# Solving Integral Equations by Common Fixed Point Theorems on Complex Partialb-Metric Spaces

Authors: Arul Joseph Gnanaprakasam; Salah Mahmoud Boulaaras; Gunaseelan Mani; Mohamed Abdalla; Asma Alharbi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we prove some common fixed point theorems for rational contraction mapping on complex partialb-metric space. The presented results generalize and expand some of the literature’s well-known results. We also explore some of the applications of our key results.


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## Body

## 1. Introduction

Introduced in 1989 by Bakhtin [1] and Czerwick [2], the concept of b-metric spaces provided a framework to extend the results already known in the classical setting of metric spaces. The concept of complex valued metric spaces was introduced in 2011 by Azam et al. [3] and given some common fixed point theorems under the condition of contraction. Rao et al. [4] introduced the definition of complex valued b-metric spaces in 2013 and provided a scheme to expand the results, as well as proved a common fixed point theorem under contraction. In 2017, Dhivya and Marudai [5] introduced the concept of complex partial metric space and suggested a plan to expand the results, as well as proved common fixed point theorems under the rational expression contraction condition. Gunaseelan [6, 7] presented the concept of complex partial b-metric space in 2019, as well as proved the fixed point theorem under the contractive condition. Many researchers have studied some intriguing concepts and applications and have shown significant results [7–23]. In this paper, we prove some common fixed point theorems for rational contraction mapping on complex partial b-metric space.


## 2. Preliminaries

LetC be the set of complex numbers and 1,2,3∈C. Define a partial order ≤ on C as follows: 1≤2 if and only if R1≤R2 and I1≤I2.Consequently, one can infer that1≤2 if one of the following conditions is satisfied:
(i)
R1=R2, I1<I2(ii)
R1<R2, I1=I2(iii)
R1<R2, I1<I2(iv)
R1=R2, I1=I2In particular, we write1⋨2 if 1≠2 and one of (i), (ii), and (iii) is satisfied and we write 1≺2 if only (iii) is satisfied. Notice that
(a)
If0≤1⋨2, then 1<2(b)
If1≤2 and 2≺3, then 1≺3(c)
Ifη,γ∈ℝ and η≤γ, then η1≤γ1 for all 0≤1∈CHere,C+=ℵ,y∣ℵ,y≥0,ℵ,y∈ℝ+ denote the set of nonnegative complex numbers.Now, let us recall some basic concepts and notations, which will be used in the sequel.Definition 1 [6].
A complex partialb-metric on a nonvoid set W is a function φcb:W×W⟶ℂ+ such that for all ℵ,ϑ,Z∈W,
(i)
0≤φcbℵ,ℵ≤φcbℵ,ϑsmallself‐distances(ii)
φcbℵ,ϑ=φcbϑ,ℵsymmetry(iii)
φcbℵ,ℵ=φcbℵ,ϑ=φcbϑ,ϑ⇔ℵ=ϑequality(iv)
∃ a real number s≥1 and s is an independent of ℵ,ϑ,Z such that φcbℵ,ϑ≤sφcbℵ,Z+φcbZ,ϑ−φcbZ,Ztriangularity
A complex partialb-metric space is a pair W,φcb such that W is a nonvoid set and φcb is the complex partial b-metric on W. The number s is called the coefficient of W,φcb.Definition 2 [6].
LetW,φcb be a complex partial b-metric space with coefficient s. Let ℵn be any sequence in W and ℵ∈W. Then,
(1)
The sequenceℵn is said to converge to ℵ, if limn⟶∞φcbℵn,ℵ=φcbℵ,ℵ(2)
The sequenceℵn is said to be Cauchy sequence in W,φcb if limn,m⟶∞φcbℵn,ℵm exists and is finite(3)
W,φcb is said to be a complete complex partial b-metric space if for every Cauchy sequence ℵn in W, there exists ℵ∈W such that limn,m⟶∞φcbℵn,ℵm=limn⟶∞φcbℵn,ℵ=φcbℵ,ℵ.Definition 3.
LetS and T be self-mappings of nonempty set W. A point ℵ∈W is called a common fixed point of S and T if ℵ=Sℵ=Tℵ.In 2019, Gunaseelan [6] proved some fixed point theorems on complex partial b-metric space as follows.Theorem 4.
LetW,φcb be any complete complex partial b-metric space with coefficient s≥1 and S:W⟶W be a mapping satisfying
(1)φcbSℵ,Sϑ≤λmaxφcbℵ,ϑ,φcbℵ,Sℵ,φcbϑ,Sϑ,for all ℵ,ϑ∈W, where λ∈0,1/s. Then, S has a unique fixed point ℵ∗∈W and φcbℵ∗,ℵ∗=0.Inspired by the study made by Gunaseelan [6], here, we prove some common fixed point theorems for rational contraction mapping on complex partial b-metric space with an application.


## 3. Main Results

In this section, we will give our main result of this paper, where some common fixed point theorems for rational contraction mapping on complex partialb-metric space are given.Theorem 5.
LetW,φcb be a complete complex partial b-metric space with the coefficient s≥1 and S,T:W⟶W be mappings satisfying
(2)φcbSℵ,Tϑ≤á1φcbℵ,ϑ+á2φcbℵ,Sℵφcbϑ,Tϑ1+φcbℵ,Tϑ+φcbℵ,ϑ,for all ℵ,ϑ∈W, where á1,á2 are nonnegative reals with á1+sá2<1. Then, S and T have a unique common fixed point in W.Proof.
Letℵ0 be arbitrary point in W, and define a sequence ℵn in W such that
(3)ℵ2n+1=Sℵ2n,ℵ2n+2=Tℵ2n+1,∀n≥0.
Next, show that the sequenceℵn is Cauchy. By using (3), we get
(4)φcbℵ2n+1,ℵ2n+2=φcbSℵ2n,Tℵ2n+1≤á1φcbℵ2n,ℵ2n+1+á2φcbℵ2n,Sℵ2nφcbℵ2n+1,Tℵ2n+11+φcbℵ2n,Tℵ2n+1+φcbℵ2n,ℵ2n+1,so that
(5)φcbℵ2n+1,ℵ2n+2≤á1φcbℵ2n,ℵ2n+1+á2φcbℵ2n,ℵ2n+1φcbℵ2n+1,ℵ2n+2φcbℵ2n,ℵ2n+2+φcbℵ2n,ℵ2n+1.
By the hypothesis of theorem, we get(6)φcbℵ2n+1,ℵ2n+2≤sφcbℵ2n+1,ℵ2n+φcbℵ2n,ℵ2n+2−φcbℵ2n,ℵ2nφcbℵ2n+1,ℵ2n+2≤sφcbℵ2n+1,ℵ2n+φcbℵ2n,ℵ2n+2.
Hence,(7)φcbℵ2n+1,ℵ2n+2≤á1φcbℵ2n,ℵ2n+1+sá2φcbℵ2n,ℵ2n+1=á1+sá2φcbℵ2n,ℵ2n+1φcbℵ2n+1,ℵ2n+2≤á1+sá2φcbℵ2n,ℵ2n+1.
Similarly,(8)φcbℵ2n+2,ℵ2n+3≤á1+sá2φcbℵ2n+1,ℵ2n+2,since á1+sá2<1. Therefore, with δ=á1+sá2<1 and for all n≥0, consequently, we have
(9)φcbℵ2n+1,ℵ2n+2≤δφcbℵ2n,ℵ2n+1≤δ2φcbℵ2n−1,ℵ2n⋮≤δ2n+1φcbℵ0,ℵ1.
That is,(10)φcbℵn+1,ℵn+2≤δφcbℵn,ℵn+1≤δ2φcbℵn−1,ℵn⋮≤δn+1φcbℵ0,ℵ1.
For anym>n,m,n∈ℕ, we have
(11)φcbℵn,ℵm≤sφcbℵn,ℵn+1+φcbℵn+1,ℵm−φcbℵn+1,ℵn+1≤sφcbℵn,ℵn+1+φcbℵn+1,ℵm≤sφcbℵn,ℵn+1+s2φcbℵn+1,ℵn+2+s2φcbℵn+2,ℵm−φcbℵn+2,ℵn+2≤sφcbℵn,ℵn+1+s2φcbℵn+1,ℵn+2+s2φcbℵn+2,ℵm≤sφcbℵn,ℵn+1+s2φcbℵn+1,ℵn+2+s3φcbℵn+2,ℵn+3+s3φcbℵn+3,ℵm−φcbℵn+3,ℵn+3≤sφcbℵn,ℵn+1+s2φcbℵn+1,ℵn+2+s3φcbℵn+2,ℵn+3+s3φcbℵn+3,ℵm⋮≤sφcbℵn,ℵn+1+s2φcbℵn+1,ℵn+2+s3φcbℵn+2,ℵn+3+⋯sm−n−2φcbℵm−3,ℵm−2+sm−n−1φcbℵm−2,ℵm−1+sm−nφcbℵm−1,ℵm.
From (10), we get
(12)φcbℵn,ℵm≤sδnφcbℵ0,ℵ1+s2δn+1φcbℵ0,ℵ1+s3δn+2φcbℵ0,ℵ1+⋯+sm−n−2δm−3φcbℵ0,ℵ1+sm−n−1δm−2φcbℵ0,ℵ1+sm−nδm−1φcbℵ0,ℵ1=∑i=1m−nsiδi+n−1φcbℵ0,ℵ1.
Hence,(13)φcbℵn,ℵm≤∑i=1m−nsi+n−1δi+n−1φcbℵ0,ℵ1=∑t=nm−1stδtφcbℵ0,ℵ1≤∑t=n∞sδtφcbℵ0,ℵ1=sδn1−sδφcbℵ0,ℵ1,
and hence,(14)φcbℵn,ℵm≤sδn1−sδφcbℵ0,ℵ1⟶0asm,n⟶∞.
Thus,ℵn is a Cauchy sequence in W. Since W is complete, there exists some u∈W such that ℵn⟶u as n⟶∞ and
(15)φcbu,u=limn⟶∞φcbu,ℵn=limn⟶∞φcbℵn,ℵn=0.
Assume on the contrary that there existsz∈W such that
(16)φcbu,Su=z>0.
By using the triangular inequality and (2), we obtain
(17)z=φcbu,Su≤sφcbu,ℵ2n+2+φcbℵ2n+2,Su−φcbℵ2n+2,ℵ2n+2≤sφcbu,ℵ2n+2+φcbℵ2n+2,Su=sφcbu,ℵ2n+2+φcbTℵ2n+1,Su≤sφcbu,ℵ2n+2+sá1φcbu,ℵ2n+1+sá2φcbu,Suφcbℵ2n+1,Tℵ2n+11+φcbu,Tℵ2n+1+φcbu,ℵ2n+1=sφcbu,ℵ2n+2+sá1φcbu,ℵ2n+1+sá2φcbu,Suφcbℵ2n+1,ℵ2n+21+φcbu,ℵ2n+2+φcbu,ℵ2n+1,which implies that
(18)z=φcbu,Su≤sφcbu,ℵ2n+2+sá1φcbu,ℵ2n+1+sá2φcbu,Suφcbℵ2n+1,ℵ2n+21+φcbu,ℵ2n+2∣+φcbu,ℵ2n+1.
Asn⟶∞ in (18), we obtain that z=φcbu,Su≤0, a contradiction with (16). Therefore z=0. Hence, Su=u. Similarly, we obtain Tu=u.
Assume thatu∗ is another common fixed point of S and T. Then,
(19)φcbu,u∗=φcbSu,Tu∗≤á1φcbu,u∗+á2φcbu,Suφcbu∗,Tu∗1+φcbu,Tu∗+φcbu,u∗,so that
(20)φcbu,u∗≤á1φcbu,u∗+á2φcbu,Suφcbu∗,Tu∗1+φcbu,Tu∗+φcbu,u∗≤á1φcbu,u∗.
Hence,u=u∗, which proves the uniqueness. This completes the proof of the theorem.☐Theorem 6.
LetW,φcb be a complete complex partial b-metric space with the coefficient s≥1 and S,T:W⟶W be mappings satisfying
(21)φcbSℵ,Tϑ≤á1φcbℵ,ϑ+á2φcbℵ,Tϑ+á3φcbℵ,Sℵ+φcbϑ,Tϑ,for all ℵ,ϑ∈W, where á1,á2, and á3 are nonnegative reals with á1+2sá2+2á3<1. Then, S and T have a unique common fixed point in W.Proof.
Letℵ0 be arbitrary point in W, and define a sequence ℵn in W such that
(22)ℵ2n+1=Sℵ2n,ℵ2n+2=Tℵ2n+1,∀n≥0.
Next, show that the sequenceℵn is Cauchy. By using (22), we get
(23)φcbℵ2n+1,ℵ2n+2=φcbSℵ2n,Tℵ2n+1≤á1φcbℵ2n,ℵ2n+1+á2φcbℵ2n,Tℵ2n+1+á3φcbℵ2n,Sℵ2n+φcbℵ2n+1,Tℵ2n+1=á1φcbℵ2n,ℵ2n+1+á2φcbℵ2n,Tℵ2n+1+á3φcbℵ2n,ℵ2n+1+φcbℵ2n+1,ℵ2n+2,so that
(24)∣φcbℵ2n+1,ℵ2n+2∣≤á1∣φcbℵ2n,ℵ2n+1∣+á2∣φcbℵ2n,ℵ2n+2∣+á3∣φcbℵ2n,ℵ2n+1∣+∣φcbℵ2n+1,ℵ2n+2∣.
By the notion of complex partialb-metric space, we get
(25)φcbℵ2n,ℵ2n+2≤sφcbℵ2n,ℵ2n+1+φcbℵ2n+1,ℵ2n+2−φcbℵ2n+1,ℵ2n+1φcbℵ2n,ℵ2n+2≤sφcbℵ2n,ℵ2n+1+φcbℵ2n+1,ℵ2n+2.
Hence,(26)φcbℵ2n+1,ℵ2n+2≤á1φcbℵ2n,ℵ2n+1+sá2φcbℵ2n,ℵ2n+1+φcbℵ2n+1,ℵ2n+2+á3φcbℵ2n,ℵ2n+1+φcbℵ2n+1,ℵ2n+2≤á1+sá2+á31−sá2−á3φcbℵ2n,ℵ2n+1.
Similarly,(27)φcbℵ2n+2,ℵ2n+3≤á1+sá2+á31−sá2−á3φcbℵ2n+1,ℵ2n+2.
Setδ=á1+sá2+á3/1−sá2−á3. Since á1+2sá2+2á3<1 and for all n≥0, consequently, we have
(28)φcbℵ2n+1,ℵ2n+2≤δφcbℵ2n,ℵ2n+1≤δ2φcbℵ2n−1,ℵ2n⋮≤δ2n+1φcbℵ0,ℵ1.
That is,(29)φcbℵn+1,ℵn+2≤δφcbℵn,ℵn+1≤δ2φcbℵn−1,ℵn⋮≤δn+1φcbℵ0,ℵ1.
For anym>n,m,n∈ℕ, we have
(30)∣φcbℵn,ℵm∣≤s∣φcbℵn,ℵn+1∣+∣φcbℵn+1,ℵm∣−∣φcbℵn+1,ℵn+1∣≤s∣φcbℵn,ℵn+1∣+∣φcbℵn+1,ℵm∣≤s∣φcbℵn,ℵn+1∣+s2∣φcbℵn+1,ℵn+2∣+s2∣φcbℵn+2,ℵm∣−∣φcbℵn+2,ℵn+2∣≤s∣φcbℵn,ℵn+1∣+s2∣φcbℵn+1,ℵn+2∣+s2∣φcbℵn+2,ℵm∣≤s∣φcbℵn,ℵn+1∣+s2∣φcbℵn+1,ℵn+2∣+s3∣φcbℵn+2,ℵn+3∣+s3∣φcbℵn+3,ℵm∣−∣φcbℵn+3,ℵn+3∣≤s∣φcbℵn,ℵn+1∣+s2∣φcbℵn+1,ℵn+2∣+s3∣φcbℵn+2,ℵn+3∣+s3φcbℵn+3,ℵm⋮≤s∣φcbℵn,ℵn+1∣+s2∣φcbℵn+1,ℵn+2∣+s3∣φcbℵn+2,ℵn+3∣+⋯sm−n−2∣φcbℵm−3,ℵm−2∣+sm−n−1∣φcbℵm−2,ℵm−1∣+sm−nφcbℵm−1,ℵm.
From (29), we get
(31)φcbℵn,ℵm≤sδnφcbℵ0,ℵ1+s2δn+1φcbℵ0,ℵ1+s3δn+2φcbℵ0,ℵ1+⋯+sm−n−2δm−3φcbℵ0,ℵ1+sm−n−1δm−2φcbℵ0,ℵ1+sm−nδm−1φcbℵ0,ℵ1=∑i=1m−nsiδi+n−1φcbℵ0,ℵ1.
Hence,(32)∣φcbℵn,ℵm∣≤∑i=1m−nsi+n−1δi+n−1∣φcbℵ0,ℵ1∣=∑t=nm−1stδt∣φcbℵ0,ℵ1∣≤∑t=n∞sδtφcbℵ0,ℵ1=sδn1−sδφcbℵ0,ℵ1,and hence,
(33)φcbℵn,ℵm≤sδn1−sδφcbℵ0,ℵ1⟶0asm,n⟶∞.
Thus,ℵn is a Cauchy sequence in W. Since W is complete, there exists some u∈W such that ℵn⟶u as n⟶∞ and
(34)φcbu,u=limn⟶∞φcbu,ℵn=limn⟶∞φcbℵn,ℵn=0.
Assume on the contrary that there existsz∈W such that
(35)∣φcbu,Su∣=∣z∣>0.
By using the triangular inequality and (21), we obtain
(36)z=φcbu,Su≤sφcbu,ℵ2n+2+φcbℵ2n+2,Su−φcbℵ2n+2,ℵ2n+2≤sφcbu,ℵ2n+2+φcbℵ2n+2,Su=sφcbu,ℵ2n+2+φcbSu,Tℵ2n+1≤sφcbu,ℵ2n+2+sá1φcbu,ℵ2n+1+sá2φcbu,Tℵ2n+1+sá3φcbu,Su+φcbℵ2n+1,Tℵ2n+1=sφcbu,ℵ2n+2+sá1φcbu,ℵ2n+1+sá2φcbu,ℵ2n+2+sá3z+φcbℵ2n+1,ℵ2n+2,which implies that
(37)∣z∣=∣φcbu,Su∣≤s∣φcbu,ℵ2n+2∣+sá1∣φcbu,ℵ2n+1∣+sá2∣φcbu,ℵ2n+2∣+sá3∣z∣+∣φcbℵ2n+1,ℵ2n+2∣.
Asn⟶∞ in (37), we obtain that ∣z∣=∣φcbu,Su∣≤0, a contradiction with (35). Therefore, ∣z∣=0. Hence, Su=u. Similarly, we obtain Tu=u.
Assume thatu∗ is another common fixed point of S and T. Then,
(38)φcbu,u∗=φcbSu,Tu∗≤á1φcbu,u∗+á2φcbu,Tu∗+á3φcbu,Su+φcbu∗,Tu∗,which implies that ∣φcbu,u∗∣≤0, a contradiction. So u=u∗, which proves the uniqueness.☐Example 1.
LetW=1,2,3,4 be endowed with the order ℵ≤ϑ if and only if ϑ≤ℵ. Then, ≤ is a partial order in W. Define the complex partial b-metric space φcb:W×W⟶ℂ+ as follows (Table 1):
It is easy to verify thatW,φcb is a complete complex partial b-metric space with the coefficient s≥1 for x∈0,π/2. Define S,T:W⟶W by Sℵ=1,
(39)Tℵ=1ifℵ∈1,2,3,2ifℵ=4.
Letλ1=1/9 and λ2=1/8; we consider the following cases:
(1)
Ifℵ=1 and ϑ∈W−4, then Sℵ=Tϑ=1 and the conditions of Theorem 5 are satisfied(2)
Ifℵ=1 and ϑ=4, then Sℵ=1, Tϑ=2,(40)φcbSℵ,Tϑ=e2ix≤9λ1ei2x=λ1φcbℵ,ϑ+λ2φcbℵ,Sℵφcbϑ,Tϑ1+φcbℵ,Tϑ+φcbℵ,ϑ(3)
Ifℵ=2 and ϑ=4, then Sℵ=1, Tϑ=2(41)φcbSℵ,Tϑ=e2ix≤1+18ei2x=λ19ei2x+λ2ei2x9ei2x1+9ei2x=λ1φcbℵ,ϑ+λ2φcbℵ,Sℵφcbϑ,Tϑ1+φcbℵ,Tϑ+φcbℵ,ϑ(4)
Ifℵ=3 and ϑ=4, then Sℵ=1, Tϑ=2(42)φcbSℵ,Tϑ=e2ix≤1+980ei2x=λ19ei2x+λ2ei2x9ei2x1+10ei2x=λ1φcbℵ,ϑ+λ2φcbℵ,Sℵφcbϑ,Tϑ1+φcbℵ,Tϑ+φcbℵ,ϑ(5)
Ifℵ=4 and ϑ=4, then Sℵ=1, Tϑ=2(43)φcbSℵ,Tϑ=e2ix≤1+916ei2x=λ19ei2x+λ29ei2x9ei2x1+9ei2x+9ei2x=λ1φcbℵ,ϑ+λ2φcbℵ,Sℵφcbϑ,Tϑ1+φcbℵ,Tϑ+φcbℵ,ϑ
Moreover forá1+sá2<1, the conditions of Theorem 5 are satisfied. Therefore, 1 is the unique common fixed point of S and T.Table 1ℵ,ϑφcbℵ,ϑ(1,1), (2,2)0(1,2), (2,1), (1,3), (3,1), (2,3), (3,2), (3,3)e2ix(1,4), (4,1), (2,4), (4,2), (3,4), (4,3), (4,4)9e2ix


## 4. Application

Consider the following systems of integral equations:(44)ws=bs+∫abT1s,p,wpdp,(45)ws=bs+∫abT2s,p,wpdp,where
(1)
[label=](2)
ws is an unknown variable for each s∈J=a,b, b>a≥0(3)
bs is the deterministic free term defined for s∈a,b(4)
T1s,p and T2s,p are deterministic kernels defined for s,p∈J=a,bIn this section, we present an existence theorem for a common solution to (44) and (45) that belongs to W=CJ,ℝn (the set of continuous functions defined on J) by using the obtained result in Theorem 5. We consider the continuous mappings S,T:W⟶W given by
(46)Sws=bs+∫abT1s,p,wpdp,w∈W,s∈J,Tws=bs+∫abT2s,p,wpdp,z∈W,s∈J.Then, the existence of a common solution to the integral equations (44) and (45) is equivalent to the existence of a common fixed point of T1 and T2. It is well known that W, endowed with the metric φcb defined by
(47)φcbw,z=ws−zs2+2,for all w,z∈W, is a complete complex partial b-metric space. W can also be equipped with the partial order ≤ given by
(48)w,z∈W,w≤zifandonlyws≥zs,foralls∈J.Further, let us consider that a system of integral equation as (44) and (45) under the following condition holds:
(1)
T1,T2:J×J×ℝn⟶ℝn are continuous functions satisfying
(49)∣T1s,p,wp−T2s,p,zp∣≤φcbw,zb−aet−2b−a,∀t>0Theorem 7.
LetCJ,ℝn,φcb be a complete complex partial b-metric space; then, the systems (44) and (45) under condition (3) have a unique common solution.Proof.
Forw,z∈CJ,ℝn and s∈J, we define the continuous mappings S,T:W⟶W by
(50)Sws=bs+∫abT1s,p,wpdp,Tws=bs+∫abT2s,p,wpdp.
Then, we have(51)φcbSws,Tzs=Sws−Tzs2+2=∫abT1s,p,wp−T2s,p,zp2dp+2≤∫abφcbw,zb−aet−2b−adp+2=φcbw,zet=λ1φcbw,z=λ1φcbw,z+λ2φcbw,Swφcbz,Tz1+φcbw,Tz+φcbw,z.
Hence, all the conditions of Theorem5 are satisfied for á1+sá2=0<1, with t>0. Therefore, the system of integral equations (44) and (45) has a unique common solution.


## 5. Conclusion

In this paper, we proved some common fixed point theorems for rational contraction mapping on complex partialb-metric space. An illustrative example and application on complex partial b-metric space is given.



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