# Application to Coupled Fixed-Point Theorems on Complex Partial b-Metric Space

Authors: Gunaseelan Mani; Lakshmi Narayan Mishra; Vishnu Narayan Mishra; Imran Abbas Baloch; Manuel De La Sen
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we prove coupled fixed-point theorems in complex partial b-metric space. 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

The notion of b-metric space was introduced by Backhtin [1] in 1989, and Czerwik [2] extended the results of metric spaces. The notion of complex valued metric spaces was introduced by Azam et al. [3] in 2011 who also proved some common fixed-point theorems under the contraction condition. In 2013, Rao et al. [4] introduced the concept of complex valued b-metric space which is more general than the well-known complex valued metric space and also proved common fixed-point theorem under the contraction condition. In 2017, Dhivya and Marudai [5] introduced the notion of complex partial metric space and also proved common fixed-point theorems under the contraction condition of rational expression. In 2019, Gunaseelan [6] introduced the notion of complex partial b-metric space and also proved fixed-point theorem under the contractive condition. Some interesting concepts and applications have been studied by many authors, and important results have been obtained in [7–12]. In this paper, we prove coupled fixed-point theorems in complex partial b-metric space.In the next section, we give basic definitions, examples, and primary results for the better understanding of our major results presented in this research paper.


## 2. Preliminaries

Letℂ be the set of complex numbers and η1,η2,η3∈ℂ. Define a partial order ⪯ on ℂ as follows: η1⪯η2 if and only if Reη1≤Reη2, Imη1≤Imη2.Consequently, one can infer thatη1⪯η2 if one of the following conditions is satisfied:(i)
Reη1=Reη2, Imη1<Imη2(ii)
Reη1<Reη2, Imη1=Imη2(iii)
Reη1<Reη2, Imη1<Imη2(iv)
Reη1=Reη2, Imη1=Imη2In particular, we writeη1⋨η2 if η1≠η2 and one of (i), (ii), and (iii) is satisfied and we write η1≺η2 if only (iii) is satisfied. Notice that(a)
If0≺¯η1⋨η2, then η1<η2(b)
Ifη1≺¯η2 and η2≺η3, then η1≺η3(c)
Ifλ,β∈ℝ, and λ≤β, then λa1≺¯βa1 for all 0≺¯a1∈ℂDefinition 1 (see [4]).
LetY be a nonvoid set and let s≥1 be a given real number. A function d:Y×Y⟶ℂ is called a complex valued b-metric on Y if for all a,b,c∈Y the following conditions are satisfied:(i)
0≺¯da,b and da,b=0 if and only if a=b(ii)
da,b=db,a(iii)
da,b≺¯sda,c+dc,b
The pairY,d is called a complex valued b-metric space.
Here,ℂ+=a,ba,b∈ℝ+ and ℝ+=a∈ℝa≥0 denote the set of non-negative complex numbers and the set of non-negative real numbers, respectively. We now give the complex partial metric space.Definition 2 (see [5]).
A complex partial metric on a nonvoid set△ is a function ζc:△×△⟶ℂ+ such that for all e,f,g∈△:(i)
0≺¯ζce,e≺¯ζce,fsmall self−distances(ii)
ζce,f=ζcf,esymmetry(iii)
ζce,e=ζce,f=ζcf,f if and only if e=fequality(iv)
ζce,f≺¯ζce,g+ζcg,f−ζcg,gtriangularity
A complex partial metric space is a pair△,ζc such that △ is a nonvoid set and ζc is the complex partial metric on △.Definition 3 (see [6]).
A complex partial b-metric on a nonvoid set△ is a function θcb:△×△⟶ℂ+ such that for all e,f,g∈△:(i)
0≺¯θcbe,e≺¯θcbe,fsmall self−distances(ii)
θcbe,f=θcbf,esymmetry(iii)
θcbe,e=θcbe,f=θcbf,f⟺e=fequality(iv)
∃ a real number s≥1 such that θcbe,f≺¯sθcbe,g+θcbg,f−θcbg,gtriangularity
A complex partial b-metric space is a pair△,θcb such that △ is a nonvoid set and θcb is the complex partial b-metric on △. The number s is called the coefficient of △,θcb.Remark 1 (see [6]).
In a complex partial b-metric space△,θcb, if e,f∈△ and θcbe,f=0, then e=f, but the converse may not be true.Remark 2 (see [6]).
It is clear that every complex partial metric space is a complex partial b-metric space with coefficients=1 and every complex valued b-metric is a complex partial b-metric space with the same coefficient and zero self-distance. However, the converse of this fact need not hold.Example 1 (see [6]).
Let△=ℝ+ and θcb:△×△⟶ℂ+ be defined by θcbe,f=maxe,f3+e−f3+imaxe,f3+e−f3,∀e,f∈△. Then, △,θcb is a complex partial b-metric space with coefficient s=23, but it is neither a complex valued b-metric nor a complex partial metric.Proposition 1 (see [6]).
Let△ be a nonvoid set such that ζc is a complex partial metric and d is a complex valued b-metric with coefficient s>1 on △. Then, the function θcb:△×△⟶ℂ+ defined by θcbe,f=ζce,f+de,f,∀e,f∈△ is a complex partial b-metric on △, that is, △,θcb is a complex partial b-metric space.Proposition 2 (see [6]).
Let△,ζc be a complex partial metric space, r≥1; then, △,θcb is a complex partial b-metric space with coefficient s=2r−1, where θcb is defined by θcbe,f=ζce,fr.Every complex partial b-metricθcb on a nonvoid set △ generates a topology τcb on △ whose base is the family of open θcb-balls Bθcbe,ε where τcb=Bθcbe,ε:e∈△,ε>0 and Bθcbe,ε=f∈△:θcbe,f<ε+θcbe,e. Now, we define Cauchy sequence and convergent sequence in complex partial b-metric spaces.Definition 4 (see [6]).
Let△,θcb be a complex partial b-metric space with coefficient s. Let en be any sequence in △ and e∈△. Then,(i)
The sequenceen is said to be convergent with respect to τcb and converges to e, if limn⟶∞θcben,e=θcbe,e.(ii)
The sequenceen is said to be Cauchy sequence in △,θcb if limn,m⟶∞θcben,em exists and is finite.(iii)
△,θcb is said to be a complete complex partial b-metric space if for every Cauchy sequence en in △ there exists e∈△ such that limn,m⟶∞θcben,em=limn⟶∞θcben,e=θcbe,e.(iv)
A mappingT:△⟶△ is said to be continuous at e0∈△ if for every ε>0, there exists δ>0 such that TBθcbe0,δ⊂BδcbTe0,ε.
Let△ be a complex partial b-metric space and B⊆△. A point e∈△ is called an interior of set B, if there exists 0<r∈ℂ such that Bθcbe,r=f∈△:θcbe,f<θcbe,e+r⊆B. A subset B is called open, if each point of B is an interior point of B. A point e∈△ is said to be a limit point of B, for every 0<r∈ℂ, Bθcbe,r∩B−e≠ϕ. A subset C⊆△ is called closed if B contains all its limit points.Example 2 (see [6]).
Let△=ℝ+, a>0 be any constant and define θcb:△×△⟶ℂ+ by θcbe,f=maxe,f+a1+i∀e,f∈△.
Then,△,θcb is a complex partial b-metric space with arbitrary coefficient s≥1. Now, define a sequence en in △ by en=1 for all n∈ℕ. Note that, if f≥1, we have θcben,f=f+a1+i=θcbf,f.Therefore, limn⟶∞θcben,f=θcbf,f for all f≥1. Thus, the limit of convergent sequence in complex partial b-metric space need not be unique.Example 3 (see [6]).
Let△=0,∞ be endowed with complex partial b-metric θcb:△×△⟶ℂ+ with θcb=maxe,f2+imaxe,f2∀e,f∈△.
It is easy to verify that△,θcb is a complex partial b-metric space and note that self-distance need not be zero, for example, θcb1,1=1+i≠0. Now, the complex valued b-metric not induced by θcb is as follows: dθcbe,f=2θcbe,f−θcbe,e−θcbf,f; without loss of generality, suppose that e≥f; then, dθcbe,f=2maxe,f2+imaxe,f2−e2+ie2−f2+if2. Therefore, dθcbe,f=e2−f2+ie2−f2.
Thus, we have the following proposition.Proposition 3 (see [6]).
There exists a complex partial b-metricθcb which does not define a complex b-metric dθcb, where dθcbe,f=2θcbe,f−θcbe,e−θcbf,f∀e,f∈△.Definition 5 (see [13]).
Let△,⪯ be a partially ordered set and Λ:△×△⟶△. We say that Λ has the mixed monotone property if Λi,j is monotone nondecreasing in i and is monotone nonincreasing in j, that is, for any i,j∈△,(1)i1,i2∈△,i1≺¯i2⟹Λi1,j⪯Λi2,j,j1,j2∈△,j1≺¯j2⟹Λi,j1≽Λi,j2.Definition 6.
Let△,θcb be a complex partial b-metric space. An element i,j∈△×△, is called a coupled fixed point of the mapping Λ:△×△⟶△ if Λi,j=i and Λj,i=j.Example 4.
Let△=0,∞ be endowed with complex partial b-metric θcb:△×△⟶ℂ+ defined by θcbe,j=maxe,j2+imaxe,j2∀e,j∈△. Consider the mapping Λ:△×△⟶△ with Λe,j=ie+j/4. Here, 0,0 is the coupled fixed point of Λ.
In 2019, Gunaseelan and Mishra [14] proved the following theorem.Theorem 1 (see [14]).
LetU,ξc be a complete complex partial metric space. Suppose that the mapping ϕ:U×U⟶U satisfies the following contractive condition for all α,β,γ,δ∈U:(2)ξcϕα,β,ϕγ,δ≺¯kξcϕα,β,α+lξcϕγ,δ,γ,where k,l are nonnegative constants with k+l<1. Then, ϕ has a unique coupled fixed point.Inspired by Theorem1, we prove coupled fixed-point theorem on partially ordered complex partial b-metric space using mixed monotone property.In the next section, we firstly prove that a continuous mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space has a coupled fixed point under certain conditions. Secondly, we give result of a coupled fixed point for a mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space by losing the property of continuity. Then, we find the condition under which a continuous mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space has a unique coupled fixed point under certain conditions. We also give an example of continuous mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space and show that it has unique coupled fixed point under said conditions.


## 3. Main Results

Let△,≺¯ be a partially ordered set and θcb be a complex partial b-metric space on △. Further, we endow the product space △×△ with the following partial order:(3)fori,j,g,h∈△×△,g,h⪯i,j⟺i≽g,j⪯h.We begin with the following theorem that establishes the existence of a fixed-point theorem for a functionΛ on the product △×△.Theorem 2.
Let△,θcb,⪯ be a partially ordered complete complex partial b-metric space with the coefficient s≥1. Let Λ:△×△⟶△ be a continuous mapping having the mixed monotone property on △. Assume that there exists a 2sα∈0,1 with(4)θcbΛi,j,Λg,h≺¯αθcbi,g+θcbj,h,∀i≽g,j⪯h.If there existsi0,j0∈△ such that(5)i0⪯Λi0,j0,j0≽Λj0,i0,then Λ has a coupled fixed point.Proof.
Sincei0≺¯Λi0,j0=i1 (say) and j0≽Λj0,i0=j1 (say), letting i2=Λi1,j1 and j2=Λj1,i1, we denote(6)Λ2i0,j0=ΛΛi0,j0,Λj0,i0=Λi1,j1=i2,Λ2j0,i0=ΛΛj0,i0,Λi0,j0=Λj1,i1=j2.
Due to the mixed monotone property ofΛ,(7)i2=Λ2i0,j0=Λi1,j1≽Λi0,j0=i1,j2=Λ2j0,i0=Λj1,i1⪯Λj0,i0=j1.
Further, forp=1,2,…, we let(8)ip+1=Λp+1i0,j0=ΛΛpi0,j0,Λpj0,i0,jp+1=Λp+1j0,i0=ΛΛpj0,i0,Λpi0,j0.
We can easily verify that(9)i0⪯Λi0,j0=i1⪯Λ2i0,j0=i2⪯⋯⪯Λp+1i0,j0⪯⋯,(10)j0≽Λj0,i0=j1≽Λ2j0,i0=j2≽⋯≽Λp+1j0,i0≽⋯.
Ifip+1=ip and jp+1=jp for some p, then Λip,jp=ip and Λjp,ip=jp, and hence ip,jp is a coupled fixed point of Λ. Suppose, further, that(11)ip≠ip+1,orjp≠jp+1,for eachp∈ℕ0.
Now, we claim that, forp∈ℕ0,(12)θcbip+1,ip+θcbjp+1,jp≤2pαpθcbi1,i0+θcbj1,j0.
Indeed, forp=1, using i1≽i0, j1⪯j0, we get(13)θcbi2,i1=θcbΛi1,j1,Λi0,j0⪯αθcbi1,i0+θcbj1,j0,which implies that(14)θcbi2,i1≤αθcbi1,i0+θcbj1,j0.
Similarly,(15)θcbj2,j1=θcbΛj1,i1,Λj0,i0⪯αθcbj1,j0+θcbi1,i0,which implies that(16)θcbj2,j1≤αθcbj1,j0+θcbi1,i0.
Adding (12) and (16), we have(17)θcbi2,i1+θcbj2,j1≤2αθcbi0,i1+θcbj0,j1.
In a similar way, proceeding by induction, if we assume that (12) holds, we get that(18)θcbip+2,ip+1+θcbjp+2,jp+1≤2αθcbip+1,ip+θcbjp+1,jp≤2p+1αp+1θcbi0,i1+θcbj0,j1.
Hence, by induction, (12) is proved. Set(19)lp≔θcbip,ip+1+θcbjp,jp+1,p∈ℕ.
Then, the sequencelp is decreasing and(20)lp≤2pαpl0.
By assumption (9), lp>0 for p∈ℕ0. Then, for each p≥q, we have(21)θcbiq,ip⪯sθcbiq,iq+1+s2θcbiq+1,iq+2+⋯+spθcbip−1,ip−θcbiq+1,iq+1−θcbiq+2,iq+2−θcbiq+3,iq+3−⋯−θcbip−1,ip−1⪯sθcbiq,iq+1+s2θcbiq+1,iq+2+⋯+spθcbip−1,ip,θcbjq,jp⪯sθcbjq,jq+1+s2θcbjq+1,jq+2+⋯+spθcbjp−1,jp−θcbjq+1,jq+1−θcbjq+2,jq+2−θcbjq+3,jq+3−⋯−θcbjp−1,jp−1⪯sθcbjq,jq+1+s2θcbjq+1,jq+2+⋯+spθcbjp−1,jp,which implies that(22)θcbiq,ip≤sθcbiq,iq+1+s2θcbiq+1,iq+2+⋯+spθcbip−1,ip,θcbjq,jp≤sθcbjq,jq+1+s2θcbjq+1,jq+2+⋯+spθcbjp−1,jp.
Therefore,(23)lq=θcbiq,ip+θcbjq,jp≤sθcbiq,iq+1+θcbjq,jq+1+s2θcbiq+1,iq+2+θcbjq+1,jq+2+⋯⋯+spθcbip−1,ip+θcbjp−1,jp≤slq+s2lq+1+⋯+splp−1≤s2qαq1+s2α+⋯+sp−12αp−q−1l0≤s2αq1−s2αl0⟶0,asq⟶∞.
Therefore,ip and jp are Cauchy sequences in △. Since △ is complete complex partial b-metric space, there exists t,r∈△×△ such that(24)limp⟶∞ip=t,limp⟶∞jp=r,and θcbt,t=limp⟶∞θcbt,tp=limp,q⟶∞θcbtp,tq=0 and θcbr,r=limp⟶∞θcbr,rp=limp,q⟶∞θcbrp,rq=0. Finally, we claim that t,r is a coupled fixed point of Λ. Indeed, from ip+1=Λip,jp and jp+1=Λjp,ip, using (24) and the continuity of Λ, it immediately follows that t=Λt,r and r=Λr,t.
In the next theorem, we will substitute the continuity hypothesis onΛ by an additional property satisfied by the space △,θcb,≺¯.Theorem 3.
Let△,θcb,≺¯ be a partially ordered complete complex partial b-metric space with the coefficient s≥1. Let Λ:△×△⟶△ be a mapping having the mixed monotone property on △. Assume that there exists 2sα∈0,1 with(25)θcbΛi,j,Λg,h≺¯αθcbi,g+θcbj,h,∀i≽g,j⪯h.Finally, assume that△ has the following properties:(i)
If a nondecreasing sequenceip in △ converges to i∈△, then ip≺¯i for all p.(ii)
If a nonincreasing sequencejp in △ converges to j∈△, then jp≽j for all p.Then,Λ has a coupled fixed point.Proof.
Following the proof of Theorem2, we only have to show t,r is a coupled fixed point of Λ. We have(26)θcbΛt,r,t⪯sθcbΛt,r,ip+1+θcbip+1,t−θcbip+1,ip+1⪯sθcbΛt,r,ip+1+θcbip+1,t=sθcbΛt,r,ip+1+θcbip+1,t.
Since the nondecreasing sequenceip converges to t and the nonincreasing sequence jp converges to r, by (i)–(iii), we have(27)t≽ip,r⪯jp,∀p.
Now, from the contractive condition (25), we have(28)θcbΛt,r,Λip,jp≺¯αθcbt,ip+θcbr,jp.
Then, from (26), we get(29)θcbΛt,r,t⪯sαθcbt,ip+θcbr,jp+θcbip+1,t,which implies that(30)θcbΛt,r,t≤sαθcbt,ip+sαθcbr,jp+sθcbip+1,t.
Taking limit asp⟶∞, we have(31)θcbΛt,r,t≤0.
Therefore,Λt,r=t. Similarly, we can prove that Λr,t=r. Hence, t,r is a coupled fixed point of Λ.Theorem 4.
Assume that(32)∀e,f,e∗,f∗∈△×△,∃x1,x2∈△×△that is comparable toe,fande∗,f∗.Adding (32) to the hypotheses of Theorem 2, we obtain the uniqueness of the coupled fixed point of Λ.Proof.
From Theorem2, we know that there exists a coupled fixed point t,r of Λ, which is obtained as t=limp⟶∞Λpi0,j0 and r=limp⟶∞Λpj0,i0. Suppose that e∗,f∗ is another coupled fixed point, i.e.,(33)Λe∗,f∗=e∗,Λf∗,e∗=f∗.
Let us claim that(34)θcbt,e∗+θcbr,f∗=0.
We discuss two cases.Case 1:t,r is comparable with e∗,f∗ with respect to the ordering in △×△. Let, e.g., t≽e∗ and r≺¯f∗. Then, we can apply the contractive condition (4) to obtain(35)θcbt,e∗=θcbΛt,r,Λe∗,f∗⪯αθcbt,e∗+θcbr,f∗,which implies that(36)θcbt,e∗≤αθcbt,e∗+θcbr,f∗.(37)θcbr,f∗=θcbΛr,t,Λf∗,e∗≤¯αθcbr,f∗+θcbt,e∗,which implies that(38)θcbr,f∗≤αθcbr,f∗+θcbt,e∗.Adding (36) and (38), we get(39)θcbt,e∗+θcbr,f∗≤2αθcbt,e∗+θcbr,f∗.Since2α∈0,1/s, (34) holds.Case 2:t,r is not comparable with e∗,f∗. In this case, there exists x1,x2∈△×△ that is comparable both to t,r and e∗,f∗. Then, for all p∈ℕ, Λpx1,x2,Λpx2,x1 is comparable both to Λpt,r,Λpr,t=t,r and Λpe∗,f∗,Λpf∗,e∗=e∗,f∗. We have(40)θcbt,e∗+θcbr,f∗=θcbΛpt,r,Λpe∗,f∗+θcbΛpr,t,Λpf∗,e∗⪯θcbΛpt,r,Λpx1,x2+θcbΛpx1,x2,Λpe∗,f∗+θcbΛpr,t,Λpx2,x1+θcbΛpx2,x1,Λpf∗,e∗⪯2pαpθcbt,x1+θcbr,x2+θcbe∗,x1+θcbf∗,x2,which implies that(41)θcbt,e∗+θcbr,f∗≤2pαpθcbt,x1+θcbr,x2+θcbe∗,x1+θcbf∗,x2.
Since2α∈0,1/s, (34) holds.
We deduce that in all cases, (34) holds. This implies that t,r=e∗,f∗ and the uniqueness of the coupled fixed point of Λ is proved.Theorem 5.
In addition to the hypotheses of Theorem2 (resp. Theorem 3), suppose that i0,j0 in △ are comparable. Then, t=r.Proof.
Suppose thati0⪯j0. We claim that(42)ip⪯jp,∀p∈ℕ.
From the mixed monotone property ofΛ, we have(43)i1=Λi0,j0⪯Λj0,j0⪯Λj0,i0=j1.
Assume thatip⪯jp for some p. Now,(44)ip+1=Λp+1i0,j0=ΛΛpi0,j0,Λpj0,i0=Λip,jp⪯Λjp,jp⪯¯Λjp,ip=jp+1.
Hence, (42) holds.
Now, using (42) and the contractive condition, we get(45)θcbt,r⪯sθcbt,ip+1+θcbip+1,r−θcbip+1,ip+1⪯sθcbt,ip+1+θcbip+1,r⪯sθcbt,ip+1+sθcbip+1,jp+1+θcbjp+1,r−θcbjp+1,jp+1⪯sθcbt,ip+1+s2θcbip+1,jp+1+s2θcbjp+1,r=sθcbt,ip+1+s2θcbΛip,jp,Λjp,ip+s2θcbjp+1,r⪯sθcbt,ip+1+s2αθcbip,jp+θcbjp,ip+s2θcbjp+1,r,which implies that(46)θcbt,r≤sθcbt,ip+1+s2αθcbip,jp+θcbjp,ip+s2θcbjp+1,r.
Passing to the limit asp⟶∞, we get(47)θcbt,r≤2s2αθcbt,r.
Since2s2α<1, this implies that θcbt,r=0, i.e., t=r.Example 5.
Let△=1,∞ be equipped with the partial order ≼ defined by(48)e≺¯f⟺e≤f,and with the functional θcb:△×△⟶ℂ+ defined by θcbe,f=e−f2+2+ie−f2+2 for all e,f∈△. Clearly, △,θcb is a partially ordered complete complex partial b-metric space with s=2. Define the mapping Λ:△×△⟶△ by(49)Λe,f=0,ife<f,e−f2,ife≥f.
Obviously, the mappingΛ has the mixed monotone property and is continuous. Let e,f,t,r∈△ be such that e≤t and f≥r. We have considered the following cases.Case 1:e≥f. Since e≤t, we have t≥e≥f≥r:(50)θcbΛe,f,Λt,r=θcbe−f2,t−r2=e−f2−t−r22+2+ie−f2−t−r22+2=e−t2+r−f22+2+ie−t2+r−f22+2⪯2e−t22+r−f22+2+i2e−t22+r−f22+2=12e−t2+r−f2+4+ie−t2+r−f2+4=αθcbe,t+θcbr,t.Case 2:e<f, t≥r, and e>r:(51)θcbΛe,f,Λt,r=0−t−r22+2+i0−t−r22+2=t−r24+2+it−r24+2⪯t−r+f−e24+2+it−r+f−e24+2⪯12e−t2+f−r2+4+i12e−t2+f−r2+4=αθcbe,t+θcbr,t.Case 3:e<f, t≥r, and e<r:(52)θcbΛe,f,Λt,r=0−t−r22+2+i0−t−r22+2=t−r22+2+it−r22+2⪯t−r+f−e24+2+it−r+f−e24+2⪯12e−t2+f−r2+4+i12e−t2+f−r2+4=αθcbe,t+θcbr,t.Case 4:e<f and t<r:(53)θcbΛe,f,Λt,r=θcb0,0=21+i≺¯αθcbe,t+θcbr,t.
Thus,Λ satisfies all assumptions of Theorem 4 and it has a unique coupled fixed point (which is 0,0).
Next, we present a result for the existence of a unique solution for a particular system of integral equations.


### 3.1. Applications to Integral Equations

We study the existence of solutions for the following system of integral equations:(54)eu=∫abT1u,s+T2u,sHs,es+Ks,fsds+lu,(55)fu=∫abT1u,s+T2u,sHs,fs+Ks,esds+lu,where u∈I=a,b.We assume thatT1, T2, H,K satisfy the following conditions:(i)
T1u,s≽0 and T2u,s≺¯0 for all u,s∈a,b.(ii)
There existe,f∈ℝ, e≺¯f such that(56)0≼Hu,e−Hu,f≼e−fand−e−f≼Ku,e−Ku,f≼0.(iii)
∫abT1u,s−T2u,s2ds⪯α/4.Theorem 6.
Consider integral equations (54) and (55) with T1,T2∈CI,ℝ,H,K∈CI×ℝ,ℝ, and l∈CI,ℝ. Under assumptions (i)–(iii), equations (54) and (55) have a unique solution.Proof.
Consider the natural order relation on△=CI,ℝ; that is, for e,f∈CI,ℝ,(57)e≺¯f⟺eu≤fu,∀u∈I.
It is well known that△ is a complete complex partial b-metric space with respect to(58)θcbe,f=e−f2+2+ie−f2+2,e,f∈CI,ℝ.
Suppose thattp is a monotone nondecreasing sequence in △ that converges to a point t∈△. Then, for every u∈I, the sequence of real numbers(59)t1u≤t2u≤⋯≤tpu≤⋯converges to tu. Therefore, for all u∈I, p∈ℕ, tp≤tu. Hence, tp≤t, for all p. Similarly, it can be verified that, if for all u∈I, ru is a limit of a monotone nondecreasing sequence rp in △, then ru≤rpu for all p, and hence r≤rp for all p.
Also,△×△=CI,ℝ×CI,ℝ is a partially ordered set under the following order relation in △×△:(60)e,f,t,r∈△×△,e,f≤t,r⟺eu≺¯tu,fu≽ru,∀u∈I.
For anye,f∈△, maxeu,fu and mineu,fu, for each u∈I, are in △ and are the upper and lower bounds of e,f, respectively. Therefore, for every e,f,t,r∈△×△, there exists maxe,t,minf,r∈△×△ that is comparable to e,f and t,r. Define Λ:△×△⟶△ by(61)Λe,fu=∫abT1u,sHs,es+Ks,fsds+∫abT2u,sHs,fs+Ks,esds+lu,for allu∈a,b.
We now claim thatΛ has the mixed monotone property. If t1,r≺¯t2,r, then(62)Λt1,ru=∫abT1u,sHs,t1s+Ks,rsds+∫abT2u,sHs,rs+Ks,t1sds+lu⪯∫abT1u,sHs,t2s+Ks,rsds+∫abT2u,sHs,rs+Ks,t2sds+lu=Λt2,ru.
Similarly, ift,r1⪯t,r2, then(63)Λt,r1u=∫abT1u,sHs,ts+Ks,r1sds+∫abT2u,sHs,r1s+Ks,tsds+lu⪯∫abT1u,sHs,ts+Ks,r2sds+∫abT2u,sHs,r2s+Ks,tsds+lu=Λt,r2u.
Thus,Λt,r is monotone nondecreasing in t and Λt,r is monotone nonincreasing in r. Also, for e,f⪯t,r, that is, t≽e, r≺¯f, it follows that(64)θcbΛt,r,Λe,f=Λt,r−Λe,f2+2+iΛt,r−Λe,f2+2=∫abT1u,sHs,ts+Ks,rsds+∫abT2u,sHs,rs+Ks,tsds+lu−∫abT1u,sHs,es+Ks,fsds−∫abT2u,sHs,fs+Ks,esds−lu2+2+i∫abT1u,sHs,ts+Ks,rsds+T2u,sHs,rs+Ks,tsds+lu−∫abT1u,sHs,es+Ks,fsds−∫abT2u,sHs,fs+Ks,esds−lu2+2θcbΛt,r,Λe,f=∫abT1u,sHs,ts−Hs,es+Ks,rs−Ks,fsds−∫abT2u,sHs,fs−Hs,rs+Ks,es−Ks,ts2+2+i∫abT1u,sHs,ts−Hs,es+Ks,rs−Ks,fsds−∫abT2u,sHs,fs−Hs,rs+Ks,es−Ks,ts2+2⪯∫abT1u,sts−es+fs−rsds−∫abT2u,sfs−rs+ts−esds2+2+i∫abT1u,sts−es+fs−rsds−∫abT2u,sfs−rs+ts−esds2+2⪯4∫abT1u,s−T2u,s2fs−rs2+ts−es2ds+4+i4∫abT1u,s−T2u,s2fs−rs2+ts−es2ds+4⪯4α4θcbf,r+θcbt,e=αθcbf,r+θcbt,e.
Now, all the hypotheses of Theorem4 are satisfied. Therefore, Λ has a unique coupled fixed point.


## 3.1. Applications to Integral Equations

We study the existence of solutions for the following system of integral equations:(54)eu=∫abT1u,s+T2u,sHs,es+Ks,fsds+lu,(55)fu=∫abT1u,s+T2u,sHs,fs+Ks,esds+lu,where u∈I=a,b.We assume thatT1, T2, H,K satisfy the following conditions:(i)
T1u,s≽0 and T2u,s≺¯0 for all u,s∈a,b.(ii)
There existe,f∈ℝ, e≺¯f such that(56)0≼Hu,e−Hu,f≼e−fand−e−f≼Ku,e−Ku,f≼0.(iii)
∫abT1u,s−T2u,s2ds⪯α/4.Theorem 6.
Consider integral equations (54) and (55) with T1,T2∈CI,ℝ,H,K∈CI×ℝ,ℝ, and l∈CI,ℝ. Under assumptions (i)–(iii), equations (54) and (55) have a unique solution.Proof.
Consider the natural order relation on△=CI,ℝ; that is, for e,f∈CI,ℝ,(57)e≺¯f⟺eu≤fu,∀u∈I.
It is well known that△ is a complete complex partial b-metric space with respect to(58)θcbe,f=e−f2+2+ie−f2+2,e,f∈CI,ℝ.
Suppose thattp is a monotone nondecreasing sequence in △ that converges to a point t∈△. Then, for every u∈I, the sequence of real numbers(59)t1u≤t2u≤⋯≤tpu≤⋯converges to tu. Therefore, for all u∈I, p∈ℕ, tp≤tu. Hence, tp≤t, for all p. Similarly, it can be verified that, if for all u∈I, ru is a limit of a monotone nondecreasing sequence rp in △, then ru≤rpu for all p, and hence r≤rp for all p.
Also,△×△=CI,ℝ×CI,ℝ is a partially ordered set under the following order relation in △×△:(60)e,f,t,r∈△×△,e,f≤t,r⟺eu≺¯tu,fu≽ru,∀u∈I.
For anye,f∈△, maxeu,fu and mineu,fu, for each u∈I, are in △ and are the upper and lower bounds of e,f, respectively. Therefore, for every e,f,t,r∈△×△, there exists maxe,t,minf,r∈△×△ that is comparable to e,f and t,r. Define Λ:△×△⟶△ by(61)Λe,fu=∫abT1u,sHs,es+Ks,fsds+∫abT2u,sHs,fs+Ks,esds+lu,for allu∈a,b.
We now claim thatΛ has the mixed monotone property. If t1,r≺¯t2,r, then(62)Λt1,ru=∫abT1u,sHs,t1s+Ks,rsds+∫abT2u,sHs,rs+Ks,t1sds+lu⪯∫abT1u,sHs,t2s+Ks,rsds+∫abT2u,sHs,rs+Ks,t2sds+lu=Λt2,ru.
Similarly, ift,r1⪯t,r2, then(63)Λt,r1u=∫abT1u,sHs,ts+Ks,r1sds+∫abT2u,sHs,r1s+Ks,tsds+lu⪯∫abT1u,sHs,ts+Ks,r2sds+∫abT2u,sHs,r2s+Ks,tsds+lu=Λt,r2u.
Thus,Λt,r is monotone nondecreasing in t and Λt,r is monotone nonincreasing in r. Also, for e,f⪯t,r, that is, t≽e, r≺¯f, it follows that(64)θcbΛt,r,Λe,f=Λt,r−Λe,f2+2+iΛt,r−Λe,f2+2=∫abT1u,sHs,ts+Ks,rsds+∫abT2u,sHs,rs+Ks,tsds+lu−∫abT1u,sHs,es+Ks,fsds−∫abT2u,sHs,fs+Ks,esds−lu2+2+i∫abT1u,sHs,ts+Ks,rsds+T2u,sHs,rs+Ks,tsds+lu−∫abT1u,sHs,es+Ks,fsds−∫abT2u,sHs,fs+Ks,esds−lu2+2θcbΛt,r,Λe,f=∫abT1u,sHs,ts−Hs,es+Ks,rs−Ks,fsds−∫abT2u,sHs,fs−Hs,rs+Ks,es−Ks,ts2+2+i∫abT1u,sHs,ts−Hs,es+Ks,rs−Ks,fsds−∫abT2u,sHs,fs−Hs,rs+Ks,es−Ks,ts2+2⪯∫abT1u,sts−es+fs−rsds−∫abT2u,sfs−rs+ts−esds2+2+i∫abT1u,sts−es+fs−rsds−∫abT2u,sfs−rs+ts−esds2+2⪯4∫abT1u,s−T2u,s2fs−rs2+ts−es2ds+4+i4∫abT1u,s−T2u,s2fs−rs2+ts−es2ds+4⪯4α4θcbf,r+θcbt,e=αθcbf,r+θcbt,e.
Now, all the hypotheses of Theorem4 are satisfied. Therefore, Λ has a unique coupled fixed point.


## 4. Conclusion

In 2019, Gunaseelan and Mishra [14] proved coupled fixed-point theorem on complex partial metric space. In this paper, we proved coupled fixed-point theorems on complex partial b-metric space using partially ordered set and mixed monotone property. An illustrative application in partially ordered complex partial b-metric space is given.



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