑓_𝑛]⁺_𝛼=[𝑓]⁺_𝛼 hold for every 𝛼∈(0,1] [20].

A sequence (𝑥_𝑛) of generalized fuzzy numbers converges weakly to a generalized fuzzy number 𝑥 (and we write 𝑥−→_𝑛^𝑤𝑥) if distribution functions (𝑥^𝑙_𝑛) converge weakly to 𝑥^𝑙 and (𝑥^𝑟_𝑛) converge weakly to 𝑥^𝑟 [21].

A double sequence (𝑋_𝑛𝑘) of fuzzy real numbers is said to be convergent in Pringsheim’s sense to the fuzzy real number 𝐿 if, for every 𝜀>0, there exists 𝑛₀,  𝑘₀∈𝑁 such that 𝑑(𝑋_𝑛𝑘,𝐿)<𝜀, for all 𝑛≥𝑛₀, 𝑘≥𝑘₀.

A double sequence (𝑋_𝑛𝑘) of fuzzy real numbers is said to be regularly convergent if it converges in Pringsheim’s sense, and the following limits exist: lim_𝑛𝑑𝑋_𝑛𝑘,𝐿_𝑘=0,forsome𝐿_𝑘∈𝑅(𝐼),foreach𝑘∈𝑁,lim_𝑘𝑑𝑋_𝑛𝑘,𝐽_𝑛=0,forsome𝐽_𝑛∈𝑅(𝐼),foreach𝑛∈𝑁. (6)

A fuzzy real number sequence (𝑋_𝑘) is said to be bounded if sup_𝑘|𝑋_𝑘|≤𝜇, for some 𝜇∈𝑅^∗(𝐼).

For 𝑟∈𝑅 and 𝑋∈𝑅(𝐼), we define 𝑟𝑋(𝑡)=𝑋𝑟⁻¹𝑡(𝑡)=if𝑟≠0,𝑟𝑋0if𝑟=0. (7)

Throughout the paper ₂𝑤_𝐹, (₂ℓ_∞)_𝐹, ₂𝑐_𝐹, (₂𝑐₀)_𝐹,  ₂𝑐^𝑅_𝐹, and (₂𝑐^𝑅₀)_𝐹 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 𝐸_𝐹 is said to be solid (or normal) if ⟨𝑌_𝑛𝑘⟩∈𝐸_𝐹, whenever |𝑌_𝑛𝑘|≤|𝑋_𝑛𝑘|, for all 𝑛, 𝑘∈𝑁, for some ⟨𝑋_𝑛𝑘⟩∈𝐸_𝐹.

Let 𝐾={(𝑛_𝑖,𝑘_𝑖)∶𝑖∈𝑁;𝑛₁<𝑛₂<𝑛₃<⋯and𝑘₁<𝑘₂<𝑘₃<⋯}⊆𝑁×𝑁, and let 𝐸_𝐹 be a double sequence space. A K-step space of 𝐸_𝐹 is a sequence space 𝜆^𝐸_𝐾=𝑋_{𝑛𝑖𝑘𝑖}∈₂𝑤_𝐹∶𝑋_𝑛𝑘∈𝐸_𝐹. (8)

A canonical preimage of a sequence ⟨𝑋_{𝑛𝑖𝑘𝑖}⟩∈𝐸_𝐹 is a sequence ⟨𝑌_𝑛𝑘⟩ defined as follows: 𝑌_𝑛𝑘=𝑋_𝑛𝑘if(𝑛,𝑘)∈𝐾,0,otherwise. (9)

A canonical preimage of a step space 𝜆^𝐸_𝐾 is a set of canonical preimages of all elements in 𝜆^𝐸_𝐾.

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

From the above definitions, we have the following remark.

Remark 1. A sequence space 𝐸_𝐹 is solid ⇒𝐸_𝐹 is monotone.
A double sequence space 𝐸_𝐹 is said to be symmetric if (𝑋_{𝜋(𝑛,𝑘)})∈𝐸_𝐹, whenever (𝑋_𝑛𝑘)∈𝐸_𝐹, where 𝜋 is a permutation of 𝑁×𝑁.
A double sequence space 𝐸_𝐹 is said to be sequence algebra if (𝑋_𝑛𝑘⊗𝑌_𝑛𝑘)∈𝐸_𝐹, whenever (𝑋_𝑛𝑘), (𝑌_𝑛𝑘)∈𝐸_𝐹.
A double sequence space 𝐸_𝐹 is said to be convergence-free if (𝑌_𝑛𝑘)∈𝐸_𝐹, whenever (𝑋_𝑛𝑘)∈𝐸_𝐹, and 𝑋_𝑛𝑘=0 implies that 𝑌_𝑛𝑘=0.
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 𝑝=⟨𝑝_𝑛𝑘⟩ be a sequence of positive real numbers ₂ℓ_∞𝐹𝑝=𝑋_𝑛𝑘∈₂𝑤_𝐹∶sup_𝑛,𝑘𝑑𝑋_𝑛𝑘,0^𝑝𝑛𝑘<∞₂,𝑐_𝐹𝑝=𝑋_𝑛𝑘∈₂𝑤_𝐹∶lim_𝑛,𝑘𝑑𝑋_𝑛𝑘,𝐿^𝑝𝑛𝑘(𝐼.=0,forsome𝐿∈𝑅) (10)
For 𝐿=0, we get the class (₂𝑐_𝐹)₀(𝑝).
Also a fuzzy sequence ⟨𝑋_𝑛𝑘⟩∈₂𝑐^𝑅_𝐹(𝑝) if ⟨𝑋_𝑛𝑘⟩∈₂𝑐_𝐹(𝑝), and the following limits exist: lim_𝑛𝑑𝑋_𝑛𝑘,𝐿_𝑘^𝑝𝑛𝑘=0,forsome𝐿_𝑘∈𝑅(𝐼),foreach𝑘∈𝑁,lim_𝑘𝑑𝑋_𝑛𝑘,𝐽_𝑛^𝑝𝑛𝑘=0,forsome𝐽_𝑛∈𝑅(𝐼),foreach𝑛∈𝑁. (11)
For the class of sequences (₂𝑐^𝑅_𝐹)₀(𝑝), 𝐿=𝐿_𝑘=𝐽_𝑛=0.
We define 𝑚_𝐹(𝑝)=₂𝑐^𝑅_𝐹(𝑝)∩(₂ℓ_∞)_𝐹(𝑝), (𝑚₀)_𝐹(𝑝)=(₂𝑐^𝑅_𝐹)₀(𝑝)∩(₂ℓ_∞)_𝐹(𝑝).

3. Main Results

Theorem 2. Let ⟨𝑝_𝑛𝑘⟩ be bounded. Then, the classes of sequences (₂ℓ_∞)_𝐹(𝑝), ₂𝑐^𝑅_𝐹(𝑝),  (₂𝑐^𝑅_𝐹)₀(𝑝), 𝑚_𝐹(𝑝), and (𝑚₀)_𝐹(𝑝) are complete metric spaces with respect to the metric defined by 𝑓(𝑋,𝑌)=sup_𝑛,𝑘𝑑𝑋_𝑛𝑘,𝑌_𝑛𝑘^{𝑝𝑛𝑘/𝐻},𝑤ℎ𝑒𝑟𝑒𝐻=max1,sup𝑝_𝑛𝑘. (12)

Proof. We prove the result for ₂ℓ_∞(𝑝). Let ⟨𝑋^𝑖_𝑛𝑘⟩ be a Cauchy sequence in (₂ℓ_∞)_𝐹(𝑝). Then, for a given 𝜀>0, there exists 𝑚₀∈𝑁 such that 𝑓𝑋^𝑖_𝑛𝑘,𝑋^𝑗_𝑛𝑘<𝜀<𝜀^{𝑝𝑛𝑘/𝐻},∀𝑖,𝑗≥𝑚₀,⇒𝑑𝑋^𝑖_𝑛𝑘,𝑋^𝑗_𝑛𝑘<𝜀,∀𝑖,𝑗≥𝑚₀⇒𝑋^𝑗_𝑛𝑘^∞_𝑗=1isaCauchysequenceoffuzzyrealnumberforeach𝑛,𝑘∈𝑁. (13)
Since 𝑅(𝐼) is complete, there exist fuzzy numbers 𝑋_𝑛𝑘 such that lim_𝑗→∞𝑋^𝑗_𝑛𝑘=𝑋_𝑛𝑘, for each 𝑛, 𝑘∈𝑁.
Taking 𝑗→∞ in (13), we have 𝑓𝑋^𝑖_𝑛𝑘,𝑋_𝑛𝑘<𝜀. (14)
Using the triangular inequality 𝑓𝑋_𝑛𝑘,0≤𝑓𝑋_𝑛𝑘,𝑋^𝑗_𝑛𝑘+𝑓𝑋^𝑗_𝑛𝑘,0, (15) we have ⟨𝑋_𝑛𝑘⟩∈(₂ℓ_∞)_𝐹(𝑝). Hence, (₂ℓ_∞)_𝐹(𝑝) is complete.

Property 1. The space (₂ℓ_∞)_𝐹(𝑝) is symmetric, but the spaces ₂𝑐_𝐹(𝑝), ₂𝑐^𝑅_𝐹(𝑝), (₂𝑐_𝐹)₀(𝑝), (₂𝑐^𝑅_𝐹)₀(𝑝), 𝑚_𝐹(𝑝), and (𝑚₀)_𝐹(𝑝) are not symmetric.

Proof. Obviously the space (₂ℓ_∞)_𝐹(𝑝) is symmetric. For the other spaces, consider the following example.
Example  3. Consider the sequence space ₂𝑐_𝐹(𝑝). Let 𝑝_1𝑘=2, for all 𝑘∈𝑁 and 𝑝_𝑛𝑘=3, otherwise. Let the sequence ⟨𝑋_𝑛𝑘⟩ be defined by 𝑋_1𝑘=1∀𝑘∈𝑁, (16) and for 𝑛>1, 𝑋_𝑛𝑘(𝑡)=𝑡+2,for−2≤𝑡≤−1,−𝑡,for−1≤𝑡≤0,0,otherwise. (17)
Let ⟨𝑌_𝑛𝑘⟩ be a rearrangement of ⟨𝑋_𝑛𝑘⟩ defined by 𝑌_𝑛𝑛=1, (18) and for 𝑛≠𝑘, 𝑌_𝑛𝑘(𝑡)=𝑡+2,for−2≤𝑡≤−1,−𝑡,for−1≤𝑡≤0,0,otherwise. (19)
Then, ⟨𝑋_𝑛𝑘⟩∈₂𝑐_𝐹(𝑝), but ⟨𝑌_𝑛𝑘⟩∉₂𝑐_𝐹(𝑝). Hence, ₂𝑐_𝐹(𝑝) is not symmetric. Similarly, it can be established that the other spaces are also not symmetric.

Theorem 4. The spaces (₂ℓ_∞)_𝐹(𝑝), (₂𝑐₀)_𝐹(𝑝), (₂𝑐^𝑅₀)_𝐹(𝑝),  and (𝑚₀)_𝐹(𝑝) are solid.

Proof. Consider the sequence space (₂ℓ_∞)_𝐹(𝑝). Let ⟨𝑋_𝑛𝑘⟩∈(₂ℓ_∞)_𝐹(𝑝), and let ⟨𝑌_𝑛𝑘⟩ be such that 𝑑(𝑌_𝑛𝑘,0)≤𝑑(𝑋_𝑛𝑘,0).
The result follows from the inequality 𝑑𝑌_𝑛𝑘,0^𝑝𝑛𝑘≤𝑑𝑋_𝑛𝑘,0^𝑝𝑛𝑘. (20)
Hence, the space (₂ℓ_∞)_𝐹(𝑝) is solid. Similarly, the other spaces are also solid.

Property 2. The spaces ₂𝑐_𝐹(𝑝), (₂𝑐^𝑅)_𝐹(𝑝), and 𝑚_𝐹(𝑝) are not monotone and hence are not solid.

Proof. The result follows from the following example.
Example 5. Consider the sequence space ₂𝑐_𝐹(𝑝). Let 𝑝_𝑛𝑘=3 for 𝑛+𝑘 even and 𝑝_𝑛𝑘=2, otherwise. Let 𝐽={(𝑛,𝑘)∶𝑛+𝑘iseven}⊆𝑁×𝑁. Let ⟨𝑋_𝑛𝑘⟩ be defined by the following:
for all 𝑛, 𝑘∈𝑁, 𝑋_𝑛𝑘(𝑡)=𝑡+3,for−3≤𝑡≤−2,𝑛𝑡(3𝑛−1)⁻¹+3𝑛(3𝑛−1)⁻¹,for−2≤𝑡≤−1+𝑛⁻¹,0,otherwise. (21) Then, ⟨𝑋_𝑛𝑘⟩∈₂𝑐_𝐹(𝑝). Let ⟨𝑌_𝑛𝑘⟩ be the canonical preimage of ⟨𝑋_𝑛𝑘⟩_𝐽 for the subsequence 𝐽 of 𝑁. Then, 𝑌_𝑛𝑘=𝑋_𝑛𝑘,for(𝑛,𝑘)∈𝐽,0,otherwise. (22) Then, ⟨𝑌_𝑛𝑘⟩∉₂𝑐_𝐹(𝑝). Thus, ₂𝑐_𝐹(𝑝) is not monotone. Similarly, the other spaces are also not monotone. Hence, the spaces ₂𝑐_𝐹(𝑝), (₂𝑐^𝑅)_𝐹(𝑝), and 𝑚_𝐹(𝑝) are not solid.

Property 3. The spaces (₂ℓ_∞)_𝐹(𝑝), ₂𝑐_𝐹(𝑝), (₂𝑐₀)_𝐹(𝑝), (₂𝑐^𝑅)_𝐹(𝑝), (₂𝑐^𝑅₀)_𝐹(𝑝), 𝑚_𝐹(𝑝),  and (𝑚₀)_𝐹(𝑝)   are not convergence-free.

The result follows from the following example.

Example 6. Consider the sequence space ₂𝑐_𝐹(𝑝). Let 𝑝_1𝑘=1, for all 𝑘∈𝑁, 𝑝_𝑛𝑘=3, otherwise. Consider the sequence ⟨𝑋_𝑛𝑘⟩ defined by 𝑋_1𝑘=0, (23) and for other values, 𝑋_𝑛𝑘(𝑡)=()𝑡+2,for−2≤𝑡≤−1,−𝑛𝑡𝑛+1⁻¹+(𝑛+1)⁻¹,for−1≤𝑡≤𝑛⁻¹,0,otherwise. (24)
Let the sequence ⟨𝑌_𝑛𝑘⟩ be defined by 𝑌_1𝑘=0, (25) and for other values, 𝑌_𝑛𝑘(𝑡)=()1,for0≤𝑡≤1,𝑛−1(𝑛−1)⁻¹,for1≤𝑡≤𝑛,0,otherwise. (26) Then, ⟨𝑋_𝑛𝑘⟩∈₂𝑐_𝐹(𝑝), but ⟨𝑌_𝑛𝑘⟩∉₂𝑐_𝐹(𝑝). Hence, the space ₂𝑐_𝐹(𝑝) is not convergence-free. Similarly, the other spaces are also not convergence-free.

Theorem 7. 𝑍(𝑝)⊆(₂ℓ_∞)_𝐹(𝑝),  for 𝑍=(₂𝑐^𝑅)_𝐹, (₂𝑐^𝑅₀)_𝐹,  𝑚_𝐹, (𝑚₀)_𝐹. The inclusions are strict.

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

Theorem 8. Let 0<𝑞_𝑖𝑗≤𝑝_𝑖𝑗<∞,  for all 𝑖,𝑗∈𝑁. Then, 𝑍(𝑝)⊆𝑍(𝑞)   for  𝑍=₂𝑐_𝐹,  (₂𝑐^𝑅)_𝐹, (₂𝑐_𝐹)₀, (₂𝑐^𝑅₀)_𝐹,  𝑚_𝐹, (𝑚₀)_𝐹.

Proof. Consider the sequence spaces ₂𝑐_𝐹(𝑝) and ₂𝑐_𝐹(𝑞). Let ⟨𝑋_𝑛𝑘⟩∈₂𝑐_𝐹(𝑝).
Then, {𝑑(𝑋_𝑛𝑘,𝐿)}^𝑝𝑛𝑘<𝜀, for all 𝑛≥𝑛₀,𝑘≥𝑘₀.
The result follows from the inequality {𝑑(𝑋_𝑛𝑘,𝐿)}^𝑞𝑛𝑘≤{𝑑(𝑋_𝑛𝑘,𝐿)}^𝑝𝑛𝑘.

Theorem 9. The spaces (₂ℓ_∞)_𝐹(𝑝), ₂𝑐_𝐹(𝑝), (₂𝑐₀)_𝐹(𝑝), (₂𝑐^𝑅)_𝐹(𝑝), (₂𝑐^𝑅₀)_𝐹(𝑝), 𝑚_𝐹(𝑝),   and (𝑚₀)_𝐹(𝑝)   are sequence algebras.

Proof. Consider the sequence space (₂𝑐₀)_𝐹(𝑝). Let ⟨𝑋_𝑛𝑘⟩, ⟨𝑌_𝑛𝑘⟩∈(₂𝑐₀)_𝐹(𝑝). Then, the result follows immediately from the inequality 𝑑𝑋_𝑛𝑘𝑌_𝑛𝑘,0^𝑝𝑛𝑘≤𝑑𝑋_𝑛𝑘,0^𝑝𝑛𝑘𝑑𝑌_𝑛𝑘,0^𝑝𝑛𝑘. (27)

Acknowledgment

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