Some Double Sequence Spaces of Fuzzy Real Numbers of

Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 627047, 4 pages http://dx.doi.org/10.1155/2013/627047

Research Article Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type Bipul Sarma Department of Mathematics, Madhab Choudhury College-Gauhati University, Barpeta, Assam 781301, India Correspondence should be addressed to Bipul Sarma; [email protected] Received 7 November 2012; Revised 3 January 2013; Accepted 16 January 2013 Academic Editor: Pierpaolo D’Urso 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. 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 ⟨𝑋𝑛𝑘 ⟩, a double infinite array of elements 𝑋𝑛𝑘 , where each 𝑋𝑛𝑘 is a fuzzy real number. The initial work on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], M´oricz [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) = max (󵄨󵄨󵄨𝑎1 − 𝑏1 󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑎2 − 𝑏2 󵄨󵄨󵄨) ,

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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 = [𝑎1𝛼 , 𝑎2𝛼 ], [𝑌]𝛼 = [𝑏1𝛼 , 𝑏2𝛼 ], and 𝛼 ∈ [0, 1]; the product of 𝑋 and 𝑌 is defined by [𝑋 ⊗ 𝑌]𝛼 = [ min 𝑎𝑖𝛼 ⋅ 𝑏𝑗𝛼 , max 𝑎𝑖𝛼 ⋅ 𝑏𝑗𝛼 ] . 𝑖,𝑗∈{1,2}

𝑖,𝑗∈{1,2}

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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, for 𝑡 = 𝑟, 0, for 𝑡 ≠ 𝑟.

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

if 𝑡 ≥ 0,

if 𝑡 < 0.

0≤𝛼≤1

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Then 𝑑 defines a metric on 𝑅(𝐼). The additive identity and multiplicative identity in 𝑅(𝐼) are denoted by 0 and 1, respectively. A sequence (𝑋𝑘 ) 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(𝑋𝑘 , 𝐿) < 𝜀, for all 𝑘 ≥ 𝑛0 . A sequence of fuzzy numbers (𝑓𝑛 ) converges to a fuzzy number 𝑓 if both lim𝑛 → ∞ [𝑓𝑛 ]−𝛼 = [𝑓]−𝛼 and lim𝑛 → ∞ [𝑓𝑛 ]+𝛼 = [𝑓]+𝛼 hold for every 𝛼 ∈ (0, 1] [20]. A sequence (𝑥𝑛 ) of generalized fuzzy numbers converges 𝑤 → weakly to a generalized fuzzy number 𝑥 (and we write 𝑥𝑛 󳨀 𝑥) if distribution functions (𝑥𝑛𝑙 ) converge weakly to 𝑥𝑙 and (𝑥𝑛𝑟 ) converge weakly to 𝑥𝑟 [21]. A double sequence (𝑋𝑛𝑘 ) 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(𝑋𝑛𝑘 , 𝐿) < 𝜀, for all 𝑛 ≥ 𝑛0 , 𝑘 ≥ 𝑘0 . A double sequence (𝑋𝑛𝑘 ) of fuzzy real numbers is said to be regularly convergent if it converges in Pringsheim’s sense, and the following limits exist: 𝑑 (𝑋𝑛𝑘 , 𝐿 𝑘 ) = 0, lim 𝑛

𝑘

for some 𝐽𝑛 ∈ 𝑅 (𝐼) ,

𝑌𝑛𝑘 = {

𝑟𝑋 (𝑡) = 0 if 𝑟 = 0.

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Remark 1. A sequence space 𝐸𝐹 is solid ⇒ 𝐸𝐹 is monotone. A double sequence space 𝐸𝐹 is said to be symmetric if (𝑋𝜋(𝑛,𝑘) ) ∈ 𝐸𝐹 , whenever (𝑋𝑛𝑘 ) ∈ 𝐸𝐹 , where 𝜋 is a permutation of 𝑁 × 𝑁. A double sequence space 𝐸𝐹 is said to be sequence algebra if (𝑋𝑛𝑘 ⊗ 𝑌𝑛𝑘 ) ∈ 𝐸𝐹 , whenever (𝑋𝑛𝑘 ), (𝑌𝑛𝑘 ) ∈ 𝐸𝐹 . A double sequence space 𝐸𝐹 is said to be convergence-free if (𝑌𝑛𝑘 ) ∈ 𝐸𝐹 , whenever (𝑋𝑛𝑘 ) ∈ 𝐸𝐹 , and 𝑋𝑛𝑘 = 0 implies that 𝑌𝑛𝑘 = 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 = ⟨𝑝𝑛𝑘 ⟩ be a sequence of positive real numbers (2 ℓ∞ )𝐹 (𝑝) = {⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑤𝐹 : 𝑝𝑛𝑘

𝑛,𝑘

2 𝑐𝐹 (𝑝)

< ∞} , (10)

= {⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑤𝐹 : lim{𝑑 (𝑋𝑛𝑘 , 𝐿)}

𝑝𝑛𝑘

𝑛,𝑘

= 0,

for some 𝐿 ∈ 𝑅 (𝐼) } .

A fuzzy real number sequence (𝑋𝑘 ) is said to be bounded if sup𝑘 |𝑋𝑘 | ≤ 𝜇, for some 𝜇 ∈ 𝑅∗ (𝐼). For 𝑟 ∈ 𝑅 and 𝑋 ∈ 𝑅(𝐼), we define if 𝑟 ≠ 0,

if (𝑛, 𝑘) ∈ 𝐾, otherwise.

A canonical preimage of a step space 𝜆𝐸𝐾 is a set of canonical preimages of all elements in 𝜆𝐸𝐾 . A double sequence space 𝐸𝐹 is said to be monotone if 𝐸𝐹 contains the canonical preimage of all its step spaces. From the above definitions, we have the following remark.

for each 𝑛 ∈ 𝑁.

𝑟𝑋 (𝑡) = 𝑋 (𝑟−1 𝑡)

𝑋𝑛𝑘 0,

sup{𝑑 (𝑋𝑛𝑘 , 0)} (6)

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A canonical preimage of a sequence ⟨𝑋𝑛𝑖 𝑘𝑖 ⟩ ∈ 𝐸𝐹 is a sequence ⟨𝑌𝑛𝑘 ⟩ defined as follows:

for some 𝐿 𝑘 ∈ 𝑅 (𝐼) , for each 𝑘 ∈ 𝑁,

lim 𝑑 (𝑋𝑛𝑘 , 𝐽𝑛 ) = 0,

𝜆𝐸𝐾 = {⟨𝑋𝑛𝑖 𝑘𝑖 ⟩ ∈ 2 𝑤𝐹 : ⟨𝑋𝑛𝑘 ⟩ ∈ 𝐸𝐹 } .

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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 𝑑 ([𝑋]𝛼 , [𝑌]𝛼 ) .

Let 𝐾 = {(𝑛𝑖 , 𝑘𝑖 ) : 𝑖 ∈ 𝑁; 𝑛1 < 𝑛2 < 𝑛3 < ⋅ ⋅ ⋅ and 𝑘1 < 𝑘2 < 𝑘3 < ⋅ ⋅ ⋅} ⊆ 𝑁 × 𝑁, and let 𝐸𝐹 be a double sequence space. A K-step space of 𝐸𝐹 is a sequence space

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Throughout the paper 2 𝑤𝐹 , (2 ℓ∞ )𝐹 , 2 𝑐𝐹 , (2 𝑐0 )𝐹 , 2 𝑐𝐹𝑅 , and 𝑅 (2 𝑐0 )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 𝐸𝐹 is said to be solid (or normal) if ⟨𝑌𝑛𝑘 ⟩ ∈ 𝐸𝐹 , whenever |𝑌𝑛𝑘 | ≤ |𝑋𝑛𝑘 |, for all 𝑛, 𝑘 ∈ 𝑁, for some ⟨𝑋𝑛𝑘 ⟩ ∈ 𝐸𝐹 .

For 𝐿 = 0, we get the class (2 𝑐𝐹 )0 (𝑝). Also a fuzzy sequence ⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑐𝐹𝑅 (𝑝) if ⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑐𝐹 (𝑝), and the following limits exist: lim {𝑑 (𝑋𝑛𝑘 , 𝐿 𝑘 )} 𝑛

𝑝𝑛𝑘

= 0,

for some 𝐿 𝑘 ∈ 𝑅 (𝐼) , for each 𝑘 ∈ 𝑁,

lim{𝑑 (𝑋𝑛𝑘 , 𝐽𝑛 )} 𝑘

𝑝𝑛𝑘

= 0,

for some 𝐽𝑛 ∈ 𝑅 (𝐼) ,

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for each 𝑛 ∈ 𝑁. For the class of sequences (2 𝑐𝐹𝑅 )0 (𝑝), 𝐿 = 𝐿 𝑘 = 𝐽𝑛 = 0. We define 𝑚𝐹 (𝑝) = 2 𝑐𝐹𝑅 (𝑝) ∩ (2 ℓ∞ )𝐹 (𝑝), (𝑚0 )𝐹 (𝑝) = 𝑅 (2 𝑐𝐹 )0 (𝑝) ∩ (2 ℓ∞ )𝐹 (𝑝).

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3 and for 𝑛 ≠ 𝑘,

3. Main Results Theorem 2. Let ⟨𝑝𝑛𝑘 ⟩ be bounded. Then, the classes of sequences (2 ℓ∞ )𝐹 (𝑝), 2 𝑐𝐹𝑅 (𝑝), (2 𝑐𝐹𝑅 )0 (𝑝), 𝑚𝐹 (𝑝), and (𝑚0 )𝐹 (𝑝) are complete metric spaces with respect to the metric defined by 𝑓 (𝑋, 𝑌) = sup{𝑑 (𝑋𝑛𝑘 , 𝑌𝑛𝑘 )}

𝑝𝑛𝑘 /𝐻

𝑛,𝑘

, (12)

where 𝐻 = max (1, sup 𝑝𝑛𝑘 ) . 𝑖 Proof. We prove the result for 2 ℓ∞ (𝑝). Let ⟨𝑋𝑛𝑘 ⟩ be a Cauchy sequence in (2 ℓ∞ )𝐹 (𝑝). Then, for a given 𝜀 > 0, there exists 𝑚0 ∈ 𝑁 such that 𝑗

𝑖 𝑓 (𝑋𝑛𝑘 , 𝑋𝑛𝑘 ) < 𝜀 < 𝜀𝑝𝑛𝑘 /𝐻,

∀𝑖, 𝑗 ≥ 𝑚0 ,

𝑡 + 2, { { 𝑌𝑛𝑘 (𝑡) = {−𝑡, { {0,

Theorem 4. The spaces (2 ℓ∞ )𝐹 (𝑝), (2 𝑐0 )𝐹 (𝑝), (2 𝑐0𝑅 )𝐹 (𝑝), and (𝑚0 )𝐹 (𝑝) are solid. Proof. Consider the sequence space (2 ℓ∞ )𝐹 (𝑝). Let ⟨𝑋𝑛𝑘 ⟩ ∈ (2 ℓ∞ )𝐹 (𝑝), and let ⟨𝑌𝑛𝑘 ⟩ be such that 𝑑(𝑌𝑛𝑘 , 0) ≤ 𝑑(𝑋𝑛𝑘 , 0). The result follows from the inequality 𝑝𝑛𝑘

(13)

∞

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Then, ⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑐𝐹 (𝑝), but ⟨𝑌𝑛𝑘 ⟩ ∉ 2 𝑐𝐹 (𝑝). Hence, 2 𝑐𝐹 (𝑝) is not symmetric. Similarly, it can be established that the other spaces are also not symmetric.

𝑗

𝑖 󳨐⇒ 𝑑 (𝑋𝑛𝑘 , 𝑋𝑛𝑘 ) < 𝜀, ∀𝑖, 𝑗 ≥ 𝑚0

for − 2 ≤ 𝑡 ≤ −1, for − 1 ≤ 𝑡 ≤ 0, otherwise.

{𝑑 (𝑌𝑛𝑘 , 0)}

𝑝𝑛𝑘

≤ {𝑑 (𝑋𝑛𝑘 , 0)}

.

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󳨐⇒ ⟨𝑋𝑛𝑘 ⟩𝑗=1 is a Cauchy sequence of fuzzy

Hence, the space (2 ℓ∞ )𝐹 (p) is solid. Similarly, the other spaces are also solid.

real number for each 𝑛, 𝑘 ∈ 𝑁.

Property 2. The spaces 2 𝑐𝐹 (𝑝), (2 𝑐𝑅 )𝐹 (𝑝), and 𝑚𝐹 (𝑝) are not monotone and hence are not solid.

𝑗

Since 𝑅(𝐼) is complete, there exist fuzzy numbers 𝑋𝑛𝑘 𝑗 such that lim𝑗 → ∞ 𝑋𝑛𝑘 = 𝑋𝑛𝑘 , for each 𝑛, 𝑘 ∈ 𝑁. Taking 𝑗 → ∞ in (13), we have 𝑖 , 𝑋𝑛𝑘 ) < 𝜀. 𝑓 (𝑋𝑛𝑘

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Using the triangular inequality 𝑗

𝑗

𝑓 (⟨𝑋𝑛𝑘 ⟩ , 0) ≤ 𝑓 (⟨𝑋𝑛𝑘 ⟩ , ⟨𝑋𝑛𝑘 ⟩) + 𝑓 (⟨𝑋𝑛𝑘 ⟩ , 0) , (15) we have ⟨𝑋𝑛𝑘 ⟩ ∈ (2 ℓ∞ )𝐹 (𝑝). Hence, (2 ℓ∞ )𝐹 (𝑝) is complete. Property 1. The space (2 ℓ∞ )𝐹 (𝑝) is symmetric, but the spaces 𝑅 𝑅 2 𝑐𝐹 (𝑝), 2 𝑐𝐹 (𝑝), (2 𝑐𝐹 )0 (𝑝), (2 𝑐𝐹 )0 (𝑝), 𝑚𝐹 (𝑝), and (𝑚0 )𝐹 (𝑝) are not symmetric. Proof. Obviously the space (2 ℓ∞ )𝐹 (𝑝) is symmetric. For the other spaces, consider the following example. Example 3. Consider the sequence space 2 𝑐𝐹 (𝑝). Let 𝑝1𝑘 = 2, for all 𝑘 ∈ 𝑁 and 𝑝𝑛𝑘 = 3, otherwise. Let the sequence ⟨𝑋𝑛𝑘 ⟩ be defined by 𝑋1𝑘 = 1 ∀𝑘 ∈ 𝑁,

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𝑡 + 2, for − 2 ≤ 𝑡 ≤ −1, { { 𝑋𝑛𝑘 (𝑡) = {−𝑡, for − 1 ≤ 𝑡 ≤ 0, { 0, otherwise. {

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and for 𝑛 > 1,

Example 5. Consider the sequence space 2 𝑐𝐹 (𝑝). Let 𝑝𝑛𝑘 = 3 for 𝑛 + 𝑘 even and 𝑝𝑛𝑘 = 2, otherwise. Let 𝐽 = {(𝑛, 𝑘) : 𝑛 + 𝑘 is even} ⊆ 𝑁 × 𝑁. Let ⟨𝑋𝑛𝑘 ⟩ be defined by the following: for all 𝑛, 𝑘 ∈ 𝑁, 𝑋𝑛𝑘 (𝑡) 𝑡 + 3, { { = {𝑛𝑡(3𝑛 − 1)−1 + 3𝑛(3𝑛 − 1)−1 , { {0,

for − 3 ≤ 𝑡 ≤ −2, for − 2 ≤ 𝑡 ≤ −1 + 𝑛−1 , otherwise. (21)

Then, ⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑐𝐹 (𝑝). Let ⟨𝑌𝑛𝑘 ⟩ be the canonical preimage of ⟨𝑋𝑛𝑘 ⟩𝐽 for the subsequence 𝐽 of 𝑁. Then, 𝑋 , for (𝑛, 𝑘) ∈ 𝐽, 𝑌𝑛𝑘 = { 𝑛𝑘 0, otherwise.

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Then, ⟨𝑌𝑛𝑘 ⟩ ∉ 2 𝑐𝐹 (𝑝). Thus, 2 𝑐𝐹 (𝑝) is not monotone. Similarly, the other spaces are also not monotone. Hence, the spaces 𝑅 2 𝑐𝐹 (𝑝), (2 𝑐 )𝐹 (𝑝), and 𝑚𝐹 (𝑝) are not solid. Property 3. The spaces (2 ℓ∞ )𝐹 (𝑝), 2 𝑐𝐹 (𝑝), (2 𝑐0 )𝐹 (𝑝), (2 𝑐𝑅 )𝐹 (𝑝), (2 𝑐0𝑅 )𝐹 (𝑝), 𝑚𝐹 (𝑝), and (𝑚0 )𝐹 (𝑝) are not convergence-free. The result follows from the following example. Example 6. Consider the sequence space 2 𝑐𝐹 (𝑝). Let 𝑝1𝑘 = 1, for all 𝑘 ∈ 𝑁, 𝑝𝑛𝑘 = 3, otherwise. Consider the sequence ⟨𝑋𝑛𝑘 ⟩ defined by

Let ⟨𝑌𝑛𝑘 ⟩ be a rearrangement of ⟨𝑋𝑛𝑘 ⟩ defined by 𝑌𝑛𝑛 = 1,

Proof. The result follows from the following example.

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𝑋1𝑘 = 0,

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and for other values, 𝑡 + 2, { { 𝑋𝑛𝑘 (𝑡) = {−𝑛𝑡(𝑛 + 1)−1 + (𝑛 + 1)−1 , { {0,

for − 2 ≤ 𝑡 ≤ −1, for − 1 ≤ 𝑡 ≤ 𝑛−1 , otherwise. (24)

Let the sequence ⟨𝑌𝑛𝑘 ⟩ be defined by 𝑌1𝑘 = 0,

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and for other values, 1, { { 𝑌𝑛𝑘 (𝑡) = {(𝑛 − 1) (𝑛 − 1)−1 , { {0,

for 0 ≤ 𝑡 ≤ 1, for 1 ≤ 𝑡 ≤ 𝑛, otherwise.

(26)

Then, ⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑐𝐹 (𝑝), but ⟨𝑌𝑛𝑘 ⟩ ∉ 2 𝑐𝐹 (𝑝). Hence, the space is not convergence-free. Similarly, the other spaces are also not convergence-free. 2 𝑐𝐹 (𝑝)

Theorem 7. 𝑍(𝑝) ⊆ (2 ℓ∞ )𝐹 (𝑝), for 𝑍 = (2 𝑐𝑅 )𝐹 , (2 𝑐0𝑅 )𝐹 , 𝑚𝐹 , (𝑚0 )𝐹 . The inclusions are strict. Proof. Since convergent sequences are bounded, the proof is clear. Theorem 8. Let 0 < 𝑞𝑖𝑗 ≤ 𝑝𝑖𝑗 < ∞, for all 𝑖, 𝑗 ∈ 𝑁. Then, 𝑍(𝑝) ⊆ 𝑍(𝑞) for 𝑍 = 2 𝑐𝐹 , (2 𝑐𝑅 )𝐹 , (2 𝑐𝐹 )0 , (2 𝑐0𝑅 )𝐹 , 𝑚𝐹 , (𝑚0 )𝐹 . Proof. Consider the sequence spaces 2 𝑐𝐹 (𝑝) and 2 𝑐𝐹 (𝑞). Let ⟨𝑋𝑛𝑘 ⟩ ∈ 2 𝑐𝐹 (𝑝). Then, {𝑑(𝑋𝑛𝑘 , 𝐿)}𝑝𝑛𝑘 < 𝜀, for all 𝑛 ≥ 𝑛0 , 𝑘 ≥ 𝑘0 . The result follows from the inequality {𝑑(𝑋𝑛𝑘 , 𝐿)}𝑞𝑛𝑘 ≤ {𝑑(𝑋𝑛𝑘 , 𝐿)}𝑝𝑛𝑘 . Theorem 9. The spaces (2 ℓ∞ )𝐹 (𝑝), 2 𝑐𝐹 (𝑝), (2 𝑐0 )𝐹 (𝑝), (2 𝑐𝑅 )𝐹 (𝑝), (2 𝑐0𝑅 )𝐹 (𝑝), 𝑚𝐹 (𝑝), and (𝑚0 )𝐹 (𝑝) are sequence algebras. Proof. Consider the sequence space (2 𝑐0 )𝐹 (𝑝). Let ⟨𝑋𝑛𝑘 ⟩, ⟨𝑌𝑛𝑘 ⟩ ∈ (2 𝑐0 )𝐹 (𝑝). Then, the result follows immediately from the inequality 𝑝𝑛𝑘

{𝑑 (𝑋𝑛𝑘 𝑌𝑛𝑘 , 0)}

𝑝𝑛𝑘

≤ {𝑑 (𝑋𝑛𝑘 , 0)}

𝑝𝑛𝑘

{𝑑 (𝑌𝑛𝑘 , 0)}

.

(27)

Acknowledgment The author’s work is supported by UGC Project no. F. 5294/2009-10 (MRP/NERO).

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[3] F. M´oricz, “Extensions of the spaces 𝑐 and 𝑐0 from single to double sequences,” Acta Mathematica Hungarica, vol. 57, no. 1-2, pp. 129–136, 1991. [4] B. C. Tripathy, “Statistically convergent double sequences,” Tamkang Journal of Mathematics, vol. 34, no. 3, pp. 231–237, 2003. [5] M. Basarir and O. Sonalcan, “On some double sequence spaces,” Journal of Indian Academy of Mathematics, vol. 21, no. 2, pp. 193– 200, 1999. [6] H. Nakano, “Modulared sequence spaces,” Proceedings of the Japan Academy, vol. 27, pp. 508–512, 1951. [7] S. Simons, “The sequence spaces 𝑙(𝑝𝜈 ) and 𝑚(𝑝𝜈 ),” Proceedings of the London Mathematical Society 3, vol. 15, no. 1, pp. 422–436, 1965. [8] B. K. Tripathy and S. Nanda, “Absolute value of fuzzy real numbers and fuzzy sequence spaces,” Journal of Fuzzy Mathematics, vol. 8, no. 4, pp. 883–892, 2000. [9] B. Choudhary and B. C. Tripathy, “On fuzzy real-valued ℓ(𝑝)𝐹 sequence spaces,” in Proceedings of the International 8th Joint Conference on Information Sciences (10th International Conference on Fuzzy Theory and Technology), pp. 184–190, Salt Lake City, Utah, USA, July 2005. [10] B. C. Tripathy and A. Baruah, “New type of difference sequence spaces of fuzzy real numbers,” Mathematical Modelling and Analysis, vol. 14, no. 3, pp. 391–397, 2009. [11] B. C. Tripathy and A. Baruah, “N¨orlund and Riesz mean of sequences of fuzzy real numbers,” Applied Mathematics Letters, vol. 23, no. 5, pp. 651–655, 2010. [12] B. C. Tripathy and A. Baruah, “Lacunary statically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers,” Kyungpook Mathematical Journal, vol. 50, no. 4, pp. 565–574, 2010. [13] B. C. Tripathy, A. Baruah, M. Et, and M. Gungor, “On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers,” Iranian Journal of Science and Technology A, vol. 36, no. 2, pp. 147–155, 2012. [14] B. C. Tripathy and A. J. Dutta, “On fuzzy real-valued double 𝑝 sequence space 2 𝑙𝐹 ,” Mathematical and Computer Modelling, vol. 46, no. 9-10, pp. 1294–1299, 2007. [15] B. C. Tripathy and A. J. Dutta, “Bounded variation double sequence space of fuzzy real numbers,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 1031–1037, 2010. [16] B. C. Tripathy and A. J. Dutta, “On I-acceleration convergence of sequences of fuzzy real numbers,” Journal Mathematical Modelling and Analysis, vol. 17, no. 4, pp. 549–557, 2012. [17] B. C. Tripathy and S. Borgogain, “Some classes of difference sequence spaces of fuzzy real numbers defined by Orlicz function,” Advances in Fuzzy Systems, vol. 2011, Article ID 216414, 6 pages, 2011. [18] B. C. Tripathy and P. C. Das, “On convergence of series of fuzzy real numbers,” Kuwait Journal of Science & Engineering, vol. 39, no. 1, pp. 57–70, 2012. [19] O. Kaleva and S. Seikkala, “On fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 12, no. 3, pp. 215–229, 1984. [20] J. Hanˇcl, L. Miˇs´ık, and J. T. T´oth, “Cluster points of sequences of fuzzy real numbers,” Soft Computing, vol. 14, no. 4, pp. 399–404, 2010. [21] D. Zhang, “A natural topology for fuzzy numbers,” Journal of Mathematical Analysis and Applications, vol. 264, no. 2, pp. 344– 353, 2001.

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