Fixed Point Theorems in Complex-Valued b-Metric Spaces


CAUCHY – JURNAL MATEMATIKA MURNI DAN APLIKASI 
Volume 4 (4) (2017), Pages 138-145 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: 14 Nopember 2016 Reviewed: 29 Nopember 2016 Accepted: 17 May 2017 
DOI: http://dx.doi.org/10.18860/ca.v4i4.3669 
 

Fixed Point Theorems in Complex-Valued b-Metric Spaces 

Dahliatul Hasanah 

Mathematics Department, Universitas Negeri Malang 
Jl. Semarang No. 5 Malang Jawa Timur Indonesia 65145 

 

Email: dahliatul.hasanah.fmipa@um.ac.id 

ABSTRACT  

Fixed point theorems in complex-valued metric spaces have been extensively studied following the 
extension of the definition of a metric to a complex-valued metric. The definition of b-metric, which is 
considered as an extension of a metric, is also extended to complex-valued b-metric. This extension causes 
the study of existence and uniqueness of fixed point in complex-valued b-metric spaces. Fixed point 
theorems that have been studied in b-metric spaces are now of interest whether the theorems are still 
satisfied in complex-valued b-metric spaces.  
In this paper, we study the existence and uniqueness of fixed point in complex valued b-metric spaces. We 
consider some fixed point theorems in b-metric spaces to be extended to fixed point theorems in complex 
valued b-metric spaces. By using the definition of complex-valued b-metric spaces and the properties of 
partial order on complex numbers, we have obtained some fixed point theorems in complex-valued b-metric 
spaces which are derived from the corresponding theorems in b-metric spaces.  

Keywords: b-metric spaces, complex-valued b-metric spaces, fixed point theorems 

INTRODUCTION  

Fixed point theorems have been the main topic of many researches in various spaces 
including metric spaces. Fixed point theorems in metric spaces have been studied extensively by 
many researchers as in [1], [2], and [3]. After Bakhtin [4] introduced the notion of b-metric spaces 
in 1989, researchers started to develop fixed point theorems in the spaces, for example in [5], [6], 
and [7]. In 2011, Azzam et. al [8] introduced a new metric space called complex-valued metric 
space as an extension of the classical metric space in terms of the defined metric. Fixed point 
theorems are still of interest to study in the new space. This is shown by many articles talked 
about fixed point theorems in complex valued-metric spaces as in [9], [10], dan [11]. The 
extension of the notion of metric into complex-valued metric causes the extension of b-metric 
spaces into complex-valued b-metric spaces. In this paper, we observe fixed point theorems which 
have been established in b-metric space whether the theorems are still applicable in the spaces. 

 

METHODS 

First, we recall some basic definitions and properties of b-metric spaces given in [5] and 
of complex-valued b-metric spaces given in [12]. 

 

http://dx.doi.org/10.18860/ca.v4i4.3669
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Fixed Point Theorems in Complex-Valued b-Metric Spaces 

Dahliatul Hasanah 139 

b-Metric Spaces 

A 𝑏-metric space is considered as a generalization of a metric space regarding the 
definition of the metric as defined as follows. 
Definition 1 (see [5]). Let 𝑋 be a nonempty set and let 𝑠 ≥ 1 be a given real number. A function 
𝑑: 𝑋 × 𝑋 → [0, ∞) is called a b-metric space if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 the following conditions are 
satisfied. 

𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦; 
𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥); 
𝑑(𝑥, 𝑦) ≤ 𝑠[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)]. 
The pair (𝑋, 𝑑) is called a b-metric space. The number 𝑠 ≥ 1 is called the coefficient of 

(𝑋, 𝑑). 
 

Example 2. Let (𝑋, 𝑑) be a metric space. The function 𝜌(𝑥, 𝑦) is defined by 𝜌(𝑥, 𝑦) = (𝑑(𝑥, 𝑦))
2

. 

Then (𝑋, 𝜌) is a 𝑏-metric space with coefficient 𝑠 = 2. This can be seen from the nonnegativity 
property and triangle inequality of metric to prove the property (iii). 

 
Complex-Valued b-Metric Spaces 

Unlike in real numbers which has completeness property, order is not well-defined in 
complex numbers. Before giving the definition of complex-valued b-metric spaces, we define 
partial order in complex numbers (see [12]). Let ℂ be the set of complex numbers and 𝑧1, 𝑧2 ∈ ℂ. 
Define partial order ≼ on ℂ as follows. We have 𝑧1 ≼ 𝑧2 if and only if 𝑅𝑒(𝑧1) ≤ 𝑅𝑒(𝑧2) and 
𝐼𝑚(𝑧1) ≤ 𝐼𝑚(𝑧2). This means that we would have 𝑧1 ≼ 𝑧2 if and only if one of the following 
conditions 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). 

If one of the conditions (i), (ii), and (iii) holds, then we will write 𝑧1 ⋨ 𝑧2. Particularly, we have 
𝑧1 ≺ 𝑧2 if the condition (iv) is satisfied.    

From the definition of partial order on complex numbers, partial order has the following 
properties: 

(i) If 0 ≼ 𝑧1 ⋨ 𝑧2, then |𝑧1| < |𝑧2|; 

(ii) 𝑧1 ≼ 𝑧2 and 𝑧2 ≺ 𝑧3 imply 𝑧1 ≺ 𝑧3; 

(iii) If 𝑧 ∈ ℂ, 𝑎, 𝑏 ∈ ℝ and 𝑎 ≤ 𝑏, then 𝑎𝑧 ≼ 𝑏𝑧. 

A b-metric on a b-metric space is a function having real value. Based on the definition of 
partial order on complex numbers, real-valued b-metric can be generalized into complex-valued 
b-metric as follows. 
Definition 3 (see [13]). Let 𝑋 be a nonempty 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 if, for all 𝑥, 𝑦, 𝑧 ∈ ℂ, 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 4. Let 𝑋 = ℂ. Define the mapping 𝑑: ℂ × ℂ → ℂ by 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 + 𝑖|𝑥 − 𝑦|2 for all 
𝑥, 𝑦 ∈ 𝑋. Then (ℂ, 𝑑) is complex-valued b-metric space with 𝑠 = 2.  
Definition 5 (see [13]). Let (𝑋, 𝑑) be a complex-valued b-metric space.  

(i) A point 𝑥 ∈ 𝑋 is called interior point of set 𝐴 ⊆ 𝑋 if there exists 0 ≺ 𝑟 ∈ ℂ such that 
𝐵(𝑥, 𝑟) = {𝑦 ∈ 𝑌: 𝑑(𝑥, 𝑦) ≺ 𝑟} ⊆ 𝐴. 



Fixed Point Theorems in Complex-Valued b-Metric Spaces 

Dahliatul Hasanah 140 

(ii) A point 𝑥 ∈ 𝑋 is called limit point of a set 𝐴 if for every 0 ≺ 𝑟 ∈ ℂ, 𝐵(𝑥, 𝑟) ∩ (𝐴 − {𝑥}) ≠
∅. 

(iii) A subset 𝐴 ⊆ 𝑋 is open if each element of 𝐴 is an interior point of 𝐴. 
(iv) A subset 𝐴 ⊆ 𝑋 is closed if each limit point of 𝐴 is contained in 𝐴. 

Definition 6 (see [13]). Let (𝑋, 𝑑) be a complex valued b-metric space, (𝑥𝑛 ) be a sequence in 𝑋 
and 𝑥 ∈ 𝑋. 

(i) (𝑥𝑛 ) is convergent to 𝑥 ∈ 𝑋 if for every 0 ≺ 𝑟 ∈ ℂ there exists 𝑁 ∈ ℕ such that for all 𝑛 ≥
𝑁, 𝑑(𝑥𝑛 , 𝑥) ≺ 𝑟. Thus 𝑥 is the limit of (𝑥𝑛 ) and we write lim(𝑥𝑛 ) = 𝑥 or (𝑥𝑛 ) → 𝑥 as 𝑛 →

∞. 
(ii) (𝑥𝑛 ) is said to be Cauchy sequence if for every 0 ≺ 𝑟 ∈ ℂ there exists 𝑁 ∈ ℕ such that for 

all 𝑛 ≥ 𝑁, 𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 ) ≺ 𝑟, where 𝑚 ∈ ℕ.  
(iii)  If for every Cauchy sequence in 𝑋 is convergent, then (𝑋, 𝑑) is said to be a complete 

complex valued b-metric space. 
Lemma 7 (see [13]). Let (𝑋, 𝑑) be a complex valued b-metric space ad let (𝑥𝑛 ) be a sequence in 𝑋. 
Then (𝑥𝑛 ) converges to 𝑥 if and only if |𝑑(𝑥, 𝑦)| → 0 as 𝑛 → ∞. 
Lemma 8 (see [13]). Let Let (𝑋, 𝑑) be a complex valued b-metric space ad let (𝑥𝑛 ) be a sequence 
in 𝑋. Then (𝑥𝑛 ) is a Cauchy sequence if and only if  |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚)| → 0 as 𝑛 → ∞, where 𝑚 ∈ ℕ.    

 

 

RESULTS AND DISCUSSIONS 

The following fixed point theorem is studied by Mishra,et.al [5] in b-metric spaces: 
Theorem 9. Let (𝑋, 𝜌) be a complete b-metric space and let 𝑇: 𝑋 → 𝑋 be a function with the 
following  

𝜌(𝑇𝑥, 𝑇𝑦) ≤ 𝑎𝜌(𝑥, 𝑇𝑥) + 𝑏𝜌(𝑦, 𝑇𝑦) + 𝑐𝜌(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋 
 where 𝑎, 𝑏, and 𝑐 are non-negative real numbers and satisfy 𝑎 + 𝑠(𝑏 + 𝑐) < 1 for 𝑠 ≥ 1 then 𝑇 has 
a uniqe fixed point. 
 This theorem is extended to a fixed point theorem in complex valued b-metric spaces as 
follows. 
 
Theorem 10. Let (𝑋, 𝑑) be a complete complex valued b-metric space and let 𝑇: 𝑋 → 𝑋 be a 
function with the following  

𝑑(𝑇𝑥, 𝑇𝑦) ≼ 𝑎𝑑(𝑥, 𝑇𝑥) + 𝑏𝑑(𝑦, 𝑇𝑦) + 𝑐𝑑(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋 
 where 𝑎, 𝑏, and 𝑐 are non-negative real numbers and satisfy 𝑎 + 𝑠(𝑏 + 𝑐) < 1 for 𝑠 ≥ 1 then 𝑇 has 
a uniqe fixed point. 
Proof. Let 𝑥0 ∈ 𝑋 and (𝑥𝑛 ) be a sequence in X such that 

𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇
𝑛𝑥0. 

Note that for all 𝑛 ∈ ℕ, we have 
𝑑(𝑥𝑛+2, 𝑥𝑛+1) = 𝑑(𝑇𝑥𝑛+1, 𝑇𝑥𝑛 ) 

≼ 𝑎𝑑(𝑥𝑛+1, 𝑇𝑥𝑛+1) + 𝑏𝑑(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑐𝑑(𝑥𝑛+1, 𝑥𝑛 ) 
≼ 𝑎𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑏𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑐𝑑(𝑥𝑛+1, 𝑥𝑛 ) 
≼ 𝑎𝑑(𝑥𝑛+2, 𝑥𝑛+1) + (𝑏 + 𝑐)𝑑(𝑥𝑛+1, 𝑥𝑛 ) 

(1 − 𝑎)𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ (𝑏 + 𝑐)𝑑(𝑥𝑛+1, 𝑥𝑛 ) 

𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ (
𝑏 + 𝑐

1 − 𝑎
) 𝑑(𝑥𝑛+1, 𝑥𝑛 ). 

If we take 𝛾 =
𝑏+𝑐

1−𝑎
 and continuing this process, then we have 

𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ 𝛾
𝑛+1𝑑(𝑥1, 𝑥0). 

For 𝑚 ∈ ℕ, 
𝑑(𝑥𝑛 , 𝑥𝑛+𝑚) ≼ 𝑠[𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑑(𝑥𝑛+1, 𝑥𝑛+𝑚 )] 

 ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠
2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠

2𝑑(𝑥𝑛+2, 𝑥𝑛+𝑚 ) 
 ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠

2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠
3𝑑(𝑥𝑛+2, 𝑥𝑛+3) + ⋯ + 𝑠

𝑚 𝑑(𝑥𝑛+𝑚−1, 𝑥𝑛+𝑚) 
 ≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1) + 𝑠

2𝛾𝑛+1𝑑(𝑥0, 𝑥1) + 𝑠
3𝛾𝑛+2𝑑(𝑥0, 𝑥1) + ⋯ + 𝑠

𝑚 𝛾𝑛+𝑚−1𝑑(𝑥0, 𝑥1) 
≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾)

2 + ⋯ + (𝑠𝛾)𝑚−1] 



Fixed Point Theorems in Complex-Valued b-Metric Spaces 

Dahliatul Hasanah 141 

Now, 
|𝑑(𝑥𝑛 , 𝑥𝑛+𝑚)| ≤ |𝑠𝛾

𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾)
2 + ⋯ + (𝑠𝛾)𝑚−1]| 

 = 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾)
2 + ⋯ + (𝑠𝛾)𝑚−1] 

Since 𝑎 + 𝑠(𝑏 + 𝑐) < 1 for 𝑠 ≥ 1 then 𝛾 < 1 and 𝑠𝛾 < 1. Taking limit 𝑛 → ∞, we have 𝛾𝑛 → 0. This 
implies |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 )| → 0 as 𝑛 → ∞. By Lemma 8, the sequence (𝑥𝑛 ) is Cauchy sequence in X. 
Since 𝑋 is a complete complex valued b-metric space then (𝑥𝑛 ) is a convergent sequence.  

Let 𝑥∗ be the limit of (𝑥𝑛 ). We show that 𝑥
∗ is a fixed point of 𝑇.   

We have 
𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑥𝑛 , 𝑇𝑥

∗)] 
 = 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑇𝑥𝑛−1, 𝑇𝑥

∗)] 
 ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑏𝑑(𝑥

∗, 𝑇𝑥∗) + 𝑐𝑑(𝑥𝑛−1, 𝑥
∗)] 

(1 − 𝑏𝑠)𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥 ∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑐𝑑(𝑥𝑛−1, 𝑥
∗)] 

𝑑(𝑥∗, 𝑇𝑥∗) ≼
𝑠

1 − 𝑏𝑠
[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑐𝑑(𝑥𝑛−1, 𝑥

∗)] 

𝑑(𝑥∗, 𝑇𝑥∗) ≼
𝑠

1 − 𝑏𝑠
[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑥𝑛 ) + 𝑐𝑑(𝑥𝑛−1, 𝑥

∗)] 

Taking the absolute value of both sides, we get 

|𝑑(𝑥∗, 𝑇𝑥∗)| ≤
𝑠

1 − 𝑏𝑠
|𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑥𝑛 ) + 𝑐𝑑(𝑥𝑛−1, 𝑥

∗)| 

≤
𝑠

1 − 𝑏𝑠
(|𝑑(𝑥∗, 𝑥𝑛 )| + |𝑎𝑑(𝑥𝑛−1, 𝑥𝑛 )| + |𝑐𝑑(𝑥𝑛−1, 𝑥

∗)|) 

≤
𝑠

1 − 𝑏𝑠
(|𝑑(𝑥∗, 𝑥𝑛 )| + 𝑎𝛾

𝑛 |𝑑(𝑥1, 𝑥0)| + 𝑐|𝑑(𝑥𝑛−1, 𝑥
∗)|) 

Since (𝑥𝑛 ) converges to 𝑥
∗, taking 𝑛 → ∞ implies |𝑑(𝑥∗, 𝑥𝑛 )| → 0 and |𝑑(𝑥𝑛−1, 𝑥

∗)| → 0. Thus for 
𝑛 → ∞, we have  

lim|𝑑(𝑥∗, 𝑇𝑥∗)| = 0. 

This yields  
𝑥∗ = 𝑇𝑥∗. 

Hence 𝑥∗ is a fixed point of 𝑇. 
Now we show the uniqueness of the fixed point of 𝑇. We assume that there are two fixed 

points of 𝑇 which are 𝑥 = 𝑇𝑥 and 𝑦 = 𝑇𝑦. Thus, 
𝑑(𝑥, 𝑦) = 𝑑(𝑇𝑥, 𝑇𝑦) 

≼ 𝑎𝑑(𝑥, 𝑇𝑥) + 𝑏𝑑(𝑦, 𝑇𝑦) + 𝑐𝑑(𝑥, 𝑦) 
≼ 𝑎𝑑(𝑥, 𝑥) + 𝑏𝑑(𝑦, 𝑦) + 𝑐𝑑(𝑥, 𝑦) 
≼ 𝑐𝑑(𝑥, 𝑦). 

Taking the absolute value of both sides, we have 
|𝑑(𝑥, 𝑦)| ≤ |𝑐𝑑(𝑥, 𝑦)|. 

This implies 𝑑(𝑥, 𝑦) = 0. Based on the properties of complex valued b-metric, we get 𝑥 = 𝑦. This 
completes the proof.  
 
Corollary 11. Let (𝑋, 𝑑) be a complete complex valued b-metric space with the coefficient 𝑠 ≥ 1. 
Let 𝑇: 𝑋 → 𝑋 be a mapping (for some fixed 𝑛) satisfying 

𝑑(𝑇 𝑛𝑥, 𝑇 𝑛 𝑦) ≼ 𝑎𝑑(𝑥, 𝑇 𝑛 𝑥) + 𝑏𝑑(𝑦, 𝑇 𝑛 𝑦) + 𝑐𝑑(𝑥, 𝑦) 
for all 𝑥, 𝑦 ∈ 𝑋 where 𝑎, 𝑏, 𝑐 are non-negative real numbers with 𝑎 + 𝑠(𝑏 + 𝑐) < 1. Then 𝑇 has a 
unique fixed point in 𝑋. 
Proof. Suppose 𝑆 = 𝑇 𝑛, then by the Theorem 10, there is a fixed point 𝑢 such that 

𝑆(𝑢) = 𝑢. 
Thus 𝑇𝑛 (𝑢) = 𝑢. 
We have 
𝑑(𝑇(𝑢), 𝑢) = 𝑑(𝑇(𝑇 𝑛 (𝑢)), 𝑢) 

= 𝑑 (𝑇 𝑛 (𝑇(𝑢)), 𝑇 𝑛(𝑢)) 

≼ 𝑎𝑑 (𝑇(𝑢), 𝑇 𝑛 (𝑇(𝑢))) + 𝑏𝑑(𝑢, 𝑇 𝑛 (𝑢)) + 𝑐𝑑(𝑇(𝑢), 𝑢) 

= 𝑎𝑑 (𝑇(𝑢), 𝑇(𝑇 𝑛 (𝑢))) + 𝑏𝑑(𝑢, 𝑢) + 𝑐𝑑(𝑇(𝑢), 𝑢) 



Fixed Point Theorems in Complex-Valued b-Metric Spaces 

Dahliatul Hasanah 142 

= 𝑎𝑑(𝑇(𝑢), 𝑇(𝑢)) + 𝑏𝑑(𝑢, 𝑢) + 𝑐𝑑(𝑇(𝑢), 𝑢) 
= 𝑐𝑑(𝑇(𝑢), 𝑢). 

Taking the absolute value of both sides, we have 
|𝑑(𝑇(𝑢), 𝑢)| ≤ 𝑐|𝑑(𝑇(𝑢), 𝑢)|. 

Hence 
(1 − 𝑐)|𝑑(𝑇(𝑢), 𝑢)| ≤ 0. 

This must be  
|𝑑(𝑇(𝑢), 𝑢)| = 0, 

which is 
𝑇(𝑢) = 𝑢 = 𝑇 𝑛(𝑢). 

So 𝑇 has a fixed point. 
To prove its uniqueness, suppose 𝑣 is also a fixed point of 𝑇. We need to prove that 𝑢 = 𝑣. 
We see that 

𝑑(𝑢, 𝑣) = 𝑑(𝑇 𝑛 (𝑢), 𝑇 𝑛 (𝑣)) 

≼ 𝑎𝑑(𝑢, 𝑇 𝑛 (𝑢)) + 𝑏𝑑(𝑣, 𝑇 𝑛(𝑣)) + 𝑐𝑑(𝑢, 𝑣) 

= 𝑎𝑑(𝑢, 𝑢) + 𝑏𝑑(𝑣, 𝑣) + 𝑐𝑑(𝑢, 𝑣) 
= 𝑐𝑑(𝑢, 𝑣). 

Taking the absolute value of both sides we get 
(1 − 𝑐)𝑑(𝑢, 𝑣) ≤ 0. 

Since 𝑐 < 1, this yields  
𝑑(𝑢, 𝑣) = 0. 

Thus 𝑢 = 𝑣. 
This completes the uniqueness of the fixed point. 
 
Theorem 12. Let (𝑋, 𝑑) be a complete complex-valued b-metric space with constant 𝑠 ≥ 1 and let 
𝑇: 𝑋 → 𝑋 be a mapping such that  

𝑑(𝑇𝑥, 𝑇𝑦) ≼ 𝛼𝑑(𝑥, 𝑇𝑦) + 𝛽𝑑(𝑦, 𝑇𝑥) 

for every 𝑥, 𝑦 ∈ 𝑋, where 𝛼, 𝛽 ≥ 0 with 𝛼𝑠 <
1

1+𝑠
 or 𝛽𝑠 <

1

1+𝑠
. Then 𝑇 has a fixed point in 𝑋. 

Moreover, if 𝛼 + 𝛽 < 1, then 𝑇 has a unique fixed point in 𝑋. 
Proof. Let 𝑥0 ∈ 𝑋 and (𝑥𝑛 ) be a sequence in X such that 

𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇
𝑛𝑥0. 

Now, for every 𝑛 ∈ ℕ we have 
𝑑(𝑥𝑛+2, 𝑥𝑛+1) = 𝑑(𝑇𝑥𝑛+1, 𝑇𝑥𝑛 ) 

 ≼ 𝛼𝑑(𝑥𝑛+1, 𝑇𝑥𝑛 ) + 𝛽𝑑(𝑥𝑛 , 𝑇𝑥𝑛+1) 
 = 𝛼𝑑(𝑥𝑛+1, 𝑥𝑛+1) + 𝛽𝑑(𝑥𝑛 , 𝑥𝑛+2) 
 = 𝛽𝑑(𝑥𝑛 , 𝑥𝑛+2) 
≼ 𝑠𝛽𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠𝛽𝑑(𝑥𝑛+1, 𝑥𝑛+2) 

(1 − 𝛽𝑠)𝑑(𝑥𝑛+1, 𝑥𝑛+2) ≼ 𝑠𝛽𝑑(𝑥𝑛 , 𝑥𝑛+1) 

𝑑(𝑥𝑛+1, 𝑥𝑛+2) ≼
𝑠𝛽

1 − 𝛽𝑠
𝑑(𝑥𝑛 , 𝑥𝑛+1). 

Taking 𝛾 =
𝑠𝛽

1−𝛽𝑠
  and continuing the process yields 

𝑑(𝑥𝑛+1, 𝑥𝑛+2) ≼ 𝛾𝑑(𝑥𝑛 , 𝑥𝑛+1) 
 ≼ 𝛾2𝑑(𝑥𝑛−1, 𝑥𝑛 ) 

⋮ 
≼ 𝛾𝑛+1𝑑(𝑥0, 𝑥1). 

On the other hand, 
𝑑(𝑥𝑛 , 𝑥𝑛+𝑚) ≼ 𝑠[𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑑(𝑥𝑛+1, 𝑥𝑛+𝑚 )] 

 ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠
2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠

2𝑑(𝑥𝑛+2, 𝑥𝑛+𝑚 ) 
 ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠

2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠
3𝑑(𝑥𝑛+2, 𝑥𝑛+3) + ⋯ + 𝑠

𝑚 𝑑(𝑥𝑛+𝑚−1, 𝑥𝑛+𝑚) 
 ≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1) + 𝑠

2𝛾𝑛+1𝑑(𝑥0, 𝑥1) + 𝑠
3𝛾𝑛+2𝑑(𝑥0, 𝑥1) + ⋯ + 𝑠

𝑚 𝛾𝑛+𝑚−1𝑑(𝑥0, 𝑥1) 
 ≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾)

2 + ⋯ + (𝑠𝛾)𝑚−1]. 



Fixed Point Theorems in Complex-Valued b-Metric Spaces 

Dahliatul Hasanah 143 

Since 𝛽𝑠 <
1

1+𝑠
 then 𝛾 ∈ (0,

1

𝑠
) and 𝑠𝛾 < 1. Taking limit 𝑛 → ∞, we have 𝛾𝑛 → 0. This implies 

|𝑑(𝑥𝑛 , 𝑥𝑛+𝑚)| → 0 as 𝑛 → ∞. By Lemma 8, the sequence (𝑥𝑛 ) is Cauchy sequence in X. Since 𝑋 is a 
complete complex valued b-metric space then (𝑥𝑛 ) is a convergent sequence. 
Suppose that (𝑥𝑛 ) converges to 𝑥

∗ ∈ 𝑋. We show that 𝑥∗ is a fixed point of 𝑇. 
𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑥𝑛 , 𝑇𝑥

∗)] 
= 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝑑(𝑇𝑥𝑛−1, 𝑇𝑥

∗) 
= 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝛼𝑑(𝑥𝑛−1, 𝑇𝑥

∗) + 𝑠𝛽𝑑(𝑇𝑥𝑛−1, 𝑥
∗) 

= 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝛼𝑑(𝑥𝑛−1, 𝑇𝑥
∗) + 𝑠𝛽𝑑(𝑥𝑛 , 𝑥

∗) 
= (𝑠 + 𝑠𝛽)𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠

2𝛼𝑑(𝑥𝑛−1, 𝑥
∗) + 𝑠2𝛼𝑑(𝑥∗, 𝑇𝑥∗) 

(1 − 𝛼𝑠2)𝑑(𝑥∗, 𝑇𝑥∗) ≼ (𝑠 + 𝑠𝛽)𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠
2𝛼𝑑(𝑥𝑛−1, 𝑥

∗). 
Since (𝑥𝑛 ) converges to 𝑥

∗ that is |𝑑(𝑥∗, 𝑥𝑛 )| → 0 as 𝑛 → ∞. and taking the absolute value of both 
sides, we have  

|𝑑(𝑥∗, 𝑇𝑥∗)| ≤ 0. 
This means that |𝑑(𝑥∗, 𝑇𝑥∗)| = 0, i.e. 𝑥∗ = 𝑇𝑥∗. So the limit of the sequence is a fixed point of 𝑇. 
Now we suppose that 𝛼 + 𝛽 < 1. Assume that there are 𝑢, 𝑣 ∈ 𝑋 such that 𝑢 = 𝑇𝑢 and 𝑣 = 𝑇𝑣. 

𝑑(𝑢, 𝑣) = 𝑑(𝑇𝑢, 𝑇𝑣) 
 ≼ 𝛼𝑑(𝑢, 𝑇𝑣) + 𝛽𝑑(𝑣, 𝑇𝑢) 
 = 𝛼𝑑(𝑢, 𝑣) + 𝛽𝑑(𝑣, 𝑢) 
 = (𝛼 + 𝛽)𝑑(𝑢, 𝑣). 

This means that |𝑑(𝑢, 𝑣)| ≤ (𝛼 + 𝛽)|𝑑(𝑢, 𝑣)|. Since 𝛼 + 𝛽 < 1, then we must have |𝑑(𝑢, 𝑣)| = 0, 
i.e. 𝑢 = 𝑣. The proof of the uniqueness of the fixed point of 𝑇 when 𝛼 + 𝛽 < 1 is complete. 

Mir  [14] studied the following fixed point theorem in b-metric spaces and we generalize 
it into complex-valued b-metric spaces. 
 
Theorem 13. Let (𝑋, 𝑑) be a complete b-metric space with constant 𝑠 ≥ 1 and define the sequence 
(𝑥𝑛 ) by the recursion  

𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇
𝑛𝑥0, 𝑛 = 1,2, … 

Let 𝑇: 𝑋 → 𝑋 be a mapping for which there exists 𝜇 ∈ [𝑜,
1

2
) such that  

𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜇[𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦)] 
for all 𝑥, 𝑦 ∈ 𝑋. Then there exists 𝑥∗ ∈ 𝑋 such that 𝑥𝑛 → 𝑥

∗ and 𝑥∗ is unique fixed point of 𝑇. 
 
Theorem 14. Let (𝑋, 𝑑) be a complete complex-valued b-metric space with constant 𝑠 ≥ 1. Let 

𝑇: 𝑋 → 𝑋 be a mapping for which there exists 𝜇 ∈ [𝑜,
1

2
) such that  

𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜇[𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦)] 
for all 𝑥, 𝑦 ∈ 𝑋. Then there exists unique fixed point of 𝑇. 
Proof. We construct a sequence (𝑥𝑛 ) in 𝑋 by  

𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇
𝑛𝑥0, 𝑛 = 1,2, … 

for any 𝑥0 ∈ 𝑋. We show that this sequence (𝑥𝑛 ) is Cauchy sequence in 𝑋 and hence is convergent 
sequence. 
For every 𝑛 ∈ ℕ we have 
𝑑(𝑥𝑛+2, 𝑥𝑛+1) = 𝑑(𝑇𝑥𝑛+1, 𝑇𝑥𝑛 ) 

 ≼ 𝜇(𝑑(𝑥𝑛+1, 𝑇𝑥𝑛+1) + 𝑑(𝑥𝑛 , 𝑇𝑥𝑛 )) 

 = 𝜇(𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑑(𝑥𝑛 , 𝑇𝑥𝑛 )) 

(1 − 𝜇)𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ 𝜇𝑑(𝑥𝑛 , 𝑇𝑥𝑛 )   

𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼
𝜇

1 − 𝜇
𝑑(𝑥𝑛+1, 𝑥𝑛 ) 

If this process continues, for all 𝑛 ∈ ℕ we have 

𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ (
𝜇

1 − 𝜇
)

𝑛+1

𝑑(𝑥1, 𝑥0). 

Therefore, 
𝑑(𝑥𝑛 , 𝑥𝑛+𝑚) ≼ 𝑠[𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑑(𝑥𝑛+1, 𝑥𝑛+𝑚 )] 

 ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠
2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠

2𝑑(𝑥𝑛+2, 𝑥𝑛+𝑚 ) 



Fixed Point Theorems in Complex-Valued b-Metric Spaces 

Dahliatul Hasanah 144 

 ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠
2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠

3𝑑(𝑥𝑛+2, 𝑥𝑛+3) + ⋯ + 𝑠
𝑚 𝑑(𝑥𝑛+𝑚−1, 𝑥𝑛+𝑚) 

 ≼ 𝑠 (
𝜇

1 − 𝜇
)

𝑛

𝑑(𝑥0, 𝑥1) + 𝑠
2 (

𝜇

1 − 𝜇
)

𝑛+1

𝑑(𝑥0, 𝑥1) + 𝑠
3 (

𝜇

1 − 𝜇
)

𝑛+2

𝑑(𝑥0, 𝑥1) + ⋯

+ 𝑠𝑚 (
𝜇

1 − 𝜇
)

𝑛+𝑚−1

𝑑(𝑥0, 𝑥1). 

≼ 𝑠 (
𝜇

1 − 𝜇
)

𝑛

𝑑(𝑥0, 𝑥1) [1 + 𝑠 (
𝜇

1 − 𝜇
) + (𝑠 (

𝜇

1 − 𝜇
))

2

+ ⋯ + (𝑠 (
𝜇

1 − 𝜇
))

𝑚−1

]. 

Note that 𝜇 ∈ [𝑜,
1

2
) implies (

𝜇

1−𝜇
) ∈ [0,1). Taking the absolute value of both sides gives  

|𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 )| → 0 as 𝑛 → ∞. 
Thus (𝑥𝑛 ) is Cauchy sequence in 𝑋 and is therefore a convergent sequence. Suppose 𝑥

∗ ∈ 𝑋 is the 
limit of sequence (𝑥𝑛 ). We show that 𝑥

∗ is a fixed point of 𝑇. 
Using the properties of b-metric and the mapping 𝑇, we get 
𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑥𝑛 , 𝑇𝑥

∗)] 
= 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝑑(𝑇𝑥𝑛−1, 𝑇𝑥

∗) 
= 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑠𝜇𝑑(𝑥

∗, 𝑇𝑥∗) 
= 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥𝑛−1, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥

∗, 𝑇𝑥∗) 
(1 − 𝑠𝜇)𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠𝑑(𝑥 ∗, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥𝑛−1, 𝑥𝑛 ) 

= 𝑠𝑑(𝑥 ∗, 𝑥𝑛 ) + 𝑠𝜇 (
𝜇

1 − 𝜇
)

𝑛−1

𝑑(𝑥1, 𝑥0) 

𝑑(𝑥∗, 𝑇𝑥∗) ≼
1

(1 − 𝑠𝜇)
(𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝜇 (

𝜇

1 − 𝜇
)

𝑛−1

𝑑(𝑥1, 𝑥0)). 

Taking the absolute value of both sides and taking 𝑛 → ∞, we have  
lim 𝑑(𝑥∗, 𝑇𝑥∗) = 0. 

This implies 𝑥∗ is a fixed point of 𝑇, i.e. 𝑥∗ = 𝑇𝑥∗. 
Now we assume that there exists another fixed point 𝑣 of 𝑇, that is 𝑣 = 𝑇𝑣. 
We have 
𝑑(𝑥∗, 𝑣) = 𝑑(𝑇𝑥∗, 𝑇𝑣) 

≼ 𝜇(𝑑(𝑥∗, 𝑇𝑥∗) + 𝑑(𝑣, 𝑇𝑣)) 

= 𝜇(𝑑(𝑥∗, 𝑥∗) + 𝑑(𝑣, 𝑣)) 

= 0. 
Taking the absolute value of both sides we obtain |𝑑(𝑥∗, 𝑣)| ≤ 0 which is a contradiction to the 
distinct fixed points. Hence 𝑥∗ is the unique fixed point of 𝑇. 

 
 

CONCLUSION  

Complex valued b-metric spaces are considered as a generalisation of b-metric spaces by 
changing the definition of real valued metric into complex valued metric. This change is expected 
to bring wider applications of fixed point theorems. In this paper, we extend fixed point theorems 
in b-metric spaces into fixed point theorems in complex valued b-metric spaces. By using the 
properties of partial order on complex numbers, a mapping 𝑇 on a real valued b-metric space 
which has unique fixed point still has unique fixed point in a complex valued b-metric space.   

 
 

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