# Results on Complex Partial b-Metric Space with an Application

Authors: Gunaseelan Mani; Arul Joseph Gnanaprakasam; Ramakrishnan Kalaichelvan; Yaé Ulrich Gaba
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we prove a fixed point theorem in complex partialb-metric space under new contraction mapping. The proved results generalize and extend some of the well-known results in the literature. We also give some applications of our main results.


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## Body

## 1. Introduction

Introduced in 1989 by Bakhtin [1] and Czerwik [2], the concept of b-metric spaces provided a framework to extend the results already known in the classical setting of metric spaces. About two decades later, more precisely in 2011, Azamet al. [3] came up with the notion of complex-valued metric spaces and provided some common fixed point theorems under some contractive conditions. Two years after, it was in [4] Rao et al. discussed for the first time the idea of complex-valued b-metric spaces.It was just very recently, in 2017, that Dhivya and Marudai [5] extended all the preceding results in the setting of complex partial metric spaces making use of a rational type contraction.This was followed by Gunaseelan [6], who introduced the concepts of complex partial b-metric spaces and discussed some results of fixed point theory for self-mappings in these new spaces.Many authors have studied related interesting metric such as structures along with some applications. And, in this line, significant results have been obtained and can be read in [7–23]. In this paper, under new contraction condition, we prove a fixed point theorem in complex partial b-metric space. Although there have been a significant amount of scientific contributions to the theory of partial b-metric space, very few address that of complex valued, and even less, the applicability of complex partial b-metrics in the resolution integral equations. This, however, in one the main contributions of the present work. We begin by recalling basic facts about complex partial b-metric spaces.


## 2. Preliminaries

Letℂ be the set of complex numbers and ϖ1,ϖ2,ϖ3∈ℂ. Define a partial order ≺¯ on ℂ as follows:ϖ1≺¯ϖ2 if and only if Re.partϖ1≤Re.partϖ2, Im.partϖ1≤Im.partϖ2.Consequently, one can infer thatϖ1≺¯ϖ2 if one of the following conditions is satisfied:(i)
Re.partϖ1=Re.partϖ2 and Im.partϖ1<Im.partϖ2(ii)
Re.partϖ1<Re.partϖ2 and Im.partϖ1=Im.partϖ2(iii)
Re.partϖ1<Re.partϖ2 and Im.partϖ1<Im.partϖ2(iv)
Re.partϖ1=Re.partϖ2 and Im.partϖ1=Im.partϖ2In particular, we writeϖ1⋨ϖ2 if ϖ1≠ϖ2 and one of (i), (ii), and (iii) is satisfied, and we write ϖ1≺ϖ2 if only (iii) is satisfied. Notice that(a)
If0≺¯ϖ1⋨ϖ2, then ϖ1<ϖ2(b)
Ifϖ1≺¯ϖ2 and ϖ2≺ϖ3, then ϖ1≺ϖ3(c)
Ifτ,γ∈ℝ and τ≤γ, then τϖ1≺¯γϖ1 for all 0≺¯ϖ1∈ℂDefinition 1.
(see [4]). Let Θ be a nonvoid set and let s≥1 be a given real number. A function d:Θ×Θ⟶ℂ is called a complex-valued b-metric on Θ if, for all e,f,g∈Θ, the following conditions are satisfied:(i)
0≺¯de,f and de,f=0 if and only if e=f(ii)
de,f=df,g(iii)
de,f≺¯sde,g+dg,fThe pairΘ,d is called a complex-valued b-metric space.Here,ℂ+=e,g|e,g∈ℝ+ and ℝ+=e∈ℝ|e≥0 denote the set of nonnegative complex numbers and the set of nonnegative real numbers, respectively. We now give the complex partial metric space.Definition 2.
(see [5]). A complex partial metric on a nonvoid set Θ is a function ξc:Θ×Θ⟶ℂ+ such that, for all α,β,γ∈Θ,(i)
0≺¯ξcα,α≺¯ξcα,βsmall self−distances(ii)
ξcα,β=ξcβ,αsymmetry(iii)
ξcα,α=ξcα,β=ξcβ,β if and only if α=βequality(iv)
ξcα,β≺¯ξcα,γ+ξcγ,β−ξcβ,βtriangularity
A complex partial metric space is a pairΘ,ξc such that Θ is a nonvoid set and ξc is the complex partial metric on Θ.Definition 3.
(see [6]). A complex partial b-metric on a nonvoid set Θ is a function ⋎cb:Θ×Θ⟶ℂ+ such that, for all α,β,γ∈Θ,(i)
0≺¯⋎cbα,α≺¯⋎cbα,βsmall self−distances(ii)
⋎cbα,β=⋎cbβ,αsymmetry(iii)
⋎cbα,α=⋎cbα,β=⋎cbβ,β⇔α=βequality(iv)
∃ a real number s≥1 and s is an independent of α,β,γ such that ⋎cbα,β≺¯s⋎cbα,γ+⋎cbγ,β−⋎cbγ,γtriangularity
A complex partialb-metric space is a pair Θ,⋎cb such that Θ is a nonvoid set and ⋎cb is the complex partial b-metric on Θ. The number s is called the coefficient of Θ,⋎cb.Remark 1.
(see [6]). In a complex partial b-metric space Θ,⋎cb if α,β∈Θ and ⋎cbα,β=0, then α=β, but the converse may not be true.Every complex partialb-metric ⋎cb on a nonvoid set Θ generates a topology τcb on Θ whose base is the family of open ⋎cb-balls B⋎cbα,ɛ, where τcb=B⋎cbα,ɛ:α∈Θ,ɛ>0 and B⋎cbα,ɛ=β∈Θ:⋎cbα,β<ɛ+⋎cbα,α. Now, we define Cauchy sequence and convergent sequence in complex partial b-metric spaces.Definition 4.
(see [6]). Let Θ,⋎cb be a complex partial b-metric space with coefficient s. Let αn be any sequence in Θ and α∈Θ. Then,(i)
The sequenceαn is said to be convergent with respect to τcb and converges to α if limn⟶∞⋎cbαn,α=⋎cbα,α(ii)
The sequenceαn is said to be Cauchy sequence in Θ,⋎cb if limn,m⟶∞⋎cbαn,αm exists and is finite(iii)
Θ,⋎cb is said to be a complete complex partial b-metric space if, for every Cauchy sequence αn in Θ, there exists α∈Θ such that limn,m⟶∞⋎cbαn,αm=limn⟶∞⋎cbαn,α=⋎cbα,α(iv)
A mappingΞ:Θ⟶Θ is said to be continuous at α0∈Θ if, for every ɛ>0, there exists δ>0 such that ΞB⋎cbα0,δ⊂B⋎cbΞα0,ɛLetΘ,⋎cb be a complex partial b-metric space and D⊆Θ. A point α∈Θ is called an interior of set a D if there exists 0<r∈ℂ such that D⋎cbα,r=β∈Θ:⋎cbα,β<⋎cbα,α+r⊆D. A subset D is called open if each point of D is an interior point of D. A point α∈Θ is said to be a limit point of D, for every 0<r∈ℂ, D⋎cbα,r∩D−α≠ϕ. A subset E⊆Θ is called closed iff E contains all its limit points.Example 1.
(see [6]). Let Θ=0,∞ be endowed with complex partial b-metric ⋎cb:Θ×Θ⟶ℂ+ with ⋎cb=maxα,β2+imaxα,β2∀α,β∈Θ.In 2019, Gunaseelan [6] proved the following theorem.Theorem 1.
(see [6]). Let Θ,⋎cb be a complete complex partial b-metric space with coefficient s≥1 and Ξ:Θ⟶Θ be a mapping satisfying the following condition:(1)⋎cbΞα,Ξβ≺¯a⋎cbα,Ξα+⋎cbβ,Ξβ,∀α,β∈Θ,where a∈0,1/s. Then, Ξ has a unique fixed point γ∈Θ and ⋎cbγ,γ=0.Inspired by Theorem1, we prove a fixed point theorem on complex partial b-metric space under new contraction mapping.In Section3, we first prove, under new contraction mapping, a fixed point theorem on complete complex partial b-metric space. We also provide an example of the complete complex partial b-metric space and clarify that, under certain conditions, it has a unique fixed point.


## 3. Main Results

Theorem 2.
LetΘ,⋎cb be a complete complex partial b-metric space with constant s≥1 and let Ξ be a self-mapping on Θ. Suppose that there exist functions τi, i=1,2,3,4,5, of ℂ+ into ℂ+ such thatC1 Each τi is upper semicontinuous from the rightC2τ1v+τ2v+τ3v+τ4v+τ5v≺1/2s2v,  ∀v>0C3 For any distinct α,β∈Θ,(2)12s⋎cbα,Ξα≺⋎cbα,β⇒⋎cbα,β⋎cbΞα,Ξβ≺¯τ1⋎cbα,β⋎cbα,β+τ2⋎cbα,β⋎cbα,Ξβ+τ3⋎cbα,β⋎cbΞα,β+τ4⋎cbα,β⋎cbα,Ξα+τ5⋎cbα,β⋎cbβ,Ξβ.
Then,Ξ has a unique fixed point.Proof.
Let us first prove that if fixed points ofΞ exists, then it is unique. Let α,β∈Θ be two distinct fixed points of Ξ, that is, Ξα=α≠β=Ξβ. Therefore, ⋎cbα,β≻0. If ⋎cbα,α=0, we have 0=1/2s⋎cbα,α=1/2s⋎cbα,Ξα≺⋎cbα,β. If ⋎cbα,α≻0. By the definition of complex partial b-metric space, we obtain(3)12s⋎cbα,Ξα=12s⋎cbα,α≺⋎cbα,β.
From conditionC3, we derive(4)⋎cbα,β2=⋎cbα,β⋎cbΞα,Ξβ≼¯τ1⋎cbα,β⋎cbα,β+τ2⋎cbα,β⋎cbα,Ξβ+τ3⋎cbα,β⋎cbΞα,β+τ4⋎cbα,β⋎cbα,Ξα+τ5⋎cbα,β⋎cbβ,Ξβ=τ1⋎cbα,β⋎cbα,β+τ2⋎cbα,β⋎cbα,β+τ3⋎cbα,β⋎cbα,β+τ4⋎cbα,β⋎cbα,α+τ5⋎cbα,β⋎cbβ,β≼¯τ1⋎cbα,β⋎cbα,β+τ2⋎cbα,β⋎cbα,β+τ3⋎cbα,β⋎cbα,β+τ4⋎cbα,β⋎cbα,β+τ5⋎cbα,β⋎cbα,β=τ1⋎cbα,β+τ2⋎cbα,β+τ3⋎cbα,β+τ4⋎cbα,β+τ5⋎cbα,β⋎cbα,β≼¯12s⋎cbα,β≺⋎cbα,β,which implies that(5)⋎cbα,β2<⋎cbα,β,which is impossible. Therefore, α=β. Choose α1∈Θ. Set(6)α2=Ξα1,…,αm+1=Ξαm=Ξm+1α1.
Let⋎m=⋎cbαm,Ξαm. If there exists m∈ℕ such that ⋎m=0, the proof is complete. So, we presume that, for every m∈ℕ, 0≺⋎m=⋎cbαm,Ξαm=⋎cbαm,αm+1. Therefore, we have(7)12s⋎cbαm,Ξαm≺⋎cbαm,Ξαm,∀m∈ℕ.
From (2), we derive(8)⋎cbαm,Ξαm⋎cbΞαm,Ξ2αm≼¯τ1⋎cbαm,Ξαm⋎cbαm,Ξαm+τ2⋎cbαm,Ξαm⋎cbαm,Ξ2αm+τ3⋎cbαm,Ξαm⋎cbΞαm,Ξαm+τ4⋎cbαm,Ξαm⋎cbαm,Ξαm+τ5⋎cbαm,Ξαm⋎cbΞαm,Ξ2αm≼¯τ1⋎cbαm,Ξαm⋎cbαm,Ξαm+sτ2⋎cbαm,Ξαm⋎cbαm,Ξαm+sτ2⋎cbαm,Ξαm⋎cbΞαm,Ξ2αm−τ2⋎cbαm,Ξαm⋎cbΞαm,Ξαm+sτ3⋎cbαm,Ξαm⋎cbΞαm,αm+sτ3⋎cbαm,Ξαm⋎cbαm,Ξαm−τ3αm,Ξαm⋎cbαm,αm+τ4⋎cbαm,Ξαm⋎cbαm,Ξαm+τ5⋎cbαm,Ξαm⋎cbΞαm,Ξ2αm≼¯τ1⋎cbαm,Ξαm⋎cbαm,Ξαm+sτ2⋎cbαm,Ξαm⋎cbαm,Ξαm+sτ2⋎cbαm,Ξαm⋎cbΞαm,Ξ2αm+sτ3⋎cbαm,Ξαm⋎cbΞαm,αm+sτ3⋎cbαm,Ξαm⋎cbαm,Ξαm+τ4⋎cbαm,Ξαm⋎cbαm,Ξαm+τ5⋎cbαm,Ξαm⋎cbΞαm,Ξ2αm=τ1⋎cbαm,Ξαm+sτ2⋎cbαm,Ξαm+2sτ3⋎cbαm,Ξαm+τ4⋎cbαm,Ξαm⋎cbαm,Ξαm+sτ2⋎cbαm,Ξαm+τ5⋎cbαm,Ξαm⋎cbΞαm,Ξ2αm,and consequently,(9)⋎m⋎m+1≺¯τ1⋎m+sτ2⋎m+2sτ3⋎m+τ4⋎m⋎m+sτ2⋎m+τ5⋎m⋎m+1,which implies that(10)⋎m+1≺¯τ1⋎m+sτ2⋎m+2sτ3⋎m+τ4⋎m⋎m−sτ2⋎m−τ5⋎m⋎m.
FromC3, we obtain(11)τ1v+2sτ2v+2sτ3v+τ4v+τ5v≺¯2sτ1v+τ2v+τ3v+τ4v+τ5v≺2s12s2v=1sv≺¯v,∀v>0.
Consequently, we derive(12)τ1v+sτ2v+2sτ3v+τ4vv−sτ2v−τ5v≺1,∀v>0.
Therefore, from (10), we obtain(13)⋎cbαm+1,Ξαm+1=⋎m+1≺⋎m=⋎cbαm,Ξαm,∀m∈ℕ,which implies that(14)⋎cbαm+1,Ξαm+1<⋎cbαm,Ξαm,∀m∈ℕ.
Consequently,Re.part⋎m+1<Re.part⋎m and Im.part⋎m+1<Im.part⋎m. Therefore, Re.part⋎m is a decreasing sequence of real numbers which is bounded from below. So, Re.part⋎m converges to some point ⋎∈0,∞. Similarly, Im.part⋎m converges to some point ⋎∈0,∞. Hence, ⋎m=⋎cbαm,Ξαm converges to some point ⋎∈0,∞. If ⋎>0, then using condition C1, we obtain(15)⋎=limm⟶∞⋎m+1=limm⟶∞sup⋎m+1≼¯limm⟶∞supτ1⋎m+sτ2⋎m+2sτ3⋎m+τ4⋎m⋎m−sτ2⋎m−τ5⋎m⋎m=τ1⋎+sτ2⋎+2sτ3⋎+τ4⋎⋎−sτ2⋎−τ5⋎⋎≺⋎,which implies that ⋎<⋎, which is impossible. Therefore,(16)limm⟶∞⋎m=limm⟶∞⋎cbαm,Ξαm=0.
Next, we prove that(17)limm,r⟶∞⋎cbαm,αr=0.
Suppose not, we assume that there existɛ>0 and sequence ζm and ξm of natural numbers such that(18)ζm≻ξm≻m,⋎cbαζm,αξm≽ɛ,⋎cbαζm−1,αξm≺ɛ,∀n∈ℕ.
Therefore, we derive(19)ɛ≺¯⋎cbαζm,αξm≺¯s⋎cbαζm,αζm−1+⋎cbαζm−1,αξm−⋎cbαζm−1,αζm−1≺¯s⋎cbαζm,αζm−1+s⋎cbαζm−1,αξm≺¯s⋎cbαζm,αζm−1+sɛ=s⋎cbαζm−1,Ξαζm−1+sɛ.
Using (16), we derive(20)ɛ≺¯limm⟶∞inf⋎cbαζm,αξm≺¯limm⟶∞sup⋎cbαζm,αξm≺¯sɛ.
From (16), there exists M∈ℕ such that 1/2s⋎cbαζm,Ξαζm≺ɛ,∀m≻M, and using (18), we derive(21)12s⋎cbαζm,Ξαζm≺⋎cbαζm+1,αξm+1=⋎cbΞαζm,Ξαξm,∀m≻M.
Therefore, fromC3, for every m≻M, we obtain(22)⋎cbαζm,αξm⋎cbΞαζm,Ξαξm≼¯τ1⋎cbαζm,αξm⋎cbαζm,αξm+τ2⋎cbαζm,αξm⋎cbαζm,Ξαξm+τ3⋎cbαζm,αξm⋎cbΞαζm,αξm+τ4⋎cbαζm,αξm⋎cbαζm,Ξαζm+τ5⋎cbαζm,αξm⋎cbαξm,Ξαξm≼¯τ1⋎cbαζm,αξm⋎cbαζm,αξm+τ2⋎cbαζm,αξm⋎cbαζm,Ξαξm+τ3⋎cbαζm,αξm⋎cbΞαζm,αξm+sτ4⋎cbαζm,αξm⋎cbαζm,αξm+sτ4⋎cbαζm,αξm⋎cbαξm,Ξαζm−τ4⋎cbαζm,αξm⋎cbαξm,αξm+sτ5⋎cbαζm,αξm⋎cbΞαξm,αζm+sτ5⋎cbαζm,αξm⋎cbαζm,αξm−τ5⋎cbαζm,αξm⋎cbαζm,αζm≼¯τ1⋎cbαζm,αξm⋎cbαζm,αξm+τ2⋎cbαζm,αξm⋎cbαζm,Ξαξm+τ3⋎cbαζm,αξm⋎cbαξm,Ξαζm+sτ4⋎cbαζm,αξm⋎cbαζm,αξm+sτ4⋎cbαζm,αξm⋎cbαξm,Ξαζm+sτ5⋎cbαζm,αξm⋎cbΞαξm,αζm+sτ5⋎cbαζm,αξm⋎cbαζm,αξm≼¯τ1⋎cbαζm,αξm+sτ4⋎cbαζm,αξm+sτ5⋎cbαζm,αξm⋎cbαζm,αξm+τ2⋎cbαζm,αξm+sτ5⋎cbαζm,αξm⋎cbΞαξm,αζm+τ3⋎cbαζm,αξm+sτ4⋎cbαζm,αξm⋎cbαξm,Ξαζm.
Hence, fromC2, we derive(23)⋎cbαζm,αξm⋎cbΞαζm,Ξαξm≼¯12s2⋎cbαζm,αξm2+12s2⋎cbαζm,αξm⋎cbΞαζm,αζm+12s2⋎cbαζm,αξm⋎cbαξm,Ξαξm.
From (16)–(23), we obtain(24)ɛ2≺¯limm⟶∞sup⋎cbαζm,αξm⋎cbαζm+1,αξm+1≺¯limm⟶∞sup12s2⋎cbαζm,αξm2n+12s2⋎cbαζm,αξm⋎cbΞαζm,αζmn+12s2⋎cbαζm,αξm⋎cbαξm,Ξαξmn≺¯12s2sɛ2=12ɛ2≺ɛ2,which is impossible. Hence, limm,r⟶∞⋎cbαm,αr=0. By completeness of Θ,⋎cb, there exists α∈Θ such that(25)⋎cbα,α=limm⟶∞⋎cbαm,α=limm,r⟶∞⋎cbαm,αr=0.
We shall prove that, for everym∈ℕ,(26)12s⋎cbαm,Ξαm≺⋎cbαm,α,or(27)12s⋎cbΞαm,Ξ2αm≺⋎cbΞαm,α.
Suppose not, we assume that there existsq∈ℕ such that(28)12s⋎cbαq,Ξαq≽⋎cbαq,α,12s⋎cbΞαq,Ξ2αq≽⋎cbΞαq,α.
By the definition of complex partialb-metric space and (28), we derive(29)⋎cbαq,Ξαq≺¯s⋎cbαq,α+⋎cbα,Ξαq−⋎cbα,α=s⋎cbαq,α+s⋎cbα,Ξαq≺¯s12s⋎cbαq,Ξαq+s12s⋎cbΞαq,Ξ2αq≺12⋎cbαq,Ξαq+12⋎cbαq,Ξαq=⋎cbαq,Ξαq,which is impossible. Hence, (25) and (26) holds. From (25), we obtain(30)⋎cbαm,α⋎cbΞαm,Ξα≼¯τ1⋎cbαm,α⋎cbαm,α+τ2⋎cbαm,α⋎cbαm,Ξα+τ3⋎cbαm,α⋎cbΞαm,α+τ4⋎cbαm,α⋎cbαm,Ξαm+τ5⋎cbαm,α⋎cbα,Ξα≼¯τ1⋎cbαm,α⋎cbαm,α+sτ2⋎cbαm,α⋎cbαm,Ξαm+sτ2⋎cbαm,α⋎cbΞαm,Ξα+τ3⋎cbαm,α⋎cbΞαm,α+τ4⋎cbαm,α⋎cbαm,Ξαm+sτ5⋎cbαm,α⋎cbα,Ξαm+sτ5⋎cbαm,α⋎cbΞαm,Ξα≼¯τ1⋎cbαm,α⋎cbαm,α+sτ2⋎cbαm,α+τ4⋎cbαm,α⋎cbαm,Ξαm+sτ2⋎cbαm,α+sτ5⋎cbαm,α⋎cbΞαm,Ξα+τ3⋎cbαm,α+sτ5⋎cbαm,α⋎cbΞαm,α.
From (12), we derive(31)⋎cbΞαm,Ξα≼¯⋎cbαm,α+⋎cbαm,Ξαm+⋎cbΞαm,Ξα+⋎cbΞαm,α.
Using (25) and (26), we derive(32)limm⟶∞⋎cbΞαm,Ξα=0.
Since(33)⋎cbα,Ξα≺¯s⋎cbα,Ξαm+⋎cbΞαm,Ξα−⋎cbΞαm,Ξαm≺¯s⋎cbα,Ξαm+⋎cbΞαm,Ξα=s⋎cbα,αm+1+s⋎cbΞαm,Ξα,using (25) and (32), we get α=Ξα. From (27), we obtain(34)⋎cbΞαm,α⋎cbΞ2αm,Ξα≼¯τ1⋎cbΞαm,α⋎cbΞαm,α+τ2⋎cbΞαm,α⋎cbΞαm,Ξα+τ3⋎cbΞαm,α⋎cbΞ2αm,α+τ4⋎cbΞαm,α⋎cbΞαm,Ξ2αm+τ5⋎cbΞαm,α⋎cbα,Ξα≼¯τ1⋎cbΞαm,α⋎cbΞαm,α+sτ2⋎cbΞαm,α⋎cbΞαm,Ξ2αm+sτ2⋎cbΞαm,α⋎cbΞ2αm,Ξα−τ2⋎cbΞαm,α⋎cbΞ2αm,Ξ2αm+τ3⋎cbΞαm,α⋎cbΞ2αm,α+τ4⋎cbΞαm,α⋎cbΞαm,Ξ2αm+sτ5⋎cbΞαm,α⋎cbα,Ξ2αm+sτ5⋎cbΞαm,α⋎cbΞ2αm,Ξατ5⋎cbΞαm,α⋎cbΞ2αm,Ξ2αm≼¯τ1⋎cbΞαm,α⋎cbΞαm,α+sτ2⋎cbΞαm,α⋎cbΞαm,Ξ2αm+sτ2⋎cbΞαm,α⋎cbΞ2αm,Ξα+τ3⋎cbΞαm,α⋎cbΞ2αm,α+τ4⋎cbΞαm,α⋎cbΞαm,Ξ2αm+sτ5⋎cbΞαm,α⋎cbα,Ξ2αm+sτ5⋎cbΞαm,α⋎cbΞ2αm,Ξα≼¯τ1⋎cbΞαm,α⋎cbΞαm,α+sτ2⋎cbΞαm,α+τ4⋎cbΞαm,α⋎cbΞαm,Ξ2αm+sτ2⋎cbΞαm,α+sτ5⋎cbΞαm,α⋎cbΞ2αm,Ξα+τ3⋎cbΞαm,α+sτ5⋎cbΞαm,α⋎cbΞ2αm,α,which means that(35)⋎cbΞ2αm,Ξα≺τ1Ξαm,α⋎cbΞαm,α⋎cbΞαm,α−sτ2⋎cbΞαm,α−sτ5⋎cbΞαm,α+sτ2⋎cbΞαm,α+τ4⋎cbΞαm,α⋎cbΞαm,Ξ2α⋎cbΞαm,α−sτ2⋎cbΞαm,α−sτ5⋎cbΞαm,α+τ3⋎cbΞαm,α+sτ5⋎cbΞαm,α⋎cbΞ2αm,α⋎cbΞαm,α−sτ2⋎cbΞαm,α−sτ5⋎cbΞαm,α.
From (12) and (35), we derive that(36)0≺¯⋎cbΞ2αm,Ξα≺¯⋎cbΞαm,α+⋎cbΞαm,Ξ2α+⋎cbα,Ξ2αm.
Using (16) and (25), we derive(37)limm⟶∞⋎cbΞ2αm,Ξα=0.
Since(38)⋎cbα,Ξα≺¯s⋎cbα,Ξ2αm+⋎cbΞ2αm,Ξα−⋎cbΞ2αm,Ξ2αm≺¯s⋎cbα,Ξ2αm+⋎cbΞ2αm,Ξα,using (25) and (37), we get α=Ξα.Example 2.
LetΘ=ℕ∪r+1/j+3:r,j∈ℕ. Define ⋎cb:Θ×Θ⟶ℂ+ by(39)⋎cbr,j=maxr,j2+r−j2+imaxr,j2+r−j2.
Define a mappingΞ on Θ by(40)−10ptΞr=2,r∈ℕ,8r+1j+3,r∈r+1j+3:r,j∈ℕ.
Then,Ξ satisfies in the assumption of Theorem 2.Proof.
It is clear thatΞ,⋎cb is a complete complex partial b-metric space with coefficient s=22 and 2 is a unique fixed point of Ξ. Let j<r:(41)12⋎cbr+1j+3,Ξr+1j+3=12⋎cbr+1j+3,8r+1j+3=12maxr+1j+3,8r+1j+32+r+1j+3−8r−1j+32+imaxr+1j+3,8r+1j+32+r+1j+3−8r−1j+32=128r+1j+32+49r2+i8r+1j+32+49r2=12113r2+1j+32+16rj+3+i113r2+1j+32+16rj+3,⋎cbr+1j+3,j+1j+3=maxr+1j+3,j+1j+32+r+1j+3−j−1j+32+imaxr+1j+3,j+1j+32+|r+1j+3−j−1j+3|2=r+1j+32+r−j2+ir+1j+32+r−j2≺¯r2+1j+32+2rj+3+r−j2+ir2+1j+32+2rj+3+r−j2≺r2+1j+32+2rj+3+r+j2+ir2+1j+32+2rj+3+r+j2.
So, forj<r, 1/2⋎cbr+1/j+3,Ξr+1/j+3⊀⋎cbr+1/j+3,j+1/j+3. Therefore, Ξ satisfies all the conditions of Theorem 2.


### 3.1. Nonlinear Integral Equations

In this section, we prove the existence and uniqueness of a solution for nonlinear Fredholm integral equations by using Theorem2.Theorem 3.
Consider the nonlinear Fredholm integral equation:(42)rt=jt+∫abKt,s,rsds,where a,b∈ℝ with a<b and j:a,b⟶ℝ and K:a,b2×ℝ⟶ℝ are given continuous mappings. Suppose that the following condition holds:(a)
The mappingΞ:Ca,b⟶Ca,b defined by Ξrt=jt+∫abKt,s,rsds for all r∈Ca,b and t∈a,b.
Then, the nonlinear integral equation (42) has a unique solution.Proof.
LetΘ=Ca,b. Clearly, Θ with the complex partial b-metric ⋎cb:Θ×Θ⟶C+ given by(43)⋎cbrt,vt=maxrt,vt2+imaxrt,vt2,t∈a,b,for all r,v∈Θ is a complete complex partial b-metric space. Without loss generality, we may assume that(44)r≺¯v⇔rt≤vt,∀t∈a,b.
Now,(45)12sjt+∫abKt,s,vs2+ijt+∫abKt,s,vs2≺jt+∫abKt,s,vs2+ijt+∫abKt,s,vs2.
Therefore,(46)12s⋎cbvt,Ξvt≺⋎cbrt,vt.
Hence,Ξ satisfies all the conditions of Theorem 2.


## 3.1. Nonlinear Integral Equations

In this section, we prove the existence and uniqueness of a solution for nonlinear Fredholm integral equations by using Theorem2.Theorem 3.
Consider the nonlinear Fredholm integral equation:(42)rt=jt+∫abKt,s,rsds,where a,b∈ℝ with a<b and j:a,b⟶ℝ and K:a,b2×ℝ⟶ℝ are given continuous mappings. Suppose that the following condition holds:(a)
The mappingΞ:Ca,b⟶Ca,b defined by Ξrt=jt+∫abKt,s,rsds for all r∈Ca,b and t∈a,b.
Then, the nonlinear integral equation (42) has a unique solution.Proof.
LetΘ=Ca,b. Clearly, Θ with the complex partial b-metric ⋎cb:Θ×Θ⟶C+ given by(43)⋎cbrt,vt=maxrt,vt2+imaxrt,vt2,t∈a,b,for all r,v∈Θ is a complete complex partial b-metric space. Without loss generality, we may assume that(44)r≺¯v⇔rt≤vt,∀t∈a,b.
Now,(45)12sjt+∫abKt,s,vs2+ijt+∫abKt,s,vs2≺jt+∫abKt,s,vs2+ijt+∫abKt,s,vs2.
Therefore,(46)12s⋎cbvt,Ξvt≺⋎cbrt,vt.
Hence,Ξ satisfies all the conditions of Theorem 2.


## 4. Conclusion

In the present work, we presented a new fixed point theorem for self-mappings defined on complex partialb-metric. We illustrated our main theorem by an example and showed, moreover, that the main theorem can be easily used to solve a nonlinear integral equation.



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*Source: 5565470-2021-04-10.xml*