Communications on Applied Nonlinear Analysis 

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Vol 31 No. 5s (2024) 

 

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A Common Fixed Point Result in Menger Space 

*1 Ajay Kumar Chaudhary 
1Department of Mathematics, Trichandra Multiple Campus 

Tribhuvan University, Kathmandu, Nepal 

* Corresponding author E-mail: akcsaurya81@gmail.com 

 

Article History: 

Received: 13-05-2024 

Revised: 23-06-2024 

Accepted: 10-07-2024 

Abstract:  

By using compatibility condition type (P) in probabilistic metric space, establish common 

fixed point results for four self-mappings with control function in [0,1]. The result of 

Chaudhary et. al [5] is a particular case of this new result and it extends and generalizes 

other similar results in the literature. 

Keywords:  Common fixed point, Menger Space, Compatible mappings, Compatible 

mappings of type (P). 

Mathematics Subject Classification: 47H10, 54H25 

  

1. Introduction:  

Probabilistic metric space (PM space) is the idea of Karl’s Menger [11], a significant generalization of 

M. Frechet's [3] metric space. If PM space includes Menger inequality, then it is called Menger space. 

This space becomes active after the significant work of B. Schweizer and A. Skalar [13], [16] and 

V.M. Sehgal and A.T. Barucha Reid [14]. 

In 1991, S. N. Mishra [12] introduced the notion of compatible mapping in the Menger space and then 

so many researchers worked in this space, defining weakly compatible mappings, different compatible 

mappings types like (A), (K), (P) etc. see references [[2], [6], [7], [8], [9], [10], [14], [16]]. Recently, 

Chaudhary et. al [5-6] have given notions of compatible mapping of type (P) and weakly compatible 

mappings of type (P). 

This paper gives the new results in Menger space by using a control function  φ: [0,1] → [0,1] in four 

self-mappings and also deduces some consequences. 

2. Preliminaries: 

Definition 2.1 [16]: If a function 𝑀: ℝ → ℝ+ is 

(i) a non-decreasing function,                                   

(ii) left continuous and   

(iii) inf { 𝐹 (𝑥): 𝑥 ∈  ℝ}  =  0, sup{𝐹 (𝑥): 𝑥 ∈ ℝ}  =  1    

      then 𝑀 is said to be a distribution function.      

Definition 2.2 [4]: Let 𝑀: 𝑌 × 𝑌 →  𝐿 be a distribution function, 𝐿 be the set of all distribution 

functions and 𝑌 be a non-empty set. Then, a  pair (𝑌, 𝑀) is said to be probabilistic metric space 

(abbreviated as pm-Space) if the distribution function 𝑀 (𝑝, 𝑞), where (𝑝, 𝑞) ∈ 𝑌 × 𝑌, also denoted by 

𝑀𝑝,𝑞 satisfies following conditions: 

             (M1)      𝑀𝑝,𝑞(𝑥) = 1 for every 𝑥 >  0 if and only if  𝑝 =  𝑞, 



Communications on Applied Nonlinear Analysis 

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 (M2)       𝑀𝑝,𝑞(0) = 0  for every  𝑝, 𝑞 ∈ 𝐾,  

            (M3)      𝑀𝑝,𝑞(𝑥) = 𝑀𝑞,𝑝(𝑥)   for every 𝑝, 𝑞 ∈ 𝐾, and  

 (M4)      𝑀𝑝,𝑞(𝑥 + 𝑦) = 1 if and only if 𝑀𝑝,𝑟(𝑥) = 1 and 𝑀𝑟,𝑞(𝑦) = 1. 

Here, 𝑀𝑝,𝑞(𝑥) represents the value of distribution function 𝑀𝑝,𝑞 𝑎𝑡 𝑥 ∈ ℝ. 

Definition 2.3 [4]: A function  𝑡 ∶  [0, 1]  ×  [0, 1]  → [0, 1] is referred to as a triangular norm  

(shortly t-norm)   

 if it satisfies the following conditions: 

            T1:      𝑡 (0, 0) =  0,  

            T2:     𝑡 (𝑎, 1) = 𝑎  for all 𝑎 ∈ [0, 1], 

            T3:     𝑡 (𝑎, 𝑏) =  𝑡 (𝑏, 𝑎) for all 𝑎, 𝑏 ∈  [0, 1], 

            T4:     𝑖𝑓 𝑎 ≤  𝑐, 𝑏 ≤  𝑑 then 𝑡 (𝑎, 𝑏) ≤  𝑡 (𝑐, 𝑑), and  

 T5: 𝑡 (𝑡 (𝑎, 𝑏), 𝑐) =  𝑡 (𝑎, 𝑡 (𝑏, 𝑐)), where 𝑎, 𝑏, 𝑐, 𝑑 ∈ [0, 1]. 

Definition 2.4 [2]:  A probabilistic metric space (𝑌, 𝑀)  is said to be  Menger space (𝑌, 𝑀, 𝑡),   

                                 where t is a t-norm satisfying the following conditions: 

           (M5)     𝑀𝑝,𝑞(𝑥 + 𝑦) ≥ 𝑡 (𝑀𝑝,𝑟(𝑥), 𝑀𝑟,𝑞(𝑦)) for every 𝑝, 𝑞, 𝑟 ∈ 𝑌 and 𝑥, 𝑦 ∈ ℝ > 0. 

Definition 2.5 [2]: A mapping 𝐴: 𝑌 →  𝑌 in Menger Space (𝐾, 𝐹, 𝑡), is said to be continuous  

 at a point 𝑝 ∈ 𝑌 if for every  >  0  and  >  0, there exist 1 >  0 and 1 > 0 such that  

if     𝑀𝑝,𝑞 (1)  >  1 – 1 then 𝑀𝐴𝑝,𝐴𝑞()  >  1 − . 

Definition 2.6 [2]: Let (𝑌, 𝑀, 𝑡) be a Menger space and 𝑡 be a continuous t-norm. Then, 

(a) A sequence {𝑦𝑛} in 𝑌 is said to converge to a point 𝑦 in 𝑌 if and only if  for every  

 >  0 and  >  0, there exist an integer 𝑁 =  𝑁 (, ) such that 𝑀𝑦𝑛,𝑦() >  1 −   for all 

 𝑛 ≥ 𝑁. In this case, we write,  𝑙𝑖𝑚
𝑛→∞

  𝑦𝑛 = y. 

(b) A sequence {𝑦𝑛}  in 𝑌 is said to be a Cauchy sequence if for every  >  0  and  >  0, there exists 

an integer 𝑁 =  𝑁 (, )  >  0 such that 𝑀𝑦𝑛,𝑦𝑚
() >  1 −    for all 𝑚, 𝑛 ≥ 𝑁. 

(c) A Menger space (𝑌, 𝑀, 𝑡) is said to be complete if every Cauchy sequence in  𝑌 converges to a 

point in  𝑌. 

Definition 2.7:[7]  Common fixed point of self-mapping functions 𝐴, 𝐵: 𝑌 → 𝑌  is a point 𝑦 Î 𝑌 if     

                                𝐴(𝑦)  =  𝐵(𝑦)  =  𝑦. 

Example 2.1: Let 𝐴, 𝐵: ℝ → ℝ be functions such that 𝐴(𝑦)  =  
𝑦2

4
 and 𝐵(𝑦) =   2𝑦 − 4,  

                        then 𝑦 =  4 is a common fixed point of 𝐴 and 𝐵. 



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Definition 2.8:[12] Two mappings 𝐴, 𝐵: 𝑌 → 𝑌 are said to be compatible mappings in Menger   

 space (𝑌, 𝑀, 𝑡) iff  𝑙𝑖𝑚
𝑛→∞

  𝐹𝐴𝐵𝑥𝑛 ,   𝐵𝐴𝑥𝑛 
(𝑥)  =  1  for all 𝑥 > 0, whenever sequence {𝑥𝑛} in Y 

  such that    𝑙𝑖𝑚
𝑛→∞

𝐴𝑥𝑛 = 𝑙𝑖𝑚
𝑛→∞

 𝐵𝑥𝑛 = 𝑦 for some 𝑦 in 𝑌.    

Definition 2.10: [15] Two mappings 𝐴, 𝐵: 𝑌 → 𝑌 are said to be weakly compatible (or coincidently 

commuting) in Menger space (𝑌, 𝐹, 𝑡) if they commute at their coincidence points, that is, if 𝐴𝑥 = 𝐵𝑥  

for some 𝑥 ∈ 𝑌 then 𝐴𝐵𝑥 = 𝐵𝐴𝑥.   

Definition 2.11:[6] Two mappings 𝐴, 𝐵: 𝑌 → 𝑌 are said to be compatible mappings of type (𝑷) in 

Menger space (𝑌, 𝑀, 𝑡) iff 𝑙𝑖𝑚
𝑛→∞

  𝑀𝐴𝐴𝑥𝑛,   𝐵𝐵𝑥𝑛
(𝑥)  =  1 ∀ 𝑥 > 0 whenever {𝑥𝑛} is a sequence in 𝑌 such 

that 𝑙𝑖𝑚
𝑛→∞

 𝐴𝑥𝑛 = 𝑙𝑖𝑚
𝑛→∞

 𝐵𝑥𝑛 = 𝑦  for some 𝑦 𝑖𝑛 𝑌. 

Definition 2.12: [5]Two mappings 𝐴, 𝐵: 𝑌 → 𝑌 are said to be weakly compatible mapping of type(𝑷) 

in Menger Space (𝑌, 𝑀, 𝑡) iff 𝑙𝑖𝑚
𝑛→∞

  𝑀𝐴𝐴𝑥𝑛,   𝐵𝐵𝑥𝑛
(𝑥)  ≥ 𝑀𝐴𝑥𝑛,   𝐵𝑥𝑛

(𝑥)  ∀ 𝑥 > 0,  

 whenever {𝑥𝑛}  is a sequence in 𝑌 such that 𝑙𝑖𝑚
𝑛→∞

 𝐴𝑥𝑛 = 𝑙𝑖𝑚
𝑛→∞

 𝐵𝑥𝑛 = 𝑦  for some 𝑦 𝑖𝑛 𝑌. 

Example 2.2: Let (𝑌, 𝑑) be metric space where 𝑌 = [0, 2]with usual metric 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|  

  and (𝑌, 𝑀) be PM space with 

                          𝑀𝑥,𝑦(𝑡) =  {𝑒 
𝑑(𝑥,𝑦)

𝑡 ,   𝑖𝑓 𝑡 > 0,
0,   𝑖𝑓 𝑡 = 0.

   for all 𝑥, 𝑦 ∈ 𝑌. 

We define 𝐴 and 𝐵 as: 

                           𝐴(𝑥) =  {
1 − 𝑥,    𝑓𝑜𝑟  𝑥 ∈ [0, 1/2)

1 ,   𝑓𝑜𝑟 𝑥 ∈ [ 
1

2
, 2]

 and  𝐵(𝑥) =  {
𝑥, 𝑓𝑜𝑟  𝑥 ∈ [0, 1/2)

1,   𝑓𝑜𝑟 𝑥 ∈ [
1

2
, 2] .

 

Taking sequence {𝑥𝑛}  in 𝑌 where 𝑥𝑛 =
1

2
−

1

𝑛
, 𝑛 ∈ 𝑁. Then, (𝐴, 𝐵) are weakly compatible mappings 

of type (𝑃) and it is neither compatible mappings of type (𝑃) nor compatible mappings. 

Theorem 2.1[2]: Let (𝑌, 𝑀, 𝑡)  be Menger space with the continuous 𝑡 − 𝑛𝑜𝑟𝑚 𝑡 and 𝐴: 𝑌 → 𝑌. Then, 

𝐴 is continuous at a point 𝑦 ∈ 𝑌 if and only if for every sequence {𝑦𝑛} in 𝑌 converging to a point  𝑦, 

then sequence {𝐴𝑦𝑛} converges to the point 𝐴𝑦, i.e. if  {𝑦𝑛}  → 𝑦 then it implies   {𝐴𝑦𝑛}  → 𝐴𝑦. 

Proposition 2.1[9]: In Menger Space(𝑌, 𝑀, 𝑡), if 𝑡 (𝑘, 𝑘) ≥ 𝑘 for all 𝑘 ∈  [0, 1] 

 then 𝑡(𝑎, 𝑏)  =  𝑚𝑖𝑛 {𝑎, 𝑏} for all 𝑎, 𝑏 ∈ [0, 1]. 

Lemma 2.1[15]: Let (𝑌, 𝑀, 𝑡)  be a Menger space. If there exists 𝑘 ∈ (0, 1)  such that 

 for all   𝑝, 𝑞 ∈  𝑌,  𝑀𝑝,𝑞(𝑘𝑥) ≥ 𝑀𝑝,𝑞(𝑥) then 𝑝 = 𝑞. 

Proposition 2.2:[5] Let (𝑌, 𝑀, 𝑡) be a Menger space such that the t-norm 𝑡 is continuous and 𝑡 (𝑥, 𝑥)  ≥

𝑥   for all  𝑥 ∈ [0, 1] and 𝐴, 𝐵: 𝑌 → 𝑌 be continuous mappings. Then, 𝐴 and 𝐵 are weakly compatible 

mappings of type (P) if they are compatible mappings of type(𝑃). 



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Proposition 2.3: [5]Let (𝑌, 𝑀, 𝑡) be a Menger space such that the t-norm 𝑡 is continuous and 𝑡 (𝑥, 𝑥)  ≥

𝑥   for all  𝑥 ∈ [0, 1] and 𝐴, 𝐵: 𝑌 → 𝑌 be continuous mappings. Then, A and B are compatible mappings 

of type (𝑃) if they are weakly compatible mappings of type (𝑃). 

Proposition 2.4: [5]Let (𝑌, 𝑀, 𝑡) be a Menger space such that the t-norm 𝑡 is continuous and 𝑡 (𝑥, 𝑥)  ≥

𝑥  for all  𝑥 ∈ [0, 1] and 𝐴, 𝐵: 𝑌 → 𝑌 be mappings. If 𝐴 and 𝐵 are weakly compatible mappings of type 

(𝑃) and 𝐴𝑘 =  𝐵𝑘 for some 𝑘 ∈ 𝐾, then, 𝐴𝐴𝑘 =  𝐴𝐵𝑘 =  𝐵𝐴𝑘 =  𝐵𝐵𝑘. 

Proposition 2.5:[5] Let  (𝑌, 𝑀, 𝑡) be a Menger space such that the t-norm 𝑡 is continuous and 

𝑡 (𝑥, 𝑥)  ≥ 𝑥 for all  𝑥 ∈ [0, 1]  and 𝐴, 𝐵: 𝑌 →  𝑌 be mappings. Let 𝐴 and 𝐵 be weakly compatible 

mappings of type (𝑃) and 𝑙𝑖𝑚
𝑛→∞

 A𝑘𝑛 =   𝑙𝑖𝑚
𝑛→∞

 B𝑘𝑛 = k for some 𝑘 ∈ 𝑌. Then 

             We have, 

              (𝑖) 𝑙𝑖𝑚
𝑛→∞

 BB𝑘𝑛 = Ak  𝑖𝑓 𝐴 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑘,    

             (𝑖𝑖) 𝑙𝑖𝑚
𝑛→∞

 AA𝑘𝑛 = Bk  𝑖𝑓 𝐵 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑘,    

 (𝑖𝑖𝑖) 𝐴𝐵𝑘 =  𝐵𝐴𝑘 𝑎𝑛𝑑 𝐴𝑘 =  𝐵𝑘 𝑖𝑓 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑘. 

The following lemma needs to prove the main theorem: 

Lemma 2.2[15]: Let {𝑥𝑛} be a sequence in Menger space (𝑌, 𝑀, 𝑡), where t is continuous 𝑡 −norm 

and 𝑡 (𝑥, 𝑥)  ≥  𝑥 for all 𝑥 ∈ [0, 1]. If there exists a constant 𝑘 ∈  [0, 1] such that 

  𝑀𝑥𝑛 , 𝑥𝑛+1
 (𝑘𝑥)  ≥ 𝑀𝑥𝑛−1 , 𝑥𝑛

(𝑥)  for all 𝑥 >  0 and 𝑛 ∈ 𝑁,  

then {𝑥𝑛} is a Cauchy sequence in 𝑌.       

3. Main Theorem: 

Now, we prove our main theorem for weakly compatible mappings of type (𝑃) in complete Menger 

space: 

Theorem 3.1: Let (𝑌, 𝑀, 𝑡) be a complete Menger space with 𝑡 (𝑥, 𝑦)  =  𝑚𝑖𝑛 {𝑥, 𝑦} for all  

 𝑥, 𝑦 ∈  [0, 1] 𝑎𝑛𝑑 𝐴, 𝐵, 𝑆, 𝑇: 𝑌 → 𝑌  be mappings such that 

(3.1.1)         𝐴 (𝑌) ⊂ 𝑇 (𝑌)𝑎𝑛𝑑   𝐵 (𝑌) ⊂  𝑆 (𝑌),   

(3.1.2)          the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are weakly compatible mappings of type (𝑃), 

(3.1.3)          One of 𝐴, 𝑆, 𝐵, 𝑇 be continuous, and 

(3.1.4)          there exists a constant  ξ ∈  (0, 1) such that  

             𝑀(𝐴𝑥, 𝐵𝑦, ξ 𝑞) ≥ φ{min{𝑀(𝑆𝑥, 𝐴𝑥, 𝑞), 𝑀(𝑇𝑦, 𝐵𝑦, 𝑞), 𝑀( 𝑇𝑦, 𝐴𝑥, 𝑟𝑞), 

                          𝑀(𝑆𝑥, 𝐵𝑦(2 − 𝑟)𝑞, 𝑀(𝑆𝑥, 𝑇𝑦, 𝑞)}} 

                for all   𝑥, 𝑦 ∈ 𝑌, 𝑟 ∈  (0, 2) and 𝑞 >  0,  𝑤ℎ𝑒𝑟𝑒 φ: [0,1] → [0,1] satisfies 

(i)   φ is continuous and non-decreasing on [0,1] 

(ii) φ(n) > n for all n in [0,1] 



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noting that if φ ∈  Φ, class of all mappings φ: [0,1] → [0,1] then φ(0) = 0, φ(1) = 1 and φ(n) ≥ n 

for all n in [0,1]. 

Then, 𝐴, 𝐵, 𝑆, 𝑇 have a unique common fixed point in 𝑌. 

Proof:  Consider  𝑢0  ∈  𝑌. Since 𝐴(𝑌)  ⊂  𝑇 (𝑌), so there exists a point 𝑢1𝑖𝑛 𝑌 such that 

 𝐴𝑢0  = 𝑇𝑢 1 =  𝑣0. Again, since 𝐵(𝑌)  ⊂  𝑆 (𝑌), so for 𝑢1, we may choose 𝑢2 in 𝑌 such that 

 𝐵𝑢1  =  𝑆𝑢2  =  𝑣1 and so on. 

And inductively, we may construct sequence {𝑢𝑛} and {𝑣𝑛}  in 𝑌 such that 

 𝐴𝑢2𝑛 =  𝑇𝑢 2𝑛+1 =  𝑣2𝑛 , and  𝐵𝑢2𝑛+1 =  𝑆𝑢 2𝑛+2 =  𝑣2𝑛+1,    𝑓𝑜𝑟   𝑛 = 0, 1, 2, …   

Putting  𝑥 =  𝑢2𝑛   𝑎𝑛𝑑  𝑦 =  𝑢2𝑛+1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑞 > 0 𝑎𝑛𝑑 𝑟 = 1 − 𝑝 𝑤𝑖𝑡ℎ 𝑝 𝜖 (0, 1) 𝑖𝑛    (3. 1.4), 

we get   

𝑀(𝐴𝑢2𝑛, 𝐵𝑢2𝑛+1,    ξ 𝑞) ≥ φ{min{𝑀(𝑆𝑢2𝑛, 𝐴𝑢2𝑛,   𝑞), 𝑀(𝑇𝑢2𝑛+1, 𝐵𝑢2𝑛+1,   𝑞),

𝑀(𝑇𝑢2𝑛+1, 𝐴𝑢2𝑛 , ((1 − 𝑝))𝑞), 𝑀(𝑆𝑢2𝑛, 𝐵𝑢2𝑛+1,((1 + 𝑝)𝑞), 𝑀(𝑆𝑢2𝑛, 𝑇𝑢2𝑛+1,   𝑞)}} 

or, 𝑀(𝑣2𝑛, 𝑣2𝑛+1,    ξ 𝑞) ≥ 

 φ{min{𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛 , ((1 − 𝑝))𝑞), 𝑀(𝑣2𝑛−1, 𝑣2𝑛+1,((1 +

𝑝)𝑞), 𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞)}} 

    ≥ φ{min{𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1,   𝑞),   𝑀(𝑣2𝑛−1, 𝑣2𝑛+1,((1 + 𝑝)𝑞), 𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞)}}         

 ≥ φ{min{𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1,   𝑞),   𝑀(𝑣2𝑛−1, 𝑣2𝑛, 𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1, 𝑝𝑞),

𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞)}}          ≥ φ{min{𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1, 𝑝𝑞)}}         

As 𝑝 → 1, we obtain 

 𝑀(𝑣2𝑛, 𝑣2𝑛+1,    ξ 𝑞) ≥  φ{min{𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1, 𝑞)}} 

                                      ≥  φ{min{𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞), 𝑀(𝑣2𝑛, 𝑣2𝑛+1,   𝑞)}  

Or, 𝑀(𝑣2𝑛, 𝑣2𝑛+1,    ξ 𝑞) ≥  φ{𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞)} >  𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞), by property of  φ 

Hence, we get       𝑀(𝑣2𝑛, 𝑣2𝑛+1,    ξ 𝑞) ≥  𝑀(𝑣2𝑛−1, 𝑣2𝑛,   𝑞) 

Similarly, we obtain 

                              𝑀(𝑣2𝑛+1, 𝑣2𝑛+2,    ξ 𝑞) ≥  𝑀(𝑣2𝑛, 𝑣2𝑛+1,   𝑞) 

Therefore, for every 𝑛 ∈ 𝑁,  𝑀(𝑣𝑛, 𝑣𝑛+1,    ξ 𝑞) ≥  𝑀(𝑣𝑛−1, 𝑣𝑛,   𝑞)  

So, using Lemma (2.2), {𝑣𝑛}  is a Cauchy sequence in 𝐾. 

Since the Menger space (𝑌, 𝑀, 𝑡) is complete, so {𝑣𝑛} converges to a point 𝑧 in 𝑌 and consequently 

the subsequences {𝐴𝑢2𝑛
} , {𝐵𝑢2𝑛+1

}, {𝑆𝑢2𝑛
}, {𝑇𝑢2𝑛+1

}of {𝑣𝑛}   also converges to 𝑧. 



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Now, suppose that 𝑇 is continuous. Then, since 𝐵 & 𝑇 are weakly compatible mappings of type (𝑃) 

then by proposition 2.5, 𝐵𝐵𝑢2𝑛+1
, 𝑇𝐵𝑢2𝑛+1

→ 𝑇𝑧 𝑎𝑠 𝑛 → ∞. Putting 𝑥 = 𝑢2𝑛  and 𝑦 =  𝐵𝑢2𝑛+1
 in 

relation (3.1.4) , we get  

𝑀(𝐴𝑢2𝑛, 𝐵𝐵𝑢2𝑛+1,    ξ 𝑞) ≥ 

φ{min{𝑀(𝑆𝑢2𝑛, 𝐴𝑢2𝑛,   𝑞), 𝑀(𝑇𝐵𝑢2𝑛+1, 𝐵𝐵𝑢2𝑛+1,   𝑞), 𝑀(𝑇𝐵𝑢2𝑛+1, 𝐴𝑢2𝑛 , 𝑟𝑞), 𝑀(𝑆𝑢2𝑛, 𝐵𝐵𝑢2𝑛+1,(2 −

𝑟)𝑞), 𝑀(𝑆𝑢2𝑛, 𝑇𝐵𝑢2𝑛+1,   𝑞)}} 

Taking 𝑛 →  ∞, we have 

𝑀(𝑧, 𝑇𝑧, ξ 𝑞) ≥ φ{min{𝑀(𝑧, 𝑧, 𝑞), 𝑀(𝑇𝑧, 𝑇𝑧, 𝑞), 𝑀(𝑇𝑧, 𝑧, 𝑟𝑞), 𝑀(𝑧, 𝑇𝑧(2 − 𝑟)𝑞), 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

Letting 𝑟 =  1 − 𝑝 with 𝑝 ∈  (0, 1) then 

𝑀(𝑧, 𝑇𝑧, ξ 𝑞) ≥ φ{min{ 𝑀(𝑇𝑧, 𝑧, (1 − 𝑝)𝑞), 𝑀(𝑧, 𝑇𝑧(2 − (1 − 𝑝)𝑞), 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

Or,  𝑀(𝑧, 𝑇𝑧, ξ 𝑞) ≥ φ{min{ 𝑀(𝑇𝑧, 𝑧, (1 − 𝑝)𝑞), 𝑀(𝑧, 𝑇𝑧(1 + 𝑝)𝑞), 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

               ≥ φ{min{ 𝑀(𝑇𝑧, 𝑧, (1 − 𝑝 + 1 + 𝑝)𝑞), 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

               ≥ φ{min{ 𝑀(𝑇𝑧, 𝑧, 2𝑞), 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

                           ≥ φ{min{ 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

   Therefore,    𝑀(𝑧, 𝑇𝑧, ξ 𝑞) ≥ φ{𝑀(𝑧, 𝑇𝑧, 𝑞)}                                

                      Or, 𝑀(𝑧, 𝑇𝑧, ξ 𝑞) ≥ 𝑀(𝑧, 𝑇𝑧, 𝑞), by property of  φ 

which implies 𝑧 =  𝑇𝑧  by Lemma 2.1. 

Similarly, replacing 𝑥 by 𝑢2𝑛 and 𝑦 𝑏𝑦 𝑧 in relation ( 3.1.4), we have 

 𝑀(𝐴𝑢2𝑛, 𝐵𝑧, ξ 𝑞) ≥  φ{min{𝑀(𝑆𝑢2𝑛, 𝐴𝑢2𝑛,   𝑞), 𝑀(𝑇𝑧, 𝐵𝑧, 𝑞), 𝑀(𝑇𝑧, 𝐴𝑢2𝑛, 𝑟𝑞),

𝑀(𝑆𝑢2𝑛, 𝐵𝑧, (2 − 𝑟)𝑞), 𝑀(𝑆𝑢2𝑛, 𝑇𝑧, 𝑞)}} 

Taking 𝑛 →  ∞, we get 

𝑀(𝑧, 𝐵𝑧, ξ 𝑞) ≥ φ{min{𝑀(𝑧, 𝑧, 𝑞), 𝑀(𝑧, 𝐵𝑧, 𝑞), 𝑀(𝑧, 𝑧, 𝑟𝑞), 𝑀(𝑧, 𝐵𝑧(2 − 𝑟)𝑞), 𝑀(𝑧, 𝑧, 𝑞)}} 

          ≥ φ{min{ 𝑀(𝑧, 𝐵𝑧, 𝑞), 𝑀(𝑧, 𝐵𝑧(2 − (1 − 𝑝))𝑞)}} 

                      ≥ φ{min{ 𝑀(𝑧, 𝐵𝑧, 𝑞), 𝑀(𝑧, 𝐵𝑧(1 + 𝑝))𝑞)}} 

                      ≥ φ{min{ 𝑀(𝑧, 𝐵𝑧, 𝑞), 𝑀(𝑧, 𝑧, 𝑞), 𝑀(𝑧, 𝐵𝑧, 𝑝𝑞)}} 

                      ≥ φ{min{ 𝑀(𝑧, 𝐵𝑧, 𝑞), 𝑀(𝑧, 𝑧, 𝑞), 𝑀(𝑧, 𝐵𝑧, 𝑝𝑞)}} 

                      ≥ φ{min{ 𝑀(𝑧, 𝐵𝑧, 𝑞), 𝑀(𝑧, 𝐵𝑧, 𝑞)}}, as 𝑝 → 1 

So that 𝑀(𝑧, 𝐵𝑧, ξ 𝑞) ≥ φ{𝑀(𝑧, 𝐵𝑧, 𝑞)} 

Or, 𝑀(𝑧, 𝐵𝑧, ξ 𝑞) ≥ 𝑀(𝑧, 𝐵𝑧, 𝑞), by property of  φ 

which implies 𝑧 =  𝐵𝑧  by Lemma 2.1. 

Since, 𝐵(𝑌) ⊂  𝑆 (𝑌), so there exists a point 𝑤 in 𝑌 such that 𝐵𝑧 =  𝑆𝑤 =  𝑧. 

By using relation ( 3.1.4) with 𝑥 = 𝑤, 𝑦 = 𝑧,  we have 



Communications on Applied Nonlinear Analysis 

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   𝑀(𝐴𝑤, 𝑧, ξ 𝑞) ≥ φ{min{𝑀(𝑆𝑤, 𝐴𝑤, 𝑞), 𝑀(𝑇𝑧, 𝐵𝑧, 𝑞), 𝑀( 𝑇𝑧, 𝐴𝑧, 𝑟𝑞), 𝑀(𝑆𝑤, 𝐵𝑧(2 −

𝑟)𝑞, 𝑀(𝑆𝑤, 𝑇𝑧, 𝑞)}} 

                      ≥ φ{min{𝑀(𝑧, 𝐴𝑤, 𝑞), 𝑀(𝑇𝑧, 𝑧, 𝑞), 𝑀( 𝑧, 𝐴𝑤, (1 − 𝑝)𝑞), 𝑀(𝑆𝑤, 𝑧(1 +

𝑝)𝑞, 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

                     ≥ φ{min{𝑀(𝑧, 𝐴𝑤, 𝑞), 𝑀(𝑇𝑧, 𝑧, 𝑞), 𝑀( 𝐴𝑤, 𝑧, (1 − 𝑝)𝑞), 𝑀(𝑆𝑤, 𝑧(1 +

𝑝)𝑞, 𝑀(𝑧, 𝑇𝑧, 𝑞)}} 

                     ≥ φ{min{𝑀(𝑧, 𝐴𝑤, 𝑞), 𝑀(𝑧, 𝑧, 𝑞), 𝑀( 𝐴𝑤, 𝑆𝑤, (1 − 𝑝 + 1 + 𝑝)𝑞)}} 

                   ≥ φ{min{𝑀(𝑧, 𝐴𝑤, 𝑞), 𝑀( 𝐴𝑤, 𝑧, 2𝑞) }} 

Therefore, 𝑀(𝐴𝑤, 𝑧, ξ 𝑞) ≥ φ{𝑀(𝑧, 𝐴𝑤, 𝑞)} 

Or, 𝑀(𝐴𝑤, 𝑧, ξ 𝑞) ≥ 𝑀(𝑧, 𝐴𝑤, 𝑞), by property of  φ 

which implies 𝐴𝑤 =  𝑧  by Lemma 2.1. 

Again, since 𝐴 𝑎𝑛𝑑 𝑆 are weakly compatible mappings of type (𝑃) and 𝐴𝑤 =  𝑆𝑤 =  𝑧, by 

proposition 2.4, we have for every  > 0 

1 = 𝑀(𝐴𝐴𝑤, 𝑆𝑆𝑤, 𝜖) ≥ 𝑀(𝐴𝑤, 𝑆𝑤, 𝜖)  

Hence   𝐴𝑤 = 𝐴𝐴𝑤 = 𝑆𝑆𝑤 = 𝑆𝑤 

Finally, by relation (3.1.4) with  𝑥 =  𝑧, 𝑦 = 𝐵𝑧 = 𝑧, we have 

  𝑀(𝐴𝑧, 𝑧, ξ 𝑞) =  𝑀(𝐴𝑧, 𝐵𝑧, ξ 𝑞) ≥ φ{min{𝑀(𝑆𝑧, 𝐴𝑧, 𝑞), 𝑀(𝑇𝑧, 𝑧, 𝑞), 𝑀( 𝑇𝑧, 𝐴𝑧, 𝑟𝑞), 

𝑀(𝑆𝑧, 𝑧(2 − 𝑟)𝑞, 𝑀(𝑆𝑧, 𝑇𝑧, 𝑞)} 

                                        ≥ φ{min{𝑀(𝐴𝑧, 𝐴𝑧, 𝑞), 𝑀(𝑧, 𝑧, 𝑞), 𝑀( 𝑧, 𝐴𝑧, 𝑟𝑞), 𝑀(𝐴𝑧, 𝑧(2 −

𝑟)𝑞, 𝑀(𝐴𝑧, 𝑧, 𝑞)} 

                                       ≥ φ{min{ 𝑀( 𝐴𝑧, 𝑧, 𝑟𝑞), 𝑀(𝑧, 𝐴𝑧(2 − 𝑟)𝑞, 𝑀(𝐴𝑧, 𝑧, 𝑞)} 

                                      ≥ φ{min{ 𝑀(𝐴𝑧, 𝐴𝑧, 𝑟𝑞 + (2 − 𝑟)𝑞, 𝑀(𝐴𝑧, 𝑧, 𝑞)} 

                                       ≥ φ{min{ 𝑀(𝐴𝑧, 𝑧, 𝑞)}                                             

                                        ≥ φ{𝑀(𝐴𝑧, 𝑧, 𝑞)}   

Or, 𝑀(𝐴𝑧, 𝑧, ξ 𝑞) ≥ 𝑀(𝐴𝑧, 𝑧, 𝑞), by property of φ                                                                                                 

 𝐴𝑧 =  𝑧, by Lemma 2.1. 

 Hence, 𝐴𝑧 =  𝐵𝑧 =  𝑆𝑧 =  𝑇𝑧 =  𝑧 . 

That is, 𝑧 is a common fixed point of given mappings 𝐴, 𝐵, 𝑆 & 𝑇. 

Uniqueness: Suppose 𝑧1 is another point in 𝑌 such that  

                                  𝑧1  =  𝐴𝑧1  =  𝐵𝑧1  =  𝑆𝑧1  =  𝑇𝑧1. 

Then, putting 𝑥 =  𝑧 𝑎𝑛𝑑 𝑦 =  𝑧1, 𝑟 =  1 in (3.1.4), we get 

  𝑀(𝐴𝑧, 𝐵𝑧1, ξ 𝑞) =  𝑀(𝑧, 𝑧1, ξ 𝑞) ≥ φ{min{𝑀(𝑆𝑧, 𝐴𝑧, 𝑞), 𝑀(𝑇𝑧1, 𝐵𝑧1, 𝑞), 𝑀( 𝑇𝑧1, 𝐴𝑧, 𝑞), 

𝑀(𝑆𝑧, 𝑇𝑧1, 𝑞)} 



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 Or,   𝑀(𝑧, 𝑧1, ξ 𝑞) ≥ φ{min{𝑀(𝑧, 𝑧1, 𝑞), 𝑀( 𝑧, 𝑧, 𝑞)} 

Or,   𝑀(𝑧, 𝑧1, ξ 𝑞) ≥ φ{𝑀(𝑧, 𝑧1, 𝑞)} 

        𝑀(𝑧, 𝑧1, ξ 𝑞) ≥ 𝑀(𝑧, 𝑧1, 𝑞), by property of φ                                                                                                 

 𝑧 =  𝑧1, by Lemma 2.1. 

Hence, 𝑧 =  𝐴𝑧 =  𝐵𝑧 =  𝑆𝑧 =  𝑇𝑧, and 𝑧 is a unique common fixed point for A, B, S, and T in 𝑌. 

This completes the proof. 

4. Conclusion: In conclusion, the result of Chaudhary et. al. [5] is a particular case of this 

theorem. Also, this theorem may apply to consequences results in metric space in four self-mappings 

and generalizes and improves other similar results in the literature.  

5. Acknowledgments: 

I am very thankful to the editor and to the anonymous reviewers for their careful suggestions, and 

also thanks to the University Grant Commission, Nepal for their financial contribution for my 

research publication. 

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