Microsoft Word - 157-166


 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

 ContractiveiMappings Having Mixed Finite MonotoneiProperty in 
GeneralizediMetric Spaces 

   Zena Hussein Maibed 
mrs_ zena.hussein@yahoo.com 

Department of Mathematics, College of Education for Pure Science, Ibn Al-Haitham, University of 
Baghdad, Baghdad, Iraq. 

  
Article history: Received 25 October 2018, Accepted 11 November 2018, Publish January 

2019 

 
  

Abstract     
      The concepts of the modified tuple coincidence points and the mixed finite monotone 
property is introduced in this paper. Also, the existence and uniqueness of modified 
tupled coincidence point is discusses without continuous condition for mappings having 
mixed finite monotone property in generalized metric spaces. 
 
Keywords: generalized metric spaces, fixed point, coincidence point, monotone property. 
 

1. Introduction  
        In 2000,iDhage [1] introduced𝐷–metric space as a generalization of metric space 
and he proved many results in this spaceibut in 2005, Mustafa and Sims [2] proved 
thatithe results presented by Dhage are invalid in topological structure and hence they 
introducedi𝐺–metric space and as a generalized of metric space. On other Bhashkar 
andiLakshmikantham in [3] introduced the conceptiof coupled fixed point and proved the 
existence of a coupledifixed theoremiin partiallyiordered complete metric space. In 2009, 
Lakshmikantham and Ciric [4] definedimixedi𝑔–monotone propertyiand coupled 
coincidenceipoint in partiallyiordered metric spaces, also in 2011, Berinde and Borcut [5] 
introduced the concept ofitriple fixed pointiand proved some resultsia roundiand in 2012, 
Berinde and Borcut [6] defined the concept of triple Coincidenceipoint and 
establishedisome triple Coincidenceipoint theoremsiin partially orderedimetric space. In 
this paper, we will give a mixed finite monotone property and modified tupled 
coincidence point with study the existence of modified tupled coincidence point in 
partially ordered generalized metric space. 
 
2.Background 
   In this section, we recall some definitions and properties introduced by Mustafa and 
Simis [2]  
 
Definition 1 
    Let 𝑋 beia nonemptyiset, 𝒢: 𝑋 𝑋 𝑋 → 𝑅 be a function satisfying: 
1. 𝒢 𝑥, 𝑦, 𝑧 0  𝑖𝑓  𝑥 𝑦 𝑧. 
2.0 𝒢 𝑥, 𝑥, 𝑦    for all 𝑥, 𝑦 ∈ 𝑋 𝑤𝑖𝑡ℎ  𝑥 𝑦. 



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

3. 𝒢 𝑥, 𝑥, 𝑦 𝒢 𝑥, 𝑦, 𝑧    for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 𝑤𝑖𝑡ℎ 𝑦 𝑧   
4. 𝒢 𝑥, 𝑦, 𝑧  𝒢 𝑥, 𝑧, 𝑦 𝐺 𝑦, 𝑧, 𝑥     …. 

 𝑠ymmetry in all hree variable                                                                   
5. 𝒢 𝑥, 𝑦, 𝑧  𝒢 𝑥, 𝑎, 𝑎 𝒢 𝑎, 𝑦, 𝑥, 𝑧 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦, 𝑧, 𝑎 ∈ 𝑋 . 
 
    Then the function 𝒢 is called generalized metric and the pair (X, 𝒢) is called a  
generalized metric space or more specially 𝒢_ metric space. 
 
Definition 2 
    Let (X, 𝒢) be ai𝒢_ metricispace, and let 𝑥  be a sequence of points of  𝑋.We say that 
𝑥  is 𝒢_convergentito 𝑥 if lim

, →
𝒢 𝑥, 𝑥 , 𝑥 0, thatiis, for any  𝜀 0 there exist 

𝑁 ∈ ℕ such that 
 𝒢 𝑥, 𝑥 , 𝑥 𝜀,   for all 𝑛, 𝑚 𝑁.We call 𝑥 the limit of sequence and write  𝑥 → 𝑥 or 
lim
,→

𝑥 𝑥. 

 
Definition 3  
     Let (X, 𝒢) be a 𝒢_ metric space, A sequence 𝑥  is called 𝒢_cauchy sequence  if for 
any 𝜀 0 there exist 𝑁 ∈ ℕ such that 𝒢 𝑥 , 𝑥 , 𝑥 𝜀 
  for all 𝑛, 𝑚, 𝑙 𝑁, that is, 𝒢 𝑥 , 𝑥 , 𝑥 → 0 𝑎𝑠 𝑛, 𝑚, 𝑙 → ∞ . 
 
Proposition 4 
    Let (X, 𝒢) be a 𝐺_ metricispace. A mapping is called 𝒢_continuousiat 𝑥 ∈ 𝑋 if and if it 
is 𝒢_sequentiallycontinuous at 𝑥, that is ,wheneveri 𝑥  is  𝒢_convergentito 
𝑥,  𝑓 𝑥  𝑖𝑠  𝒢_convergentito 𝑓 𝑥 . 
Proposition 5 
   A 𝒢_ metric space (X, 𝒢) is called 𝐺_completeiif every 𝒢_cauchyisequence 
is 𝒢_convergentiin (X, 𝒢). 
 
3. Main Results  
     In this section, the modification of tupled coincidence point is proposed as the follows: 

Definition 6 
       Let 𝑋,  be a partiallyiordered set. If 𝒯 : 𝑋 → 𝑋, 𝒯 , 𝒯 , … , 𝒯 , 𝒰, 𝒱: 𝑋 → 𝑋 are 
there mappings. An element 𝑥 , 𝑥 , … … , 𝑥 ∈ 𝑋  is called modified tupled coincidence 
point of 𝒯 , 𝒯 , …  𝒯 , 𝒰 and 𝒱 if: 

𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥 𝒰𝒱 𝑥  

𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥 , 𝑥 𝒰𝒱 𝑥  

⋮ 

𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥 𝒰𝒱 𝑥  



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

Definition 7 

    Let 𝑋,  be a partiallyiordered set. If 𝒯 : 𝑋 → 𝑋, 𝒯 , 𝒯 , … , 𝒯 , 𝒰, 𝒱: 𝑋 → 𝑋 are 
there mappingsiwe say that 𝑓 have  mixedifinite monotone property if 

𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥  is monotoneifinite  increasing if 𝑛 is odd, and 

𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥  is monotone finite decreasingiif 𝑛 is even 

That is, forieach   𝑥 , 𝑥 , … … , 𝑥 ∈ 𝑋 

𝑦 , 𝑧 ∈ 𝑋, 𝒰𝒱 𝑦 𝒰𝒱 𝑧 ⟹ 𝒯 𝒯  …  𝒯 𝑦 , 𝑥 , 𝑥 , … , 𝑥

𝒯 𝒯  …  𝒯 𝑧 , 𝑥 , 𝑥 , … , 𝑥  

𝑦 , 𝑧 ∈ 𝑋,    𝒰𝒱 𝑦 𝒰𝒱 𝑧 ⟹ 𝒯 𝒯  …  𝒯 𝑥 , 𝑦 , 𝑥 , … , 𝑥

𝒯 𝒯  …  𝒯 𝑥 , 𝑧 , 𝑥 , … , 𝑥  

 

⋮ 
 

𝑦 , 𝑧 ∈ 𝑋, 𝒰𝒱 𝑦 𝒰𝒱 𝑧 ⟹ 

𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … , 𝑦 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … , 𝑧   𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑  

𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … , 𝑦 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … , 𝑧   𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛  

Let 

i. 𝒜  is the setiof all mappings 𝒯 : 𝑋 → 𝑋 and 𝒯 , 𝒯 , … , 𝒯 , 𝒰 and 𝒱 ∶ 𝑋 → 𝑋 
such that: 

1. 𝒰𝒱 𝑋  is completeisubspace of 𝑋, containing 𝒯 𝒯  …  𝒯 𝑋  

2. 𝒯 , 𝒯 , … , 𝒯 , 𝒰 and 𝒱 are continuousiand commuteimappings. 
3. 𝒯 , 𝒯 , … , 𝒯 , have  mixed finite monotone property. 

ii. 𝔅 is the set of all mappings 𝜔: 0, ∞ → 0, ∞  increasing mapping 
Such that: 

1. 𝜔 𝑡 𝑡   ∀ 𝑡 0 
2. 𝜔 0 0   𝑎𝑛𝑑   lim

→
𝜔 𝑡 0 , where 𝜔  denotes the 𝑛 the iterate 

of 𝜔. 

From the above definition, we show the following modification  

 



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

Theorem 8 

    Let 𝑋, 𝒢, be a partiallyiordered generalized metric space, 𝒯 : 𝑋 → 𝑋 and 
𝒯 , 𝒯 , … , 𝒯 , 𝒰 and 𝒱 ∶ 𝑋 → 𝑋  are mappingsilies in 𝒜 and hold that following 
conditions, ∀  𝑥 , 𝑥 , … , 𝑥 , 𝑦 , 𝑦 , … , 𝑦 ∈ 𝑋 𝑎𝑛𝑑 ℒ 0 

𝒢 𝒯 𝒯 ∗ … ∗  𝒯 𝑥 , 𝑥 , … , 𝑥 , 𝒯 𝒯  …  𝒯 𝑦 , 𝑦 , … 𝑦 , ℒ

𝓌 max
𝜔 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ , 𝜔 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ , … ,

𝜔 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ
1  

Where is 𝓌 upper semicontinuous from 𝑅  into itself satisfying 𝓌 𝑥 𝑥 for all 𝑥 0  
.If  

 𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥   

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥 , 𝑥   

                                           ⋮                                                                                                       2  

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , 𝑥 , … … , 𝑥    if 𝑛 𝑖𝑠 𝑜𝑑𝑑  

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , 𝑥 , … … , 𝑥    if 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛  

Then 𝒯 , 𝒯 , … , 𝒯 , 𝒰 𝑎𝑛𝑑𝒱 have a modified tupled coincidence point. 

Proof 

Define                     𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥  

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥 , 𝑥  

⋮ 

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , 𝑥 , … … , 𝑥  

⟹ 𝒰𝒱 𝑥 𝒰𝒱 𝑥  

𝒰𝒱 𝑥 𝒰𝒱 𝑥  

⋮ 

𝒰𝒱 𝑥 𝒰𝒱 𝑥         if 𝑛 is odd 

𝒰𝒱 𝑥 𝒰𝒱 𝑥         if 𝑛 is even 

Also, we define,      𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥  

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥 , 𝑥  

⋮ 



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥  

Since 𝑓 has mixed finite monotone property 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝒰𝒱 𝑥  

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝒰𝒱 𝑥  

⋮ 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝒰𝒱 𝑥      if 𝑛 is odd 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝒰𝒱 𝑥      if 𝑛 is even 

Continue process until we get to 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥  

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥  

⋮ 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥      if 𝑛 is odd 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥      if 𝑛 is even 

In general: 

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥  

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , … … , 𝑥 , 𝑥  

⋮ 

𝒰𝒱 𝑥 𝒯 𝒯  …  𝒯 𝑥 , 𝑥 , 𝑥 , … … , 𝑥  

And 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … … .. 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … … .. 

⋮ 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … ..   if 𝑛 is odd 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … . . 𝒰𝒱 𝑥 𝒰𝒱 𝑥 . … ..  if 𝑛 is even 

Now, we have 

𝒰𝒱 𝑥  , 𝒰𝒱 𝑥  , … … 𝑎𝑛𝑑 𝒰𝒱 𝑥  are sequence in 𝑔𝑇 𝑋  

𝒰𝒱 𝑥 → 𝑟 , 𝒰𝒱 𝑥 → 𝑟 , ………, 𝒰𝒱 𝑥 → 𝑟  



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

Since  𝑟 , 𝑟 , … … , 𝑟 ∈ 𝒰𝒱 𝑋 , then there exists 𝑥 , 𝑥 , … … , 𝑥 ∈ 𝑋. 

Such that,          𝑟 𝒰𝒱 𝑥 , 𝑟 𝒰𝒱 𝑥 , … … . , 𝑟 𝒰𝒱 𝑥  

Considering the hypotheses (i) and (ii) give in the theorem we get 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝑟  

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝑟  

⋮ 

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝑟   𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑  

𝒰𝒱 𝑥 𝒰𝒱 𝑥 𝑟   𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛  

Since 𝒰 and 𝒱 are continuous mapping, then we have: 

𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟  

𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟  

⋮ 

𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟  

And hence,                             𝒰𝒱 𝒰𝒱 𝑥 𝒰𝒱 𝑟  

𝒰𝒱 𝒰𝒱 𝑥 𝒰𝒱 𝑟  

⋮ 

𝒰𝒱 𝒰𝒱 𝑥 𝒰𝒱 𝑟   𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 

𝒰𝒱 𝒰𝒱 𝑥 𝒰𝒱 𝑟    𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 

Choose ℒ satisfy: 

𝒢 𝒰𝒱 𝑟 , 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , ℒ 𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , ℒ, 𝒰𝒱 𝒰𝒱 𝑥  

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ  

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , 𝒯 𝒯  …  𝒯 𝒰𝒱 𝑥 , 𝒰𝒱 𝑥 , … 𝒰𝒱 𝑥 , ℒ  

𝓌 max
𝜔 𝒢 𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ , 𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ , … ,

𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ
  

𝓌 max
𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝑥 , ℒ , 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ , … ,

𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ
 

But,  𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟 ,………..and 



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟  

Which is implies, by definition 𝒢-convergentiin 𝒢-metric space, 

𝒢 𝒰𝒱 𝑟 , 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , ℒ 0  ⟹   𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟
𝒰𝒱 𝑟  

Also, choose ℒ  satisfy: 

𝒢 𝒰𝒱 𝑟 , 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … , 𝑟 , ℒ  

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … , 𝑟 , ℒ  , 𝒰𝒱 𝒰𝒱 𝑥  

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … , 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ   

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … , 𝑟 , 𝒯 𝒯  …  𝒯 𝒰𝒱 𝑥 , 𝒰𝒱 𝑥 , … , 𝒰𝒱 𝑥 , ℒ   

𝓌 max
𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ  , 𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ  , … ,

𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ  
 

𝓌 max
𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ  , 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ  , … ,

𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ  
 

But,  𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟 ,………..   and 

𝒰𝒱 𝒰𝒱 𝑥 → 𝒰𝒱 𝑟  

Which is implies, by definition 𝒢 -convergent in 𝒢-metric space, 

𝒢 𝒰𝒱 𝑟 , 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , ℒ 0  ⟹   𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟
𝒰𝒱 𝑟  

Continue these processes 

Choose ℒ ∗ satisfy: 𝒢 𝒰𝒱 𝑟 , 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , ℒ∗  

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , ℒ∗, 𝒰𝒱 𝒰𝒱 𝑥  

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ∗  

𝒢 𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 , 𝒯 𝒯  …  𝒯 𝒰𝒱 𝑥 , 𝒰𝒱 𝑥 , … … , 𝒰𝒱 𝑥 , ℒ∗  

𝓌 max
𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ ∗ , 𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ∗

, … … , 𝜔 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ∗
 

𝓌 max
𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ∗ , 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ∗

, … … , 𝒢 𝒰𝒱 𝑟 , 𝒰𝒱 𝒰𝒱 𝑥 , ℒ∗
 

But ,                 𝒰𝒱 𝒰𝒱 𝑥  is 𝒢–convergent to 𝒰𝒱 𝑟                           



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

𝒰𝒱 𝒰𝒱 𝑥  is 𝒢–convergent to 𝒰𝒱 𝑟  

⋮ 

       𝒰𝒱 𝒰𝒱 𝑥  is 𝒢–convergent to 𝒰𝒱 𝑟  

𝒰𝒱 𝒰𝒱 𝑥  is 𝒢–convergent to 𝒰𝒱 𝑟  

Which is implies, by definition of 𝒢–convergent in 𝐺–metric space, 

𝒢 𝒰𝒱 𝑟 , 𝑓 𝑟 , 𝑟 , … … , 𝑟 , ℒ∗ 0 
 

⇒  𝒯 𝒯  …  𝒯 𝑟 , 𝑟 , … … , 𝑟 𝒰𝒱 𝑟  

So 𝑟 , 𝑟 , … … , 𝑟  is a modified tupled coincidence point of 𝒯 , 𝒯 , … , 𝒯 , 𝒰 𝑎𝑛𝑑 𝒱. 

From theorem(8),we can get the following corollaries 

Corollary 9 

      Let 𝑋, 𝒢, be a partiallyiordered generalizedimetric space .Under the same 
assumptionsiof theorem(8) but 

𝒢 𝒯 𝒯  ∗ … ∗  𝒯 𝑥 , 𝑥 , … , 𝑥 , 𝒯 𝒯  …  𝒯 𝑦 , 𝑦 , … 𝑦 , ℒ

𝓌
1
𝑛

𝜔 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ 𝜔 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ ⋯
𝜔 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ

 

Then 𝒯 , 𝒯 , … , 𝒯 , 𝒰 𝑎𝑛𝑑𝒱 have a modified tupled coincidence point. 

Corollary 10 

   Let 𝑋, 𝐺, be a partiallyiordered generalized metric space 

𝒢 𝒯 𝒯  ∗ … ∗  𝒯 𝑥 , 𝑥 , … , 𝑥 , 𝒯 𝒯  …  𝒯 𝑦 , 𝑦 , … 𝑦 , ℒ

𝓌
𝑘 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ 𝑘 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ ⋯

𝑘 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ
  ,such that 𝑘 ∈

0,1  𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 1,2, … … , 𝑛.Then 𝒯 , 𝒯 , … , 𝒯 , 𝒰 𝑎𝑛𝑑𝒱 have a modifieditupled 
coincidence point. 

Corollary 11 

    Let 𝑋, 𝒢, be a partiallyiordered generalizedimetric space .Under the same 
assumptions of theorem(8) but 

𝒢 𝒯 𝒯  ∗ … ∗  𝒯 𝑥 , 𝑥 , … , 𝑥 , 𝒯 𝒯  …  𝒯 𝑦 , 𝑦 , … 𝑦 , ℒ

𝓌 max 𝑘 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ , 𝑘 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ , 

… , 𝑘 𝒢 𝒰𝒱 𝑥 , 𝒰𝒱 𝑦 , ℒ    



 

 
 

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Ibn Al-Haitham Jour. for Pure & Appl.Sci.                                               IHJPAS 
                                https://doi.org/10.30526/32.1.1988                                                                  Vol. 32 (1) 2019 

such that 𝑘 ∈ 0,1  𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 1,2, … … , 𝑛.Then 𝒯 , 𝒯 , … , 𝒯 , 𝒰 𝑎𝑛𝑑𝒱 have a modified 
tupled tupled coincidenceipoint. 

Corollary 12 

   Let 𝑋, 𝒢, be a partiallyiordered generalizedimetric space .Under the same 
assumptions of theorem(8) but 

𝒢 𝒯 𝒯  ∗ … ∗  𝒯 𝑥 , 𝑥 , … , 𝑥 , 𝒯 𝒯  …  𝒯 𝑦 , 𝑦 , … 𝑦 , ℒ
𝒢 𝒰𝒱 ,𝒰𝒱 ℒ 𝒢 𝒰𝒱 ,𝒰𝒱 ,ℒ ⋯ 𝒰𝒱 ,𝒰𝒱 ,ℒ

. 

Then 𝒯 , 𝒯 , … , 𝒯 , 𝒰 𝑎𝑛𝑑𝒱 have a modified tupled coincidence point . 

4.Conclusion 

    The new concepts of modified tupledicoincidence points and mixed finite monotone 
property are introduced. Also, we established some modified tupled coincidence 
theorems in partially ordered generalized metric space. 
 

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