Communications on Applied Nonlinear Analysis 

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Common Fixed Point Theorem In Complex Valued Extended 𝑩-Metric 

Space 

 

Jitender Kumar1 and Rajesh Kumar2 
1Department of Mathematics, Govt. College for Girls, Palwal, Kurukshetra 136118, India.  

Email: kumar.jmaths@gmail.com 

2Department of Mathematics, Hindu College, University of Delhi, Delhi 110007, India. 

Email: rajeshhinducollege@gmail.com 

 

Article History: 

Received: 27-09-2024 

Revised: 29-11-2024 

Accepted: 09-12-2024 

Abstract:  

In this paper, we proved a common fixed point theorem for generalized contractive type 

maps in complex valued extended 𝑏-metric space, which generalized many results in the 

literature. 

Keywords: Fixed point theorem, Contractive type mapping, Complex valued extended 

𝑏-metric space. 

 

1. Introduction 

In 2011, Azam et al. [1] introduced the notion of complex valued metric spaces and proved a common 

fixed point theorem for a pair of contractive type maps involving rational expressions which is a 

generalization of the classification Banach fixed point theorem. In 2013, Rao et al. [7] introduced the 

concept of complex valued 𝑏-metric space. 

Subsequently, many authors have studied the existence and uniqueness of common fixed point of self-

mappings in view of contractive conditions. Some of these observations are described in [1,4-6,8,9]. 

In 2019, N. Ullah et al. [11] extended the concept of complex valued 𝑏-metric space to complex valued 

extended 𝑏-metric space. 

The main purpose of this paper is to present a common fixed point result for two self maps satisfying 

a rational inequality in complex valued extended 𝑏-metric space. 

2. Preliminaries 

Let ℂ be the set of complex number and 𝑧1, 𝑧2 ∈ ℂ. Define a partial order ≾ on ℂ as follows: 

  
 𝑧1 ≾ 𝑧2

 iff Re(𝑧1) ≤ Re(𝑧2) ,  Im(𝑧1) ≤ Im(𝑧2)
 (2.1) 

Thus 𝑧1 ≾ 𝑧2 if one of the following holds: 

(i)Re(𝑧1) = Re(𝑧2) and Im(𝑧1) = Im(𝑧2), 

(ii)Re(𝑧1) < Re(𝑧2) and Im(𝑧1) = Im(𝑧2), 

(iii)Re(𝑧1) = Re(𝑧2) and Im(𝑧1) < Im(𝑧2), 

(iv)Re(𝑧1) < Re(𝑧2) and Im(𝑧1) < Im(𝑧2). 



Communications on Applied Nonlinear Analysis 

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We will write if 𝑧1 ≺ 𝑧2 if 𝑧1 ≠ 𝑧2 and one of (ii), (iii) and (iv) is satisfied: Also we will write 𝑧1 ≺ 𝑧2 

if only (iv) is satisfied. 

We can easily check that the following statements are held: 

(i)If 𝑎, 𝑏 ∈ 𝑅 and 𝑎 ≤ 𝑏 then 𝑎𝑧 ≾ 𝑏𝑧 for all 𝑧 ∈ ℂ; 

(ii)if 0 ≾ 𝑧1 ≺ 𝑧2, then |𝑧1| < |𝑧2|; 

(iii)if 𝑧1 ≾ 𝑧2 and 𝑧2 ≺ 𝑧3, then 𝑧1 ≺ 𝑧3. 

Definition 2.1 ([1]). Let 𝑋 be a nonempty set. A function 𝑑: 𝑋 × 𝑋 → ℂ is called a complex valued 

metric on 𝑋 if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 the following conditions are satisfied: 

(i)0 ≾ 𝑑(𝑥, 𝑦) and 𝑑(𝑥, 𝑦) = 0 if and only for 𝑥 = 𝑦; 

(ii)𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥); 

(iii)𝑑(𝑥, 𝑦) ≾ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦). 

The pair (𝑋, 𝑑) is called a complex valued metric space. 

Example 2.1 ([5]). Let 𝑋 = ℂ Define the mapping 𝑑: 𝑋 × 𝑋 → ℂ by 

  𝑑(𝑧1, 𝑧2) = 𝑖|𝑧1 − 𝑧2| with 𝑧1 = 𝑥1 + 𝑖𝑦1, 𝑧2 = 𝑥2 + 𝑖𝑦2 (2.2) 

(𝑋, 𝑑) is complex valued metric space. 

 

Example 2.2 ([8]). Let 𝑋 = ℂ. Define the mapping 𝑑: 𝑋 × 𝑋 → ℂ by 

  𝑑(𝑥, 𝑦) = 𝑐𝑖𝑘|𝑥 − 𝑦|,   where 𝑘 ∈ 𝑅,  ∀𝑥, 𝑦 ∈ 𝑋 (2.3) 

Then (𝑋, 𝑑) is complex valued metric space. 

Definition 2.2 ([7]). Let 𝑋 be a non-empty set and let 𝑠 ≥ 1 be a given real number. A function 

𝑑: 𝑋 × 𝑋 → ℂ is called a complex valued 𝑏-metric on 𝑋 it for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 the following conditions 

are satisfied: 

(i)0 ≾ 𝑑(𝑥, 𝑦) and 𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦; 

(ii)𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥); 

(iii)𝑑(𝑥, 𝑦) ≾ 𝑠[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)]. 

The pair (𝑋, 𝑑) is called a complex valued 𝑏-metric space. 

 

Example 2.3 ([7]). Let 𝑋 = [0,1]. Define mapping 𝑑: 𝑋 × 𝑋 → ℂ by 

  𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 + 𝑖|𝑥 − 𝑦|2,  ∀𝑥, 𝑦 ∈ 𝑋 (2.4) 

Then (𝑋, 𝑑) is complex valued 𝑏-metric space with 𝑠 = 2. 

Definition 2.3 ([7]). Let (𝑋, 𝑑) be a complex valued 𝑏-metric space. Consider the following 

(i)  A point 𝑥 ∈ 𝑋 is called interior point of a set 𝐴 ⊆ 𝑋 whenever there exists 0 ≺ 𝑠 ∈ C such that 

𝐵(𝑥, 𝑟) ≔ {𝑦 ∈ 𝑋: 𝑑(𝑥, 𝑦) ≺ 𝑠} ⊆ 𝐴. 



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(ii)  A point 𝑥 ∈ 𝑋 is called a limit point of a set 𝐴 whenever, for every 0 ≺ 𝑟 ∈ ℂ, 𝐵(𝑥, 𝑟) ∩ 𝐴 −

{𝑥} ≠ ∅. 

(iii)  A subset 𝐴 ⊆ 𝑋 is called open whenever each element of 𝐴 is an interior point of 𝐴. 

(iv)  A subbasis for a Hausdorff topology 𝜏 on 𝑋 is a family 

𝐹 = {𝐵(𝑥, 𝑟): 𝑥 ∈ 𝑋 and 0 ≺ 𝑟}. 

Definition 2.4 ([11]). Let 𝑋 be a non-empty set and 𝜙: 𝑋 × 𝑋 → [1, ∞]. If a mapping 𝑑: 𝑋 × 𝑋 → C 

satisfy: 

(i)0 ≾ 𝑑(𝑥, 𝑦) and 𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦; 

(ii)𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥); 

(iii)𝑑(𝑥, 𝑦) ≾ 𝜙(𝑥, 𝑦)[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)]; 

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 then (𝑋, 𝑑) is called a complex valued extended 𝑏-metric space. 

Example 2.4 ([11]). Let 𝑋 = [0, ∞) and 𝜙: 𝑋 × 𝑋 → [1, ∞) be a function defined by 𝜙(𝑥, 𝑦) = 1 +

𝑥 + 𝑦 and 𝑑: 𝑋 × 𝑋 → ℂ by 

  𝑑(𝑥, 𝑦) = {
0  if 𝑥 = 𝑦
𝑖  if 𝑥 ≠ 𝑦

 (2.5) 

Then (𝑋, 𝑑) is a complex valued extended 𝑏-metric space. 

Theorem 2.1 ([1]). Let (𝑋, 𝑑) be a complete complex valued metric space and 𝜆, 𝜇 be nonnegative 

real numbers such that 𝜆 + 𝜇 < 1. Suppose that 𝑆, 𝑇: 𝑋 → 𝑋 are mapping satisfying: 

  𝑑(𝑆𝑥, 𝑇𝑦) ≾ 𝜆𝑑(𝑥, 𝑦) +
𝜇⋅𝑑(𝑥,𝑆𝑥)⋅𝑑(𝑦,𝑇𝑦)

1+𝑑(𝑥,𝑦)
 (2.6) 

for all 𝑥, 𝑦 ∈ 𝑋. Then 𝑆, 𝑇 have a unique Common fired point in 𝑋. 

Theorem 2.2 ([9]). Let (𝑋, 𝑑) be a complete complex valued b-metric space with the coefficient 𝑠 ≥ 1 

and 𝑓, 𝑔: 𝑋 → 𝑋 be mapping satisfying: 

  𝑑(𝑓𝑥, 𝑔𝑦) ≾ 𝜆𝑑(𝑥, 𝑦) +
𝜇⋅𝑑(𝑥,𝑓𝑥)⋅𝑑(𝑦,𝑔𝑦)

1+𝑑(𝑥,𝑦)
+

𝛿⋅𝑑(𝑦,𝑓𝑥)⋅𝑑(𝑥,𝑔𝑦)

1+𝑑(𝑥,𝑦)
 (2.7) 

where 𝜆, 𝜇, 𝛿 nonnegative real numbers with 𝑠𝜆 + 𝜇 + 𝛿 < 1. Then 𝑓, 𝑔 have a unique common fixed 

point in 𝑋. 

3. Main Result 

Theorem 3.1. Let (𝑋, 𝑑) be a complete CVEbMS with 𝜙: 𝑋 × 𝑋 → [1, ∞) and 𝑓, 𝑔: 𝑋 → 𝑋 be two self-

maps satisfying 

𝑑(𝑓𝑥, 𝑔𝑦) ≾ 𝐴 ⋅ 𝑑(𝑥, 𝑦) + 𝐵 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐶 ⋅

𝑑(𝑦, 𝑓𝑥) ⋅ 𝑑(𝑥, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)
 

  +𝐷 ⋅
𝑑(𝑥,𝑓𝑥)⋅𝑑(𝑥,𝑔𝑦)

1+𝑑(𝑥,𝑦)
+ 𝐸 ⋅

𝑑(𝑦,𝑓𝑥)⋅𝑑(𝑦,𝑔𝑦)

1+𝑑(𝑥,𝑦)
 (3.1) 



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where 𝐴, 𝐵, 𝐶, 𝐷, 𝐸 nonnegative real numbers, with 𝐴 + 𝐵 + 𝐶 + 2𝐷 + 2𝐸 < 1. Then 𝑓 and 𝑔 have a 

unique common fixed point in 𝑋. 

Proof. For any arbitrary point, 𝑥0 ∈ 𝑋. Define a sequence {𝑥𝑛} in 𝑋 such that 

  𝑥2𝑛+1 = 𝑓𝑥2𝑛, 𝑥2𝑛+2 = 𝑔𝑥2𝑛+1,   for 𝑛 = 0,1,2,3, … (3.2) 

Now, we show that the sequence {𝑥𝑛} is a Cauchy sequence. 

Let 𝑥 = 𝑥2𝑛 and 𝑦 = 𝑥2𝑛+1 in (3.1), we have 

𝑑(𝑓𝑥2𝑛, 𝑔𝑥2𝑛+1) = 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

≾ 𝐴 ⋅ 𝑑(𝑥2𝑛, 𝑥2𝑛+1) + 𝐵 ⋅
𝑑(𝑥2𝑛, 𝑓𝑥2𝑛) ⋅ 𝑑(𝑥2𝑛+1, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 +𝐶 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥2𝑛) ⋅ 𝑑(𝑥2𝑛, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 +𝐷 ⋅
𝑑(𝑥2𝑛, 𝑓𝑥2𝑛) ⋅ 𝑑(𝑥2𝑛, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 +𝐸 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥2𝑛) ⋅ 𝑑(𝑥2𝑛+1, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 

i.e., 

𝑑(𝑥2𝑛+1, 𝑥2𝑛+2) ≾ 𝐴 ⋅ 𝑑(𝑥2𝑛, 𝑥2𝑛+1) + 𝐵 ⋅
𝑑(𝑥2𝑛, 𝑥2𝑛+1) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 +𝐶 ⋅
𝑑(𝑥2𝑛+1, 𝑥2𝑛+1) ⋅ 𝑑(𝑥2𝑛, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 +𝐷 ⋅
𝑑(𝑥2𝑛, 𝑥2𝑛+1) ⋅ 𝑑(𝑥2𝑛, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 +𝐸 ⋅
𝑑(𝑥2𝑛+1, 𝑥2𝑛+1) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

⇒  𝑑(𝑥2𝑛+1, 𝑥2𝑛+2) ≾ 𝐴 ⋅ 𝑑(𝑥2𝑛, 𝑥2𝑛+1) + 𝐵 ⋅
𝑑(𝑥2𝑛, 𝑥2𝑛+1) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)

 +𝐷 ⋅
𝑑(𝑥2𝑛, 𝑥2𝑛+1) ⋅ 𝑑(𝑥2𝑛, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)
                         (3.3)

 

which implies that 

|𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| ≤ 𝐴 ⋅ |𝑑(𝑥2𝑛, 𝑥2𝑛+1)| + 𝐵 ⋅
|𝑑(𝑥2𝑛, 𝑥2𝑛+1)| ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)|

|1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)|

 +𝐷 ⋅
|𝑑(𝑥2𝑛, 𝑥2𝑛+1)| ⋅ |𝑑(𝑥2𝑛, 𝑥2𝑛+2)|

|1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)|
                          (3.4)

 

Since 

|1 + 𝑑(𝑥2𝑛, 𝑥2𝑛+1)| > |𝑑(𝑥2𝑛, 𝑥2𝑛+1)| 

We get 



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|𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| ≤ 𝐴 ⋅ |𝑑(𝑥2𝑛, 𝑥2𝑛+1)| + 𝐵 ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| + 𝐷 ⋅ |𝑑(𝑥2𝑛, 𝑥2𝑛+1)|

 +𝐷. |𝑑2𝑛+1, 𝑥2𝑛+2|

|𝑑(𝑥2𝑛, 𝑥2𝑛+2)| ≤ |𝑑(𝑥2𝑛, 𝑥2𝑛+1)|+∣ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

 ⇒  (1 − 𝐵 − 𝐷) ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| ≤ (𝐴 + 𝐷) ⋅ |𝑑(𝑥2𝑛, 𝑥2𝑛+1)|

 ⇒ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| ≤
𝐴 + 𝐷

1 − 𝐵 − 𝐷
|𝑑(𝑥2𝑛, 𝑥2𝑛+1)|.                                                              (3.5)

 

Similarly, we get 

  |𝑑(𝑥2𝑛+2, 𝑥2𝑛+3)| ≤
𝐴+𝐷

1−𝐵−𝐷
|𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)|. (3.6) 

𝑑(𝑥2𝑛+2, 𝑥2𝑛+3) = 𝑑(𝑥2𝑛+3, 𝑥2𝑛+2)

= 𝑑(𝑓𝑥2𝑛+2, 𝑔𝑥2𝑛+1)

≾ 𝐴 ⋅ 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1) + 𝐵 ⋅
𝑑(𝑥2𝑛+2, 𝑓𝑥2𝑛+2) ⋅ 𝑑(𝑥2𝑛+1, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)

 +𝐶 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥2𝑛+2) ⋅ 𝑑(𝑥2𝑛+2, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)

 +
𝐷 ⋅ 𝑑(𝑥2𝑛+2, 𝑓𝑥2𝑛+2) ⋅ 𝑑(𝑥2𝑛+2, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)

 +𝐸 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥2𝑛+2) ⋅ 𝑑(𝑥2𝑛+1, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)

⇒  𝑑(𝑥2𝑛+2, 𝑥2𝑛+3) = 𝑑(𝑓𝑥2𝑛+2, 𝑔𝑥2𝑛+1)

 ≾ 𝐴 ⋅ 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1) + 𝐵 ⋅
𝑑(𝑥2𝑛+2, 𝑥2𝑛+3) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)

 +𝐶 ⋅
𝑑(𝑥2𝑛+1, 𝑥2𝑛+3) ⋅ 𝑑(𝑥2𝑛+2, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)

 +𝐷 ⋅
𝑑(𝑥2𝑛+2, 𝑥2𝑛+3) ⋅ 𝑑(𝑥2𝑛+2, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)

 +𝐸 ⋅
𝑑(𝑥2𝑛+1, 𝑥2𝑛+3) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)
.                                   (3.7)

 

which implies that 

|𝑑 ⋅ (𝑥2𝑛+2, 𝑥2𝑛+3)| ≤ 𝐴 ⋅ |𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)| + 𝐵 ⋅
|𝑑(𝑥2𝑛+2, 𝑥2𝑛+3)| ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)|

|1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)|

 +𝐸 ⋅
|𝑑(𝑥2𝑛+1, 𝑥2𝑛+3)| ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)|

|1 + 𝑑(𝑥2𝑛+2, 𝑥2𝑛+1)|
                                   (3.8)

 

Since 

|1 + 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| > |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)|. 

So, we get 

|𝑑(𝑥2𝑛+2, 𝑥2𝑛+3)| < 𝐴 ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| + 𝐵 ⋅ |𝑑(𝑥2𝑛+2, 𝑥2𝑛+3)| 



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 +𝐸 ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| + 𝐸 ⋅ |𝑑(𝑥2𝑛+2, 𝑥2𝑛+3)|

⇒  (1 − 𝐵 − 𝐸)|𝑑. (𝑥2𝑛+2, 𝑥2𝑛+3)| ≤ 𝐴 ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)| + 𝐸 ⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)|

⇒ |𝑑. (𝑥2𝑛+2, 𝑥2𝑛+3)| ≤
𝐴 + 𝐸

1 − 𝐵 − 𝐸
⋅ |𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)|

 

Putting 

𝜆 = max {
𝐴 + 𝐷

1 − 𝐵 − 𝐷
,

𝐴 + 𝐸

1 − 𝐵 − 𝐸
} 

we obtain that 

  |𝑑(𝑥𝑘,, 𝑥𝑘+1)| ≤ 𝜆𝑘|𝑑(𝑥0, 𝑥1)|,   for some 𝑘 ∈ 𝑁 (3.9) 

Now, for 𝑚 > 𝑛 and by triangular inequality, we have 

𝑑(𝑥𝑛, 𝑥𝑚)  ≾ 𝜙(𝑥𝑛, 𝑥𝑚)[𝑑(𝑥𝑛, 𝑥𝑛+1) + 𝑑(𝑥𝑛+1, 𝑥𝑚)]

 ≾ 𝜙(𝑥𝑛, 𝑥𝑚)𝜆𝑛𝑑(𝑥0, 𝑥1) + 𝜙(𝑥𝑛, 𝑥𝑚)𝑑(𝑥𝑛+1, 𝑥𝑚)

 ≾ 𝜙(𝑥𝑛, 𝑥𝑚)𝜆𝑛𝑑(𝑥0, 𝑥1) + 𝜙(𝑥𝑛, 𝑥𝑚)𝜙(𝑥𝑛+1, 𝑥𝑚)

 ⋅ [𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑑(𝑥𝑛+2, 𝑥𝑚)]

 ≾ 𝜙(𝑥𝑛, 𝑥𝑚)𝜆𝑛𝑑(𝑥0, 𝑥1) + 𝜙(𝑥𝑛, 𝑥𝑚)𝜙(𝑥𝑛+1, 𝑥𝑚)

 ⋅ 𝜆𝑛+1 ⋅ 𝑑[(𝑥0, 𝑥1) + 𝜙(𝑥𝑛, 𝑥𝑚)𝜙(𝑥𝑛+1, 𝑥𝑚) ⋅ 𝜆𝑛+1 ⋅ 𝑑(𝑥𝑛+2, 𝑥𝑚)].

 

This implies that 

𝑑(𝑥𝑛, 𝑥𝑚) ≾ 𝜙(𝑥𝑛, 𝑥𝑚)𝜆𝑛𝑑(𝑥0, 𝑥1) + 𝜙(𝑥𝑛, 𝑥𝑚)𝜙(𝑥𝑛+1, 𝑥𝑚)𝜆𝑛+1 ⋅ 𝑑(𝑥0, 𝑥1) + ⋯

 +𝜙(𝑥𝑛, 𝑥𝑚)𝜙(𝑥𝑛+1, 𝑥𝑚) … 𝜙(𝑥𝑚−1, 𝑥𝑚)𝜆𝑚−1 ⋅ 𝑑(𝑥0, 𝑥1)                                     (3.10)
 

which implies that 

|𝑑(𝑥𝑛, 𝑥𝑚)| ≤ |𝑑(𝑥0, 𝑥1)|[𝜙(𝑥𝑛, 𝑥𝑚)𝜆𝑛 + 𝜙(𝑥𝑛, 𝑥𝑚)𝜙(𝑥𝑛+1, 𝑥𝑚)𝜆𝑛+1 + ⋯

+𝜙(𝑥𝑛, 𝑥𝑚)𝜙(𝑥𝑛+1, 𝑥𝑚) … 𝜙(𝑥𝑚−1, 𝑥𝑚)𝜆𝑚−1].
 

Since limit 𝜙(𝑥𝑛, 𝑥𝑚)𝜆 < 1, 𝑛, 𝑚 → ∞, so the series 

∑  

∞

𝑛=1

𝜆𝑛 ∏  

𝐾

𝑖=1

𝜙(𝑥𝑖, 𝑥𝑚) converges by ratio test for each 𝑚 ∈ 𝑁 

Let 

  𝑥 = ∑  ∞
𝑛=1 𝜆𝑛 ∏  𝐾

𝑖=1 𝜙(𝑥𝑖, 𝑥𝑚),  𝑥𝑛 = ∑  𝑛
𝑗=1 𝜆𝑗 ∏  𝐾

𝑖=1 𝜙(𝑥𝑖 , 𝑥𝑚), (3.11) 

Thus for 𝑚 > 𝑛, the above inequality can be written as 

|𝑑(𝑥𝑛, 𝑥𝑚)| ≤ |𝑑(𝑥0, 𝑥1)| ⋅ |𝑥𝑚−1 − 𝑥𝑛|. 

Now, by taking the limit as 𝑛, 𝑚 → ∞ we get 

|𝑑(𝑥𝑛, 𝑥𝑚)| → 0 as 𝑛, 𝑚 → ∞ 

Thus {𝑥𝑛} is a Cauchy sequence in 𝑋. But 𝑋 is a complete metric space, so this Cauchy sequence 

convergent and say converges to 𝑥. 



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i.e., lim𝑛→∞  𝑥𝑛 = 𝑥. 

Now, we prove that 𝑓𝑥 = 𝑥 i.e., 𝑑(𝑥, 𝑓𝑥) = 0. 

On the contrary, suppose that 

 0 ≺ 𝑣 = 𝑑(𝑥, 𝑓𝑥) = 𝑑(𝑓𝑥, 𝑥),

 0 ≺ 𝑣 ≾ 𝑑(𝑓𝑥, 𝑥)

     ≾ 𝜙(𝑓𝑥, 𝑥)[𝑑(𝑓𝑥, 𝑥2𝑛+2) + 𝑑(𝑥2𝑛+2, 𝑥)]

     ≾ 𝜙(𝑓𝑥, 𝑥) ⋅ [𝑑(𝑓𝑥, 𝑔𝑥2𝑛+1) + 𝑑(𝑥2𝑛+2, 𝑥)]

 

i.e., 

0 ≺ 𝑣 ≾ 𝜙(𝑓𝑥, 𝑥) [𝐴 ⋅ 𝑑(𝑥, 𝑥2𝑛+1) + 𝐵 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑥2𝑛+1, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥, 𝑥2𝑛+1)

 +𝐶 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥) ⋅ 𝑑(𝑥, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥, 𝑥2𝑛+1)
+ 𝐷 ⋅

𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑥2𝑛+1, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥, 𝑥2𝑛+1)

+𝐸 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥) ⋅ 𝑑(𝑥2𝑛+1, 𝑔𝑥2𝑛+1)

1 + 𝑑(𝑥, 𝑥2𝑛+1)
+ 𝑑(𝑥2𝑛+2, 𝑥)]

 

i.e. 

0 ≺ 𝑣 ≾ 𝜙(𝑓𝑥, 𝑥) [𝐴 ⋅ 𝑑(𝑥, 𝑥2𝑛+1) + 𝐵 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥, 𝑥2𝑛+1)

 +𝐶 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥) ⋅ 𝑑(𝑥, 𝑥2𝑛+2)

1 + 𝑑(𝑥, 𝑥2𝑛+1)
+ 𝐷 ⋅

𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥, 𝑥2𝑛+1)

+𝐸 ⋅
𝑑(𝑥2𝑛+1, 𝑓𝑥) ⋅ 𝑑(𝑥2𝑛+1, 𝑥2𝑛+2)

1 + 𝑑(𝑥, 𝑥2𝑛+1)
+ 𝑑(𝑥2𝑛+2, 𝑥)] .

 

Now, taking the limit as 𝑛 → ∞ we get 0 ≺ 𝑣 ≺ 0, which implies that 𝑣 = 0. 

i.e. 𝑑(𝑓𝑥, 𝑥) = 0 ⇒  𝑓𝑥 = 𝑥. 

Similarly, we can prove that 𝑔𝑥 = 𝑥. 

Now, we can show that 𝑓 and 𝑔 have unique common fixed point. 

On the contrary, suppose that 𝑥 and 𝑦 be two common fixed point of 𝑓 and 𝑔. 

Now, 

𝑑(𝑥, 𝑦) = 𝑑(𝑓𝑥, 𝑔𝑦) ≾ 𝐴 ⋅ 𝑑(𝑥, 𝑦) + 𝐵 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐶 ⋅

𝑑(𝑦, 𝑓𝑥) ⋅ 𝑑(𝑥, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)

 +𝐷 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐸 ⋅

𝑑(𝑦, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)

 

 

 

 

 



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i.e. 

𝑑(𝑥, 𝑦) = 𝑑(𝑓𝑥, 𝑔𝑦) ≾ 𝐴 ⋅ 𝑑(𝑥, 𝑦) + 𝐵 ⋅
𝑑(𝑥, 𝑥) ⋅ 𝑑(𝑦, 𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐶 ⋅

𝑑(𝑦, 𝑥) ⋅ 𝑑(𝑥, 𝑦)

1 + 𝑑(𝑥, 𝑦)

 +𝐷 ⋅
𝑑(𝑥, 𝑥) ⋅ 𝑑(𝑦, 𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐸 ⋅

𝑑(𝑦, 𝑥) ⋅ 𝑑(𝑦, 𝑦)

1 + 𝑑(𝑥, 𝑦)

 

which implies that 

|𝑑(𝑥, 𝑦)| =  |𝑑(𝑓𝑥, 𝑔𝑦)|

≤ 𝐴 ⋅ |𝑑(𝑥, 𝑦)| + 𝐵 ⋅
|𝑑(𝑥, 𝑥) ⋅ 𝑑(𝑦, 𝑦)|

|1 + 𝑑(𝑥, 𝑦)|
+ 𝐶 ⋅

|𝑑(𝑦, 𝑥) ⋅ 𝑑(𝑥, 𝑦)|

|1 + 𝑑(𝑥, 𝑦)|

 +𝐷 ⋅
|𝑑(𝑥, 𝑥) ⋅ 𝑑(𝑦, 𝑦)|

|1 + 𝑑(𝑥, 𝑦)|
+ 𝐸 ⋅

|𝑑(𝑦, 𝑥) ⋅ 𝑑(𝑦, 𝑦)|

|1 + 𝑑(𝑥, 𝑦)|
.

 

Since 

|1 + 𝑑(𝑥, 𝑦)| > |𝑑(𝑥, 𝑦)| 

i.e. 

|𝑑(𝑥, 𝑦)|

|1 + 𝑑(𝑥, 𝑦)|
< 1, 

so we get 

|𝑑(𝑥, 𝑦)| = |𝑑(𝑓𝑥, 𝑔𝑦)| < 𝐴. |𝑑(𝑥, 𝑦)| + 𝐵. 0 + 𝐶. |𝑑(𝑥, 𝑦)| + 𝐷. 0 + 𝐸. 0 

i.e. 

|𝑑(𝑥, 𝑦)| = |𝑑(𝑓𝑥, 𝑔𝑦)| < 𝐴 ⋅ |𝑑(𝑥, 𝑦)| + 𝐶. |𝑑(𝑥, 𝑦)| 

i.e. 

 (1 − 𝐴 − 𝐶)|𝑑(𝑥, 𝑦)| < 0, a contradiction since 𝐴 + 𝐵 + 𝐶 + 2𝐷 + 2𝐸 < 1

⇒ 𝐴 + 𝐶 < 1
 

So, 𝑥 = 𝑦, which proves the uniqueness of common fixed point of 𝑓 and 𝑔 in 𝑋. 

Corollary 3.1. Let (𝑋, 𝑑) be a complete CVEbMS with 𝜙: 𝑋 × 𝑋 → [1, ∞) and 𝑓: 𝑋 → 𝑋 be self-map 

satisfying 

𝑑(𝑓𝑥, 𝑓𝑦) ≾ 𝐴 ⋅ 𝑑(𝑥, 𝑦) + 𝐵 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑓𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐶 ⋅

𝑑(𝑦, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑓𝑦)

1 + 𝑑(𝑥, 𝑦)

 +𝐷 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑥, 𝑓𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐸 ⋅

𝑑(𝑦, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑓𝑦)

1 + 𝑑(𝑥, 𝑦)

 

where 𝐴, 𝐵, 𝐶, 𝐷, 𝐸 nonnegative real numbers, with 𝐴 + 𝐵 + 𝐶 + 2𝐷 + 2𝐸 < 1. Then 𝑓 has a unique 

fires point in 𝑋. 

Proof. Taking 𝑔 = 𝑓 in Theorem 3.1. 

 



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Corollary 3.2. Let (𝑋, 𝑑) be a complete CVEbMS with 𝜙: 𝑋 × 𝑋 → [1, ∞) and 𝑓, 𝑔: 𝑋 → 𝑋 be self-

maps satisfying 

𝑑(𝑓𝑥, 𝑔𝑦) ≾ 𝐴 ⋅ 𝑑(𝑥, 𝑦) + 𝐵 ⋅
𝑑(𝑥, 𝑓𝑥) ⋅ 𝑑(𝑦, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)
+ 𝐶 ⋅

𝑑(𝑦, 𝑓𝑥) ⋅ 𝑑(𝑥, 𝑔𝑦)

1 + 𝑑(𝑥, 𝑦)
 

where 𝐴, 𝐵, 𝐶 nonnegative real numbers, with 𝐴 + 𝐵 + 𝐶 < 1. Then 𝑓 and 𝑔 have a unique common 

fixed point in 𝑋. 

Proof. Taking 𝐷 = 𝐸 = 0 in Theorem 3.1. 

Remark 3.1. (i) Theorem 3.1 generalized Theorem 15 of [6] after substituting 𝐶 = 𝐷 = 𝐸 = 0 and 

𝜙(𝑥, 𝑦) = 𝑠 ≥ 1, 

(ii) Theorem 3.1 generalized Theorem 10 of [9] after substituting 𝐷 = 𝐸 = 0 and 𝜙(𝑥, 𝑦) = 𝑠 ≥ 1. 

(iii) Theorem 3.1 generalized Theorem 1 of [4] after substituting 𝐷 = 𝐸 = 0 and 𝜙(𝑥, 𝑦) = 1. 

(iv) Theorem 3.1 generalized Theorem 4 of [1] after substituting 𝐶 = 𝐷 = 𝐸 = 0 and 𝜙(𝑥, 𝑦) = 1. 

(v) Theorem 3.1 generalized Theorem 2 of [10] after substituting 𝐵 = 𝐶 = 𝐷 = 𝐸 = 0 and ℂ = 𝑅. 

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