SOME PROPERTIES OF A TRIPLE SEQUENCE SPACES OF FUZZY REAL NUMBERS BY DOUBLE ORLICZ FUNCTIONS

Аннотация

INTRODUCTION Zadeh established the idea of fuzzy set theory [1]. Based on this, other authors have introduced and examined sequences of fuzzy numbers, studying significant properties. In Bromwich, the work on the double sequence of real numbers was discovered [2]. In addition, a number of people, including Moricz [3], Basarir and Sonalncan [4], and others, exacted the double sequence. 𝐸3, denoting the family of all ℝ or 𝕔 triple sequences, is used throughout this article. 𝐹 A triple sequence in [5][6] is a function from (ℕ)to( ℝ or 𝕔 ). Using a function Υ where Υ = (Υ (𝔛).Υ (𝔐)). novel outcomes of triple sequence spaces would be examined by double Orlicz function. 1 2 Assume that (𝔛.𝔐) =(𝔛 .. 𝔐 ). symbolizes a triple infinite array of elements with the name 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 (𝔛 ) .(𝔐 ). 𝔛 = (𝔛 ) be an infinite array of elements 𝔛 , and 𝔐= (𝔐 ) be an endless 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 array of elements 𝔐 . respectively. 𝔥.𝔘.𝔑 Prof. Wlayshaw Roman Orlicz, a Polish scholar, established the concept of the Orlicz function and gave it his name; consequently, he created the Orlicz space [7]. 2)Definitions and preliminaries Definition 2.1 Assume that the sequence 𝔛= (𝔛 ). 𝔐= ( 𝔐 )are Cauchy, then (𝔛.𝔐) = (𝔛 . 𝔐 ) is 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 said to be a triple Cauchy sequence if for every 𝜖 > 0 there exists 𝑁 ∈ ℕ such that 𝒹̅((𝔛 .𝔛 ).( 𝔐 .𝔐 )) 0. { } where 𝒹̅(𝔛 .0̅) (𝔛 )∈ 𝐸3:𝑠𝑢𝑝 {Υ ( 𝔥.𝔘.𝔑 ) } 0. { } and 𝒹̅(𝔐 .0̅) (𝔐 )∈ 𝐸3:𝑠𝑢𝑝 {Υ ( 𝔥.𝔘.𝔑 )} 0. { } (𝒸)3Υ= 𝐹 https://www.theamericanjournals.com/index.php/tajssei 48 THE USA JOURNALS THE AMERICAN JOURNAL OF SOCIAL SCIENCE AND EDUCATION INNOVATIONS (ISSN- 2689-100X) VOLUME 06 ISSUE10 𝒹̅(𝔛 .ℓ ) 𝒹̅(𝔐 .ℓ ) (𝔛 .𝔐 )∈𝐸3:𝑙𝑖𝑚 {Υ ( 𝔥.𝔘.𝔑 1 ) ∨Υ ( 𝔥.𝔘.𝔑 2 )}= 0 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 𝐹 𝔥.𝔘.𝔑 1 𝜌 2 𝜌 . for some 𝜌 > 0. { } where 𝒹̅(𝔛 .ℓ ) (𝔛 )∈𝐸3:𝑙𝑖𝑚 {Υ ( 𝔥.𝔘.𝔑 1 )}= 0 (𝒸)3Υ = 𝔥.𝔘.𝔑 𝐹 𝔥.𝔘.𝔑 1 𝜌 𝐹 1 for some 𝜌 > 0. { } and 𝒹̅(𝔐 .ℓ ) (𝔐 )∈ 𝐸3:𝑙𝑖𝑚 {Υ ( 𝔥.𝔘.𝔑 2 )}= 0 (𝒸)3Υ = 𝔥.𝔘.𝔑 𝐹 𝔥.𝔘.𝔑 2 𝜌 . 𝐹 2 for some 𝜌 > 0. { } (𝒸 )3Υ= 0 𝐹 𝒹̅(𝔛 .0̅) 𝒹̅(𝔐 .0̅) (𝔛 .𝔐 )∈ 𝐸3:𝑙𝑖𝑚 {Υ ( 𝔥.𝔘.𝔑 ) ∨Υ ( 𝔥.𝔘.𝔑 )}= 0 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 𝐹 𝔥.𝔘.𝔑 1 𝜌 2 𝜌 . for some 𝜌 > 0. { } where 𝒹̅(𝔛 .0̅) (𝔛 )∈ 𝐸3:𝑙𝑖𝑚 {Υ ( 𝔥.𝔘.𝔑 ) }= 0 (𝒸 )3Υ 𝔥.𝔘.𝔑 𝐹 𝔥.𝔘.𝔑 1 𝜌 . 0 𝐹 1 for some 𝜌 > 0. { } and 𝒹̅(𝔐 .0̅) (𝔐 )∈ 𝐸3:𝑙𝑖𝑚 {Υ ( 𝔥.𝔘.𝔑 )} = 0 (𝒸 )3Υ 𝔥.𝔘.𝔑 𝐹 𝔥.𝔘.𝔑 2 𝜌 . 0 𝐹 2 𝜌 > 0. { } Moreover, The triple sequence classes are defined by us. (𝑚)3 = (𝒸)3(Υ) ∩ (𝑙 )3(Υ). 𝐹 𝐹 ∞ 𝐹 (𝑚 )3 = (𝒸 )3(Υ) ∩ (𝑙 )3(Υ). 0 𝐹 0 𝐹 ∞ 𝐹 3) Main Results Theorem 3.1: Let (𝓅 ) be bounded. Then the classes of triple sequences (𝑙 )3(Υ .𝓅) is complete 𝔥.𝔘.𝔑 ∞ 𝐹 metric spaces with respect to the distance defined by 𝐺((𝔛.𝔐).(𝔅.𝔔))= https://www.theamericanjournals.com/index.php/tajssei 49 THE USA JOURNALS THE AMERICAN JOURNAL OF SOCIAL SCIENCE AND EDUCATION INNOVATIONS (ISSN- 2689-100X) VOLUME 06 ISSUE10 𝑖𝑛𝑓{𝜌 𝓅𝔥. 𝒥 𝔘.𝔑 > 0:𝑠𝑢𝑝 {(Υ ( 𝒹̅(𝔛𝔥.𝔘.𝔑.𝔐𝔥.𝔘.𝔑) )) ∨(Υ ( 𝒹̅(𝔅𝔥.𝔘.𝔑.𝔔𝔥.𝔘.𝔑) ))} ≤ 1} 𝔥.𝔘.𝔑 1 2 𝜌 𝜌 𝒥 = max (1.2𝑇−1) where 𝓅𝔥.𝔘.𝔑 𝒹̅(𝔛 𝔥.𝔘.𝔑 .𝔐 𝔥.𝔘.𝔑 ) 𝐺(𝔛.𝔐)= 𝑖𝑛𝑓{𝜌 𝒥 > 0: 𝑠𝑢𝑝 (Υ ( )) ≤ 1} 𝔥.𝔘.𝔑 1 𝜌 𝓅𝔰.𝔯.𝓋 𝒹̅(𝔅 𝔥.𝔘.𝔑 .𝔔 𝔥.𝔘.𝔑 ) 𝐺(𝔅.𝔔)= 𝑖𝑛𝑓{𝜌 𝒥 > 0: 𝑠𝑢𝑝 (Υ ( )) ≤ 1} 𝔥.𝔘.𝔑 2 𝜌 Proof : Let us consider the case (𝑙 )3(Υ.𝓅) and the other cases can be established next similar ∞ 𝐹 techniques.. Let (𝔛𝑖).(𝔐𝑖) be any Cauchy sequences in (𝑙 )3(Υ .𝓅). (𝑙 )3(Υ .𝓅) respectively, hence (𝔛𝑖.𝔐𝑖 ) = ∞ 𝐹 1 ∞ 𝐹 2 (𝔛𝑖 .𝔐𝑖 ) be a triple Cauchy sequence 𝔥.𝔘.𝔑 𝔥.𝔘.𝔑 Let 𝜖 > 0, 𝔛 .𝑟 > 0 be fixed. Then ∀ there exists a positive integer N such that 𝐺 (𝔛𝑖.𝔛𝑗)< 𝜖 and 0 Υ1 𝑟𝔛0 𝐺 (𝔐𝑖.𝔐𝑗) < 𝜖 . for 𝑖.𝑗 ≥ 𝑁. and