Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 

 

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Matthews Partial Metric Space Using 

𝓕-Contraction 

 

Bonuga Vijayabaskerreddy1,   Veladi Srinivas 2 
1 Department of Mathematics, Sreenidhi Institute of Science and Technology (SNIST), Hyderabad, India. 

Email: basker.bonuga@gmail.com (ORCID ID: 0000-0001-6254-3973) 

2 Department of mathematics, University college of Science, Osmania University, Hyderabad, India. 

Email: srinivasmaths4141@gmail.com (ORCID ID:0000-0003-1991-4569) 

 

Article History: 

Received: 12-05-2024 

Revised: 22-06-2024 

Accepted: 09-07-2024 

Abstract:  

Following the notion of partial metric space (briefly PMS) established by Matthaws [1], in 

this present research article, we proved common fixed point theorem for two pair of self 

maps using the weakly compatible mappings through  ℱ -contraction. In addition, we give 

an illustrative example.  

Keywords: Partial Metric Space (PMS), ℱ-contraction, weakly compatible (WC). 

MSC (2000): 54H25;47H10. 

1.  Introduction 

As a generalization of metric space, the concept of partial metric space emphasized by Matthews [1] 

in the year 1994.Recently, many fixed point theory researchers established fixed point theorems using 

different contractions with different weaker conditions. The PME play an vital role in study of data 

flows network and also theory of computation in the computer science. Some authors prove fixed point 

theorems in PMS like, [2],[3],[4],[6], and [8]. 

In the metric space the notion of F-contraction proposed by Wardowski [5], which is generalization of 

well known Banach contraction principle. Recently, Nazam M et.al, proved common fixed point 

theorems using one and two self mappings concerning F-contraction in PMS [7].  

On the other hand, Sessa initiated the notation of weakly commuting maps which generalized the 

concept of commuting mappings consequently Jungck G, Rhoades B E [9] generalized this idea first 

to compatible mappings and later to weakly compatible (shortly WC) mappings. 

The aim of the research article is to establish existence of unique common fixed point theorem for four 

self mappings through  F -contraction using the idea of WC mappings in PMS. 

Now we recall useful fundamental Definitions, Lemmas of PMS.  

Definition 1.1[1]: A Partial metric on non-empty set 𝔛 is a function 𝒫:𝔛 × 𝔛 → ℝ+ such that for all 

𝜆, 𝜇, 𝜉 𝑖𝑛 𝔛: 

              (𝒫ℳ𝒮1):   𝒫( 𝜆 , 𝜆) = 𝒫(𝜆, 𝜇) = 𝒫(𝜇, 𝜇) ⇔ 𝜆 = 𝜇  

              (𝒫ℳ𝒮2):    𝒫(𝜆, 𝜆) ≤ 𝒫(𝜆, 𝜇)    

mailto:basker.bonuga@gmail.com


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              (𝒫ℳ𝒮3):    𝒫(𝜆, 𝜇) = 𝒫(𝜇, 𝜆)  

               (𝒫ℳ𝒮4):   𝒫 (𝜆, 𝜇) ≤ 𝒫(𝜆, 𝜉) + 𝒫(𝜉, 𝜇) − 𝒫(𝜉, 𝜉)  .  

The pair (𝔛, 𝒫) is called partial metric space (briefly PMS) and 𝒫 is a partial metric on 𝔛. 

A mapping    𝒫𝑠: : 𝔛 × 𝔛 → ℝ+ is defined by  𝒫𝑠(𝜆, 𝜇) = 2𝒫(𝜆, 𝜇) − 𝒫(𝜆, 𝜆) − 𝒫(𝜇, 𝜇)  

is usual metric, where  𝒫  is partial metric on 𝔛. 

Example 1.2[2,3]: Suppose that 𝔛 = ℝ+⋃{ 0 }and we defined  𝒫(𝜆, 𝜇) = max {𝜆, 𝜇}  ∀𝜆, 𝜇 ∈

𝔛.Then  (𝔛, 𝒫) is PMS but not (usual) metric space.  

Definition 1.3[1]: Let (𝔛, 𝒫) be a PMS. A sequence {𝜆𝓃} ⊆ 𝒳  converges to 𝜆 ∈ 𝔛 if and only if  

𝒫(𝜆, 𝜆) = lim
𝜂→∞

𝒫(𝜆, 𝜆𝜂). Also cauchy sequence ⇔ lim
𝜂,𝜁→∞

𝒫(𝜆𝜂 , 𝜆𝜁) exists finitely. 

Lemma 1.4 [1]: Let  (𝔛, 𝒫) be a PMS and then, 

(i)  {𝜆𝜂}  is Cauchy sequence in  (𝔛, 𝒫)  if and only if it is a cauchy in metric space  (𝔛, 𝒫𝑠). 

(ii)  (𝔛, 𝒫) is complete PMS if and only if   (𝔛, 𝒫𝑠) is complete. Moreover, lim
𝜂→∞

𝒫𝑠( 𝜆,   𝜆𝜂) =

0 ⇔ 𝒫( 𝜆, 𝜆) = lim
𝜂→∞

𝒫 (𝜆, 𝜆𝜂) =  lim
𝜂,𝜁→∞

𝒫 ( 𝜆𝜂 , 𝜆𝜁) . 

Lemma1.5 [4]: Assume that 𝜆𝓃 ⟶ 𝜉 as 𝜂 → ∞ in  (𝔛, 𝒫) with  𝒫(𝜉, 𝜉) = 0 then 

lim
𝜂→∞

 𝒫(𝜆𝓃, 𝜇) =  𝒫(𝜉, 𝜇)   ∀ 𝜇 ∈ 𝔛. 

Lemma 1.6[4]: Let (𝔛, 𝒫) be a PMS.  

(i) If 𝒫 ( 𝜆, 𝜇) = 0 then 𝜆 = 𝜇.  (ii)   if 𝜆 ≠ 𝜇 then 𝒫(  𝜆, 𝜇) > 0. 

Definition 1.7[5]: A mapping ℱ: ℝ+→ℝ is said to be ℱ -contraction if it satisfying following 

conditions 

(ℱ1): if 𝜆, 𝜇 ∈ ℝ+ such that 𝜆 < 𝜇 ⇒ ℱ(𝜆) < ℱ(𝜇) 

(ℱ2): for each {𝛼𝜂}𝜂∈ℕ ∈ ℝ+,  lim
𝜂→∞

𝛼𝜂 = 0 if and only if   lim
𝜂→∞

ℱ(𝛼𝜂) = −∞. 

(ℱ3) ∃ real number 𝜃 ∈ (0,1) such that  lim
𝛼→0+

𝛼𝜃ℱ(𝛼) = 0. 

Notation 1.8[5]: We symbolize the collection of all functions which satisfy the   above specified 

constraints ℱ1 to ℱ3 by ∆ℱ . 

Definition 1.9[5]: A self mapping 𝔄: 𝔛→𝔛 is said to ℱ - contraction if there exists a 𝜏 > 0 such that 

for all 𝜆, 𝜇 ∈ 𝔛,  𝒹(𝔄𝜆, 𝔄𝜇) > 0 and we have 𝜏 + ℱ(𝒹(𝔄𝜆, 𝔄𝜇)) ≤ ℱ(𝒹(𝜆, 𝜇)). 

Example 1.10: Let ℱ: ℝ+ → ℝ be given by ℱ(𝛼) = log𝑒 𝛼 and satisfies ℱ1 to ℱ3.Each mapping 

𝔄: 𝔛→𝔛 is an ℱ -contraction such that for all 𝜆, 𝜇 ∈ 𝔛, 𝔄𝜆 ≠ 𝔄𝜇,𝒹(𝔄𝜆, 𝔄𝜇) ≤ 𝑒−𝜏𝒹(𝜆, 𝜇). 

It is clear that 𝜆, 𝜇 ∈ 𝔛 such that 𝔄𝜆 = 𝔄𝜇 then   𝒹(𝔄𝜆, 𝔄𝜇) ≤ 𝑒−𝜏𝒹(𝜆, 𝜇) also satisfying i.e. 𝔄 is 

Banach contraction. 



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Example 1.11: Defined the complete PMS  (𝔛, 𝒫) by 𝒫 ( 𝜆, 𝜇) = 𝑚𝑎𝑥{𝜆, 𝜇}  and also complete 

metric space (𝔛, 𝒹) by 𝒹(𝜆, 𝜇) = |𝜆 − 𝜇| for all 𝜆, 𝜇 ∈ 𝔛. 

Define mappings ℱ ∶ ℝ+ ⟶ ℝ and ℱ(𝛼) = log𝑒 𝛼 and 𝔄 by 𝔄(𝜆) = {
 
𝜆

2
   if 𝜆 ∈ [0,1)

3

4
   if 𝜆 = 1

 then  

𝔄 is not a ℱ -contraction in metric space certainly for  𝜆 =1 and 𝜇 =1/2, 

𝒹(𝔄𝜆, 𝔄𝜇) >0   and we have 𝜏 + ℱ(𝒹(𝔄𝜆, 𝔄𝜇)) ≤ ℱ(𝒹(𝜆, 𝜇))  

⇒ 𝜏 + ℱ (𝒹 (𝔄(1), 𝔄 (
1

2
))) ≤ ℱ (𝒹 (1,

1

2
)) 

                                                ⇒ 𝜏 + |
3

4
−

1

4
| ≤ |1 −

1

2
| 

                                                ⇒ 𝜏 +
1

2
≤

1

2
 which is a contradiction for all 𝜏 > 0.  

Now if we studying in PMS  (𝔛, 𝒫) we get, 

                                              𝜏 + ℱ(𝒫(𝔄𝜆, 𝔄𝜇)) ≤ ℱ(𝒫(𝜆, 𝜇)) 

                             ⇒ 𝜏 + ℱ (𝒫 (𝔄(1), 𝔄 (
1

2
))) ≤ ℱ (𝒫 (1,

1

2
)) 

                              ⇒ 𝜏 + ℱ (max {
3

4
,

1

4
}) ≤ ℱ (max {1,

1

2
})    

                              ⇒ 𝜏 + ℱ (
3

4
) ≤ ℱ(1)  which is true. 

In similar manner our assertion is true for every other points in 𝔛. 

Definition 1.12:  Let a pair if self mappings  𝔣 and 𝔤 are defined on a set 𝔛 is weakly compatible 

(WC) if a  point 𝜆 ∈ 𝔛 is such that     𝔣𝜆 = 𝔤𝜆   implies  𝔣𝔤𝜆 = 𝔤𝔣𝜆.   

The aim of this paper is to develop a fixed point theorem for ℱ − contraction in PMS (𝔛, 𝒫)  using 

the notation of weakly compatibility. 

In the next section we present our main result. 

3.Main Results 

Theorem 3.1: Let (𝔛, 𝒫) be a complete PMS. Suppose that 𝔣, 𝔤, 𝔖 and 𝔗: 𝔛→𝔛 are four self 

mappings satisfying  

(i) 𝔣(𝔛) 𝔗(𝔛) and  𝔤 (𝔛)  𝔖(𝔛) 

(ii) two pairs (𝔣, 𝔖) and (𝔤, 𝔗) are WC mappings 

(iii) 𝔣(𝔛) or  𝔗(𝔛) or 𝔤 (𝔛) or  𝔖(𝔛)  is closed subset of  (𝔛, 𝒫) 

(iv) assume that there exists ℱ ∆ℱ and 𝜏 > 0  for 𝜆, 𝜇 ∈ 𝔛 such that  

 𝒫(𝔣𝜆, 𝔤𝜇) > 0 ⟹  𝜏 + ℱ( 𝒫(𝔣𝜆, 𝔤𝜇)) ≤ ℱ(ℳ(𝜆, 𝜇))………….(3.1.1) 

where ℳ(𝜆, 𝜇) = 𝑚𝑎𝑥{𝒫(𝔖𝜆, 𝔗𝜇), 𝒫(𝔣𝜆, 𝔖𝜆), 𝒫(𝔤𝜇, 𝔗𝜇),
1

2
[𝒫(𝔣𝜆, 𝔗𝜇) + 𝒫(𝔤𝜇, 𝔖𝜆)].  



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Then  𝔖 , 𝔗, 𝔣  and 𝔤 have a unique common fixed point in 𝔛 . 

Proof: Let  λ0𝔛 be any point. By the (i) of (3.1), we construct sequences {𝜆𝜂} 𝑎𝑛𝑑 

 {𝜇𝜂}    𝑖𝑛 𝔛 satisfying 

  𝔗𝜆2𝜂+1 = 𝔣𝜆2𝜂 = 𝜇2𝜂+1 and 𝔖𝜆2𝜂+2 = 𝔤𝜆2𝜂+1 = 𝜇2𝜂+2 ………………… .(3.1.2) 

for  𝜂 =0,1, 2 ,……. 

Step-I: To prove that 𝒫(𝜇𝜂 , 𝜇𝜂+1) ⟶ 0 as 𝜂 ⟶ 0. 

𝜏 + ℱ(𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) ≤ ℱ(ℳ(𝜆2𝜂 , 𝜆2𝜂+1))…………………………….. .(3.1.3) 

  It follows from  (𝒫ℳ𝒮2)and  (𝒫ℳ𝒮4) that 

    ℳ(𝜆2𝜂 , 𝜆2𝜂+1) = ℳ(𝜆, 𝜇) 

= max {𝒫( 𝔖𝜆2𝜂 , 𝔗𝜆2𝜂+1), 𝒫(𝔣𝜆2𝜂 , 𝔖𝜆2𝜂), 𝒫(𝔤𝜆2𝜂+1, 𝔗𝜆2𝜂+1),
1

2
[𝒫(𝔣𝜆2𝜂 , 𝔗𝜆2𝜂+1)

+ 𝒫(𝔤𝜆2𝜂+1, 𝔖𝜆2𝜂)] 

              = max {𝒫(𝜇2𝜂𝓃, 𝜇2𝓃𝜂+1), 𝒫(𝜇2𝓃𝜂+1, 𝜇2𝜂𝓃), 𝒫(𝜇2𝑛𝜂+2, 𝜇2𝓃𝜂+1),                    

                                                       
1

2
[𝒫(𝜇2𝜂𝓃, 𝜇2𝜂+2) + 𝒫(𝜇2𝜂+1, 𝜇2𝜂+1)]} 

               ≤ 𝑚𝑎𝑥 {𝒫(𝜇2𝜂 , 𝜇2𝜂+1), 𝒫(𝜇2𝜂+2, 𝜇2𝜂+1),
1

2
[𝒫(𝜇2𝜂+1, 𝜇2𝜂+1) + 𝒫(𝜇2𝜂 , 𝜇2𝜂+1) +

                                         𝒫(𝜇2𝜂+1, 𝜇2𝜂+2) − 𝒫(𝜇2𝜂+1, 𝜇2𝜂+1)]} 

               ≤ 𝑚𝑎𝑥{𝒫(𝜇2𝜂 , 𝜇2𝜂+1), 𝒫(𝜇2𝜂+2, 𝜇2𝜂+1)} 

If  𝑚𝑎𝑥{𝒫(𝜇2𝜂 , 𝜇2𝜂+1), 𝒫(𝜇2𝜂+2, 𝜇2𝜂+1)} = 𝒫(𝜇2𝜂+1, 𝜇2𝜂+2) then  

          𝜏 + ℱ(𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) ≤ ℱ(𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) this implies   

           ℱ(𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) ≤ ℱ(𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) − 𝜏  which is congtradiction to  (ℱ -1). 

    Thus 𝑚𝑎𝑥{𝒫(𝜇2𝜂 , 𝜇2𝜂+1), 𝒫(𝜇2𝜂+2, 𝜇2𝜂+1)} = 𝒫(𝜇2𝜂 , 𝜇2𝜂+1) for all 𝜂 ∈ 𝑁. 

From (10),  𝜏 + ℱ(𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) ≤ ℱ(𝒫(𝜇2𝜂 , 𝜇2𝜂+1))… (3.1.4) 

Constituting this way, it follows that 

ℱ(𝒫(𝜇2𝜂 , 𝜇2𝜂+1)) ≤ 𝐹 (𝒫(𝜇2𝜂−1, 𝜇2𝜂)) − 𝜏 …… (3.1.5). 

Using (3.1.4) and (3.1.5) 

  ℱ(𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) ≤ ℱ(𝒫(𝜇2𝜂 , 𝜇2𝜂+1)) − 𝜏 ≤ ℱ(𝒫(𝜇2𝜂−1, 𝜇2𝜂)) − 2𝜏 

                            ≤ ℱ(𝒫(𝜇2𝜂−2, 𝜇2𝜂−1)) − 3𝜏 ≤ ℱ(𝒫(𝜇0, 𝜇1)) − (2𝜂 + 1)𝜏…….(3.1.6) 

And ℱ(𝒫(𝜇2𝜂 , 𝜇2𝜂+1)) ≤ ℱ(𝒫(𝜇2𝜂−1, 𝜇2𝜂)) − 𝜏 



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                                 ≤ ℱ (𝒫((𝜇2𝜂−2, 𝜇2𝜂−1)) − 2𝜏 

                           ≤ ℱ(𝒫((𝜇0, 𝜇1)) − (2𝜂)𝜏…..(3.1.7) 

Repeating, 

ℱ(𝒫(𝜇𝜂 , 𝜇𝜂+1)) ≤ ℱ(𝒫((𝜇0, 𝜇1)) − 𝜂𝜏 then it follows lim
𝓃𝜂→∞

ℱ(𝒫(𝜇𝜂 , 𝜇𝜂+1)) = −∞ by ℱ∆ℱ and 

 (ℱ -2) we have lim
𝜂→∞

𝒫(𝜇𝜂 , 𝜇𝜂+1) = 0..…..(3.1.8) 

Step-II 

Now we prove that {𝜇𝜂} is 𝒫 -Cauchy sequence. By  ℱ∆ℱ and (ℱ -3) thereexists 𝓀 ∈ (0,1) such 

that  

lim
𝜂→∞

𝒫(𝜇𝜂 , 𝜇𝜂+1)𝓀  ℱ(𝒫(𝜇𝜂 , 𝜇𝜂+1)) = 0…. (3.1.9) 

By (3.1.6) and (3.1.7) 

lim
𝜂→∞

𝒫(𝜇𝜂 , 𝜇𝜂+1)𝓀  ℱ(𝒫(𝜇𝜂 , 𝜇𝜂+1)) = 0 

lim
𝜂→∞

𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)𝓀  ℱ (𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)) − ℱ(𝒫(𝜇0, 𝜇1) 

                           ≤ 𝒫(𝜇2𝜂+1, 𝜇2𝜂+2)
𝓀

− (2𝜂 + 1)𝜏 ≤ 0-----(3.1.10) 

And 

lim
𝜂→∞

𝒫(𝜇2𝜂 , 𝜇2𝜂+1)𝓀  ℱ (𝒫(𝜇2𝜂 , 𝜇2𝜂+1)) − ℱ(𝒫(𝜇0, 𝜇1) 

                                             ≤ 𝒫(𝜇2𝜂 , 𝜇2𝜂+1)
𝓀

− (2𝜂)𝜏 ≤ 0----- (3.1.11) 

Using the above inequality and (3.1.9) 

lim
𝜂→∞

𝜂 𝒫(𝜇2𝜂 , 𝜇2𝜂+1)
𝓀

= 0. 

Therefore, there exists 𝜂 ∈ ℕ such that   𝜂. 𝒫(𝜇2𝜂 , 𝜇2𝜂+1)
𝓀

< 1 for all 𝜂 ≥ 𝜂1. 

(Or)  𝒫(𝜇𝓃, 𝜇𝓃+1) <
1

𝜂1/𝓀………..(3.1.12) 

Let  𝜁, 𝜂 ∈ ℕ with 𝜁 > 𝜂 > 𝜂1 using triangular inequality we have  

𝒫(𝜇𝜂 , 𝜇𝜁) = 𝒫(𝜇𝓃𝜂 , 𝜇𝜂+1) + 𝒫(𝜇𝜂+1, 𝜇𝜂+2) … . +𝒫(𝜇𝜁−1, 𝜇𝜁) 

                    −[𝒫(𝜇𝜂+1, 𝜇𝜂+1) + 𝒫(𝜇𝜂+2, 𝜇𝜂+2) + ⋯ . 𝒫(𝜇𝜁−1, 𝜇𝜁−1)] 

                   ≤ ∑ 𝒫(𝜇𝒾, 𝜇𝒾+1) ≤ ∑ 𝒫(𝜇𝒾, 𝜇𝒾+1) ≤ ∑
1

𝒾
1
𝓀

.∞
𝒾=1

∞
𝒾=1

𝜁−1
𝒾=1  

As  𝓀 ∈ (0,1) the infinite series ∑
1

𝒾
1
𝓀

  ∞
𝒾=1 converges, consequently we get lim

𝜁,𝜂→∞
𝒫(𝜇𝜂 , 𝜇𝜁) = 0.  This 



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proves that  {𝜇𝜂} is a Cauchy sequence in (𝔛, 𝒫)  and consequently we get  {𝜇𝜂} is Cauchy in 

(𝔛, 𝒹𝒫). 

Since (𝔛, 𝒫) is complete PMS, this ensures that  (𝔛, 𝒹𝒫)   is complete metric space, then ∃ 𝓊𝓅 ∈ 𝔛 

such that lim
𝜂→∞

𝒹𝒫(𝜇𝜂 , 𝓊𝓅) = 0. 

Moreover 𝒫(𝓊𝓅 , 𝓊𝓅) = lim
𝓃→∞

𝒫(𝜇𝜂 , 𝓊𝓅) = lim
𝜁,,𝜂→∞

𝒫(𝜇𝜂 , 𝜇𝜁) = 0.......(3.1.13) 

Since 𝜇𝜂 → 𝓊𝓅 then   𝔗𝜆2𝜂+1 , 𝔣𝜆2𝜂  𝔖𝜆2𝜂+2   and   𝔤𝜆2𝜂+1   converges to 𝓊𝓅. 

Step-III 

Now we claim that the mappings 𝔖 , 𝔗, 𝔣  and 𝔤  have a common fixed point. 

Suppose that the Range  𝔗(𝔛) is closed, then  ∃ 𝜈𝓅 ∈ 𝔛 such that 𝔗𝜈𝓅 =  𝓊𝓅……. (3.1.14) 

Now assuming that 𝔤𝜈𝓅 ≠  𝓊𝓅 and put 𝜆 = 𝜆2𝜂  , 𝜇 = 𝜈𝓅  in (3.1.1) then 

(𝔣𝜆2𝜂 , 𝔤𝜈𝓅) > 0 ⟹  𝜏 + ℱ( 𝒫(𝔣𝜆2𝜂 , 𝔤𝜈𝓅)) ≤ ℱ (ℳ(𝜆2𝜂 , 𝜈𝓅))…..(3.1.15) 

where 

 ℳ(𝜆2𝜂 , 𝜈𝓅) = 𝑚𝑎𝑥{𝒫(𝔖𝜆2𝜂 , 𝔗𝜈𝓅), 𝒫(𝔣𝜆2𝜂, 𝔖𝜆2𝜂), 𝒫(𝔤𝜈𝓅 , 𝔗𝜈𝓅),
1

2
[𝒫(𝔣𝜆2𝜂 , 𝔗𝜈𝓅) +

𝒫(𝔤𝜈𝓅 , 𝔖𝜆2𝜂)]} 

                             = max {𝒫(𝓊𝓅 , 𝓊𝓅), 𝒫(𝓊𝓅 , 𝔤𝜈𝓅), 𝒫(𝓊𝓅 , 𝓊𝓅),  
1

2
[𝒫(𝓊𝓅 , 𝓊𝓅) + 𝒫(𝓊𝓅 , 𝔤𝜈𝓅)]} 

 𝜏 + ℱ(𝒫(𝓊𝓅 , 𝔤𝜈𝓅)) ≤ ℱ (𝒫(𝓊𝓅 , 𝔤𝜈𝓅))   this a contradiction with 𝜏 > 0.  

Thus 𝔤𝜈𝓅=𝓊𝓅.  ……(3.1.15) 

Therefore using (3.1.14) and (3.1.15) we obtain 𝔗𝜈𝓅 = 𝔤𝜈𝓅 = 𝓊𝓅 .Since 𝔤 and 𝔗 are WC mappings 

then, 𝔤𝓊𝓅 = 𝔤𝔗𝜈𝓅 = 𝔗𝔤𝜈𝓅 = 𝔗𝓊𝓅 . ………….(3.1.16) 

Now we show that 𝔤𝓊𝓅 = 𝜈𝓅 . 

On contrary, let 𝔤𝓊𝓅 ≠ 𝜈𝓅 and use 𝜆 = 𝜆2𝜂 , 𝜇 = 𝜈𝓅 in (3.1.1) then 

𝜏 + ℱ(𝒫(𝔣𝜆2𝜂 , 𝔤𝜈𝓅)) ≤ ℱ (ℳ(𝜆2𝜂𝓊𝓅)) 

ℳ(𝜆2𝜂 , 𝓊𝓅) = 𝑚𝑎𝑥{𝒫(𝔖𝜆2𝜂 , 𝔗𝓊𝓅), 𝒫(𝔣𝓍𝜆2𝓃, 𝔖𝜆2𝜂), 𝒫(𝔤𝓊𝓅 , 𝔗𝓊𝓅),
1

2
[𝒫(𝔣𝜆2𝜂 , 𝔗𝓊𝓅)

+ 𝒫(𝔤𝓊𝓅 , 𝔖𝜆2𝓃)]} 

                                        = 𝒫(𝓊𝓅 , 𝔤𝓊𝓅). 

 By permitting, continuity of ℱ and applying the limit as 𝜂→∞ ,we have  

𝜏 + ℱ(𝒫(𝓊𝓅 , 𝔤𝓊𝓅)) ≤ ℱ (ℳ(𝓊𝓅 , 𝔤𝓊𝓅)) which is contradiction. 



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Therefore  𝒫(𝓊𝓅 , 𝔤𝓊𝓅) = 0, this yields 

 𝔗𝓊𝓅 = 𝔤𝓊𝓅 = 𝓊𝓅 …….(3.1.17) 

Now we show that  𝓊𝓅 is a fixed pint of the mappings  𝔣  and  𝔖 . Since 𝔤 (𝒳)  𝔖(𝒳)  ∃ a point 

𝔷𝓅 ∈ 𝒳  such that 𝔤𝓊𝓅 = 𝔖𝔷𝓅. 

Suppose that    𝔣𝔷𝓅 ≠ 𝔖𝔷𝓅, then 

𝜏 + ℱ(𝒫(𝔣𝔷𝓅., 𝔤𝓊𝓅)) ≤ ℱ (ℳ(𝔷𝓅., 𝓊𝓅)) 

ℳ(𝔷𝓅., 𝓊𝓅) = 𝑚𝑎𝑥{𝒫(𝔖𝔷𝓅., 𝔗𝓊𝓅), 𝒫(𝔣𝔷𝓅., 𝔖𝔷𝓅.), 𝒫(𝔤𝓊𝓅 , 𝔗𝓊𝓅),
1

2
[𝒫(𝔣𝔷𝓅., 𝔗𝓊𝓅)

+ 𝒫(𝔤𝓊𝓅 , 𝔖𝔷𝓅.)]} 

= 𝑚𝑎𝑥{𝒫(𝔤𝓊𝓅 , 𝔤𝓊𝓅), 𝒫(𝔤𝓊𝓅 , 𝔤𝓊𝓅), 𝒫(𝔤𝓊𝓅 , 𝔣𝔷𝓅.),
1

2
[𝒫(𝔣𝔷𝓅., 𝔤𝓊𝓅) + 𝒫(𝔤𝓊𝓅 , 𝔤𝓊𝓅)]} 

                = 𝒫(𝔤𝓊𝓅 , 𝔣𝔷𝓅.) 

  𝜏 + ℱ(𝒫(𝔣𝔷𝓅., 𝔤𝓊𝓅)) ≤ 𝒫(𝔤𝓊𝓅 , 𝔣𝔷𝓅)   this a contradiction with 𝜏 > 0.  

Thus 𝔤𝓊𝓅 = 𝔣𝔷𝓅 = 𝔖𝔷𝓅 …. (3.1.18). 

By using weakly compatible nature of  𝔣  and 𝔖, we get   

𝔖𝓊𝓅 = 𝔣 𝔖𝔷𝓅 = 𝔖𝔣𝔷𝓅 = 𝔣𝓊𝓅. …………(3.1.19). 

Finally we show that 𝔣𝓊𝓅 = 𝓊𝓅 .Assume 𝔣𝓊𝓅 ≠ 𝓊𝓅 and put  𝜆 = 𝜇 = 𝓊𝓅 in (3.1.1) we get 

𝜏 + ℱ(𝒫(𝔣𝓊𝓅, 𝓊𝓅)) ≤ ℱ(𝒫(𝔣𝓊𝓅 , 𝔤𝓊𝓅)) ≤ ℱ (ℳ(𝓊𝓅 , 𝓊𝓅)) 

ℳ(𝓊𝓅 , 𝓊𝓅) = 𝑚𝑎𝑥{𝒫(𝔖𝓊𝓅, 𝔗𝓊𝓅), 𝒫(𝔣𝓊𝓅 , 𝔖𝓊𝓅), 𝒫(𝔤𝓊𝓅 , 𝔗𝓊𝓅),
1

2
[𝒫(𝔣𝓊𝓅 , 𝔗𝓊𝓅)

+ 𝒫(𝔤𝓊𝓅 , 𝔖𝓊𝓅)]} 

                      = 𝒫(𝓊𝓅 , 𝔣𝓊𝓅) 

𝜏 + ℱ(𝒫(𝔣𝓊𝓅 , 𝓊𝓅)) ≤ 𝒫(𝔣𝓊𝓅 , 𝓊𝓅) this a contradiction with 𝜏 > 0.  

Thus 𝔣𝓊𝓅 = 𝔖𝓊𝓅 = 𝓊𝓅…..(3.1.20). 

Using (3.1.17) and (3.1.20)  𝓊𝓅 is fixed point of   𝔣 , 𝔤, 𝔖  and  𝔗 . 

Step-IV: 

Let 𝜔𝓅(≠𝓊𝓅) be the another fixed point   of   𝔣 , 𝔤, 𝔖  and  𝔗. 

Put    𝜆 = 𝓊𝓅 , 𝜇 = 𝜔𝓅 in the contraction (3.1.1), we get 

𝜏 + ℱ (𝒫(𝔣𝓊𝓅, 𝜔𝓅)) = 𝜏 + ℱ (𝒫(𝔣𝓊𝓅 , 𝔤𝜔𝓅)) ≤ ℱ (ℳ(𝓊𝓅 , 𝜔𝓅)) 



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ℳ(𝓊𝓅 , , 𝜔𝓅) = 𝑚𝑎𝑥{𝒫(𝔖𝓊𝓅 , 𝔗𝜔𝓅), 𝒫(𝔣𝓊𝓅, 𝔖𝜔𝓅), 𝒫(𝔤𝜔𝓅 , 𝔗𝜔𝓅),
1

2
[𝒫(𝔣𝓊𝓅 , 𝔗𝜔𝓅)

+ 𝒫(𝔤𝜔𝓅 , 𝔖𝓊𝓅)]} 

                       = 𝒫(𝓊𝓅 , 𝜔𝓅) 

𝜏 + ℱ (𝒫(𝔣𝓊𝓅 , 𝜔𝓅)) ≤ 𝒫(𝓊𝓅 , 𝜔𝓅) this a contradiction with 𝜏 > 0. Thus 𝓊𝓅 = 𝜔𝓅. 

In conclusion, these four mappings have a unique common fixed point.       

Example 3.2: Let (𝔛, 𝒫) with 𝔛 = [0,1] is a complete PMS and 𝒫 (𝜆, 𝜇)=max{ 𝜆, 𝜇 }  ∀ 𝜆, 𝜇 ∈ 𝔛. 

Define mappings ℱ: ℝ+ ⟶ ℝ and ℱ(𝛼) = log𝑒 𝛼.  

Let  𝔣 , 𝔤, 𝔖  and  𝔗: 𝔛 → 𝔛 be defined as 𝔣(𝜆) =
𝜆

4
, 𝔤 (𝜆) = 0, 𝔖(𝜆) =

3𝜆

2
, 𝔗(𝜆) = 𝜆,  then 

𝔣(𝔛) = [0,
1

4
]  ,    𝔤 (𝔛) = {0},       𝔖(𝔛) = [0,

3

4
] , and    𝔗(𝔛) = [0,1]  these  ranges of mappings 

satisfying the inclusion inequalities (i) of theorem 3.1. 

Now clearly 𝔣(0) = 𝔖(0) = 0 gives 𝔣𝔖(0) = 𝔖𝔣(0)  which gives the pair (𝔣, 𝔖) is weakly 

compatible and  𝔤(0) = 𝔗(0) = 0 gives 𝔤𝔗(0) = 𝔗𝔤(0) this implies the pair (𝔤, 𝔗) is weakly 

compatible at the coincident point zero.  

We discuss the existence of contraction condition (3.1.1) under some conditions 

Now 𝒫(𝔣𝜆, 𝔤𝜇) = max {
𝜆

4
, 0} ,𝒫(𝔖𝜆, 𝔗𝜇) = 𝑚𝑎𝑥 {

3𝜆

2
, 𝜇},𝒫(𝔣𝜆, 𝔖𝜆) = max {

𝜆

4
,

3𝜆

2
}, 

𝒫(𝔤𝜇, 𝔗𝜇) = max{0, 𝜇} , 𝒫(𝔣𝜆, 𝔗𝜇) = max {
𝜆

4
, 𝜇}   and 𝒫(𝔤𝜇, 𝔖𝜆) = max {0,

3𝜆

2
} .  

Case I: If  
3𝜆

2
> 𝜇 then  

ℳ(𝜆, 𝜇) = 𝑚𝑎𝑥 {
3𝜆

2
,
3𝜆

2
, 𝜇,

1

2
[
𝜆

4
+

3𝜆

2
]} =

3𝜆

2
 

𝒫(𝔣𝜆, 𝔤𝜇) = 𝑚𝑎𝑥 {
𝜆

4
, 0} =

𝜆

4
> 0 ⟹  𝜏 + ℱ (

𝜆

4
) ≤ ℱ (

3𝜆

2
) ⟹ 𝜏 + log𝑒 (

𝜆

4
) ≤  log𝑒 (

6𝜆

4
) . 

Case II: If  
3𝜆

2
< 𝜇 then  

ℳ(𝜆, 𝜇) = 𝑚𝑎𝑥 {𝜇,
3𝜆

2
, 𝜇,

1

2
[𝜇 +

3

4
]} = 𝜇 

𝒫(𝔣𝜆, 𝔤𝜇) = 𝑚𝑎𝑥 {
𝜆

4
, 0} =

𝜆

4
> 0 ⟹  𝜏 + 𝐹 (

𝜆

4
) ≤ 𝐹(𝜇) ⟹ 𝜏 + log𝑒 (

𝜆

4
) ≤  log𝑒(𝜇). 

This gives 𝜏 + log𝑒
1

6
(

3𝜆

2
) ≤  log𝑒(𝜇). 

In both the cases, inequality (3.1.1)  is satified  for all   λ, μ .   

In this illustration, it is observed that zero is the unique common fixed point.  



Communications on Applied Nonlinear Analysis 

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4.Conclusion: In this research article, we have proved a fixed point result for  ℱ − contraction in      

PMS via weakly compatible mappings. Finally, we have provided one example to support our main 

result with unique common fixed point. 

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https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
https://doi.org/10.1016/j.amc.2011.07.005
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https://www.researchgate.net/publication/277880136_A_Focus_on_a_Common_Fixed_Point_Theorem_using_Weakly_Compatible_Mappings