139 

Ibn Al-Haitham Jour. for Pure & Appl. Sc i.                                                     IHJPAS                                
  https://doi.org/10.30526/32.1.1997                                                                                                
Vol. 32 (1) 2019 

𝐅𝐢𝐱𝐞𝐝  𝐏𝐨𝐢𝐧𝐭𝐬  𝐑𝐞𝐬𝐮𝐥𝐭𝐬 𝐢𝐧  𝑮 − 𝐌𝐞𝐭𝐫𝐢𝐜  𝐒𝐩𝐚𝐜𝐞𝐬  

Anaam Neamah Faraj 
anaamnema1@gmail.com 

Salwa Salman Abed 

 salwaalbundi@yahoo.com 

Departmen of Mathmatics, College of Education for pure sciences, Ibn Al Haitham, University of Baghdad,  
Baghdad, Iraq. 

 

Article history: Received 11 October 2018,Accepted 26 November 2018,Publish February 2019 

 

  Abstract  

     In this paper, the concept of 𝐹 −contraction mapping on a 𝐺-metric space is extended with a 

consideration on local 𝐹 − contraction.  As a result, two fixed point theorems were proved for 

𝐹 − contraction on a closed ball in a complete 𝐺-metric space.  

Keywords:  𝐺 − metric spaces, Local fixed points, 𝐹 −  contractions. 

 
1.Introduction and Preliminaries 

    Bapure Dhage in his PhD thesis [1992] introduced a new class of generalized metric spaces, 

named D - metric spaces. Mustafa and Sims proved that most of the claims concerning the 

fundamental structures on D - metric spaces are incorrect and introduced an appropriate notion 

of D - metric space, named G-metric spaces. In fact, Mustafa, Sims and other authors introduced 

many fixed point results for self mappings in G - metric spaces under certain conditions. 

  Actually, the method is used in the study of fixed points in metric spaces,and symmetric 

spaces. In this paper, a general fixed point theorem for pairs of non weakly compatible 

mappings in G - metric space is proved. In the case of a single mapping some results. In 2012, 

Wardowski introduced a new concept for contraction mappings as called F-contraction by 

considering a class of real valued functions. 

       Let ℳ be a nonempty set and Υ : ℳ× ℳ × ℳ → ℝ+ be a function satisfying the following 
condition: 

1- 𝛶 ( 𝑞, 𝑢, 𝑣) =  0 if and only if  𝑞 = 𝑢 = 𝑣, 
2-  0 <  𝛶 ( 𝑞, 𝑞, 𝑢 ), ∀ 𝑞, 𝑢 ∈ ℳ 𝑤𝑖𝑡ℎ 𝑞 ≠ 𝑢, 
3- 𝛶 ( 𝑞, 𝑞, 𝑢) ≤ 𝛶 (𝑞, 𝑢, 𝑣)  , ∀  𝑞, 𝑢, 𝑣 ∈ ℳ  𝑤𝑖𝑡ℎ 𝑢 ≠ 𝑣, 
4- 𝛶 (𝑞, 𝑢, 𝑣) = 𝛶(𝑞, 𝑣, 𝑢) =  … , (symmetry in all three vairables), 

5- 𝛶 (𝑞, 𝑢, 𝑣) ≤ 𝛶 (𝑞, 𝑎, 𝑎 ) + 𝛶 (𝑎, 𝑢, 𝑣) , ∀  𝑞, 𝑢, 𝑣, 𝑎 ∈ ℳ.  
Then the function 𝛶 is called generalized metric on  ℳ [1] and the pair (ℳ,𝛶) is called a 𝐺-
metric space. 
A 𝐺 -metric space ℳ  is called a symmetric [2] if ∀ 𝑞, 𝑢, 𝑣 ∈ ℳ     

   𝛶  ( 𝑞, 𝑢, 𝑢 ) =  𝛶  (𝑞, 𝑞, 𝑢 )                                  
Many results and examples about 𝛶-metric space and its generalization one can found in [2-10]. 

 

Proposition 1 [5]: Let (ℳ, 𝛶 ) be a 𝐺-metric space, then the following statements are 

equivalent:  

1-(ℳ, 𝛶 ) is symmetric. 

2-𝛶  (𝑞, 𝑢, 𝑢) ≤ 𝛶 ( 𝑞, 𝑢, 𝑎 ) for all  𝑞, 𝑢, 𝑎 ∈ ℳ, 

 3-𝛶  ( 𝑞, 𝑢, 𝑣) ≤ 𝛶 (𝑞, 𝑢, 𝑎 ) +  𝛶(𝑣, 𝑢, 𝑏 )   for all  𝑞, 𝑢 , 𝑣, 𝑎, 𝑏 ∈ ℳ. 

https://doi.org/10.30526/32.1.1997
mailto:anaamnema1@gmail.com
mailto:salwaalbundi@yahoo.com


 
 

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    The Υ-ball with center  𝑟  and radius 𝜖 0 is 𝐵 (𝑟  , 𝜖 ) [10] is:        
𝐵 (𝑟 ,𝜖) = {s ∈ ℳ ∶ Υ (𝑟 , s, and s) 𝜖  . 

The sequence 𝑟  in a 𝐺  metric space  ℳ, 𝛶   is said to be  
1- 𝛶  convergent to 𝑟 if ∃𝑘 ∈ 𝑁 , 𝜖 0  for 𝑎𝑙𝑙 𝑚, 𝑛 𝑘 such that  𝛶 𝑟, 𝑟 , 𝑟 𝜖.   
2- 𝛶 – Cauchy if ∃ 𝑘 ∈ 𝑁 , 𝜖 0  for 𝑎𝑙𝑙 𝑚, 𝑛 , 𝑙 𝑘 such that   Υ 𝑟 ,𝑟 ,𝑟  ) 𝜖 .   
A 𝐺–metric space  ℳ, 𝛶  is complete if every 𝛶 -Cauchy sequence (ℳ, Υ) is Υ- convergent in 
(ℳ, Υ  1 . 

  
Proposition 2 [11]: Let ℳ, 𝛶  be a 𝐺-metric space the following statements are equivalent  

  1- 𝑟   is 𝛶-convergent to 𝑟 , if and only if 𝛶 𝑟 ,𝑟 ,𝑟  → 0   𝑎𝑠    𝑛 → ∞ ,   
                    2- Is  Υ 𝑟   , 𝑟 ,  𝑟)→ 0    𝑎𝑠    𝑛 → ∞  if and only if  Υ 𝑟 ,𝑟 ,𝑟)→ 0   𝑎𝑠 𝑚, 𝑛 → ∞.

 
Proposition 3 [6]: Let 𝑞  and 𝑢  be a sequence in a 𝐺 metric space (ℳ, Υ) if 𝑟  
converges to 𝑞 and 𝑢  converge to 𝑢. Then Υ 𝑞 , 𝑞 , 𝑢  converges to Υ 𝑞, 𝑞, 𝑢 . 
The self- mapping  𝑓 on a G-metric space  ℳ, 𝛶  is Υ - continuous at 𝑟 ∈ ℳ  [9] iff every 

sequence 𝑟 ⊂ ℳ, with  𝑟 →  𝑟 , we have   𝑓 𝑟 → 𝑓 𝑟. 
  
    A mapping 𝑓 ∶  ℳ → ℳ is said to be 𝐹-contraction if there exists 𝜏 > 0 such that for all 
 𝑞, 𝑢, 𝑣 ∈ ℳ, 

𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 0,  
                  𝜏 𝐹 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 𝐹 𝛶 𝑞, 𝑢, 𝑣 . for all 𝑞, 𝑢, 𝑣 ∈ ℳ                                  (1)  
      Let  𝐷 be the class of all functions 𝐹: 𝑅 →  𝑅 is a mapping satisfying the following 
conditions:  
(D1) 𝐹 is strictly increasing, i.e., for all 𝑞, 𝑢, 𝑣 ∈ 𝑅  such that 𝑞 𝑢 𝑣  , 𝐹 𝑞 𝐹 𝑢
𝐹 𝑣 ,  
D2  For each sequence α  

  ⊂ 0, ∞ , lim
→  

α  = 0 iff   lim
→  

𝐹 α  = -∞.  

(D3) ∃𝑘 ∈ 0, 1  such that lim
→  

𝛼 𝐹 α  = 0.  

      Every 𝐹-contraction is contractive (byD1) and then every 𝐹 contraction is 𝛶-continuous. 
Clearly, (1) and (D1) implies that every 𝐹-contraction mapping is 𝛶-continuous, since for all 
𝑞, 𝑢, 𝑣 ∈ ℳ, with𝑓 𝑞  𝑓𝑢 𝑓𝑣, 𝐹 𝑓 𝑞, 𝑓𝑢, 𝑓𝑣 𝐹 𝛶 𝑞, 𝑢, 𝑣 .  
  For illustration, we give the following example. 
 
Example 4 
a- Consider  𝐹 : 0, ∞ →  𝑅 as   𝐹  (𝛼   𝑙𝑛𝛼. It is clear that𝐹 ∈ 𝐷. Then each self mappings 
𝑓 on a 𝐺-metric space (ℳ,𝛶) is an 𝐹 -contraction  ∋ for all 𝑞, 𝑢, 𝑣 ∈ ℳ,   𝑓𝑞 𝑓𝑢 𝑓𝑣 

                                                                   𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 𝑒 𝛶 𝑞, 𝑢, 𝑣                                 

Then for 𝑞, 𝑢, 𝑣 ∈ ℳ  such that 𝑓𝑞 𝑓 𝑢 𝑓𝑣 the inequality 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 𝑒 𝛶 𝑞, 𝑢, 𝑣   

holds.       

Therefore, 𝑓 is a contraction with ℎ  𝑒 . 

 b- Let 𝐹  : 0, ∞  →  𝑅 be  𝐹  𝛼   𝛼 𝑙𝑛𝛼. It is clear that 𝐹  ∈ 𝐷. Then each self mappings 
𝑓 on a 𝐺-metric space (ℳ, 𝛶) satisfying (1.1) is an 𝐹 -contraction such that  



 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sc i.                                                     IHJPAS 
   https://doi.org/10.30526/32.1.1977                                                                     Vol. 32 (1) 2019 

                                
, ,

, ,
e , , , ,  𝑒 ,   for all𝑞, 𝑢, 𝑣 ∈ ℳ,   𝑓𝑞 𝑓 𝑢 𝑓𝑣        

 
2. Main Results  
     Throughout the following ℳ is a complete 𝐺  metric space 𝑤. 𝑟. 𝑡. distance function 𝛶. We 
can prove the following theorem  
 
Theorem 5 
      Let 𝑓: ℳ →  ℳ be 𝛶  continuous self-mapping, 𝜖  0 and 𝑞  ∈ ℳ. Suppose that ∃ ℎ ∈
0, 1 , 𝜏 0, and 𝐹 ∈ 𝐷.If for all 𝑞, 𝑢, 𝑣 ∈ 𝐵 𝑞 , 𝜖 ⊂  ℳ with 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 0 ∋ 
                            𝜏 𝐹 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 𝐹 ℎ 𝛶 𝑞, 𝑢, 𝑣 ,                                                     (2) 
 and 
                      𝛶 𝑞 , 𝑓𝑞 , 𝑓𝑞  < 1 𝑘 𝜖.                                                                               (3)     
Then ∃!  𝑟∗ in 𝐵 𝑞 , 𝜖  ∋ 𝑟∗ 𝑓𝑟∗.  
 
Proof: Suppose 𝑞 ∈ ℳ such that 𝑞 𝑓𝑞  ,  𝑞 𝑓𝑞 . Continuing in this way,  we get  

𝑞 𝑓𝑞 ,     ∀𝑛 0. 
Implies that {𝑞 } is non-increasing sequence. 
First, to prove 𝑞 ∈ 𝐵 𝑞 , 𝜖 , ∀𝑛 ∈ ℕ,  by using mathematical induction. From (3), we get 
                                              𝛶 𝑞 , 𝑞 , 𝑞  = 𝛶 𝑞 , 𝑓𝑞 , 𝑓𝑞 1 𝑘 𝜖 𝜖.                        (4)   

   

Hence, 𝑞 ∈ 𝐵 𝑞 , 𝜖 . Suppose    𝑞 ,….., 𝑞 ∈ 𝐵 𝑞 , 𝜖   for some𝑖 ∈  ℕ. Then from (2) we obtain  

𝐹  𝛶 𝑞 , 𝑞 , 𝑞 𝐹  𝛶 𝑓𝑞 , 𝑓𝑞 , 𝑓𝑞 𝐹 ℎ 𝛶 𝑞 , 𝑞 , 𝑞 𝜏 
Since 𝐹 is strictly increasing, we get  

                                         𝛶 𝑞 , 𝑞 , 𝑞 ℎ 𝛶 𝑞 , 𝑞 , 𝑞                                        (5) 
Now,  

𝛶 𝑞 , 𝑞 , 𝑞 𝛶 𝑞 ,  𝑞 , 𝑞 ⋯ 𝛶 𝑞 , 𝑞 , 𝑞  
                    𝛶 𝑞 , 𝑞 ,  𝑞 1 𝑘 ⋯ 𝑘 ] 

1 𝑘 𝜖
1 𝑘

1 𝑘
 

                                                                       𝜖. 
Thus  

𝑞 ∈ 𝐵 𝑞 , 𝜖 ,  Hence  𝑟 ∈ 𝐵 𝑞 , 𝜖       ∀ 𝑛 ∈ ℕ. 
Continuing, we have 

                          𝐹  𝛶 𝑞 , 𝑞 , 𝑟   𝐹  𝛶 𝑞 , 𝑞 , 𝑞 𝑛𝜏  
This implies that  

                                𝐹  𝛶 𝑞 , 𝑞 , 𝑞 𝐹  𝛶 𝑞 , 𝑞 , 𝑞 𝑛𝜏                       (6) 
From (6) we get   

𝑙𝑖𝑚
→

𝐹  𝛶 𝑞 , 𝑞 , 𝑞 ∞ . 

Since𝐹 ∈ 𝐷. We get  
                                           𝑙𝑖𝑚

→
 𝛶 𝑞 , 𝑞 , 𝑞 0                                                     (7) 

From (D3) there exists  𝑝 ∈ 0,1  such that  
                            𝑙𝑖𝑚  

→
 𝛶 𝑞 , 𝑞 , 𝑞  𝐹  𝛶 𝑞 , 𝑞 , 𝑞 0                       (8) 



 
 

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From (6) we have  

𝛶 𝑞 , 𝑞 , 𝑞 𝐹  𝛶 𝑞 , 𝑞 , 𝑞 𝐹 𝛶 𝑞 , 𝑞 , 𝑞  

                                       𝛶 𝑞 , 𝑞 , 𝑞 𝑛𝜏 0.                                            (9) 
By (7), (8) and letting 𝑛 → ∞, in (9) we get  

                                  𝑙𝑖𝑚
→

 𝑛  𝛶 𝑞 , 𝑞 , 𝑞 0 .                                                (10) 

we observe that from (10), then   ∃ 𝑛 ∈ ℕ ∋  𝑛 𝛶 𝑞 , 𝑞 , 𝑞 1    , ∀  𝑛 𝑛  we have  

                            𝛶 𝑞 , 𝑞 , 𝑞    ,  ∀ 𝑛 𝑛                                                (11) 

Now, 𝑚, 𝑛 ∈  ℕ ∋ 𝑚 𝑛 𝑛 . Then, by properties of  𝛶 and (11) we obtain  
𝛶 𝑞 , 𝑞 , 𝑞   𝛶 𝑞 , 𝑞 , 𝑞  𝛶 𝑞 , 𝑞 , 𝑞  …  𝛶 𝑞 , 𝑞 , 𝑞  

                                    =∑ 𝛶 𝑞 , 𝑞 , 𝑞             

                                    ∑ 𝛶 𝑞 , 𝑞 , 𝑞  

                                    ∑                                                                                       (12)  

The series ∑   is 𝛶  convergent.  

as 𝑛 → ∞, from (12) we get {𝑞  is a 𝛶-Cauchy sequence since 
lim
, →

𝛶 𝑞 , 𝑞 , 𝑞 0 . 

By completeness of ℳ,  ∃𝑟∗ ∈ 𝐵 𝑟 , 𝜖  ∋ 𝑞 → 𝑟∗ 𝑎𝑠  𝑛 → ∞.  Since   𝑓 is 𝛶  continuous. 
Then  
, 𝑞 𝑓𝑞 → 𝑓𝑟∗  as 𝑛 → ∞, that is, 𝑟∗ 𝑓𝑟∗. 
       Hence 𝑟∗ is a fixed point of 𝑓. To prove uniqueness, let 𝑞, 𝑢 ∈ 𝐵 𝑞 , 𝜖  and 𝑞 𝑢 be any 
two fixed point of  𝑓. Then from (2) we have  

𝜏 𝐹  𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑢 𝐹  ℎ 𝛶 𝑞, 𝑢, 𝑢 , 
we obtain,  

𝜏 𝐹 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑢 𝐹 𝛶 𝑞, 𝑢, 𝑢 . 
which is contradiction, so, 𝑞 𝑢.  
     For more illustration we give the following example. 
 
Example 6 

       Let ℳ 𝑅  and 𝛶 𝑞, 𝑢, 𝑣  |𝑞 𝑢|     |𝑢 𝑣|      |𝑞 𝑣|. Then (ℳ,𝛶) is a complete 
G-metric space. Define the mapping 𝑓: ℳ→ ℳ by,  

𝑓 𝑞

𝑞
4

, 𝑟 ∈ 0,1

𝑞
1
2

, 𝑟 ∈ 1, ∞ .
 

𝑟 1,   𝜖 3      , 𝐵 𝑟 , 𝜖 , .  

If 𝐹 𝛼 ln 𝛼,    𝛼 0 and 𝜏 0, then    𝛶 1, 𝑓1, 𝑓1     𝜖. 

If 𝑞, 𝑠, 𝑡 ∈  𝐵 𝑟 , 𝜖  then 
1
4

  |𝑞 𝑢|   |𝑢 𝑣|   |𝑞 𝑣|  
1
2

 |𝑞 𝑢|   |𝑢 𝑣|   |𝑞 𝑣|  

So,               𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 ℎ 𝛶 𝑞, 𝑢, 𝑣  



 
 

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Hence  

𝜏 𝐹 𝛶 𝑓 𝑞, 𝑓 𝑢, 𝑓 𝑣 𝜏 ln  𝛶 𝑓 𝑞, 𝑓 𝑢, 𝑓 𝑣  

                                       ln ℎ  𝛶 𝑞, 𝑢, 𝑣   
                                   = 𝐹 ℎ 𝛶  𝑞, 𝑢, 𝑣  
If 𝑞, 𝑢, 𝑣 ∈ 1, ∞  then  

𝑞 𝑢 𝑢 𝑣 𝑣 𝑞  |𝑟 𝑢| |𝑢 𝑣|

|𝑟 𝑣|                        𝜏 |𝑓 𝑞 𝑓 𝑢| |𝑓 𝑢 𝑓 𝑣| |𝑓 𝑞 𝑓 𝑣|   |𝑞 𝑢| |𝑢 𝑣|
|𝑞 𝑣| 
So,  𝜏 𝐹 𝛶 𝑓 𝑞, 𝑓 𝑢, 𝑓 𝑣 𝐹 𝛶 𝑞 , 𝑢, 𝑣 .    
Then the contraction does not hold on ℳ.  
     Now, we present two properties of𝑓, 𝑓: ℳ →  ℳ. We say that 𝑓 satisfies the condition:  
I-  ω 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 1,  ∀ 𝑞, 𝑢, 𝑣 ∈ ℳ whenevere  ω ∶  ℳ → 𝑅 ,   ω 𝑞, 𝑢, 𝑣 1.  
II- for given a sequence {𝑞 ⊂ ℳ with  𝑞 → 𝑞 ∈ ℳ  as 𝑛 → ∞, if 
ω  𝑞  , 𝑞 , 𝑞 φ  𝑞  , 𝑞 , 𝑞 ,  ∀𝑛 ∈ 𝑁 ⇒  𝑓 𝑞 → 𝑓𝑞, where ω, φ: ℳ → 𝑅   
are two functions. 
If  φ 𝑞, 𝑢, 𝑣 1 then (II) reduces (I).  
Let ∆𝜓   𝜓: 𝑅  →  𝑅 ∶  ∀ 𝑡 , 𝑡 , 𝑡 , 𝑡 ∈  𝑅 , 𝑡 𝑡 𝑡 𝑡   0, ∃𝜏  0 ∋  𝜓 𝑡 , 𝑡 , 𝑡 , 𝑡  
𝜏 . 
 
Definition 7 
      Let 𝑓 be a self- mapping on a 𝐺-metric space (ℳ, 𝛶) and 𝑟  ∈ℳ with 𝜖 0. Suppose that 
𝜔: ℳ  → 0, ∞ , φ: ℳ   → 𝑅  two functions. We say that 𝑓 is called 𝜔-φ- 𝜓𝐹
contraction on a closed ball if for all 𝑞, 𝑢, 𝑣 ∈ 𝐵 𝑞 , 𝜖 ⊆  ℳ, with  

𝜔  𝑞, 𝑓𝑞, 𝑓𝑞 , 𝑢, 𝑓𝑢, 𝑓𝑢 , 𝑣, 𝑓𝑣, 𝑓𝑣  φ 𝑞, 𝑢, 𝑣  
and  
𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 0, we have  

𝜓  𝛶 𝑞, 𝑓𝑞, 𝑓𝑞 , 𝛶 𝑢, 𝑓𝑢, 𝑓𝑢 , 𝛶 𝑣, 𝑓𝑣, 𝑓𝑣 𝐹 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣       

𝐹 ℎ 𝛶 𝑞, 𝑢, 𝑣 ,                                                                                               13  
and   
                                   𝛶 𝑞 , 𝑓𝑞 , 𝑓𝑞 1 𝑘 𝜖,                                                                        14  
where 0 𝑘 1, 𝜓 ∈ ∆𝜓 and 𝐹 ∈ 𝐷. 
 
Definition 8 

      Let 𝑓: ℳ →  ℳ be a self –mapping and𝜔, φ: ℳ × ℳ× ℳ → [0, +∞) be two functions.  𝑓 
Is  
called that is 𝜔- admissible mapping with respect to φ if  𝑞, 𝑢, 𝑣 ∈ ℳ, φ (𝑞, 𝑢, 𝑣) 𝜔 𝑞, 𝑢, 𝑣   
implies that φ (𝑓𝑞, 𝑓𝑢, 𝑓𝑣)    𝜔 𝑓𝑞, 𝑓𝑢, 𝑓𝑣) and 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣  0, we have  
𝜓 𝛶 𝑞, 𝑓𝑞, 𝑓𝑞 , 𝛶 𝑢, 𝑓𝑢, 𝑓𝑢 , 𝛶 𝑣, 𝑓𝑣, 𝑓𝑣 , 𝛶 𝑞, 𝑓𝑢, 𝑓𝑣 , 𝛶 𝑢, 𝑓𝑞, 𝑓𝑣 , 𝛶 𝑣, 𝑓𝑞, 𝑓𝑢
                                             𝐹 𝛶 𝑓𝑞, 𝑓𝑢, 𝑓𝑣 𝐹 𝑀 𝑞, 𝑢, 𝑣                                                 (15) 
where 𝑀 𝑞, 𝑢, 𝑣 𝑚𝑎𝑥   𝛶 𝑞, 𝑢, 𝑣 , 𝛶 𝑞, 𝑓𝑞, 𝑓𝑞 , 𝛶 𝑢, 𝑓𝑢, 𝑓𝑢 , 

 𝛶 𝑣, 𝑓𝑣, 𝑓𝑣 ,
 𝛶 𝑞, 𝑓𝑢, 𝑓𝑣  𝛶 𝑢, 𝑓𝑞, 𝑓𝑣  𝛶 𝑣, 𝑓𝑞, 𝑓𝑢

3
   

and  



 
 

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                                      ∑ 𝛶 𝑟 , 𝑓𝑟 , 𝑓𝑟   ≤,   ∀𝑗 ∈ 𝑁 and 𝜖 0.                                            (16) 

𝜓 ∈ ∆𝜓 and𝐹 ∈ 𝐷. 
 
Theorem 9 

       Let 𝑓: ℳ →  ℳ be 𝜔-φ- 𝜓𝐹 contraction  mapping on a closed ball where  
(i) 𝑓 is an 𝜔- admissible mapping with respect to φ, 
ii  there exists 𝑟 ∈ ℳ such that 𝜔 𝑟 , 𝑓𝑟 , 𝑓𝑟   φ 𝑟 , 𝑓𝑟 , 𝑓𝑟  , 

(iii) 𝑓 is an 𝜔-φ-  continuous. 
Then there exists a point 𝑟  in 𝐵 𝑟 , 𝜖  such that 𝑓𝑟 𝑟 
 
Proof: Let 𝑟  in ℳ such that 𝜔 𝑟 , 𝑓𝑟 , 𝑓𝑟   φ 𝑟 , 𝑓𝑟 , 𝑓𝑟 . For 𝑟  ∈ℳ. Let us construct a 
sequence 𝑟  such that 
 𝑟 𝑓 𝑟  , 𝑟 𝑓 𝑟 𝑓  𝑟  continuing this way, 𝑟 𝑓 𝑟  𝑓  𝑟  ,  ∀ 𝑛 ∈ 𝑁. 
Since, 𝑓 is an 𝜔- admissible mapping with respect to φ, then 

𝜔 𝑟  , 𝑟 , 𝑟  𝜔 𝑟 , 𝑓𝑟 , 𝑓𝑟   φ 𝑟 , 𝑓𝑟 , 𝑓𝑟  φ 𝑟 , 𝑟 , 𝑟  . 
Continuous we get  
                           φ 𝑟 , 𝑓 𝑟 , 𝑓 𝑟 =φ 𝑟 , 𝑟 , 𝑟  𝜔 𝑟 , 𝑟 , 𝑟    , ∀ 𝑛 ∈ 𝑁         (17   
If  ∃ 𝑛 ∈ 𝑁 ∋  𝛶 𝑟 , 𝑓 𝑟 , 𝑓 𝑟 0 , there is nothing to prove. 
So, suppose that 𝑟 𝑟  with  

𝛶 𝑓𝑟 , 𝑓𝑟 , 𝑓𝑟 𝛶 𝑓𝑟 , 𝑓𝑟 , 𝑓𝑟 0 ,    ∀ 𝑛 ∈ 𝑁.   
First, we see that 𝑟 ∈ 𝐵 𝑟 , 𝜖  ,   ∀ 𝑛 ∈ 𝑁.   
Since 𝑓 be a   𝜔-φ- 𝜓𝐹 contraction mapping on a closed ball, we get  

                               𝛶 𝑟 , 𝑟 , 𝑟 𝛶 𝑟 , 𝑓𝑟 , 𝑓𝑟 1 𝑘 𝜖 𝜖                                          (18) 
 
Thus  

𝑟 ∈ 𝐵 𝑟 , 𝜖  . Suppose 𝑟 , … , 𝑟 ∈ 𝐵 𝑟 , 𝜖   for some 𝑗 ∈ 𝑁, such that  

𝜓  𝛶 𝑟 , 𝑓𝑟 , 𝑓𝑟 , 𝛶 𝑟 , 𝑓𝑟 , 𝑓𝑟 , 𝛶 𝑟 , 𝑓𝑟 , 𝑓𝑟 , 𝛶 𝑟 , 𝑓𝑟 , 𝑓𝑟  

+𝐹   𝛶 𝑓𝑟 , 𝑓𝑟 , 𝑓𝑟 𝐹 ℎ 𝛶 𝑟 , 𝑟 , 𝑟 . 
This implies,  

𝜓  𝛶 𝑟 , 𝑟 , 𝑟 , 𝛶 𝑟 , 𝑟 , 𝑟 , 𝛶 𝑟 , 𝑟 , 𝑟 , 0  

 𝐹  𝛶 𝑓𝑟 , 𝑓𝑟 , 𝑓𝑟 𝐹 ℎ 𝛶 𝑟 , 𝑟 , 𝑟 . 

By definition of 𝜓, 
(𝛶 𝑟 , 𝑟 , 𝑟 , 𝛶 𝑟 , 𝑟 , 𝑟 , 𝛶 𝑟 , 𝑟 , 𝑟 , 0 0, 

So, ∃𝜏 0 such that, 
𝜓 𝛶 𝑟 , 𝑟 , 𝑟 , 𝛶 𝑟 , 𝑟 , 𝑟 , 𝛶 𝑟 , 𝑟 , 𝑟 . 0  𝜏. 
Therefore,   

          𝐹 𝛶 𝑟 , 𝑟 , 𝑟 𝐹 𝛶 𝑟 , 𝑟 , 𝑟 𝐹 ℎ 𝛶 𝑟 , 𝑟 , 𝑟 𝜏                             19  

       To complete, we follow the same steps of the theorem (9),since ℳ is complete 𝐺  metric 
space there exists 𝑟 ∈ 𝐵 𝑟 , 𝜖   such that  
𝑟 → 𝑟 as  𝑛 → ∞ . 𝑓 is an 𝜔-φ-  continuous and  

φ 𝑟 , 𝑟 , 𝑟  𝜔 𝑟 , 𝑟 , 𝑟 ,     ∀ 𝑛 ∈ 𝑁.   
Then 



 
 

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𝑟 𝑓 𝑟 → 𝑓 𝑟  as  𝑛 → ∞. 
That is,  𝑟  𝑓 𝑟 hence 𝑟 is a fixed point of  𝑓.   
To illustrate theorem 9, we give the following example  
 
Example 10 

      Let ℳ 𝑅  and 𝛶 be  𝐺 metric on ℳ as in Example 6  Define 𝑓: ℳ → ℳ, 
𝜔: ℳ  ℳ  ℳ → ∞ ∪ 0, ∞ ,  φ: ℳ  ℳ  ℳ → 𝑅 ,  
𝜓: 𝑅 → 𝑅 , and 𝐹: 𝑅 → 𝑅  by  

𝑓 𝑟 √
𝑟, 𝑟 ∈ 0,1 ,

2𝑟, 𝑟 ∈ 1, ∞
               , 𝜔 𝑟, 𝑢, 𝑣

𝑒 , 𝑟 ∈ 0,1

         , otherwise 
 

 φ 𝑟, 𝑢, 𝑣    for all 𝑟, 𝑢, 𝑣 ∈ ℳ, 𝜓 (𝑡 , 𝑡  , 𝑡 , 𝑡 ) = 𝜏 0  

 
and 𝐹 𝑞 ln 𝑞  with  𝑞 0. 

𝑟
1
3

, 𝜖 1 , 𝐵 𝑟 , 𝜖
1

6
,
5
6

 

then  

𝛶 ( , 𝑓 , 𝑓 0.732 𝜖. 

If 𝑟, 𝑢, 𝑣 ∈ 𝐵 𝑟 , 𝜖   then 𝜔 𝑟, 𝑢, 𝑣 𝑒 φ 𝑟, 𝑢, 𝑣 . 

      On the other hand,  𝑓 𝑟 ∈ 𝐵 𝑟 , 𝜖  , ∀  𝑟 ∈ 𝐵 𝑟 , 𝜖 . 
Then,  𝜔 𝑓𝑟, 𝑓𝑢, 𝑓𝑣  φ 𝑟, 𝑓𝑟, 𝑓𝑟  with 𝛶 𝑓 𝑟, 𝑓𝑢, 𝑓𝑣 √𝑟 √𝑢  √𝑢 √𝑣

 √𝑟 √𝑣 0. 
Clearly 𝜔 0, 𝑓1, 𝑓1  φ 0, 𝑓1, 𝑓1 , then we have  
𝛶 𝑓𝑟, 𝑓𝑢, 𝑓𝑣  

√𝑟 √𝑢  √𝑟 √𝑢

√𝑟 √𝑢

√𝑢 √𝑣  √𝑢 √𝑣

√𝑢 √𝑣

√𝑟 √𝑣   √𝑟 √𝑣

√𝑟 √𝑣
 

𝑟 𝑢

√𝑟 √𝑢
 

𝑢 𝑣

√𝑢 √𝑣
 

𝑟 𝑣

√𝑟 √𝑣
ℎ |𝑟 𝑢|  |𝑢 𝑣|  |𝑟 𝑣|  

Consequently  

𝜏 𝐹 𝛶 𝑓 𝑟, 𝑓𝑢, 𝑓𝑣 τ ln 𝛶 𝑓 𝑟, 𝑓𝑢, 𝑓𝑣 ln ℎ 𝛶 𝑟 , 𝑢, 𝑣 𝐹 ℎ 𝛶 𝑟 , 𝑢, 𝑣 . 

If  𝑟 ∉ 𝐵 𝑟 , 𝜖  or   𝑢 ∉ 𝐵 𝑟 , 𝜖  or   𝑣 ∉ 𝐵 𝑟 , 𝜖  then 

𝜔 𝑟 , 𝑢, 𝑣
1
5

≱
1
3

φ 𝑟 , 𝑢, 𝑣  

2(|𝑟 𝑢|  |𝑢 𝑣|  |𝑟 𝑣| |𝑟 𝑢|  |𝑢 𝑣|  |𝑟 𝑣|  
|𝑓 𝑟 𝑓 𝑢| |𝑓 𝑢 𝑓 𝑣| |𝑓 𝑟 𝑓 𝑣| |𝑟 𝑢|  |𝑢 𝑣|  |𝑟 𝑣| 
 𝜏 𝐹 𝛶 𝑓 𝑟, 𝑓 𝑢, 𝑓 𝑣  𝐹 𝛶 𝑟 , 𝑢, 𝑣  
The contraction does not hold. 
 
3.Conclusion and Open Problem 

    This research focus on introducing new idea of F-contraction on a closed ball which is 
different from F-contraction given in [4]. Therefore a generalization of results is very useful so 
far as it requires the F-contraction mapping only on a closed ball rather the whole space. This 
new idea however guides the researcher towards futher investigations and applications.At the 



 
 

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same time,it will be interesting to apply these concepts in a various spaces.  In future,we suggest 
study the results in [8] to verify the extent achieved in the setting of 𝐹  contraction mappings in 
modular spaces. 

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9. Phaneendra, T.; Swamy,K. K. Unique Fixed Point in G-Metric Space Through Greatest 
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