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Some Fixed Point Results for Expansive Mappings in Dislocated  

Quasi-B-Metric Spaces with an Application to Integral Equations 

 

1Dasari Ratna Babu, 2P. Sudheer Kumar,3S. Lakshmana Rao 

1*Department of Mathematics, PSCMRCET, Vijayawada-520001, India 

email: ratnababud@gmail.com 

2Department of Information Technology, Aditya Institute of Technology and Management , Tekklai -532201, India. 

e-mail: sudheerkumar9732@gmail.com 

3Lecturer in Mathematics, Govt. Degree college, Tekkali, Dr.B.R.Amedkar University, Etcherla, 

srikakulam,india,532201 

e-mail: laxmana.mat@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: In this paper, we prove some new fixed point results for expansive type 

mappings in complete dislocated quasi 𝑏-metric space. A common fixed point result is also 

established considering such mappings. Our results extend and generalize the results of Das 

et al. [9] from the dislocated quasi metric space setting to dislocated quasi-𝑏-metric spaces. 

Suitable examples are provided to demonstrate our results. The solution to a system of 

Fredholm integral equations is also established to show the applicability of our results. 

Keywords: Fixed points; dislocated qusi-𝑏-metricspace; expansive map; integral equation. 

AMS  Subject  Classification (2020): 47H10,54H25.. 

 

1. Introduction 

The development of fixed point theory is based on the generalization of contraction conditions in one 

direction or/and generalization of ambient spaces of the operator under consideration on the other. 

Using the Picard iteration approach, Polish mathematician Banach developed the Banach contraction 

mapping concept in 1922. The existence of a solution for a differential equation with initial value 

condition, the implicit function existence theorem, and fixed point theory’s elegant assertion and 

effective method of solving it have drawn the attention of academics and inspired people to conduct 

in-depth, comprehensive research. With the advent of the computer, particularly in the last few 

decades, many individuals have dealt with a large number of applications by using a range of 

iteration techniques to approach the fixed point. As a result, they made a breakthrough and gradually 

improved this subject. These days, nonlinear functional analysis relies heavily on fixed point theory. 

The notion of dislocated metric space initially surfaced in domain theory, which was proposed by 

Matthews [15] in 1986 along with various concepts of metric domains. Hitzler et al. [13] presented 

the idea of dislocated metric space later in 2000, where a point’s self-distance is not always zero. In 

this area, they also extended the Banach contraction concept. Topology, logical programming, 

computer science, electronic engineering, and other fields all heavily rely on dislocated metric space. 

Zeyadaet al. [26] expanded Hitzler’s [13] result in dislocated quasi-metric space and introduced the 

mailto:ratnababud@gmail.com
mailto:laxmana.mat@gmail.com


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entire dislocated quasi-metric space. For more details see [1, 11, 14, 16, 17, 21, 23] 

The innovative idea of expansive mapping was first presented by Wang et al. [23] in 1984. They 

carried out a comprehensive investigation and revealed complex fixed point out comes in the domain 

of entire metric space. Applications for expansive mappings can be found in nonlinear analysis, 

dynamical system theory, and chaos theory. Since then, other academics have conducted thorough 

studies, methodically developing and extending fixed point theoretical results in this specific field 

[3,10,12,18– 20, 22, 25]. 

Definition 1.1. [8] Let X be a non-empty set and 𝑠 ≥ 1 be a given real number. Let 𝑑: 𝑋 × 𝑋 →

[0,∞) be a mapping and for any 𝑎, 𝑏 , 𝑐  𝜖 𝑋:  

(i).   0 ≤ 𝑑(𝑎, 𝑏) and 𝑑(𝑎, 𝑏) = 0  if and only if 𝑎 = 𝑏; 

(ii).  𝑑(𝑎, 𝑏) = 0 implies  𝑎 = 𝑏; 

(iii). 𝑑(𝑎, 𝑏) = 0 = 𝑑(𝑏, 𝑎)  implies 𝑎 = 𝑏; 

(iv). 𝑑(𝑎, 𝑏)  = 𝑑(𝑏, 𝑎); 

(v).  𝑑(𝑎, 𝑐) ≤ 𝑑(𝑎, 𝑏) + 𝑑(𝑏, 𝑐); 

(vi). 𝑑(𝑎, 𝑐) ≤ 𝑠[𝑑(𝑎, 𝑏) + 𝑑(𝑏, 𝑐)]. 

Then 

(1) (𝑋, 𝑑) is called a metric space if  (i),  (iv), and (v)  hold;  

(2) (𝑋, 𝑑)  is called a b-metric space if (i), (iv), and (vi)  hold;  

(3) (𝑋, 𝑑) is called a quasi-metric space if  (i), and (v)  hold;  

(4) (𝑋, 𝑑) is called a quasi-b-metric space   if (i) , and  (vi)  hold; 

(5) (𝑋, 𝑑) is called a dislocated metric space (𝑑-metric space)  if  (ii) , (iv),  and  (v)  hold; 

(6) (𝑋, 𝑑) is called a dislocated 𝑏-metric space (𝑑 𝑏-metric space) if  (ii),  (iv) , and  (vi)  hold; 

(7) (𝑋, 𝑑) is called a dislocated quasi-metric space (𝑑𝑞-metric space)  if  (iii) and (v)  hold; 

(8) (𝑋, 𝑑) is called a dislocated quasi-b-metric space (𝑑𝑞 𝑏-metric space) if  (iii) and (vi)  hold. 

Even though the examples provided were well-known, we felt that providing a thorough review 

would be helpful for convenient reference. 

Example 1.2. (𝑎) Let 𝑋=ℝ and 𝑑:𝑋 × 𝑋→ℝ+ defined as 𝑑(𝑎, 𝑏) = {
𝑎 − 𝑏,   𝑎 ≥ 𝑏
1 , 𝑜𝑡ℎ𝑒𝑟𝑖𝑠𝑒.

 

 Then (𝑋, 𝑑) is a quasi-metric space, but it is not a metric space. 

(𝑏) Let 𝑋=ℝ+ and 𝑑: 𝑋 × 𝑋→ ℝ+
 defined as 𝑑(𝑎, 𝑏) = {

0,   𝑎 = 𝑏

(𝑎 + 𝑏)2  , 𝑜𝑡ℎ𝑒𝑟𝑖𝑠𝑒.
 

Then (𝑋, 𝑑) is a 𝑏-metric space, but it is not a metric space. 

(𝑐) Let 𝑋 = 𝐶([0,1],ℝ) with the usual partial ordering, and let 𝑑:𝑋 × 𝑋→ℝ+  be defined as  



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𝑑(𝑓, 𝑔) =

{
 
 

 
 
∫(𝑔(𝑡) − 𝑓(𝑡))3  𝑑𝑡,    𝑓 ≤ 𝑔,

1

0

∫(𝑓(𝑡) − 𝑔(𝑡))3 
1

0

𝑑𝑡,    𝑓 ≥ 𝑔.

 

Then (𝑋, 𝑑) is a quasi-b-metric space, but it is not a quasi-metric space and 𝑏-metric space. 

(𝑑) Let 𝑋=ℝ+ and 𝑑: 𝑋 × 𝑋 → ℝ+
 defined as 𝑑(𝑎, 𝑏) = max{𝑎, 𝑏}.Then (𝑋, 𝑑) is a 

dislocated metric space, but it is not a metric space. 

(𝑒) Let 𝑋 = [0,1] and 𝑑:𝑋 × 𝑋 → ℝ+
  be defined as  𝑑(𝑎, 𝑏) = |𝑎 − 𝑏| + 𝑎. Then (𝑋, 𝑑) is a 

dislocated  quasi-metric space, but it is not a  dislocated metric space, and it is not a quasi-

metric space. 

(𝑓) Let 𝑋 = [0,∞) and 𝑑: 𝑋 × 𝑋 →[0,∞)   be defined as  𝑑(𝑎, 𝑏) = (𝑎 + 𝑏)2. Then (𝑋, 𝑑) is 

a dislocated  𝑏-metric space, but it is not a  𝑏-metric space. 

(𝑔) Let 𝑋=ℝ and 𝑑: 𝑋 × 𝑋 → ℝ+ be 
 defined as  𝑑(𝑎, 𝑏) = |𝑎 − 𝑏|2 +

|𝑎|

𝑛
+
|𝑏|

𝑚
, where 𝑛,𝑚 ∈

ℕ\{1}, 𝑛 ≠ 𝑚.Then (𝑋, 𝑑) is a dislocated  quasi-metric space, but it is not a  quasi 𝑏- metric 

space,  dislocated 𝑏-metric space and dislocated quasi-metric space. 

Thus, we get the process diagram (refer to Figure 1), in which generalization relationships are 

represented by arrows. 

 

Figure 1:  Process diagram. 

The following lemmas are useful in proving our main results. 

Lemma 1.3. [2] Let  (𝑋, 𝑑) be a b-metric space with coefficients 𝑠 ≥1. 

Suppose that {𝑎𝑛}   and   {𝑏𝑛}   are 𝑏-convergent to  𝑥 and 𝑦 respectively. Then we have 

1

𝑠2  
 𝑑(𝑥, 𝑦) ≤ lim inf

𝑛→∞
𝑑(𝑎𝑛, 𝑏𝑛) ≤  lim sup

𝑛→∞
𝑑(𝑎𝑛, 𝑏𝑛) ≤ 𝑠2𝑑(𝑥, 𝑦). 

 In particular, if  𝑥 = 𝑦, then we have  lim
n→∞

d(an, bn ) = 0 .Moreover for each 𝑧 ∈  𝑋 we have 



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1

𝑆
 𝑑(𝑥, 𝑧) ≤ lim inf

𝑛→∞
𝑑(𝑎𝑛, 𝑧) ≤  lim sup

𝑛→∞
𝑑(𝑎𝑛, 𝑧)≤ 𝑠 𝑑(𝑥, 𝑧). 

Lemma 1.4.Let 𝑎 be a limit of some sequence {𝑎𝑛}  in a 𝑑𝑞 𝑏-metric space(𝑋, 𝑑), then 𝑑(𝑎, 𝑎) = 0. 

Proof. Let 𝑎 ∈  𝑋, {an}  ⊆ 𝑋 and a sequence which converges to 𝑎. Then 

𝑑(𝑎, 𝑏) ≤ 𝑠[𝑑(𝑎, 𝑎𝑛) + 𝑑(𝑎𝑛, 𝑎)], ∀ 𝑛 ∈ 𝑁. 

By taking limit superior as 𝑛 →  ∞  and  using Lemma 1.3, we get 𝑑(𝑎, 𝑎) = 0. 

 

Recently, Das et al. [9] established the following theorems in 𝑑𝑞-metric spaces. 

Theorem 1.5. [9] Let  (𝑋, 𝑑) be a complete  𝑑𝑞-metric space and 𝑇 be an onto self-mapping on 𝑋 

such that 

𝑑(𝑇𝑎, 𝑇𝑏)  ≥ 𝑘 min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑇𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑇𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏)

+ 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑇𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏)} 

for all 𝑎, 𝑏 ∈  𝑋 with 𝑑(𝑎, 𝑏)  ≠  0, 𝑘 >  1, nonnegative real numbers 𝛼, 𝛽𝑖 , 𝛾𝑗 , 𝛿𝑗  for 

 𝑖 =  1, 2;  𝑗 =  1, 2, 3 and 
1

𝑘
= min{𝛼, 𝛽2, 𝛾2 + 𝛾3, 𝑘

2𝛿2(𝛿1 + 𝛿2) + 𝑘𝛿3}.  

 Then T has a unique fixed  point. 

Theorem 1.6. [9] Let (𝑋, 𝑑) be a complete 𝑑𝑞 -metric space and S, T be two onto self-mapping on 𝑋 

such that 

𝑑(𝑆𝑎, 𝑇𝑏)  ≥ 𝑘 min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑆𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑆𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏)

+ 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑆𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏)} 

for all 𝑎, 𝑏 ∈  𝑋 with 𝑑(𝑎, 𝑏)  ≠  0, 𝑘 >  1, nonnegative real numbers 𝛼, 𝛽𝑖 , 𝛾𝑗 , 𝛿𝑗  for 

 𝑖 =  1, 2;  𝑗 =  1, 2, 3 and 
1

𝑘
= min{𝛼, 𝛽2, 𝛾2 + 𝛾3, 𝛿3}. 

Then 𝑆 and 𝑇 have a unique common fixed point. 

2. Main Results 

In this section, we formulate some fixed point results for onto expansive type mapping in a complete 

𝑑𝑞 𝑏-metric space. 

Theorem 2.1. Let (X,d) be a complete dq b-metric space and T be an onto self-mapping on X such 

that 



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𝑑(𝑇𝑎, 𝑇𝑏)  ≥ 𝑘 min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑇𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑇𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏)

+ 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑇𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏); 𝜆1
𝑑(𝑇𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)

+ 𝜆2
𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)}                                                                        (2.1) 

for all 𝑎, 𝑏 ∈  𝑋 with 𝑑(𝑎, 𝑏)  ≠  0, 𝑘 >  1, nonnegative real numbers 𝛼, 𝛽𝑖 , 𝛾𝑗 , 𝛿𝑗 , 𝜆𝑗 for 𝑖 =

 1, 2;  𝑗 =  1, 2, 3 and 
1

𝑘
= 𝑚𝑖𝑛{𝛼, 𝛽2, 𝛾2 + 𝛾3, 𝑘

2𝛿2(𝛿1 + 𝛿2) + 𝑘𝛿3, 𝜆3}.  Then T has a unique  fixed 

point. 

Proof.  Let us take 𝜃(𝑎, 𝑏) = min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑇𝑎,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑇𝑎, 𝑎) +

𝛾2 𝑑(𝑇𝑏, 𝑏) + 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑇𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏); 𝜆1
𝑑(𝑇𝑎,𝑏)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+

𝜆2
𝑑(𝑇𝑏,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)}. 

For 𝑎0 ∈   𝑋, since  𝑇 are onto, there exist 𝑎0 ∈  𝑋 such that 𝑎0 = 𝑇𝑎1.   Continuing this process, we 

define a sequence {𝑎𝑛} in 𝑋 with 𝑎𝑛−1 = 𝑇𝑎𝑛, for all 𝑛 ∈ ℕ. 

The cases listed below will occur. 

Case (i).  If 𝜃(𝑎, 𝑏) = 𝛼 𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑇𝑏)  ≥ 𝑘 𝛼 𝑑(𝑎, 𝑏), for all 𝑎, 𝑏 ∈  𝑋 .                                                                                             (2.2)  

Now, using (2.2), we get 

𝑑(𝑎𝑛−1, 𝑎𝑛) = 𝑑(𝑇𝑎𝑛, 𝑇𝑎𝑛+1) ≥  𝑘 𝛼 𝑑(𝑎𝑛, 𝑎𝑛+1) 

i. e. , 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤
1

𝑘 𝛼
𝑑(𝑎𝑛−1, 𝑎𝑛).  

Let 𝜏 =
1

𝑘 𝛼
< 1.  Then from the above inequality, we have 

𝑑(𝑎𝑛, 𝑎𝑛+1) ≤ 𝜏 𝑑(𝑎𝑛−1, 𝑎𝑛). 

Also,   

𝑑(𝑎𝑛+1, 𝑎𝑛+2) ≤ 𝜏 𝑑(𝑎𝑛, 𝑎𝑛+1) + 𝜏
2 𝑑(𝑎𝑛−1, 𝑎𝑛). 

From this we get, 

𝑑(𝑎𝑛, 𝑎𝑛+1) ≤ 𝜏𝑛 𝑑(𝑎0, 𝑎1). 

For 𝑗 > 𝑖,  

𝑑(𝑎𝑖 , 𝑎𝑗) ≤ 𝑠 𝑑(𝑎𝑖 , 𝑎𝑖+1) + 𝑠
2 𝑑(𝑎𝑖+1, 𝑎𝑖+2) + ⋯+ 𝑠𝑗−𝑖 𝑑(𝑎𝑗−1, 𝑎𝑗) 

≤ 𝑠 𝜏𝑖𝑑(𝑎0, 𝑎1) + 𝑠
2𝜏𝑖+1𝑑(𝑎0, 𝑎1) + ⋯+ 𝑠

𝑗−𝑖𝜏𝑗−1𝑑(𝑎0, 𝑎1) 

≤ [𝑠 𝜏𝑖 + 𝑠2𝜏𝑖+1 +⋯]𝑑(𝑎0, 𝑎1) 

= 𝑠 𝜏𝑖[1 + 𝑠𝜏 + (𝑠𝜏)2 +⋯ ]𝑑(𝑎0, 𝑎1) 



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=
𝑠 𝜏𝑖

1 − 𝑠𝜏
 𝑑(𝑎0, 𝑎1) → 0 as 𝑖, 𝑗 → ∞ 

Therefore {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑢 ∈  𝑋 such that 

lim
𝑛→∞

𝑎𝑛 = 𝑢.  Since, 𝑇 is onto, we can find 𝑝 ∈  𝑋 such that 𝑇𝑝 = 𝑢.   

Now, for all 𝑛 ∈  ℕ. 

𝑑(𝑢, 𝑎𝑛) = 𝑑(𝑇𝑝, 𝑇𝑎𝑛+1) ≥  𝑘 𝛼 𝑑(𝑝, 𝑎𝑛+1). 

Taking limit superior as 𝑛 → ∞, and using Lemma 1.3, we get 

1

𝑠
𝑑(𝑝, 𝑢) ≤

𝑘 𝛼

𝑠
lim
𝑛→∞

sup𝑑(𝑝, 𝑎𝑛+1) ≤ lim
𝑛→∞

sup𝑑(𝑢, 𝑎𝑛) ≤ 𝑠 𝑑(𝑢, 𝑢). 

From Lemma. 1.4, we get 𝑑(𝑝, 𝑢) = 0 and similarly 𝑑(𝑢, 𝑝) = 0. 

Thus, 𝑑(𝑝, 𝑢) = 𝑑(𝑢, 𝑝) = 0.  So, 𝑝 = 𝑢. 

Uniqueness.  Let 𝑣(≠ 𝑢) be another fixed point of 𝑇.  Then 

𝑑(𝑢, 𝑣) = 𝑑(𝑇𝑢, 𝑇𝑣) ≥  𝑘 𝛼 𝑑(𝑢, 𝑣), i. e. , (1 − 𝑘 𝛼)𝑑(𝑢, 𝑣) ≤ 0 

which gives us 𝑑(𝑢, 𝑣) = 0.  Similarly, we can prove that 𝑑(𝑣, 𝑢) = 0, and that 𝑢 = 𝑣. 

Case (ii).  𝜃(𝑎, 𝑏) = 𝛽1  
𝑑(𝑇𝑎,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝛽2𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑇𝑏) ≥ 𝑘 [𝛽1  
𝑑(𝑇𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏)]                                                                               (2.3) 

Now, using (2.3), we get 

𝑑(𝑎𝑛−1, 𝑎𝑛) = 𝑑(𝑇𝑎𝑛, 𝑇𝑎𝑛+1) ≥  𝑘𝛽1  
𝑑(𝑇𝑎𝑛, 𝑎𝑛)𝑑(𝑇𝑎𝑛+1, 𝑎𝑛+1)

𝑑(𝑎𝑛, 𝑎𝑛+1)
+ 𝑘𝛽2𝑑(𝑎𝑛, 𝑎𝑛+1) 

=  𝑘𝛽1  
𝑑(𝑎𝑛−1, 𝑎𝑛)𝑑(𝑎𝑛, 𝑎𝑛+1)

𝑑(𝑎𝑛, 𝑎𝑛+1)
+ 𝑘𝛽2𝑑(𝑎𝑛, 𝑎𝑛+1) 

≥ 𝑘𝛽2𝑑(𝑎𝑛, 𝑎𝑛+1). 

i. e. , 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤
1

𝑘𝛽2
𝑑(𝑎𝑛−1, 𝑎𝑛) which implies that 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤ 𝜗 𝑑(𝑎𝑛−1, 𝑎𝑛), 

where  𝜗 =
1

𝑘𝛽2
< 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, which 

converges to some 𝑢 ∈  𝑋, which can be shown to be unique fixed point of  𝑇. 

Case (iii).  𝜃(𝑎, 𝑏) = 𝛾1𝑑(𝑇𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏) + 𝛾3 𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑇𝑏) ≥ 𝑘[𝛾1𝑑(𝑆𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏) + 𝛾3 𝑑(𝑎, 𝑏)]                                                                       (2.4) 

Now, using (2.4), we get 

𝑑(𝑎𝑛−1, 𝑎𝑛) = 𝑑(𝑇𝑎𝑛, 𝑇𝑎𝑛+1) 

≥  𝑘𝛾1𝑑(𝑇𝑎𝑛, 𝑎𝑛) + 𝑘𝛾2 𝑑(𝑇𝑎𝑛+1, 𝑎𝑛+1) + 𝑘𝛾3 𝑑(𝑎𝑛, 𝑎𝑛+1) 

= 𝑘𝛾1𝑑(𝑎𝑛−1, 𝑎𝑛) + 𝑘𝛾2 𝑑(𝑎𝑛, 𝑎𝑛+1) + 𝑘𝛾3 𝑑(𝑎𝑛, 𝑎𝑛+1) 

≥ 𝑘[𝛾2 + 𝛾3]𝑑(𝑎𝑛 , 𝑎𝑛+1) 



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i.e., 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤
1

𝑘[𝛾2+𝛾3]
𝑑(𝑎𝑛−1, 𝑎𝑛) implies that 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤ 𝜌 𝑑(𝑎𝑛−1, 𝑎𝑛), where  𝜌 =

1

𝑘[𝛾2+𝛾3]
< 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, and that 

converges to some 𝑢 ∈  𝑋, which is a unique fixed point of 𝑇. 

Case (iv).  𝜃(𝑎, 𝑏) = 𝛿1𝑑(𝑇𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑆𝑏) ≥ 𝑘[𝛿1𝑑(𝑇𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏)]                                                                         (2.5) 

From (2.5), we get 

𝑑(𝑎𝑛−1, 𝑎𝑛) = 𝑑(𝑇𝑎𝑛, 𝑇𝑎𝑛+1) 

≥  𝑘𝛿1𝑑(𝑇𝑎𝑛, 𝑎𝑛+1) + 𝑘𝛿2 𝑑(𝑇𝑎𝑛+1, 𝑎𝑛) + 𝑘𝛿3 𝑑(𝑎𝑛, 𝑎𝑛+1) 

= 𝑘𝛿1𝑑(𝑎𝑛−1, 𝑎𝑛+1) + 𝑘𝛿2 𝑑(𝑎𝑛, 𝑎𝑛) + 𝑘𝛿3 𝑑(𝑎𝑛, 𝑎𝑛+1) 

 ≥ 𝑘𝛿2 𝑑(𝑎𝑛, 𝑎𝑛) + 𝑘𝛿3𝑑(𝑎𝑛 , 𝑎𝑛+1)                                                                                                            (2.6) 

where 

𝑑(𝑎𝑛 , 𝑎𝑛) = 𝑑(𝑇𝑎𝑛+1, 𝑇𝑎𝑛+1) 

≥ 𝑘𝛿1𝑑(𝑇𝑎𝑛+1, 𝑎𝑛+1) + 𝑘𝛿2 𝑑(𝑇𝑎𝑛+1, 𝑎𝑛+1) + 𝑘𝛿3 𝑑(𝑎𝑛+1, 𝑎𝑛+1) 

≥ 𝑘𝛿1𝑑(𝑎𝑛, 𝑎𝑛+1) + 𝑘𝛿2 𝑑(𝑎𝑛, 𝑎𝑛+1) 

= 𝑘[𝛿1 + 𝛿2]𝑑(𝑎𝑛, 𝑎𝑛+1). 

From the inequality (2.6), we get 

𝑑(𝑎𝑛−1, 𝑎𝑛) ≥ [𝑘2𝛿2(𝛿1 + 𝛿2) + 𝑘 𝛿3]𝑑(𝑎𝑛, 𝑎𝑛+1) which implies that 

i.e., 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤
1

𝑘2𝛿2(𝛿1+𝛿2)+𝑘 𝛿3
𝑑(𝑎𝑛−1, 𝑎𝑛) implies that 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤ 𝜔 𝑑(𝑎𝑛−1, 𝑎𝑛), where  

𝜔 =
1

𝑘2𝛿2(𝛿1+𝛿2)+𝑘 𝛿3
< 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, 

and that converges to some 𝑢 ∈  𝑋, which is a unique fixed point of  𝑇. 

Case (v).  𝜃(𝑎, 𝑏) = 𝜆1
𝑑(𝑇𝑎,𝑏)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝜆2

𝑑(𝑇𝑏,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝜆3𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑇𝑏) ≥ 𝑘 [𝜆1
𝑑(𝑇𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆2

𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)]                                     (2.7) 

Now, using (2.6), we get 

𝑑(𝑎𝑛−1, 𝑎𝑛) = 𝑑(𝑇𝑎𝑛, 𝑇𝑎𝑛+1) 

≥  𝑘 𝜆1
𝑑(𝑇𝑎𝑛, 𝑎𝑛+1)𝑑(𝑇𝑎𝑛+1, 𝑎𝑛+1)

𝑑(𝑎𝑛, 𝑎𝑛+1)
+ 𝑘 𝜆2

𝑑(𝑇𝑎𝑛+1, 𝑎𝑛)𝑑(𝑇𝑎𝑛+1, 𝑎𝑛+1)

𝑑(𝑎𝑛, 𝑎𝑛+1)

+ 𝑘 𝜆3 𝑑(𝑎𝑛, 𝑎𝑛+1) 

= 𝑘 𝜆1
𝑑(𝑎𝑛−1, 𝑎𝑛+1)𝑑(𝑎𝑛, 𝑎𝑛+1)

𝑑(𝑎𝑛 , 𝑎𝑛+1)
+ 𝑘 𝜆2

𝑑(𝑎𝑛, 𝑎𝑛)𝑑(𝑎𝑛 , 𝑎𝑛+1)

𝑑(𝑎𝑛, 𝑎𝑛+1)
+ 𝑘 𝜆3 𝑑(𝑎𝑛, 𝑎𝑛+1) 

 ≥ 𝑘𝜆3𝑑(𝑎𝑛, 𝑎𝑛+1). 



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i.e., 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤
1

𝑘𝜆3
𝑑(𝑎𝑛−1, 𝑎𝑛) which implies that 𝑑(𝑎𝑛, 𝑎𝑛+1) ≤ 𝜑 𝑑(𝑎𝑛−1, 𝑎𝑛), where  𝜑 =

1

𝑘𝜆3
< 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, and that 

converges to some 𝑢 ∈  𝑋, which is a unique fixed point of  𝑇. 

The following is an example in support of Theorem 2.1. 

Example 2.2.  Let 𝑋 = ℝ+.  We define 𝑑: 𝑋 ×  𝑋 → ℝ+ by 𝑑(𝑎, 𝑏) = |𝑎 − 𝑏|2 + |𝑎|2.  Then 

clearly, (𝑋, 𝑑) is a complete 𝑑𝑝 𝑏-metric space with 𝑠 = 2. We define self-mappings 𝑇: 𝑋 →  𝑋 by 

𝑇(𝑎) = 𝑎(𝑎 + 2), for all 𝑎 ∈  𝑋. 

We take 𝑘 =
3

2
, 𝛼 = 𝛽2 = 𝛾3 = 𝛿3 = 𝜆3 = 1, 𝛽1 = 𝛾1 = 𝛾2 = 𝛿1 = 𝛿2 = 𝜆1 = 𝜆2 = 0. 

Without loss of generality we assume that 𝑎 ≥  𝑏.  We consider 

𝑑(𝑆𝑎, 𝑇𝑏) = |𝑇𝑎 − 𝑇𝑏|2 + |𝑆𝑎|2 

= (𝑎2 + 2𝑎 − 𝑏2 − 2𝑏)2 + (𝑎2 + 2𝑎)2 

= (𝑎 − 𝑏)2(𝑎 + 𝑏 + 2)2 + 𝑎2(a + 2)2 

≥
3

2
[(𝑎 − 𝑏)2 + 𝑎2] 

= 𝑘 min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑇𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑇𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏)

+ 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑇𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏); 𝜆1
𝑑(𝑇𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)

+ 𝜆2
𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)} 

 



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Table 1 and Figure 2, illustrates the condition (2.1) of Theorem 2.1, with blue line representing the 

left part of the condition and red line representing the right part of the condition. Thus, all the 

conditions of Theorem 2.8 are satisfied. So  𝑇 has a unique common fixed point, which is clearly 0 

here. 

Corollary 2.3.  Let (𝑋, 𝑑) be a complete 𝑑𝑞 𝑏-metric space and 𝑇 be an onto self-mapping on 𝑋 

such that 𝑑(𝑇𝑎, 𝑇𝑏) ≥ 𝑘𝑑(𝑎, 𝑏) for all 𝑎, 𝑏 ∈  𝑋 with 𝑑(𝑎, 𝑏)  ≠ 0, 𝑘 > 1.  Then 𝑇 has a unique 

fixed point. 

Proof.  By taking 𝛼 = 𝛽2 = 𝛾3 = 𝛿3 = 𝜆3 = 1 and 𝛽1 = 𝛾1 = 𝛾2 = 𝛿1 = 𝛿2 = 𝜆1 = 𝜆2 = 0 in 

Theorem 2.1, the result follows easily. 

Corollary 2.4.  Let (𝑋, 𝑑) be a complete 𝑑𝑞 𝑏-metric space and 𝑇 be an onto self-mapping on 𝑋 

such that 

𝑑(𝑇𝑎, 𝑇𝑏)  ≥ 𝑘 min {𝑑(𝑎, 𝑏);  
𝑑(𝑇𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝑑(𝑎, 𝑏); 𝑑(𝑇𝑎, 𝑎) +  𝑑(𝑇𝑏, 𝑏)

+  𝑑(𝑎, 𝑏); 𝑑(𝑇𝑎, 𝑏) +  𝑑(𝑇𝑏, 𝑎) + 𝑑(𝑎, 𝑏);
𝑑(𝑇𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+
𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)

+ 𝑑(𝑎, 𝑏)} 

for all 𝑎, 𝑏 ∈  𝑋 with 𝑑(𝑎, 𝑏)  ≠ 0, 𝑘 > 1.  Then 𝑇 has a unique fixed point. 

Proof.  Putting 𝛼 = 𝛽1 = 𝛽2 = 𝛾1 = 𝛾2 = 𝛾3 = 𝛿1 = 𝛿2 = 𝛿3 = 𝜆1 = 𝜆2 = 𝜆3 = 1 in Theorem 2.1, 

the result follows easily. 

Theorem 2.5.  Let (𝑋, 𝑑) be a complete 𝑑𝑞 𝑏-metric space and 𝑇 be an onto self-mapping on 𝑋 such 

that 



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𝑑(𝑇𝑎, 𝑇𝑏)  ≥ 𝛼1𝑑(𝑎, 𝑏) + 𝛼2 [ 
𝑑(𝑇𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝑑(𝑎, 𝑏)] + 𝛼3[𝑑(𝑇𝑎, 𝑎) +  𝑑(𝑇𝑏, 𝑏) +  𝑑(𝑎, 𝑏)]

+ 𝛼4[𝑑(𝑇𝑎, 𝑏) +  𝑑(𝑇𝑏, 𝑎) + 𝑑(𝑎, 𝑏)]

+ 𝛼5 [
𝑑(𝑇𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+
𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝑑(𝑎, 𝑏)]                                            (2.8) 

for all 𝑎, 𝑏 ∈  𝑋 with 𝑑(𝑎, 𝑏) ≠ 0, 𝛼1 + 𝛼2 + 𝛼3 + 𝛼4 + 𝛼5 > 1.  Then 𝑇 has a unique fixed point. 

Proof.  Let 𝜃(𝑎, 𝑏) = min {𝑑(𝑎, 𝑏);  
𝑑(𝑇𝑎,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝑑(𝑎, 𝑏); 𝑑(𝑇𝑎, 𝑎) +  𝑑(𝑇𝑏, 𝑏) +

 𝑑(𝑎, 𝑏); 𝑑(𝑇𝑎, 𝑏) +                              𝑑(𝑇𝑏, 𝑎) + 𝑑(𝑎, 𝑏);
𝑑(𝑇𝑎,𝑏)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+
𝑑(𝑇𝑏,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+

𝑑(𝑎, 𝑏)}.                                  (2.9) 

Using (2.8) and (2.9), we get 

𝑑(𝑇𝑎, 𝑇𝑏)  ≥ 𝛼1𝜃(𝑎, 𝑏) + 𝛼2𝜃(𝑎, 𝑏) + 𝛼3𝜃(𝑎, 𝑏) + 𝛼4𝜃(𝑎, 𝑏) + 𝛼5𝜃(𝑎, 𝑏) 

= (𝛼1 + 𝛼2 + 𝛼3 + 𝛼4 + 𝛼5)𝜃(𝑎, 𝑏). 

Then the inequality becomes 𝑑(𝑇𝑎, 𝑇𝑏) ≥  𝑘 𝜃(𝑎, 𝑏), where 𝑘 > 1. Therefore by Corollary 2.4, we 

conclude the proof. 

Remark 2.6.  Theorem 2.1 and Example 2.2 extend and generalize Theorem 1.5 to 𝑑𝑞 𝑏$-metric 

spaces by taking 𝜆𝑖 = 0, 𝑖 = 1,2,3 in Theorem 2.1.  

Remark 2.7.  Corollary 2.3 extend and generalize the result of [24] in the framework of 𝑑𝑞 𝑏-metric 

space. 

 In the following, we deduce a common fixed point theorem to a pair of onto expansive type 

self-mappings. 

Theorem 2.8.  Let (𝑋, 𝑑) be a complete 𝑑𝑞 𝑏-metric space and 𝑆, 𝑇 be two onto self-mapping on 𝑋 

such that 

𝑑(𝑆𝑎, 𝑇𝑏)  ≥ 𝑘 min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑆𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑆𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏)

+ 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑆𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏); 𝜆1
𝑑(𝑆𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)

+ 𝜆2
𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)}                                                                            (2.10) 

and 

𝑑(𝑇𝑎, 𝑆𝑏)  ≥ 𝑘 min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑆𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑆𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏)

+ 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑆𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏); 𝜆1
𝑑(𝑆𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)

+ 𝜆2
𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)}                                                                            (2.11) 



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for all 𝑎, 𝑏 ∈  𝑋 with 𝑑(𝑎, 𝑏)  ≠  0, 𝑘 >  1, nonnegative real numbers 𝛼, 𝛽𝑖 , 𝛾𝑗 , 𝛿𝑗 , 𝜆𝑗 for 𝑖 =

 1, 2;  𝑗 =  1, 2, 3 and 
1

𝑘
= 𝑚𝑖𝑛{𝛼, 𝛽2, 𝛾2 + 𝛾3, 𝛿3, 𝜆3}.  Then 𝑆 and 𝑇 have a unique common fixed 

point. 

Proof.  Let us take 𝜃(𝑎, 𝑏) = min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑆𝑎,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑆𝑎, 𝑎) +

𝛾2 𝑑(𝑇𝑏, 𝑏) + 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑆𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏); 𝜆1
𝑑(𝑆𝑎,𝑏)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+

𝜆2
𝑑(𝑇𝑏,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)}. 

For 𝑎0 ∈   𝑋, since 𝑆, 𝑇 are onto, there exist 𝑎0, 𝑎1 ∈  𝑋 such that 𝑎0 = 𝑆𝑎1,, 𝑎1 = 𝑇𝑎2.   Continuing 

this process, we define a sequence {𝑎𝑛} by 

𝑆𝑎2𝑛−1 = 𝑎2𝑛−2 

𝑇𝑎2𝑛 = 𝑎2𝑛−1 , for all 𝑛 ∈ ℕ.                                                                                                                (2.12) 

The following cases will arise. 

Case (i).  If 𝜃(𝑎, 𝑏) = 𝛼 𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑆𝑏)  ≥ 𝑘 𝛼 𝑑(𝑎, 𝑏), for all 𝑎, 𝑏 ∈  𝑋 .                                                                                            (2.13)  

Now, using (2.12) and (2.13), we get 

𝑑(𝑎2𝑛, 𝑎2𝑛+1) = 𝑑(𝑆𝑎2𝑛+1, 𝑇𝑎2𝑛+2) ≥  𝑘 𝛼 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

i. e. , 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤
1

𝑘 𝛼
𝑑(𝑎2𝑛, 𝑎2𝑛+1).  

Let 𝜏 =
1

𝑘 𝛼
< 1.  Then from the above inequality, we have 

𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤ 𝜏 𝑑(𝑎2𝑛, 𝑎2𝑛+1). 

Also, from (2.11),  

𝑑(𝑎2𝑛, 𝑎2𝑛+1) ≤ 𝜏 𝑑(𝑎2𝑛−1, 𝑎2𝑛). 

So, 

𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤ 𝜏
2 𝑑(𝑎2𝑛−1, 𝑎2𝑛). 

From this, we get 

𝑑(𝑎𝑛, 𝑎𝑛+1) ≤ 𝜏𝑛 𝑑(𝑎0, 𝑎1). 

For 𝑗 > 𝑖,  

𝑑(𝑎𝑖 , 𝑎𝑗) ≤ 𝑠 𝑑(𝑎𝑖 , 𝑎𝑖+1) + 𝑠
2 𝑑(𝑎𝑖+1, 𝑎𝑖+2) + ⋯+ 𝑠𝑗−𝑖 𝑑(𝑎𝑗−1, 𝑎𝑗) 

≤ 𝑠 𝜏𝑖𝑑(𝑎0, 𝑎1) + 𝑠
2𝜏𝑖+1𝑑(𝑎0, 𝑎1) + ⋯+ 𝑠

𝑗−𝑖𝜏𝑗−1𝑑(𝑎0, 𝑎1) 

≤ [𝑠 𝜏𝑖 + 𝑠2𝜏𝑖+1 +⋯]𝑑(𝑎0, 𝑎1) 

= 𝑠 𝜏𝑖[1 + 𝑠𝜏 + (𝑠𝜏)2 +⋯ ]𝑑(𝑎0, 𝑎1) 

=
𝑠 𝜏𝑖

1 − 𝑠𝜏
 𝑑(𝑎0, 𝑎1) → 0 as 𝑖, 𝑗 → ∞ 



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Therefore {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑢 ∈  𝑋 such that 

lim
𝑛→∞

𝑎𝑛 = 𝑢.  Since, 𝑆 and 𝑇 are onto, we can find 𝑝, 𝑞 ∈  𝑋 such that 𝑆𝑝 = 𝑇𝑞 = 𝑢.   

Now, for all 𝑛 ∈  ℕ. 

𝑑(𝑢, 𝑎2𝑛+1) = 𝑑(𝑆𝑝, 𝑇𝑎2𝑛+2) ≥  𝑘 𝛼 𝑑(𝑝, 𝑎2𝑛+2). 

Taking limit superior as 𝑛 → ∞, and using Lemma 1.3, we get 

1

𝑠
𝑑(𝑝, 𝑢) ≤

𝑘 𝛼

𝑠
lim
𝑛→∞

sup𝑑(𝑝, 𝑎2𝑛+2) ≤ lim
𝑛→∞

sup𝑑(𝑢, 𝑎2𝑛+1) ≤ 𝑠 𝑑(𝑢, 𝑢). 

From Lemma. 1.4, we get 𝑑(𝑝, 𝑢) = 0 and similarly 𝑑(𝑢, 𝑝) = 0. 

Thus, 𝑑(𝑝, 𝑢) = 𝑑(𝑢, 𝑝) = 0.  So, 𝑝 = 𝑢. 

Uniqueness.  Let 𝑣(≠ 𝑢) be another common fixed point of 𝑆 and 𝑇.  Then 

𝑑(𝑢, 𝑣) = 𝑑(𝑆𝑢, 𝑇𝑣) ≥  𝑘 𝛼 𝑑(𝑢, 𝑣), i. e. , (1 − 𝑘 𝛼)𝑑(𝑢, 𝑣) ≤ 0 

which gives us 𝑑(𝑢, 𝑣) = 0.  Similarly, we can prove that 𝑑(𝑣, 𝑢) = 0, and that 𝑢 = 𝑣. 

Case (ii).  𝜃(𝑎, 𝑏) = 𝛽1  
𝑑(𝑆𝑎,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝛽2𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑆𝑏) ≥ 𝑘 [𝛽1  
𝑑(𝑆𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏)]                                                                              (2.14) 

Now, using (2.12) and (2.14), we get 

𝑑(𝑎2𝑛, 𝑎2𝑛+1) = 𝑑(𝑆𝑎2𝑛+1, 𝑇𝑎2𝑛+2) ≥  𝑘𝛽1  
𝑑(𝑎2𝑛, 𝑎2𝑛+1)𝑑(𝑎2𝑛+1, 𝑎2𝑛+2)

𝑑(𝑎2𝑛+1, 𝑎2𝑛+2)
+ 𝑘𝛽2𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

≥  𝑘𝛽2𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

i. e. , 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤
1

𝑘𝛽2
𝑑(𝑎2𝑛, 𝑎2𝑛+1) which implies that 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤ 𝜗 𝑑(𝑎2𝑛, 𝑎2𝑛+1), 

where  𝜗 < 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, which 

converges to some 𝑢 ∈  𝑋, which can be shown to be unique common fixed point of 𝑆 and 𝑇. 

Case (iii).  𝜃(𝑎, 𝑏) = 𝛾1𝑑(𝑆𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏) + 𝛾3 𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑆𝑏) ≥ 𝑘[𝛾1𝑑(𝑆𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏) + 𝛾3 𝑑(𝑎, 𝑏)]                                                                      (2.15) 

Now, using (2.12) and (2.15), we get 

𝑑(𝑎2𝑛, 𝑎2𝑛+1) = 𝑑(𝑆𝑎2𝑛+1, 𝑇𝑎2𝑛+2) 

≥  𝑘𝛾1𝑑(𝑎2𝑛, 𝑎2𝑛+1) + 𝑘𝛾2 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) + 𝑘𝛾3 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

 ≥ 𝑘[𝛾2 + 𝛾3]𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

 

i.e., 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤
1

𝑘[𝛾2+𝛾3]
𝑑(𝑎2𝑛, 𝑎2𝑛+1) implies that 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤ 𝜌 𝑑(𝑎2𝑛, 𝑎2𝑛+1), 

where  𝜌 < 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, and that 

converges to some 𝑢 ∈  𝑋, which is a unique common fixed point of 𝑆 and 𝑇. 



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Case (iv).  𝜃(𝑎, 𝑏) = 𝛿1𝑑(𝑆𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑆𝑏) ≥ 𝑘[𝛿1𝑑(𝑆𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏)]                                                                      (2.16) 

Now, using (2.12) and (2.16), we get 

𝑑(𝑎2𝑛, 𝑎2𝑛+1) = 𝑑(𝑆𝑎2𝑛+1, 𝑇𝑎2𝑛+2) 

≥  𝑘𝛿1𝑑(𝑎2𝑛, 𝑎2𝑛+2) + 𝑘𝛿2 𝑑(𝑎2𝑛+1, 𝑎2𝑛+1) + 𝑘𝛿3 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

 ≥ 𝑘𝛿3𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

i.e., 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤
1

𝑘𝛿3
𝑑(𝑎2𝑛, 𝑎2𝑛+1) implies that 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤ 𝜔 𝑑(𝑎2𝑛, 𝑎2𝑛+1), where  

𝜔 < 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, and that converges 

to some 𝑢 ∈  𝑋, which is a unique common fixed point of 𝑆 and 𝑇. 

Case (v).  𝜃(𝑎, 𝑏) = 𝜆1
𝑑(𝑆𝑎,𝑏)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝜆2

𝑑(𝑇𝑏,𝑎)𝑑(𝑇𝑏,𝑏)

𝑑(𝑎,𝑏)
+ 𝜆3𝑑(𝑎, 𝑏), then 

𝑑(𝑇𝑎, 𝑆𝑏) ≥ 𝑘 [𝜆1
𝑑(𝑆𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆2

𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)]                                    (2.17) 

Now, using (2.12) and (2.17), we get 

𝑑(𝑎2𝑛, 𝑎2𝑛+1) = 𝑑(𝑆𝑎2𝑛+1, 𝑇𝑎2𝑛+2) 

≥  𝑘𝜆1
𝑑(𝑎2𝑛, 𝑎2𝑛+2)𝑑(𝑎2𝑛+1, 𝑎2𝑛+2)

𝑑(𝑎2𝑛+1, 𝑎2𝑛+2)
+ 𝑘𝜆3 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

 ≥ 𝑘𝜆3𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) 

i.e., 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤
1

𝑘𝜆3
𝑑(𝑎2𝑛, 𝑎2𝑛+1) which implies that 𝑑(𝑎2𝑛+1, 𝑎2𝑛+2) ≤ 𝜑 𝑑(𝑎2𝑛, 𝑎2𝑛+1), 

where  𝜑 < 1.  Proceeding similar to Case (i), we get {𝑎𝑛} is a 𝑏-Cauchy sequence in 𝑋, and that 

converges to some 𝑢 ∈  𝑋, which is a unique common fixed point of 𝑆 and 𝑇. 

Example 2.9.  Let 𝑋 = ℝ+.  We define 𝑑: 𝑋 ×  𝑋 → ℝ+ by 𝑑(𝑎, 𝑏) = |𝑎 − 𝑏|2 + |𝑎|2.  Then 

clearly, (𝑋, 𝑑) is a complete 𝑑𝑝 𝑏-metric space with 𝑠 = 2. We define self-mappings 𝑆, 𝑇: 𝑋 →  𝑋 by 

𝑆(𝑎) = 𝑎(𝑎 + 2)and 𝑇(𝑎) = 2𝑎, for all 𝑎 ∈  𝑋. 

We take 𝑘 =
3

2
, 𝛼 = 𝛽2 = 𝛾3 = 𝛿3 = 𝜆3 = 1, 𝛽1 = 𝛾1 = 𝛾2 = 𝛿1 = 𝛿2 = 𝜆1 = 𝜆2 = 0. 

Without loss of generality we assume that 𝑎 ≥  𝑏.  We consider 

𝑑(𝑆𝑎, 𝑇𝑏) = |𝑆𝑎 − 𝑇𝑏|2 + |𝑆𝑎|2 

= (𝑎2 + 2𝑎 − 2𝑏2)2 + (𝑎2 + 2𝑎)2 

≥ (2𝑎 − 2𝑏)2 + 2𝑎2  

≥
3

2
[(𝑎 − 𝑏)2 + 𝑎2] 



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= 𝑘 min {𝛼 𝑑(𝑎, 𝑏); 𝛽1  
𝑑(𝑆𝑎, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝛽2𝑑(𝑎, 𝑏); 𝛾1𝑑(𝑆𝑎, 𝑎) + 𝛾2 𝑑(𝑇𝑏, 𝑏)

+ 𝛾3 𝑑(𝑎, 𝑏); 𝛿1𝑑(𝑆𝑎, 𝑏) + 𝛿2𝑑(𝑇𝑏, 𝑎) + 𝛿3𝑑(𝑎, 𝑏); 𝜆1
𝑑(𝑆𝑎, 𝑏)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)

+ 𝜆2
𝑑(𝑇𝑏, 𝑎)𝑑(𝑇𝑏, 𝑏)

𝑑(𝑎, 𝑏)
+ 𝜆3𝑑(𝑎, 𝑏)} 

 

 

Table 2 and Figure 3, illustrates the condition (2.10) and (2.11) of Theorem 2.8, with blue line 

representing the left part of the condition and red line representing the right part of the condition. 

Thus, all the conditions of Theorem 2.8 are satisfied. So,  𝑆 and 𝑇 have a unique common fixed point 

in 𝑋 which is clearly 0 here. 



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Remark  2.10.  Theorem 2.8 and Example 2.9 extend and generalize Theorem 1.6 to 𝑑𝑞 𝑏-metric 

spaces. 

Corollary 2.11.  Let (𝑋, 𝑑) be a complete 𝑑𝑞 𝑏-metric space and 𝑆, 𝑇 be two onto self-mappings on 

𝑋 such that 𝑑(𝑆𝑎, 𝑇 𝑏) ≥  𝑘 𝑑(𝑎, 𝑏), for all 𝑎, 𝑏 ∈  𝑋 with 𝑘 >  1. Then 𝑆 and 𝑇 have a unique 

common fixed point in 𝑋. 

3. Application to nonlinear integral equations 

Let Ω= 𝐶[𝑎, 𝑏] be a set of real valued continuous functions on [𝑎, 𝑏],where [𝑎, 𝑏] is closed and 

bounded integral in ℝ. We define 𝑑: Ω × Ω → ℝ+ by 𝑑(𝜉, 𝜂) = max
𝑎≤𝑡≤𝑏

{|𝜉(𝑡) − 𝜂(𝑡)|𝑝 + |𝜉(𝑡)|𝑝}, 

where 𝑝 > 1 a real number, for all 𝜉, 𝜂 𝜖 Ω. Therefore (Ω, 𝑑) is a complete 𝑏-metric space with     

𝑠 = 2𝑝. Many author's studied unique solution of a nonlinear integral equations [4-7].  In this section, 

we obtain the existence of unique solution of a nonlinear integral equation of Fredholm type defined 

by 

𝜉(𝑡) = 𝑓(𝑡) + 𝜇 ∫ ℳ(𝑡, 𝑟, 𝜉(𝑟))𝑑𝑟
𝑏

𝑎
     (3.1) 

Where   𝜉 𝜖 Ω is the unknown function, 𝜇 in ℝ, 𝑡, 𝑟 𝜖 [𝑎, 𝑏], ℳ:[𝑎, 𝑏]  × [𝑎, 𝑏] × ℝ → ℝ and 

𝑓: [𝑎, 𝑏] → ℝ are continuous functions.  Let ℱ:Ω → Ω  be a mapping defined by 

ℱ(𝜉(𝑡)) = 𝑓(𝑡) + 𝜇 ∫ ℳ(𝑡, 𝑟, 𝜉(𝑟))𝑑𝑟
𝑏

𝑎
     (3.2) 

Theorem 3.1.  Let ℱ:Ω → Ω  be a mapping defined by (3.2) and there exists a constant 𝐾 > 1 such 

that for all 𝑡, 𝑟 𝜖 [𝑎, 𝑏] and 𝜉1, 𝜉2 𝜖 Ω with |𝜇| ≥ 1, the following condition is satisfied: 

|∫ [ℳ(𝑡, 𝑟, 𝜉1(𝑟)) −ℳ(𝑡, 𝑟, 𝜉2(𝑟))]𝑑𝑟
𝑏

𝑎
|
𝑝

≥ 𝐾|𝜉1(𝑡) − 𝜉2(𝑡)|
𝑝 + 𝐾|𝜉1(𝑡)|

𝑝.  Then the system of 

nonlinear integral equations (3.1) has a unique solution in Ω . 

Proof.  Let 𝜉1, 𝜉2 𝜖 Ω and for all 𝑡 𝜖 [𝑎, 𝑏], we have  

𝑑(ℱ𝜉1, ℱ𝜉2) = max
𝑎≤𝑡≤𝑏

{|ℱ𝜉1(𝑡) − ℱ𝜉2(𝑡)|
𝑝 + |ℱ𝜉1(𝑡)|

𝑝} 

≥ |𝜇|𝑝max
𝑎≤𝑡≤𝑏

{|∫ ℳ(𝑡, 𝑟, 𝜉1(𝑟))𝑑𝑟
𝑏

𝑎

−∫ ℳ(𝑡, 𝑟, 𝜉2(𝑟))𝑑𝑟
𝑏

𝑎

|

𝑝

+ |∫ ℳ(𝑡, 𝑟, 𝜉1(𝑟))𝑑𝑟
𝑏

𝑎

|

𝑝

} 

≥ |𝜇|𝑝max
𝑎≤𝑡≤𝑏

{|∫ [ℳ(𝑡, 𝑟, 𝜉1(𝑟)) −ℳ(𝑡, 𝑟, 𝜉2(𝑟))]𝑑𝑟
𝑏

𝑎

|

𝑝

} 

≥ 𝐾max
𝑎≤𝑡≤𝑏

{|𝜉1(𝑡) − 𝜉2(𝑡)|
𝑝 + 𝐾|𝜉1(𝑡)|

𝑝} 

= 𝐾𝑑(𝜉1, 𝜉2) 

which implies that  

𝑑(ℱ𝜉1, ℱ𝜉2) ≥  𝐾𝑑(𝜉1, 𝜉2). 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

404 
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Therefore, all the conditions of Corollary 2.3 are satisfied, and hence ℱ has a unique solution for 

nonlinear integral equations defined in (3.1). 

4.  Conclusion and future work 

In this paper, we studied fixed point results for expansive mappings in dislocated quasi-𝑏-metric 

spaces.   Using similar approaches, it can be studied new fixed point results on metric and some 

generalized metric spaces.  The investigation of certain circumstances to exclude the identity map of 

𝑋 from Theorem 2.1 and Theorem 2.8 and related results is a worthwhile problem for future effort. 

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