Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

436 

 
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Bounded Closed Interval-Valued Decagonal Fuzzy Number and its Application 

 

T. Gunasekar1, J. Thiravidarani2, S. C. Premila3, Prakaash A. S4, T. N. M. Malini Mai5, M. 

Suba6 

1,2Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Chennai – 

600062, Tamil Nadu, India. 
3Department of Mathematics, Saveetha Engineering College, (Autonomous) Thandalam, Chennai, Tamil Nadu, India. 

4Department of Mathematics, Panimalar Engineering College Chennai, Tamil Nadu, India.  
5 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical 

and Technical Sciences, Chennai, Tamil Nadu, India. 
6 Department of Mathematics, S.A. Engineering College (Autonomous), Chennai, Tamil Nadu, India. 

Email: tguna84@gmail.com1, jtrrani@gmail.com2, premilac@saveetha.ac.in3, prakaashphd333@gmail.com4,  

malinimai333@gmail.com5 , suba.hari87@gmail.com6  

 

Article History: 

Received: 01-08-2024 

Revised: 22-09-2024 

Accepted: 05-10-2024 

Abstract:  

In this paper, we introduced the notation of Algebraic operations of bounded closed 

interval-valued decagonal fuzzy number. To overcome the uncertainty here we used 

fuzzy numbers. Many researchers have focused their research with Triangular and 

Trapezoidal fuzzy numbers, in some cases it is not fit to the problem when the vagueness 

arises in ten different opinions so our motto is to develop the theorems for bounded 

closed interval using Algebraic operations such as Addition, Subtraction, Multiplication 

and Division via Decagonal Fuzzy Number. 

Keywords: Fuzzy sets, Fuzzy numbers, Decagonal Fuzzy numbers, Bounded Closed 

interval 

1. Introduction 

The fuzzy set theory was first introduced by Zadeh which helps to deal uncertainty in the 

problem [1]. Gorzalczany [2] and Turksen [3] introduced the notation of fuzzy set theory. Dubois and 

Prade have defined fuzzy numbers as a fuzzy subset of the real line [9]. A fuzzy number is a multi-

valued quantity whose value is precise, rather than a single-valued quantity. Most of the researchers 

have used triangular and trapezoidal fuzzy numbers to handle imprecision in real life situations [10, 

11, 12, 15-18]. Hexagonal, heptagonal, nonagonal, decagonal fuzzy numbers have also been 

introduced to tackle the vagueness [5, 13, 14, 19, 20]. Wang and Li [4] in 1998, defined interval-valued 

fuzzy numbers and gave their extended operations. Karthik et.al [6] developed a fuzzy decision-

making system using triangular fuzzy numbers to study the impact of pesticides on human health. 

Selvaraj et.al proposed linear and nonlinear hexagonal fuzzy number with symmetry and asymmetry 

for solving transportation problem, and their respective alpha cuts also have been derived [8]. Karthik 

et.al introduced heptagonal fuzzy number for both symmetrical and asymmetrical structures, derived 

alpha cuts to solve assignment problem. Felix et.al [5] has introduced a new operation on a decagonal 

fuzzy number (DFN) under certain linguistic environment is presented. 

              This paper consists of four sections. The first section describes the introduction and basic 

definitions of fuzzy. We introducing the definition for Bounded closed interval-valued Decagonal 

fuzzy numbers in the second section. Section three seeks about the theorem for Bounded closed 

mailto:tguna84@gmail.com1
mailto:jtrrani@gmail.com2
mailto:premilac@saveetha.ac.in3
mailto:prakaashphd333@gmail.com4
mailto:malinimai333@gmail.com5
mailto:suba.hari87@gmail.com6


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

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interval-valued Decagonal fuzzy numbers. Finally, the last section is the conclusion based on theorems 

part.   

2.  Preliminaries  

                     This section describes the preliminary definitions of fuzzy sets,fuzzy numbers and 

Decagonal fuzzy numbers. 

2.1 Fuzzy Set 

A fuzzy set A% is a subset of a universe of discourse X, which is characterized by a membership 

function ( )
A

 %  representing a mapping
A

%:  0,1X → . The function value of ( )
A

 % is called the 

membership value, which represents the degree of truth that   is an element of the fuzzy set A%.  

2.2   Fuzzy numbers 

     A fuzzy set A% defined on the set of real numbers R is said to be a fuzzy number and its 

membership function : [0,1]A R →%  has the following characteristics, 

(i) A% is convex. 

 ( )1 2 1 2(1 ) min( ( ), ( )),
A A A

       + − % % %    1 2, , 0,1 .     
 

(ii) A% is normal if max ( ) 1
A

  =% .  

(iii) A% is piecewise continuous. 

2.3   Decagonal Fuzzy Number 

A Decagonal fuzzy number  D
~

=(a,b,c,d,e,f,g,h,i,j) and the membership function is defined as  

𝜇�̃�(𝜃)=

{
 
 
 
 
 
 
 

 
 
 
 
 
 
 

1

4

(𝜃−a)

(b−a)
     a ≤ 𝜃 ≤ b

1

4
+
1

4

(𝜃−b)

(c−b)
  b ≤ 𝜃 ≤ c

1

2
+
1

4

(𝜃−c)

(d−c)
  c ≤ 𝜃 ≤ d

3

4
+
1

4

(𝜃−d)

(e−d)
  d ≤ 𝜃 ≤ e

1                     e ≤ 𝜃 ≤ f

1 −
1

4

(𝜃−f)

(g−f)
  f ≤ 𝜃 ≤ g

3

4
−
1

4

(𝜃−g)

(h−g)
  g ≤ 𝜃 ≤ h

1

2
−
1

4

(𝜃−h)

(i−h)
  h ≤ 𝜃 ≤ i

1

4

(j−𝜃)

(j−i)
     i ≤ 𝜃 ≤ j

0                   otherwise}
 
 
 
 
 
 
 

 
 
 
 
 
 
 

 

3.  Bounded closed interval-valued Decagonal Fuzzy number 

The definitions of Bounded closed interval-valued Decagonal Fuzzy number are given below 

 



Communications on Applied Nonlinear Analysis 

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Definition 3.1  

𝑨 = [�̃̃�𝑳, �̃̃�𝑼] 𝒐𝒓 [�̃̃�𝟏, �̃̃�𝟐] ∈ [𝑰]𝑹 ,If A is a interval-valued Decagonal fuzzy set of R is called a 

bounded closed interval-valued decagonal fuzzy number (BCIDFN). If  �̃̃�𝐿 , �̃̃�𝑈 ∈ 𝐵𝑐[𝑅]. The 

interval-valued decagonal fuzzy number �̃̃� has two elements. Lower fuzzy number �̃̃�𝐿 and upper 

fuzzy number   �̃̃�𝑈. 

The interval valued decagonal fuzzy numbers �̃̃� can be represented as  

�̃̃� = [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿 ,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈
] 

  Where  (𝛼1
1.1 ≤ 𝛼2

1.2 ≤ 𝛼3
1.3 ≤ 𝛼4

1.4 ≤ 𝛼5
1.5 ≤ 𝛼6

1.6 ≤ 𝛼7
1.7 ≤ 𝛼8

1.8 ≤ 𝛼9
1.9 ≤ 𝛼10

1.0) 

(𝛼1
2.1 ≤ 𝛼2

2.2 ≤ 𝛼3
2.3 ≤ 𝛼4

2.4 ≤ 𝛼5
2.5 ≤ 𝛼6

2.6 ≤ 𝛼7
2.7 ≤ 𝛼8

2.8 ≤ 𝛼9
2.9 ≤ 𝛼10

2.10) 

0 < �̂��̃̃�𝐿 ≤ �̂��̃̃�𝑈 ≤ 1    dem  �̃̃�𝐿  ⊂   �̃̃�𝑈 de   

�̃̃�𝐿 , �̃̃�𝑈 ∈ 𝐵 ⊂  [𝑅] 

The set of all BCIDFNS on R is denoted by 𝐵 ⊂  [𝑅]. 

Definition 3.2 

 Let  

�̃̃� = [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿 ,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈
] ∈ 𝐵𝐶[𝑅] 

If   Ã̃  is called a positive bounded closed  interval- valued decagonal fuzzy number (BCIDFNS) if 

𝐴𝑈(𝑥) = 0  each 𝑥 ≤ 0. All positive BCIDFNS is denoted by 𝐵 ⊂  [𝑅+] 

If   Ã̃  is called  negative bounded closed  interval- valued decagonal fuzzy number (BCIDFNS) if 

𝐴𝐿(𝑥) = 0  each 𝑥 ≥ 0. All negative BCIDFNS is denoted by   𝐵 ⊂ [𝑅−]. 

Definition 3.3 (Algebraic operations on BCIDFNS)  

Let ⨷∈ {⨁,⊝,⨂, ⊘} be the binary operation on R for  𝐴1̃̃, 𝐵1̃̃ ∈ 𝐵 ⊂ [𝑅] 

𝐴1̃̃⨷𝐵1̃̃ is defined as follows, 

where   

𝐴1̃̃ = [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿 ,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈
] 

𝐵1̃̃ = [
(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ,

(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈
] 

(𝐴1̃̃⨷𝐵1̃̃) (𝑧) =
𝑉

𝑍 = 𝑋 ⨷ 𝑌
(𝐴1̃̃ (𝑋) ⩘ 𝐵1̃̃ (𝑦)) , 𝑧 ∈ 𝑅 



Communications on Applied Nonlinear Analysis 

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where    𝐴1̃̃ (𝑋) ⩘ 𝐵1̃̃ (𝑦) = [𝐴1̃̃(𝑥), 𝐴2̃̃(𝑥)] ⩘ [𝐵1̃̃(𝑥), 𝐵2̃̃(𝑥)] 

Theorem 1 

Let A1  ̃̃, B1̃̃ ∈ 𝐵 ∈ 𝐶[𝑅] then 

(𝑖)A1  ̃̃ ⨷ B1̃̃ ∈ 𝐵 𝐶[𝑅] 𝑓𝑜𝑟 ⨷∈ {⨁,⊝,⨂, }                             

(𝑖𝑖)A1  ̃̃ ∅ B1̃̃ ∈ 𝐵 𝐶 [𝑅] 𝑓𝑜𝑟 𝐵 ∈ 𝐵 𝐶 [𝑅+] 𝑜𝑟 𝐵 ∈ 𝐵 𝐶 [𝑅−] Where  

A1̃̃ = [(𝑎1
𝐿 , 𝑎2

𝐿 , 𝑎3
𝐿 , 𝑎4

𝐿 , 𝑎5
𝐿 , 𝑎6

𝐿 , 𝑎7
𝐿 , 𝑎8

𝐿 , 𝑎9
𝐿 , 𝑎10

𝐿 )(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)] 

and   

B1̃̃ = [((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)(𝑏1
𝑈, 𝑏2

𝑈, 𝑏3
𝑈 , 𝑏4

𝑈, 𝑏5
𝑈 , 𝑏6

𝑈, 𝑏7
𝑈, 𝑏, 𝑏9

𝑈, 𝑏10
𝑈 )] 

Proof: (i) For ⨷∈ {⨁,⊝,⨂,} 𝑎𝑛𝑑 𝑒𝑎𝑐ℎ 𝑍 ∈ 𝑅 

(A1̃̃  ⨷ B1̃̃) (𝑍)

=
𝑉

𝑍 = 𝑋⨷ 𝑌 ⬚

⩘((𝛽1
1.1,𝛽2

1.2,𝛽3
1.3,𝛽4

1.4,𝛽5
1.5,𝛽6

1.6,𝛽7
1.7,𝛽8

1.8,𝛽9
1.9,𝛽10

1.0)(𝛽1
2.1,𝛽2

2.2,𝛽3
2.3,𝛽4

2.4,𝛽5
2.5,𝛽6

2.6,𝛽7
2.7,𝛽8

2.8,𝛽9
2.9,𝛽10

2.10)(𝑦)

((𝛼1
1.1,𝛼2

1.2,𝛼3
1.3,𝛼4

1.4,𝛼5
1.5,𝛼6

1.6,𝛼7
1.7,𝛼8

1.8,𝛼9
1.9,𝛼10

1.0) (𝑥)

 

=

𝑉
𝑍 = 𝑋 ⨷ 𝑌

(((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)(𝑥) ⩘  ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)(𝑦)) ,

𝑉
𝑍 = 𝑋 ⨷ 𝑌

((𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)(𝑥) ⩘ (𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑦))
 

= (
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)(𝑧),

((𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑧))
) 

On the other hand  

(A1̃̃⨷B1̃̃) (𝑍)

= (
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⨷ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)(𝑧)

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨷ (𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑧)
) 

Hence  

(A1̃̃⨷B1̃̃)
𝐿

= (((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)

⨷ (𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)) 

(A2̃̃⨷B2̃̃)
𝑈

= ((𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)

⨷ (𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)) 

That 𝐴𝐿⨷ �̃̃�𝐿 , 𝐴𝑈⨷ �̃̃�𝑈 ∈ 𝐵 𝐶(𝑅)  𝑡ℎ𝑖𝑠 𝑠ℎ𝑜𝑤𝑠 𝐴 ⨷ �̃̃� ∈ 𝐵 𝐶 [𝑅] 

(𝑖𝑖) (A1̃̃⨸B1̃̃) ∈ 𝐵 𝐶 [𝑅] 𝑓𝑜𝑟 𝐵 ∈ 𝐵 𝐶[𝑅+] 𝑜𝑟 𝐵 ∈ 𝐵 𝐶[𝑅−]   



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(A1̃̃⨸B1̃̃) (𝑍)

=  
𝑉

𝑍 = 𝑋 ⨸ 𝑌
(((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)(𝑥)     

⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑦)) 

=
𝑉

𝑍 = 𝑋 ⨸ 𝑌

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)(𝑥) ⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)(𝑦)],

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)(𝑥) ⩘ (𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑦)]
 

=
𝑉

𝑍 = 𝑋 ⨸ 𝑌
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)(𝑥)

⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)(𝑦)], 

      
𝑉

𝑍 = 𝑋 ⨸ 𝑌
[(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)(𝑥)

⩘ (𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑦)] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)(𝑧),

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ (𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑧)
] 

On the other hand  

(A1̃̃⨸B1̃̃) (𝑧) = [(A1̃̃⨸B1̃̃)
𝐿

(𝑧), (A2̃̃⨸B2̃̃)
𝑈

(𝑧)] 

(A1̃̃⨸B1̃̃)
𝐿

= [((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0))

⨸ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)(𝑧)] 

(A2̃̃⨸B2̃̃)
𝑈

=      [(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)

⨸ (𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)(𝑧)] 

It follows that 

A1̃̃
𝐿

⨸B1̃̃
𝐿

, A2̃̃
𝑈

⨸B2̃̃
𝑈

∈ 𝐵 𝐶 (𝑅) 𝑡ℎ𝑖𝑠 𝑠ℎ𝑜𝑤𝑠 A1  ⨸ B1 ∈ 𝐵 𝐶 [𝑅] 

Theorem 2 

Let 𝐴1, 𝐵1, Γ1 ∈ 𝐵 𝐶 [𝑅] then 

𝐴1

= [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)] 

𝐵1

= [(𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)] 

Γ1

= [(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10), (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)] 

Proof: 𝐴1
𝐿
⩘ 𝐵1

𝐿
, 𝐴2

𝑈
⩘ 𝐵2

𝑈
 

(𝑖)𝐴1 + 𝐵1 = [(𝐴1 + 𝐵1)𝐿 , (𝐴2 + 𝐵2)𝑈]   = [𝐴1
𝐿
+ 𝐵1

𝐿
, 𝐴2

𝑈
+ 𝐵2

𝑈
] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

441 

 
https://internationalpubls.com 

                                                                   = [𝐵1
𝐿
+ 𝐴1

𝐿
, 𝐵2

𝑈
+ 𝐴2

𝑈
] 

                                                                   = [(𝐵1 + 𝐴1)𝐿 , (𝐵2 + 𝐴2)𝑈] 

                                                                   = 𝐵1 + 𝐴1 

𝐴 + 𝐵 = 𝐴 + 𝐵 = [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)

+ (𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0))]𝐿 

        [(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) +

(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈                     

[(𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)) +1 (𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0))]𝐿 

                  [(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)

+ (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10))]𝑈 

             = 𝐵 + 𝐴 

(ii) 𝐴1⨂𝐵1 = 𝐵1 ⨂ 𝐴1 

𝐴1⨂𝐵1 = [(𝐴1⨂𝐵1)𝐿 , (𝐴2⨂𝐵2)𝑈]   = [𝐴1
𝐿
⨂𝐵1

𝐿
, 𝐴2

𝑈
⨂𝐵2

𝑈
] 

                                                                   = [𝐵1
𝐿
⨂𝐴1

𝐿
, 𝐵2

𝑈
⨂𝐴2

𝑈
] 

                                                                   = [(𝐵1⨂𝐴1)𝐿, (𝐵2⨂𝐴2)𝑈] 

                                                                   = 𝐵1⨂𝐴1 

𝐴1⨂𝐵1      

= [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨂(𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)]𝐿 

                     [(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨂(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈 

                   =

[(𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)⨂⬚, (𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)]𝐿 

                  [(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)⨂⬚(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)]𝑈 

                       = 𝐵1⨂𝐴1 

(iii) (𝐴1⨁𝐵1)⨁Γ1 = 𝐴1⨁(𝐵1⨁Γ1) 

(𝐴1⨁𝐵1)⨁Γ1 = [(𝐴1⨁𝐵1)𝐿⨁Γ1
𝐿
, (𝐴2⨁𝐵2)𝑈⨁Γ2

𝑈
] 

= [
[(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⬚⨁(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0))]𝐿⨁(𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)𝐿 ,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⬚⨁(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈
] 

= [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿⨁(𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)𝐿,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈
] 

= [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)⬚⨁(𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]𝐿,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)⬚⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈
] 

= [𝐴1
𝐿
⨁(𝐵1⨁Γ1)𝐿 , 𝐴2

𝑈
⨁(𝐵2⨁Γ2)𝑈] 

(𝐴1⨂𝐵1)⨂Γ1 = 𝐴1⨂(𝐵1⨂Γ1) 

(𝐴1⨂𝐵1)⨂Γ1 = [(𝐴1⨂𝐵1)𝐿⨂Γ1
𝐿
, (𝐴2⨂𝐵2)𝑈⨂Γ2

𝑈
] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

442 

 
https://internationalpubls.com 

= [
[(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⬚⨂(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿⨂(𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10))𝐿 ,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⬚⨂(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈
] 

= [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨂(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0))𝐿⨂(𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)𝐿,

((𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10))𝑈
] 

= [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨂(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)⬚⨂(𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]𝐿,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)⬚⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈
] 

= [𝐴1
𝐿
⨂(𝐵1⨂Γ1)𝐿 , 𝐴2

𝑈
⨂(𝐵2⨂Γ2)𝑈] 

Theorem 3 

If 𝐴1, 𝐵1, Γ1 ∈ 𝐵 𝐶[𝑅+], then  

𝐴1 = [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿 , (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈] 

𝐵1 = [(((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0))𝐿 , (((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10))𝑈] 

               Γ1

= ((𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10))
𝐿

, ((𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10))
𝑈

 

𝐴1⨂(𝐵1⨁Γ1) = (𝐴1⨂𝐵1)⨁(𝐴1⨂𝐶1) 

𝐴1⨂(𝐵1⨁Γ1) = (𝐴1
𝐿
⨂(𝐵1⨁𝐶1)𝐿 , 𝐴2

𝑈
⨂(𝐵2⨁𝐶2)𝑈) 

= [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿 ⨂[(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)⨁(𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10, 𝛾5

1, 𝛾6
1, 𝛾7

1, 𝛾8
1, 𝛾9

1, 𝛾10
1 )]𝐿,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂[((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10, 𝛾8
2, 𝛾9

2, 𝛾10
2 )]𝑈

] 

 𝐴1⨂(𝐵1⨁Γ1)   = (𝐴1
𝐿
⨂𝐵1

𝐿
) (𝐴2

𝑈
⨂𝐵2

𝑈
)⨁(𝐴1

𝐿
⨂𝐴1

𝐿
) (𝐴2

𝑈
⨂Γ2

𝑈
) 

𝐴1⨂(𝐵1⨁Γ1)  = [(𝐴1⨂𝐵1)⨁(𝐴1⨂Γ1)]          

Theorem 4 

If  𝐴1, 𝐵1 ∈ 𝐵 𝐶[𝑅], 𝑡ℎ𝑒𝑛 𝐴1 ⩗ 𝐵1, 𝐴1 ⩘ 𝐵1 ∈ 𝐵 𝐶[𝑅] 

where          

𝐴1 = [((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿 , (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈] 

          𝐵1 = [((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿 , ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈] 

Proof:  only prove 𝐴1 ⩗ 𝐵1 ∈ 𝐵𝐶[𝑅] 𝑓𝑜𝑟 𝑍 ∈ [𝑅] 

(𝑖)(𝐴1 ⩗ 𝐵1)(𝑍) =
𝑉

𝑍 = 𝑋 ⩗ 𝑌
(𝐴1(𝑥) ⩘ 𝐵1(𝑦)) 

(𝐴1 ⩗ 𝐵1)(𝑍)

=
𝑉

𝑍 = 𝑋 ⩗ 𝑌

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝑥), (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈(𝑥)] ⩘

[((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿(𝑦), ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈(𝑦)]
 

(𝐴1 ⩗ 𝐵1)(𝑍)

=
𝑉

𝑍 = 𝑋 ⩗ 𝑌

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝑥) ⩘ (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈(𝑥)]

[((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿(𝑦) ⩘ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈(𝑦)]
 

(𝐴1 ⩗ 𝐵1)(𝑍)

=
𝑉

𝑍 = 𝑋 ⩗ 𝑌
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿(𝑥) ⩘ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿(𝑦))], 

  
𝑉

𝑍 = 𝑋 ⩗ 𝑌
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝑈(𝑥) ⩘ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝑈(𝑦)] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

443 

 
https://internationalpubls.com 

         

= (
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿(𝑥) ⩗ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿(𝑦))](𝑧),

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝑈(𝑥) ⩗ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝑈(𝑦))](𝑧)
) 

Note that  

(𝐴1 ⩗ 𝐵1)(𝑍) =

= (
[(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⩗ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿(𝑧)

[(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⩗ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)]𝑈(𝑧)
) 

(𝐴1 ⩗ 𝐵1)𝐿 = [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿

⩗ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)
𝐿
] 

(𝐴1 ⩗ 𝐵1)𝑈 = [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝑈

⩗ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)
𝑈
] 

Since 𝐴1
𝐿
, 𝐵1

𝐿
, 𝐴2

𝑈
, 𝐵2

𝑈
∈ 𝐵𝐶(𝑅)  𝑖𝑡 𝑓𝑜𝑙𝑙𝑜𝑤𝑠  𝑡ℎ𝑎𝑡 

𝐴1
𝐿
⩗ 𝐵1

𝐿
, 𝐴2

𝑈
⩗ 𝐵2

𝑈
∈ 𝐵𝐶 (𝑅)  𝑎𝑛𝑑 𝑠𝑜 (𝐴1 ⩗ 𝐵1)𝐿, (𝐴1 ⩗ 𝐵1)𝑈 ∈ 𝐵𝐶(𝑅) 

Hence 𝐴1  ⩗ 𝐵1 ∈ 𝐵𝐶[𝑅] 

(𝑖𝑖)(𝐴1 ⩘ 𝐵1)(𝑍) =
𝑉

𝑍 = 𝑋 ⩘ 𝑌
(𝐴1(𝑥) ⩘ 𝐵1(𝑦)) 

(𝐴1 ⩘ 𝐵1)(𝑍)

=
𝑉

𝑍 = 𝑋 ⩘ 𝑌

[(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝑥), (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈(𝑥)] ⩘

[((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿(𝑦), ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝑈(𝑦)]
 

(𝐴1 ⩘ 𝐵1)(𝑍)

=
𝑉

𝑍 = 𝑋 ⩘ 𝑌

[(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝑥) ⩘ (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈(𝑥)]

[((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿(𝑦) ⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝑈(𝑦)]
 

(𝐴1 ⩘ 𝐵1)(𝑍)

=
𝑉

𝑍 = 𝑋 ⩘ 𝑌
[(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿(𝑥) ⩘ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿(𝑦))], 

                 
𝑉

𝑍 = 𝑋 ⩘ 𝑌
[(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈(𝑥)

⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝑈(𝑦)] 

         

= (
[(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿 ⩘ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿](𝑧),

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈 ⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝑈](𝑧)
) 

Note that  

(𝐴1 ⩘ 𝐵1)(𝑍)

= (
[(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⩘ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿(𝑧)

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)]𝐿
𝑈
(𝑧)
) 

(𝐴1 ⩘ 𝐵1)𝐿 = [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿

⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)
𝐿
] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

444 

 
https://internationalpubls.com 

(𝐴1 ⩘ 𝐵1)𝑈 = [(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈

⩘ ((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)
𝑈
] 

Since 𝐴1
𝐿
, 𝐵1

𝐿
, 𝐴2

𝑈
, 𝐵2

𝑈
∈ 𝐵𝐶 (𝑅)  𝑖𝑡 𝑓𝑜𝑙𝑙𝑜𝑤𝑠  𝑡ℎ𝑎𝑡 

𝐴1
𝐿
⩘ 𝐵1

𝐿
, 𝐴2

𝑈
⩘ 𝐵2

𝑈
∈ 𝐵𝐶 (𝑅)  𝑎𝑛𝑑 𝑠𝑜 (𝐴1 ⩘ 𝐵1)𝐿 , (𝐴2 ⩘ 𝐵2)𝑈 ∈ 𝐵𝐶(𝑅) ,  

Hence 𝐴1,2  ⩘ 𝐵1,2 ∈ 𝐵𝐶[𝑅] 

Theorem 5 

If  𝐴1, 𝐵1, Γ1 ∈ 𝐵𝐶[𝑅] 

𝐴1 = [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿 , (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈] 

𝐵1 = [(((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0))𝐿 , (((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10))𝑈] 

               Γ1 =

((𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10))𝐿 , ((𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10))𝑈  

 Then  (𝑖) 𝐴1⨁(𝐵1 ⩗ Γ1) = (𝐴1⨁𝐵1) ⩗ (𝐴1⨁Γ1)  

Proof: 

𝐴1⨁(𝐵1 ⩗ Γ1) = [(𝐴1⨁(𝐵1 ⩗ Γ1))
𝐿
, (𝐴1⨁(𝐵1 ⩗ Γ1))

𝑈
] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁[((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]

𝐿
,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁[((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩗ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]
𝑈] 

 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁[((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)

𝐿
⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)

𝐿
]

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁[((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)
𝑈
⩗ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)

𝑈
]
] 

=

[
 
 
 
 
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ⩗

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿 ,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈 ⩗

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿 ⩗

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿 ,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈 ⩗

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)] ⩗

(((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10))]𝐿 ⩗,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)] ⩗

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗

(((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10))]𝐿 ⩗

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩗

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 ((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)

𝐿

((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)
𝑈
⩗ ((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈⨁

(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)
𝐿
(𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)

𝑈
]
 
 
 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

445 

 
https://internationalpubls.com 

= (𝐴1 − 𝐵1) ⩘ (𝐴1 − Γ1) 

(𝑖𝑖)𝐴1⨁(𝐵1 ⩘ Γ1) = (𝐴1⨁𝐵1) ⩘ (𝐴1⨁Γ1)  

Proof: 

𝐴1⨁(𝐵1 ⩘ Γ1) = [(𝐴1⨁(𝐵1 ⩘ Γ1))
𝐿
, (𝐴1⨁(𝐵1 ⩘ Γ1))

𝑈
] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁[((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]

𝐿
,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁[((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩘ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]
𝑈] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁[((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)

𝐿
⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10, 𝛾10

1 )
𝐿
]

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁[((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)
𝑈
⩘ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)

𝑈
]
] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿
], 

    [
(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈⨁((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10)𝑈 ⩘

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈
] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿 ⩘

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿
], 

    [
(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)⨁((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10)]𝑈 ⩘

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈
] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨁((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘

(((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10))]𝐿 ⩘
], 

    [
(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)⨁((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10) ⩘

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈
] 

= [[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)

⩘ ((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)] 

   = [((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)

⩘ ((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨁(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿 , 

                                [(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)

⩘ (𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨁(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 

= (A1⨁B1) ⩘ (A1⨁Γ1) 

(𝑖𝑖𝑖) A1⊝ (B1 ⩗ Γ1) = (A1⊝B1) ⩘ (A1⊝Γ1)  

Proof: 

A1⊝ (B1 ⩗ Γ1) = [[A1⊝ (B1 ⩗ Γ1)]𝐿 , [A1⊝ (B1 ⩗ Γ1)]𝑈] 

= [
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⊝ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10, 𝛾9

1, 𝛾10
1 )]

𝐿

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩗ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10, 𝛾10
2 )]

𝑈 ] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⊝ [((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10𝛾8

1, 𝛾9
1, 𝛾10

1 )]
𝐿

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈 ⊝ [((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩗ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10, 𝛾9
2, 𝛾10

2 )]
𝑈
]
] 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⊝ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿] ⩗

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿 ⊝ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿],

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⊝ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈] ⩗

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⊝ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈 ]
 
 
 
 

  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

446 

 
https://internationalpubls.com 

  

[
 
 
 
 
 
 [((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⊝ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0))

𝐿

] ⩗

[(((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⊝ 𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿],

[((𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10))
𝑈

] ⩗

((𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10))
𝑈

]
 
 
 
 
 
 

 

=

[
 
 
 
 
 [((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⊝ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)) ⩘]

(((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⊝ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10))
𝐿

,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)] ⩘

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾

4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)]

𝑈
]
 
 
 
 
 

  

=

[
 
 
 
 
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈⊝

((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿(𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈 ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⊝

(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈 ]
 
 
 
 

  

= (A1 − B1) ⩘ (A1 − Γ1)  

(𝑖𝑣) A1 − (B1 ⩘ Γ1) = (A1⊝B1) ⩗ (A1⊝Γ1)  

Proof: 

A1 − (B1 ⩘ Γ1) = [(A1⊝ (B1 ⩘ Γ1)𝐿 , (A1⊝ (B1 ⩘ Γ1)𝑈] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⊝ [(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]

𝐿
,

𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ [((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩘ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]
𝑈] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)

𝐿
⊝ [(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10, 𝛾10

1 )
𝐿
] ,

[𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⊝ [((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)
𝑈
⩘ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)

𝑈
]]
] 

=

[
 
 
 
 [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿 ⊝ (𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ⩗

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿⊝ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿
] ,

[
𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⊝ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈 ⩗

𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⊝ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈
]
]
 
 
 
 

 

=   

[
 
 
 
 [
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⊝ (𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿 ⩗

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⊝ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿
] ,

[
[𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10) ⊝ ((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10)]𝑈 ⩗

[𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈
]
]
 
 
 
 

 

=

[
 
 
 
 (
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⊝ (𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⊝ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)
)

𝐿

 ,

(
𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩘

𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⊝ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)
)

𝑈

]
 
 
 
 

 

= [(A1⊝B1) ⩘ (A1⊝C1)] 

Theorem 6 

If A, B, C ∈ [R+]  

 𝑤ℎ𝑒𝑟𝑒 A1 = [(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)] 

              B1 = [(𝑏1 
𝐿 , 𝑏2 

𝐿 , 𝑏3 
𝐿 , 𝑏4 

𝐿 , 𝑏5 
𝐿 , 𝑏6 

𝐿 , 𝑏7 
𝐿 , 𝑏8 

𝐿 , 𝑏9 
𝐿 , 𝑏10 

𝐿 )(𝑏1 
𝑈, 𝑏2 

𝑈, 𝑏3 
𝑈 , 𝑏4 

𝑈, 𝑏5 
𝑈, 𝑏6 

𝑈, 𝑏7 
𝑈, 𝑏8 

𝑈, 𝑏9 
𝑈, 𝑏10 

𝑈 )] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

447 

 
https://internationalpubls.com 

                𝐶 = [(𝑐1 
𝐿 , 𝑐2 

𝐿 , 𝑐3 
𝐿 , 𝑐4 

𝐿 , 𝑐5 
𝐿 , 𝑐6 

𝐿 , 𝑐7 
𝐿 , 𝑐8 

𝐿 , 𝑐9 
𝐿 , 𝑐10 

𝐿 )(𝑐1 
𝑈 , 𝑐2 

𝑈, 𝑐3 
𝑈 , 𝑐4 

𝑈, 𝑐5 
𝑈 , 𝑐6 

𝑈, 𝑐7 
𝑈 , 𝑐8 

𝑈, 𝑐9 
𝑈 , 𝑐10 

𝑈 )] 

𝑡ℎ𝑒𝑛     (𝑖)A1⨂( B1 ⩗ 𝐶) = (A1⨂ B1) ⩗ (A1⨂𝐶) 

               (𝑖𝑖)A1⨂( B1 ⩘ 𝐶) = (A1⨂ B1) ⩘ (A1⨂𝐶) 

               (𝑖𝑖𝑖)A1⨸ ( B1 ⩗ 𝐶) = (A1⨸ B1) ⩘ (A1⨸𝐶) 

               (𝑖𝑣)A1⨸ ( B1 ⩘ 𝐶) = (A1⨸ B1) ⩗ (A1⨸𝐶) 

(𝑖) A1⨂( B1 ⩗ 𝐶) = (A1⨂ B1) ⩗ (A1⨂𝐶) 

𝑃𝑟𝑜𝑜𝑓:     

A1⨂( B1 ⩗ 𝐶) = [[A1⨂( B1 ⩗ 𝐶)]𝐿 , [A1⨂( B1 ⩗ 𝐶)]𝑈] 

                      = [[A1
𝐿
⨂( B1 ⩗ 𝐶)𝐿 , ][𝐴𝑈⨂( B1 ⩗ 𝐶)𝑈]] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)

𝐿
⨂[(𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]𝐿,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)
𝑈
⨂[((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10) ⩗ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)]𝑈

] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)

𝐿
⨂[((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)

𝐿
⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]𝐿 ,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)
𝑈
⨂[((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10)

𝑈
⩗ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)]𝑈

] 

=  

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨂((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ⩗

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿⨂(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿],

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈 ⩗

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈] ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨂((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿 ⩗

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨂(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿 ,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨂((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈 ⩗

[𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨂((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨂(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨂((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩗

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈⨂

((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈]  ⩗

[(𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂

(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈] ]
 
 
 
 

 

= (𝐴⨂𝐵) ⩗ (𝐴⨂𝐶) 

(𝑖𝑖) 𝐴⨂(𝐵 ⩘ 𝐶) = (𝐴⨂𝐵) ⩘ (𝐴⨂𝐶) 

𝑃𝑟𝑜𝑜𝑓:     

𝐴⨂(𝐵 ⩘ 𝐶) = [[𝐴⨂(𝐵 ⩘ 𝐶)]𝐿 , [𝐴⨂(𝐵 ⩘ 𝐶)]𝐿] 

= [[𝐴𝐿⨂(𝐵 ⩘ 𝐶)]𝐿 , [𝐴𝑈⨂(𝐵 ⩘ 𝐶)]𝑈] 

= [
(𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0𝐿⨂[((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]𝐿,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)
𝑈
⨂[((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10) ⩘ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)]𝑈

] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)

𝐿
⨂[((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)

𝐿
⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]𝐿 ,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)
𝑈
⨂[((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10)

𝑈
⩘ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)]𝑈

] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

448 

 
https://internationalpubls.com 

=  

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨂((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿⨂(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿],

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈 ⩘

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈] ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨂((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿 ⩘

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨂(𝑐1, 𝑐2, 𝑐3, 𝑐4, 𝑐5, 𝑐6, 𝑐7, 𝑐8, 𝑐9, 𝑐10)]
𝐿 ,

[(𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6, 𝑎7, 𝑎8, 𝑎9, 𝑎10)⨂((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈 ⩘

[(𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6, 𝑎7, 𝑎8, 𝑎9, 𝑎10)⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)⨂((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)⨂(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨂((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩘

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)⨂(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈⨂

((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈]  ⩘

[((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨂

(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈] ]
 
 
 
 

 

= (𝐴⨂𝐵) ⩘ (𝐴⨂𝐶) 

(𝑖𝑖𝑖) 𝐴 ⨸ (𝐵 ⩗ 𝐶) = (𝐴 ⨸ 𝐵) ⩘ (𝐴⨸ 𝐶) 

Proof: 𝐴 ⨸ (𝐵 ⩗ 𝐶) = [[𝐴 ⨸ (𝐵 ⩗ 𝐶)]𝐿 , [𝐴 ⨸ (𝐵 ⩗ 𝐶)]𝑈] 

= [
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⨸ [((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]

𝐿
,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ [((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩗ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈
] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)

𝐿
⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)

𝐿
⩗ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)

𝐿
,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)
𝑈
⨸ ((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10)

𝑈
⩗ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)

𝑈] 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)𝐿⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)𝐿 ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿 ⨸ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿],

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨸ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈 ⩘

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨸ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈] ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⨸ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩘

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 (𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0𝐿(𝛼1

2.1, 𝛼2
2.2, 𝛼3

2.3, 𝛼4
2.4, 𝛼5

2.5, 𝛼6
2.6, 𝛼7

2.7, 𝛼8
2.8, 𝛼9

2.9, 𝛼10
2.10)𝑈⨸

((𝛽1
1.1, 𝛽2

1.2, 𝛽3
1.3, 𝛽4

1.4, 𝛽5
1.5, 𝛽6

1.6, 𝛽7
1.7, 𝛽8

1.8, 𝛽9
1.9, 𝛽10

1.0)𝐿((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)𝑈 ⩘

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0)𝐿(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)𝑈⨸

(𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)𝐿(𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)𝑈 ]
 
 
 
 

 

= (𝐴 ⨸ 𝐵) ⩘ (𝐴 ⨸ 𝐶) 

(𝑖𝑣)𝐴 ⨸ (𝐵 ⩘ 𝐶) = (𝐴 ⨸ 𝐵) ⩗ (𝐴⨸ 𝐶) 

Proof: 

𝐴⨸ (𝐵 ⩘ 𝐶) = [𝐴 ⨸ (𝐵 ⩘ 𝐶)𝐿 , 𝐴 ⨸ (𝐵 ⩘ 𝐶)𝐿] 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0 ⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10))

𝐿
,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ (((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩘ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)
𝑈] 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3 (2025) 

 

 

449 

 
https://internationalpubls.com 

= [
((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)

𝐿
⨸ [((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩘ (𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10)]𝐿 ,

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)
𝑈
⨸ [((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10) ⩘ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)]𝑈

] 

= [
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0)

𝐿
⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)

𝐿
⩘ 𝛾1

1.1, 𝛾2
1.2, 𝛾3

1.3, 𝛾4
1.4, 𝛾5

1.5, 𝛾6
1.6, 𝛾7

1.7, 𝛾8
1.8, 𝛾9

1.9, 𝛾10
1.10𝐿],

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10)
𝑈
⨸ ((𝛽1

2.1, 𝛽2
2.2, 𝛽3

2.3, 𝛽4
2.4, 𝛽5

2.5, 𝛽6
2.6, 𝛽7

2.7, 𝛽8
2.8, 𝛽9

2.9, 𝛽10
2.10)

𝑈
⩘ (𝛾1

2.1, 𝛾2
2.2, 𝛾3

2.3, 𝛾4
2.4, 𝛾5

2.5, 𝛾6
2.6, 𝛾7

2.7, 𝛾8
2.8, 𝛾9

2.9, 𝛾10
2.10)

𝑈
]
] 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0)]𝐿 ⩗

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⨸ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿 ,

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10)]𝑈 ⩗

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

=

[
 
 
 
 
[((𝛼1

1.1, 𝛼2
1.2, 𝛼3

1.3, 𝛼4
1.4, 𝛼5

1.5, 𝛼6
1.6, 𝛼7

1.7, 𝛼8
1.8, 𝛼9

1.9, 𝛼10
1.0) ⨸ ((𝛽1

1.1, 𝛽2
1.2, 𝛽3

1.3, 𝛽4
1.4, 𝛽5

1.5, 𝛽6
1.6, 𝛽7

1.7, 𝛽8
1.8, 𝛽9

1.9, 𝛽10
1.0) ⩗

((𝛼1
1.1, 𝛼2

1.2, 𝛼3
1.3, 𝛼4

1.4, 𝛼5
1.5, 𝛼6

1.6, 𝛼7
1.7, 𝛼8

1.8, 𝛼9
1.9, 𝛼10

1.0) ⨸ (𝛾1
1.1, 𝛾2

1.2, 𝛾3
1.3, 𝛾4

1.4, 𝛾5
1.5, 𝛾6

1.6, 𝛾7
1.7, 𝛾8

1.8, 𝛾9
1.9, 𝛾10

1.10)]𝐿

[(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ ((𝛽1
2.1, 𝛽2

2.2, 𝛽3
2.3, 𝛽4

2.4, 𝛽5
2.5, 𝛽6

2.6, 𝛽7
2.7, 𝛽8

2.8, 𝛽9
2.9, 𝛽10

2.10) ⩗

(𝛼1
2.1, 𝛼2

2.2, 𝛼3
2.3, 𝛼4

2.4, 𝛼5
2.5, 𝛼6

2.6, 𝛼7
2.7, 𝛼8

2.8, 𝛼9
2.9, 𝛼10

2.10) ⨸ (𝛾1
2.1, 𝛾2

2.2, 𝛾3
2.3, 𝛾4

2.4, 𝛾5
2.5, 𝛾6

2.6, 𝛾7
2.7, 𝛾8

2.8, 𝛾9
2.9, 𝛾10

2.10)]𝑈 ]
 
 
 
 

 

= (𝐴 ⨸ 𝐵) ⩗ (𝐴 ⨸ 𝐶) 

4. Conclusion 

In this paper we successfully developed decagonal fuzzy number and derived some important 

theorem for bounded closed interval-valued decagonal fuzzy numbers. Further, the alpha cuts also 

have derived for the decagonal fuzzy number, and some important theorems have also been proved 

using alpha cuts.  Application of this theorem is used in many fields such as medical, control system, 

automatic system, engineering, science and technology, etc. for handling the vagueness in the decision 

making problem. 

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