# Integral Equations Approach in Complex-Valued Generalizedb-Metric Spaces

Authors: Shahid Mehmood; Saif Ur Rehman; Ihsan Ullah; Rashad A. R. Bantan; Mohammed Elgarhy
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study a rational type common fixed-point theorem (CFP theorem) in complex-valued generalizedb-metric spaces (Gb-metric spaces) by using three self-mappings under the generalized contraction conditions. We find CFP and prove its uniqueness. To justify our result, we provide an illustrative example. Furthermore, we present a supportive application of the three Urysohn type integral equations (UTIEs) for the validity of our result. The UTIEs are


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## Body

## 1. Introduction

In 1922, Banach [1] proved a “Banach contraction principle” which is stated as “a single-valued contractive type mapping in a complete metric space has a unique FP.” Later on, this principle generalized in many directions, and some researchers contributed to the theory of FP. In [2], Nazam et al. presented the idea of weakly increasing F-contractions and established some results in ordered partial-metric space with an application. Hu and Gu [3] presented a new idea of Menger probabilistic S-metric space by using the concepts of probabilistic and S-metric spaces. They investigated its topological characteristics and established some FP theorems with illustrative examples.Bakhtin [4] introduced the idea of b-metric space. After that, Czerwik [5] established some FP results by using b-metric spaces. In [6], Boriceanu et al. extended the concept of fractal operator theory by introducing it in b-metric spaces and presented some generalized CFP results. While, Akkouchi [7] used the concept of an implicit relation in the said spaces and established CFP results for contractive type maps. Došenović at el. [8] discussed, complemented, improved, generalized, and enriched some FP results of (β−ψ1−ψ2) contraction in ordered b-metric spaces. They made a different approach of taking Picard’s sequence which help shorten the proof comparing other previous studies in literature. They also complimented and enriched CFP results for βq,ϕs,ψ contraction maps.Delfani et al. [9] contributed in ordered b-metric spaces by establishing FP theorems. In these results, they introduced the generalizations of F−ts and ψ,ϕ,F−ts contractions. They also provided a suitable example to verify their FP result. In [10], Karapinar et al. contributed in b-metric spaces by generalizing it to prove FP results in view of α,ψ-Meir-Keeler type contractions. These results improved and unified some previous results. Abdeljawad et al. [11] extended the b-metric spaces to double controlled metric spaces by improving control functions αγ1,γ2 and μγ1,γ2 on the right side of b-triangle, that is, dγ1,γ2≤αγ1,γ3dγ1,γ3+μγ3,γ2dγ3,γ2,∀γ1,γ2,γ3∈Υ. They also provided some examples in which two functions are not comparable.Mustafa and Sims [12] introduced the generalized idea of metric space and established some FP theorems by using Dhage’s theory. Later on, Mustafa et al. [13] proved some modified contractive-type FP results in this space. Abbas and Rhoades [14] started to investigate CFP results in the said spaces. In [15], Chugh et al. established the property P in G-metric space and proved some results. In this context, Mohanta et al. [16] contributed by establishing a CFP result which improved and supplemented some of the existing results. Mustafa introduced a mapping pair and obtained many CFP results for different contraction conditions. He supported these results through examples.In 2014, Aghajani et al. [17] presented the idea of Gb-metric space and proved a weakly compatible CFP theorem. Aydi [18] improved, complemented, unified, and generalized some well-known existing results in said spaces and established some coupled and tripled coincidence point results. Gupta in [19] extended some existing results from literature and obtained various FP results in Gb-metric spaces. In [20], Makran et al. proved general CFP theorem by using multivalued maps and established its application. Mebawondu et al. [21] proved FP results for a different contraction type maps which involve C-class, αsδ-admissible, Suzuki-type maps in Gb-metric spaces.Ege [22] gave the idea of complex-valued Gb-metric space and proved “Banach contraction principle and Kannan’s contraction theorems for FP.” Ege [23] used the idea of α-series to prove CFP results in said spaces. He also introduced α−ψ contraction maps to prove CFP results. Kumar and Vashistha [24] introduced the idea of coupled FP for mapping in the said space. They proved coupled FP results and supported it by providing an example satisfying their main result. Recently, Mehmood et al. [25] proved some CFP theorems by using compatible single-valued contractive type mappings on complex-valued b-metric spaces with an application.This article presents a contraction theorem in complex-valuedGb-metric spaces by using three self-maps to establish a generalized CFP-theorem. This result improves, extends, and modifies some of the existing results (e.g, [22, 26]). We present an example to support our work. We also present an application of the three UTIEs for the existence of a common solution to support our work. This study is organized as follows: Section 2 consists of preliminary concepts. In Section 3, we present a CFP theorem with an illustrative example. While in Section 4, we present an application to support our main work. Finally, in Section 5, we discuss the conclusion of our work.


## 2. Preliminaries

Let the complex numbers be denoted byC and zi,zii∈C. Now, we define ≤ as zi≤zii, iff Rezi≤Rezii and Imzi≤Imzii, where the real and imaginary parts of the complex number are denoted by Re and Im, respectively. Accordingly, zi≤zii, if any one of the following conditions holds:(1)
Rezi=Rezii and Imzi=Imzii,(2)
Rezi<Rezii and Imzi=Imzii,(3)
Rezi=Rezii and Imzi<Imzii,(4)
Rezi<Rezii and Imzi<Imzii.Particularly, we can writezi≤zii if zi≠zii and one of (2), (3), and (4) is satisfied.Remark 1 (see [27]).
The following given properties can be held and verified if(1)
β1,β2∈R and β1≤β2⇒β1y≤β2y∀y∈C,(2)
0≤zi≤zii⇒zi<zii,(3)
zi≤zii and zii<ziii⇒zi<ziii.Definition 1 (see [17]).
LetΥ≠∅ set and b≥1 be a given real number. A mapping G:Υ×Υ×Υ⟶R+ is called a Gb-metric if G holds the following axioms:(1)
Gγ1,γ2,γ3=0 if γ1=γ2=γ3(2)
0<Gγ1,γ1,γ2 for all γ1,γ2∈Υ with γ1≠γ2(3)
Gγ1,γ1,γ2≤Gγ1,γ2,γ3 for all γ1,γ2,γ3∈Υ with γ2≠γ3(4)
Gγ1,γ2,γ3=Gpγ1,γ2,γ3, where p is a permutation of γ1,γ2,γ3(5)
Gγ1,γ2,γ3≤bGγ1,a,a+Ga,γ2,γ3 for all γ1,γ2,γ3,a∈Υ
Then, a pairΥ,G is called a Gb-metric space.
Note that each G-metric space is aGb-metric space with b=1.Definition 2 (see [22]).
LetΥ≠∅ set and b≥1 be a given real number. A mapping G:Υ×Υ×Υ⟶C is called a complex-valued Gb-metric if G holds the following axioms:(1)
Gγ1,γ2,γ3=0 if γ1=γ2=γ3(2)
0<Gγ1,γ1,γ2 for all γ1,γ2∈Υ with γ1≠γ2(3)
Gγ1,γ1,γ2≤Gγ1,γ2,γ3 for all γ1,γ2,γ3∈Υ with γ2≠γ3(4)
Gγ1,γ2,γ3=Gpγ1,γ2,γ3, where p is a permutation of γ1,γ2,γ3(5)
Gγ1,γ2,γ3≤bGγ1,a,a+Ga,γ2,γ3 for all γ1,γ2,γ3,a∈Υ
Then, a pairΥ,G is called a complex-valued Gb-metric space.Example 1.
LetΥ=0,∞ and a mapping G:Υ×Υ×Υ⟶C be defined as(2)Gγ1,γ2,γ3=34γ1−γ2+34γ2−γ3+34γ3−γ12+i34γ1−γ2+34γ2−γ3+34γ3−γ12,for all γ1,γ2,γ3∈Υ. Then, G is a complex-valued Gb-metric with b=2, but it is not G-metric. To see this, let γ1=3,γ2=5,γ3=7,a=7/2, and we get G3,5,7=576/16+576/16i, G3,7/2,7/2=9/16+9/16i, and G7/2,5,7=441/16+441/16i; thus,G3,5,7=576/16+576/16i≰450/16+450/16i=G3,7/2,7/2+G7/2,5,7.
Clearly, property (5) of definition (2.3) is satisfied withb=2; hence,
Gγ1,γ2,γ3=3/4γ1−γ2+3/4γ2−γ3+3/4γ3−γ12+i3/4γ1−γ2+3/4γ2−γ3+3/4γ3−γ12 is Gb-metric.Proposition 1 (see [22]).
LetΥ,G be a complex-valued Gb-metric space. Then, ∀γ1,γ2,γ3∈Υ.(1)
Gγ1,γ2,γ3≤bGγ1,γ1,γ2+Gγ1,γ1,γ3(2)
Gγ1,γ2,γ2≤2bGγ1,γ1,γ2Definition 3 (see [22]).
LetΥ,G be a complex-valued Gb-metric space and γj be a sequence in Υ.(1)
γj is complex-valued Gb-convergent to γ if for every 0<a∈C, ∃k∈N, such that Gγ,γj,γm<a,∀j,m≥k.(2)
A sequenceγj is called complex-valued Gb-Cauchy if for every 0<a∈C, ∃k∈N, such that Gγj,γm,γl<a,∀j,m,l≥k.(3)
If every complex-valuedGb-Cauchy sequence is complex-valued Gb-convergent in Υ,G, then a pair Υ,G is called complex-valued Gb-complete.Proposition 2 (see [22]).
LetΥ,G be a complex-valued Gb-metric space and γj be a sequence in Υ. Then, γj is complex-valued Gb-convergent to γ if and only if Gγ,γj,γm⟶0 as n,m⟶∞.Theorem 1 (see [22]).
LetΥ,G be a complex-valued Gb-metric space; then, for a sequence γj in Υ and a point γ∈Υ, the following are equivalent:(1)
γj is complex-valued Gb-convergent to γ(2)
Gγj,γj,γ⟶0 as j⟶∞(3)
Gγj,γ,γ⟶0 as j⟶∞(4)
Gγm,γj,γ⟶0 as j⟶∞Proposition 3 (see [22]).
LetΥ,G be a complex-valued Gb-metric space and γj be a sequence in Υ. Then, γj is a complex-valued Gb-convergent to γ if and only if Gγ,γj,γm⟶0 as j,m⟶∞.Proof: Assume that γj is complex-valued Gb-convergent to γ and let(3)a=ε2+iε2,∀ε>0.Then, we have0<a∈C, and there is a natural number k, such that Gγ,γj,γm<a for all n,m≥k. Thus, Gγ,γj,γm<a=ε for all j,m≥k, and so, Gγ,γj,γm⟶0 as j,m⟶∞.Suppose thatGγ,γj,γm⟶0 as j,m⟶∞. For a given a∈C with 0<a, there exists a real number δ>0, such that for z∈C,(4)z<δ⇒z<a.Consideringδ, we have a natural number k, such that Gγ,γj,γm<δ for all j,m≥k. This means that Gγ,γj,γm<a for all j,m≥k, i.e., γj is complex-valued Gb-convergent to γ.


## 3. Main Result

Theorem 2.
LetΥ,G be a complete complex-valued Gb-metric space with coefficient b≥1 and F1,F2,F3:Υ⟶Υ be mappings satisfying(5)GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3+β2Wγ1,γ2,γ3,where(6)Wγ1,γ2,γ3=maxGγ1,γ2,F2γ2,GF1γ1,F1γ1,γ2,GF1γ1,γ2,γ2,GF2γ2,F2γ2,γ3,GF2γ2,γ3,γ3,Gγ1,F1γ1,F1γ1·Gγ2,F2γ2,F2γ21+Gγ1,γ2,γ2,Gγ2,F2γ2,F2γ2·Gγ3,F3γ3,F3γ31+GF1γ1,γ3,γ3,Gγ3,F3γ3,F3γ3·Gγ1,F1γ1,F1γ11+GF2γ2,F3γ3,F3γ3,for all γ1,γ2,γ3∈Υ,β1,β2∈0,1/3, such that β1+β2<1/3,bβ2<1/3, and β1+2bβ2<2/3. Then, F1,F2, and F3 have a unique CFP in Υ.
Proof. Fixγ0∈Υ, and γj be a sequence in Υ, such that(7)γ3n+1=F1γ3j,γ3j+2=F2γ3j+1,γ3j+3=F3γ3j+2∀n≥0.Now, by using (5),(8)Gγ3j+1,γ3j+2,γ3j+3=GF1γ3j,F2γ3j+1,F3γ3j+2≤β1Gγ3j,γ3j+1,γ3j+2+β2Wγ3j,γ3j+1,γ3j+2.This implies that(9)Gγ3j+1,γ3j+2,γ3j+3≤β1Gγ3j,γ3j+1,γ3j+2+β2Wγ3j,γ3j+1,γ3j+2where(10)Wγ3j,γ3j+1,γ3j+2=maxGγ3j,γ3j+1,F2γ3j+1,GF1γ3j,F1γ3j,γ3j+1,GF1γ3j,γ3j+1,γ3j+1,GF2γ3j+1,F2γ3j+1,γ3j+2,GF2γ3j+1,γ3j+2,γ3j+2,Gγ3j,F1γ3j,F1γ3j⋅Gγ3j+1,F2γ3j+1,F2γ3j+11+Gγ3j,γ3j+1,γ3j+1,Gγ3j+1,F2γ3j+1,F2γ3j+1⋅Gγ3j+2,F3γ3j+2,F3γ3j+21+GF1γ3j,γ3j+2,γ3j+2,Gγ3j+2,F3γ3j+2,F3γ3j+2⋅Gγ3j,F1γ3j,F1γ3j1+GF2γ3j+1,F3γ3j+2,F3γ3j+2=maxGγ3j,γ3j+1,γ3j+2,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+2,γ3j+2,γ3j+2,Gγ3j+2,γ3j+2,γ3j+2,Gγ3j,γ3j+1,γ3j+1⋅Gγ3j+1,γ3j+2,γ3j+21+Gγ3j,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2⋅Gγ3j+2,γ3j+3,γ3j+31+Gγ3j+1,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3⋅Gγ3j,γ3j+1,γ3j+11+Gγ3j+2,γ3j+3,γ3j+3.This implies that(11)Wγ3j,γ3j+1,γ3j+2≤maxGγ3j,γ3j+1,γ3j+2,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+2,γ3j+2,γ3j+2,Gγ3j+2,γ3j+2,γ3j+2,Gγ3j,γ3j+1,γ3j+1·Gγ3j+1,γ3j+2,γ3j+21+Gγ3j,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2·Gγ3j+2,γ3j+3,γ3j+31+Gγ3j+1,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3·Gγ3j,γ3j+1,γ3j+11+Gγ3j+2,γ3j+3,γ3j+3≤maxGγ3j,γ3j+1,γ3j+2,Gγ3j+1,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3,Gγ3j,γ3j+1,γ3j+1.By using Definition 2.3 (3) and after simplification, we get that(12)Wγ3j,γ3j+1,γ3j+2≤maxGγ3j+1,γ3j+2,γ3j+3,Gγ3j,γ3j+1,γ3j+2.Now, there are two possibilities:(i)
IfGγ3j+1,γ3j+2,γ3j+3 is a maximum term in Gγ3j+1,γ3j+2,γ3j+3,Gγ3j,γ3j+1,γ3j+2, then Wγ3j,γ3j+1,γ3j+2=Gγ3j+1,γ3j+2,γ3j+3; so, after simplification, (3.3) can be written as(13)Gγ3j+1,γ3j+2,γ3j+3≤a1Gγ3j,γ3j+1,γ3j+2,wherea1=β1/1−β2.(ii)
IfGγ3j,γ3j+1,γ3j+2 is a maximum term in Gγ3j+1,γ3j+2,γ3j+3,Gγ3j,γ3j+1,γ3j+2, then Wγ3j,γ3j+1,γ3j+2=Gγ3j,γ3j+1,γ3j+2; so, after simplification, (3.3) can be written as(14)Gγ3j+1,γ3j+2,γ3j+3≤a2Gγ3j,γ3j+1,γ3j+2,wherea2=β1+β2.Leta:=maxa1,a2<1/3; then, from (13) and (14), for all n≥0, we have(15)Gγ3j+1,γ3j+2,γ3j+3≤aGγ3j,γ3j+1,γ3j+2.Similarly,(16)Gγ3j,γ3j+1,γ3j+2≤aGγ3j−1,γ3j,γ3j+1.Now, from (16) and (15) and by induction, we have that(17)Gγ3j+1,γ3j+2,γ3j+3≤aGγ3j,γ3j+1,γ3j+2≤a2|Gγ3j−1,γ3j,γ3j+1≤⋯≤a3j+1Gγ0,γ1,γ2.Letm,n∈N and >n, and we have that(18)Gγj,γm,γm≤bGγj,γn+1,γn+1+bGγn+1,γm,γm≤bGγj,γn+1,γn+1+b2Gγn+1,γn+2,γn+2+⋯+bm−nGγm−1,γm,γm≤banGγ0,γ1,γ1+b2an+1Gγ0,γ1,γ1+⋯+bm−nam−1Gγ0,γ1,γ1≤ban+b2an+1+⋯+bm−nam−1Gγ0,γ1,γ1=ban+b2an+1+⋯+bm−nam−1Gγ0,γ1,γ1=ban1+ba+b2a2…+bm−n+1am−n+1Gγ0,γ1,γ1=ban∑t=0m−n+1btatGγ0,γ1,γ1≤ban∑t=0∞btatGγ0,γ1,γ1=ban1−baGγ0,γ1,γ1⟶0,asn⟶∞.By Proposition 2.5 (1), we haveGγj,γm,γl≤bGγj,γm,γm+Gγl,γm,γm for n,m,l∈N. If we take limit as n,m,l⟶∞, we obtain Gγj,γm,γl⟶0. It implies γj is a Gb-Cauchy sequence. Since, Υ is complete complex-valued Gb-metric space, ∃ξ∈Υ, such that, γj⟶ξ, as n⟶∞ or limn⟶∞γj=ξ. We have to show that F1ξ=ξ; by contrary case, let F1ξ≠ξ. Now by (5),(19)GF1ξ,γ3j+2,γ3j+3=GF1ξ,F2γ3j+1,F3γ3j+2≤β1Gξ,γ3j+1,γ3j+2+β2Wξ,γ3j+1,γ3j+2.This implies that(20)GF1ξ,γ3j+2,γ3j+3≤β1Gξ,γ3j+1,γ3j+2+β2Wξ,γ3j+1,γ3j+2,where(21)Wξ,γ3j+1,γ3j+2=maxGξ,γ3j+1,F2γ3j+1,GF1ξ,F1ξ,γ3j+1,GF1ξ,γ3j+1,γ3j+1,GF2γ3j+1,F2γ3j+1,γ3j+2,GF2γ3j+1,γ3j+2,γ3j+2,Gξ,F1ξ,F1ξ⋅Gγ3j+1,F2γ3j+1,F2γ3j+11+Gξ,γ3j+1,γ3j+1,Gγ3j+1,F2γ3j+1,F2γ3j+1⋅Gγ3j+2,F3γ3j+2,F3γ3j+21+GF1ξ,γ3j+2,γ3j+2,Gγ3j+2,F3γ3j+2,F3γ3j+2⋅Gξ,F1ξ,F1ξ1+GF2γ3j+1,F3γ3j+2,F3γ3j+2=maxGξ,γ3j+1,γ3j+2,GF1ξ,F1ξ,γ3j+1,GF1ξ,γ3j+1,γ3j+1,Gγ3j+2,γ3j+2,γ3j+2,Gγ3j+2,γ3j+2,γ3j+2,Gξ,F1ξ,F1ξ⋅Gγ3j+1,γ3j+2,γ3j+21+Gξ,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2⋅Gγ3j+2,γ3j+3,γ3j+31+GF1ξ,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3⋅Gξ,F1ξ,F1ξ1+Gγ3j+2,γ3j+3,γ3j+3.This implies that(22)Wξ,γ3j+1,γ3j+2≤maxGξ,γ3j+1,γ3j+2,GF1ξ,F1ξ,γ3j+1,GF1ξ,γ3j+1,γ3j+1,Gγ3j+2,γ3j+2,γ3j+2,Gγ3j+2,γ3j+2,γ3j+2,Gξ,F1ξ,F1ξ⋅Gγ3j+1,γ3j+2,γ3j+21+Gξ,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2⋅Gγ3j+2,γ3j+3,γ3j+31+GF1ξ,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3⋅Gξ,F1ξ,F1ξ1+Gγ3j+2,γ3j+3,γ3j+3.After simplification, we get that(23)Wξ,γ3j+1,γ3j+2≤maxGξ,γ3j+1,γ3j+2,GF1ξ,F1ξ,γ3j+1,GF1ξ,γ3j+1,γ3j+1,Gξ,F1ξ,F1ξ⋅Gγ3j+1,γ3j+2,γ3j+21+Gξ,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2⋅Gγ3j+2,γ3j+3,γ3j+31+GF1ξ,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3⋅Gξ,F1ξ,F1ξ1+Gγ3j+2,γ3j+3,γ3j+3.Now, there are six possibilities:(i)
fGξ,γ3j+1,γ3j+2 is a maximum term in (23), then (21) can be written as(24)GF1ξ,γ3j+2,γ3j+3≤β1Gξ,γ3j+1,γ3j+2+β2Gξ,γ3j+1,γ3j+2.Applyinglimn⟶∞ on both sides, we get(25)GF1ξ,ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ.This implies thatGF1ξ,ξ,ξ=0; thus,(26)F1ξ=ξ.(ii)
IfGF1ξ,F1ξ,γ3j+1 is a maximum term in (23), then (21) can be written as(27)GF1ξ,γ3j+2,γ3j+3≤β1Gξ,γ3j+1,γ3j+2+β2GF1ξ,F1ξ,γ3j+1.Applyinglimn⟶∞ on both sides, we get(28)GF1ξ,ξ,ξ≤β1Gξ,ξ,ξ+β2GF1ξ,F1ξ,ξ≤2bβ2GF1ξ,ξ,ξ∵Definition2.52.This implies that1−2bβ2GF1ξ,ξ,ξ≤0 is a contradiction, where 1−2bβ2≠0⇒GF1ξ,ξ,ξ=0; thus,(29)F1ξ=ξ.(iii)
IfGF1ξ,γ3j+1,γ3j+1 is a maximum term in (23), then (21) can be written as(30)GF1ξ,γ3j+2,γ3j+3≤β1Gξ,γ3j+1,γ3j+2+β2GF1ξ,γ3j+1,γ3j+1.Applyinglimn⟶∞ on both sides, we get(31)GF1ξ,ξ,ξ≤β1Gξ,ξ,ξ+β2GF1ξ,ξ,ξ.This implies that1−β2GF1ξ,ξ,ξ≤0 is a contradiction, where 1−β2≠0⇒GF1ξ,ξ,ξ=0; thus,(32)F1ξ=ξ.(iv)
IfGξ,F1ξ,F1ξ·Gγ3j+1,γ3j+2,γ3j+2/1+Gξ,γ3j+1,γ3j+1 is a maximum term in (23), then (21) can be written as(33)GF1ξ,γ3j+2,γ3j+3≤β1Gξ,γ3j+1,γ3j+2+β2Gξ,F1ξ,F1ξ·Gγ3j+1,γ3j+2,γ3j+21+Gξ,γ3j+1,γ3j+1.Applyinglimn⟶∞ on both sides, we get(34)GF1ξ,ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,F1ξ,F1ξ·Gξ,ξ,ξ1+Gξ,ξ,ξ.This implies thatGF1ξ,ξ,ξ=0; thus,(35)F1ξ=ξ.(v)
IfGγ3j+1,γ3j+2,γ3j+2·Gγ3j+2,γ3j+3,γ3j+3/1+GF1ξ,γ3j+2,γ3j+2 is a maximum term in (23), then (21) can be written as(36)GF1ξ,γ3j+2,γ3j+3≤β1Gξ,γ3j+1,γ3j+2+β2Gγ3j+1,γ3j+2,γ3j+2·Gγ3j+2,γ3j+3,γ3j+31+GF1ξ,γ3j+2,γ3j+2Applyinglimn⟶∞ on both sides, we get(37)GF1ξ,ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ·Gξ,ξ,ξ1+GF1ξ,ξ,ξ.This implies thatGF1ξ,ξ,ξ=0; thus,(38)F1ξ=ξ.(vi)
IfGγ3j+2,γ3j+3,γ3j+3·Gξ,F1ξ,F1ξ/1+Gγ3j+2,γ3j+3,γ3j+3 is a maximum term in (23), then (21) can be written as(39)GF1ξ,γ3j+2,γ3j+3≤β1Gξ,γ3j+1,γ3j+2+β2Gγ3j+2,γ3j+3,γ3j+3·Gξ,F1ξ,F1ξ1+Gγ3j+2,γ3j+3,γ3j+3.Applyinglimn⟶∞ on both sides, we get(40)GF1ξ,ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ·Gξ,F1ξ,F1ξ1+Gξ,ξ,ξ.This implies thatGF1ξ,ξ,ξ=0; thus,(41)F1ξ=ξ.From (26)–(41), we get that ξ is a FP of F1, that is,(42)F1ξ=ξ.Next, we have to show thatF2ξ=ξ; by contrary case, let F2ξ≠ξ. Now by (5),(43)Gγ3j+1,F2ξ,γ3j+3=GF1γ3j,F2ξ,F3γ3j+2≤β1Gγ3j,ξ,γ3j+2+β2Wγ3j,ξ,γ3j+2.This implies that(44)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2Wγ3j,ξ,γ3j+2,where(45)Wγ3j,ξ,γ3j+2=maxGγ3j,ξ,F2ξ,GF1γ3j,F1γ3j,ξ,GF1γ3j,ξ,ξ,GF2ξ,F2ξ,γ3j+2,GF2ξ,γ3j+2,γ3j+2,Gγ3j,F1γ3j,F1γ3j·Gξ,F2ξ,F2ξ1+Gγ3j,ξ,ξ,Gξ,F2ξ,F2ξ·Gγ3j+2,F3γ3j+2,F3γ3j+21+GF1γ3j,γ3j+2,γ3j+2,Gγ3j+2,F3γ3j+2,F3γ3j+2·Gγ3j,F1γ3j,F1γ3j1+GF2ξ,F3γ3j+2,F3γ3j+2=maxGγ3j,ξ,F2ξ,Gγ3j+1,γ3j+1,ξ,Gγ3j+1,ξ,ξ,GF2ξ,F2ξ,γ3j+2,GF2ξ,γ3j+2,γ3j+2,Gγ3j,γ3j+1,γ3j+1·Gξ,F2ξ,F2ξ1+Gγ3j,ξ,ξ,Gξ,F2ξ,F2ξ·Gγ3j+2,γ3j+3,γ3j+31+Gγ3j+1,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3·Gγ3j,γ3j+1,γ3j+11+GF2ξ,γ3j+3,γ3j+3.This implies that(46)Wγ3j,ξ,γ3j+2=maxGγ3j,ξ,F2ξ,Gγ3j+1,γ3j+1,ξ,Gγ3j+1,ξ,ξ,GF2ξ,F2ξ,γ3j+2,GF2ξ,γ3j+2,γ3j+2,Gγ3j,γ3j+1,γ3j+1·Gξ,F2ξ,F2ξ1+Gγ3j,ξ,ξ,Gξ,F2ξ,F2ξ·Gγ3j+2,γ3j+3,γ3j+31+Gγ3j+1,γ3j+2,γ3j+2,Gγ3j+2,γ3j+3,γ3j+3·Gγ3j,γ3j+1,γ3j+11+GF2ξ,γ3j+3,γ3j+3.Now, there are eight possibilities:(i)
IfGγ3j,ξ,F2ξ is a maximum term in (46), then after simplification, (44) can be written as(47)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2Gγ3j,ξ,F2ξ.Applyinglimn⟶∞ on both sides, we get(48)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,F2ξ.This implies that1−β2Gξ,F2ξ,ξ≤0 is a contradiction, where 1−β2≠0⇒Gξ,F2ξ,ξ=0; thus,(49)F2ξ=ξ.(ii)
IfGγ3j+1,γ3j+1,ξ is a maximum term in (46), then (44) can be written as(50)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2Gγ3j+1,γ3j+1,ξ.Applyinglimn⟶∞ on both sides, we get(51)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ.This implies thatGξ,F2ξ,ξ=0; thus,(52)F2ξ=ξ.(iii)
IfGγ3j+1,ξ,ξ is a maximum term in (46), then (44) can be written as(53)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2Gγ3j+1,ξ,ξ.Applyinglimn⟶∞ on both sides, we get(54)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ.This implies thatGξ,F2ξ,ξ=0; thus,(55)F2ξ=ξ.(iv)
IfGF2ξ,F2ξ,γ3j+2 is a maximum term in (46), then (44) can be written as(56)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2GF2ξ,F2ξ,γ3j+2.Applyinglimn⟶∞ on both sides, we get(57)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2GF2ξ,F2ξ,ξ≤2bβ2Gξ,F2ξ,ξ∵Proposition2.52.This implies that1−2bβ2Gξ,F2ξ,ξ≤0 is a contradiction, where 1−2bβ2≠0⇒Gξ,F2ξ,ξ=0; thus,(58)F2ξ=ξ.(v)
IfGF2ξ,γ3j+2,γ3j+2 is a maximum term in (46), then (44) can be written as(59)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2GF2ξ,γ3j+2,γ3j+2.Applyinglimn⟶∞ on both sides, we get(60)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2GF2ξ,ξ,ξ.This implies that1−β2Gξ,F2ξ,ξ≤0 is a contradiction, where 1−β2≠0⇒Gξ,F2ξ,ξ=0; thus,(61)F2ξ=ξ.(vi)
IfGγ3j,γ3j+1,γ3j+1·Gξ,F2ξ,F2ξ/1+Gγ3j,ξ,ξ is a maximum term in (46), then (44) can be written as(62)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2Gγ3j,γ3j+1,γ3j+1·Gξ,F2ξ,F2ξ1+Gγ3j,ξ,ξ.Applyinglimn⟶∞ on both sides, we get(63)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ·Gξ,F2ξ,F2ξ1+Gξ,ξ,ξ.This implies thatGξ,F2ξ,ξ=0; thus,(64)F2ξ=ξ.(vii)
IfGξ,F2ξ,F2ξ·Gγ3j+2,γ3j+3,γ3j+3/1+Gγ3j+1,γ3j+2,γ3j+2 is a maximum term in (46), then (44) can be written as(65)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2Gξ,F2ξ,F2ξ·Gγ3j+2,γ3j+3,γ3j+31+Gγ3j+1,γ3j+2,γ3j+2.Applyinglimn⟶∞ on both sides, we get(66)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,F2ξ,F2ξ·Gξ,ξ,ξ1+Gξ,ξ,ξ.This implies thatGξ,F2ξ,ξ=0; thus,(67)F2ξ=ξ.(viii)
IfGγ3j+2,γ3j+3,γ3j+3·Gγ3j,γ3j+1,γ3j+1/1+GF2ξ,γ3j+3,γ3j+3 is a maximum term in (46), then (44) can be written as(68)Gγ3j+1,F2ξ,γ3j+3≤β1Gγ3j,ξ,γ3j+2+β2Gγ3j+2,γ3j+3,γ3j+3·Gγ3j,γ3j+1,γ3j+11+GF2ξ,3j+3,γ3j+3.Applyinglimn⟶∞ on both sides, we get(69)Gξ,F2ξ,ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ·Gξ,ξ,ξ1+GF2ξ,ξ,ξ.This implies thatGξ,F2ξ,ξ=0; thus,(70)F2ξ=ξ.From (49) to (70), we find that ξ is a FP of F2, that is,(71)F2ξ=ξ.Now, we have to show thatF3ξ=ξ; by contrary case, let F3ξ≠ξ. Now, by (5),(72)Gγ3j+1,γ3j+2,F3ξ=GF1γ3j,F2γ3j+1,F3ξ≤β1Gγ3j,γ3j+1,ξ+β2Wγ3j,γ3j+1,ξ.This implies that(73)Gγ3j+1,γ3j+2,F3ξ≤β1Gγ3j,γ3j+1,ξ+β2Wγ3j,γ3j+1,ξ,where(74)Wγ3j,γ3j+1,ξ=maxGγ3j,γ3j+1,F2γ3j+1,GF1γ3j,F1γ3j,γ3j+1,GF1γ3j,γ3j+1,γ3j+1,GF2γ3j+1,F2γ3j+1,ξ,GF2γ3j+1,ξ,ξ,Gγ3j,F1γ3j,F1γ3j·Gγ3j+1,F2γ3j+1,F2γ3j+11+Gγ3j,γ3j+1,γ3j+1,Gγ3j+1,F2γ3j+1,F2γ3j+1·Gξ,F3ξ,F3ξ1+GF1γ3j,ξ,ξ,Gξ,F3ξ,F3ξ·Gγ3j,F1γ3j,F1γ3j1+GF2γ3j+1,F3ξ,F3ξ=maxGγ3j,γ3j+1,γ3j+2,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+2,γ3j+2,ξ,Gγ3j+2,ξ,ξ,Gγ3j,γ3j+1,γ3j+1·Gγ3j+1,γ3j+2,γ3j+21+Gγ3j,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2·Gξ,F3ξ,F3ξ1+Gγ3j+1,ξ,ξ,Gξ,F3ξ,F3ξ·Gγ3j,γ3j+1,γ3j+11+Gγ3j+2,F3ξ,F3ξ.This implies that(75)Wγ3j,γ3j+1,ξ=maxGγ3j,γ3j+1,γ3j+2,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+1,γ3j+1,γ3j+1,Gγ3j+2,γ3j+2,ξ,Gγ3j+2,ξ,ξ,Gγ3j,γ3j+1,γ3j+1·Gγ3j+1,γ3j+2,γ3j+21+Gγ3j,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2·Gξ,F3ξ,F3ξ1+Gγ3j+1,ξ,ξ,Gξ,F3ξ,F3ξ·Gγ3j,γ3j+1,γ3j+11+Gγ3j+2,F3ξ,F3ξAfter simplification, we get that(76)Wγ3j,γ3j+1,ξ=maxGγ3j,γ3j+1,γ3j+2,Gγ3j+2,γ3j+2,ξ,Gγ3j+2,ξ,ξ,Gγ3j,γ3j+1,γ3j+1·Gγ3j+1,γ3j+2,γ3j+21+Gγ3j,γ3j+1,γ3j+1,Gγ3j+1,γ3j+2,γ3j+2·Gξ,F3ξ,F3ξ1+Gγ3j+1,ξ,ξ,Gξ,F3ξ,F3ξ·Gγ3j,γ3j+1,γ3j+11+Gγ3j+2,F3ξ,F3ξ.Now, there are six possibilities:(i)
IfGγ3j,γ3j+1,γ3j+2 is a maximum term in (76), then (73) can be written as(77)Gγ3j+1,γ3j+2,F3ξ≤β1Gγ3j,γ3j+1,ξ+β2Gγ3j,γ3j+1,γ3j+2.Applyinglimn⟶∞ on both sides, we get(78)Gξ,ξ,F3ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ.This implies thatGξ,ξ,F3ξ=0; thus,(79)F3ξ=ξ.(ii)
IfGγ3j+2,γ3j+2,ξ is a maximum term in (76), then (73) can be written as(80)Gγ3j+1,γ3j+2,F3ξ≤β1Gγ3j,γ3j+1ξ+β2Gγ3j+2,γ3j+2,ξ.Applyinglimn⟶∞ on both sides, we get(81)Gξ,ξ,F3ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ.This implies thatGξ,ξ,F3ξ=0; thus,(82)F3ξ=ξ.(iii)
IfGγ3j+2,ξ,ξ is a maximum term in (76), then (73) can be written as(83)Gγ3j+1,γ3j+2,F3ξ≤β1Gγ3j,γ3j+1,ξ+β2Gγ3j+2,ξ,ξ,Applyinglimn⟶∞ on both sides, we get(84)Gξ,ξ,F3ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ.This implies thatGξ,ξ,F3ξ=0; thus,(85)F3ξ=ξ.(iv)
IfGγ3j,γ3j+1,γ3j+1·Gγ3j+1,γ3j+2,γ3j+2/1+Gγ3j,γ3j+1,γ3j+1 is a maximum term in (76), then (73) can be written as(86)Gγ3j+1,γ3j+2,F3ξ≤β1Gγ3j,γ3j+1,ξ+β2Gγ3j,γ3j+1,γ3j+1·Gγ3j+1,γ3j+2,γ3j+21+Gγ3j,γ3j+1,γ3j+1.Applyinglimn⟶∞ on both sides, we get(87)Gξ,ξ,F3ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ·Gξ,ξ,ξ1+Gξ,ξ,ξ.This implies thatGξ,ξ,F3ξ=0; thus,(88)F3ξ=ξ.(v)
IfGγ3j+1,γ3j+2,γ3j+2/1+Gγ3j+1,ξ,ξ is a maximum term in (76), then (73) can be written as(89)Gγ3j+1,γ3j+2,F3ξ≤β1Gγ3j,γ3j+1,ξ+β2Gγ3j+1,γ3j+2,γ3j+2·Gξ,F3ξ,F3ξ1+Gγ3j+1,ξ,ξ.Applyinglimn⟶∞ on both sides, we get(90)Gξ,ξ,F3ξ≤β1Gξ,ξ,ξ+β2Gξ,ξ,ξ·Gξ,F3ξ,F3ξ1+Gξ,ξ,ξ.This implies thatGξ,ξ,F3ξ=0; thus,(91)F3ξ=ξ.(vi)
IfGξ,F3ξ,F3ξ/1+Gγ3j+2,F3ξ,F3ξ is a maximum term in (76), then (73) can be written as(92)Gγ3j+1,γ3j+2,F3ξ≤β1Gγ3j,γ3j+1,ξ+β2Gξ,F3ξ,F3ξ·Gγ3j,γ3j+1,γ3j+11+Gγ3j+2,F3ξ,F3ξ.Applyinglimn⟶∞ on both sides, we get(93)Gξ,ξ,F3ξ≤β1Gξ,ξ,ξ+β2Gξ,F3ξ,F3ξ·Gξ,ξ,ξ1+Gξ,F3ξ,F3ξ.This implies thatGξ,ξ,F3ξ=0; thus,(94)F3ξ=ξ.From (79)–(94), we find that ξ is a FP of F3, that is,(95)F3ξ=ξ.Hence proved thatξ is a CFP of F1,F2 and F3, that is,(96)F1ξ=F2ξ=F3ξ=ξ.Uniqueness: assume thatξ∗∈Υ is another CFP of the mappings F1,F2, and F3, such that(97)F1ξ=F2ξ=F3ξ=ξ,F1ξ∗=F2ξ∗=F3ξ∗=ξ∗.Then, from (5), we have that(98)Gξ,ξ∗,ξ∗=GF1ξ,F2ξ∗,F3ξ∗≤β1Gξ,ξ∗,ξ∗+β2Wξ,ξ∗,ξ∗.This implies that(99)Gξ,ξ∗,ξ∗≤β1Gξ,ξ∗,ξ∗+β2Wξ,ξ∗,ξ∗,where(100)Wξ,ξ∗,ξ∗=maxGξ,ξ∗,F2ξ∗,GF1ξ,F1ξ,ξ∗,GF1ξ,ξ∗,ξ∗,GF2ξ∗,F2ξ∗,ξ∗,GF2ξ∗,ξ∗,ξ∗,Gξ,F1ξ,F1ξ⋅Gξ∗,F2ξ∗,F2ξ∗1+Gξ,ξ∗,ξ∗,Gξ∗,F2ξ∗,F2ξ∗⋅Gξ∗,F3ξ∗,F3ξ∗1+GF1ξ,ξ∗,ξ∗,Gξ∗,F3ξ∗,F3ξ∗⋅Gξ,F1ξ,F1ξ1+GF2ξ∗,F3ξ∗,F3ξ∗=maxGξ,ξ∗,ξ∗,Gξ,ξ,ξ∗,Gξ,ξ∗,ξ∗,Gξ∗,ξ∗,ξ∗,Gξ∗,ξ∗,ξ∗,Gξ,ξ,ξ⋅Gξ∗,ξ∗,ξ∗1+Gξ,ξ∗,ξ∗,Gξ∗,ξ∗,ξ∗⋅Gξ∗,ξ∗,ξ∗1+Gξ,ξ∗,ξ∗,Gξ∗,ξ∗,ξ∗⋅Gξ,ξ,ξ1+Gξ∗,ξ∗,ξ∗.This implies that(101)Wξ,ξ∗,ξ∗=maxGξ,ξ∗,ξ∗,Gξ,ξ,ξ∗,Gξ,ξ∗,ξ∗,Gξ∗,ξ∗,ξ∗,Gξ∗,ξ∗,ξ∗,Gξ,ξ,ξ·Gξ∗,ξ∗,ξ∗1+Gξ,ξ∗,ξ∗,Gξ∗,ξ∗,ξ∗·Gξ∗,ξ∗,ξ∗1+Gξ,ξ∗,ξ∗,ξ∗,ξ∗,ξ∗·Gξ,ξ,ξ1+Gξ∗,ξ∗,ξ∗.Now, by using Proposition 2.5 (2) and after simplifying, we get that(102)Wξ,ξ∗,ξ∗≤maxGξ,ξ∗,ξ∗,2bGξ,ξ∗,ξ∗.Clearly,2bGξ,ξ∗,ξ∗ is a maximum term in (102), so (99) can be written as(103)Gξ,ξ∗,ξ∗≤β1Gξ,ξ∗,ξ∗+2bβ2Gξ,ξ∗,ξ∗.This implies that1−β1−2bβ2Gξ,ξ∗,ξ∗≤0 is a contradiction, where 1−β1−2bβ2≠0⇒Gξ,ξ∗,ξ∗=0⇒ξ=ξ∗. Hence proved that F1,F2, and F3 have a unique CFP in Υ.Corollary 1.
LetΥ,G be a complete complex-valued Gb-metric space with coefficient b≥1 and F1,F2,F3:Υ⟶Υ be mappings satisfying(104)GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3+β2Wγ1,γ2,γ3,where(105)Wγ1,γ2,γ3=maxGγ1,γ2,F2γ2,GF1γ1,F1γ1,γ2,GF1γ1,γ2,γ2,GF2γ2,F2γ2,γ3,GF2γ2,γ3,γ3,Gγ1,F1γ1,F1γ1·Gγ2,F2γ2,F2γ21+Gγ1,γ2,γ2,,for all γ1,γ2,γ3∈Υ,β1,β2∈0,1/3, such that β1+β2<1/3,bβ2<1/3, and β1+2bβ2<2/3; then, F1,F2, and F3 have a unique CFP in Υ.Corollary 2.
LetΥ,G be a complete complex-valued Gb-metric space with coefficient b≥1 and F1,F2,F3:Υ⟶Υ be mappings satisfying(106)GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3+β2Wγ1,γ2,γ3,where(107)Wγ1,γ2,γ3=maxGγ1,γ2,F2γ2,GF1γ1,F1γ1,γ2,GF1γ1,γ2,γ2,GF2γ2,F2γ2,γ3,GF2γ2,γ3,γ3,Gγ2,F2γ2,F2γ2·Gγ3,F3γ3,F3γ31+GF1γ1,γ3,γ3,,for all γ1,γ2,γ3∈Υ,β1,β2∈0,1/3, such that β1+β2<1/3,bβ2<1/3, and β1+2bβ2<2/3; then, F1,F2, and F3 have a unique CFP in Υ.Corollary 3.
LetΥ,G be a complete complex-valued Gb-metric space with coefficient b≥1 and F1,F2,F3:Υ⟶Υ be mappings satisfying(108)GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3+β2Wγ1,γ2,γ3,where(109)Wγ1,γ2,γ3=maxGγ1,γ2,F2γ2,GF1γ1,F1γ1,γ2,GF1γ1,γ2,γ2,GF2γ2,F2γ2,γ3,GF2γ2,γ3,γ3,Gγ3,F3γ3,F3γ3·Gγ1,F1γ1,F1γ11+GF2γ2,F3γ3,F3γ3,for all γ1,γ2,γ3∈Υ,β1,β2∈0,1/3, such that β1+β2<1/3,bβ2<1/3, and β1+2bβ2<2/3; then, F1,F2, and F3 have a unique CFP in Υ.Remark 2.
If we putβ2=0 and F1=F2=F3 in Theorem 2, we get (Theorem 3.7 [22]).Example 2.
LetΥ,G be a complex-valued Gb-metric space, where Υ=0,1 and G:Υ×Υ×Υ⟶C, with Gγ1,γ2,γ3=3/4γ1−γ2+3/4γ2−γ3+3/4γ3−γ12+i3/4γ1−γ2+3/4γ2−γ3+3/4γ3−γ12, for all γ1,γ2,γ3∈Υ. Now, we define F1,F2,F3:Υ⟶Υ as(110)F1γ=F2γ=F3γ=ln1+γ5+γ.
Notice that(111)Gγ1,γ2,γ3,Wγ1,γ2,γ3≥0.
In all regards, it is enough to show thatGF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3, for all γ1,γ2,γ3∈0,1 and β1,β2∈0,1/3, with β1+β2<1/3, bβ2<1/3, and β1+2bβ2<2/3.(112)GF1γ1,F2γ2,F3γ3=34F1γ1−F2γ2+34F2γ2−F3γ3+34F3γ3−F1γ12+i34F1γ1−F2γ2+34F2γ2−F3γ3+34F3γ3−F1γ12=34ln1+γ15+γ1−ln1+γ25+γ2+34ln1+γ25+γ2−ln1+γ35+γ3+34ln1+γ35+γ3−ln1+γ15+γ12+i34ln1+γ15+γ1−ln1+γ25+γ2+34ln1+γ25+γ2−ln1+γ35+γ3+34ln1+γ35+γ3−ln1+γ15+γ12≤34γ15+γ1−γ25+γ2+34γ25+γ2−γ35+γ3+34γ35+γ3−γ15+γ12+i34γ15+γ1−γ25+γ2+34γ25+γ2−γ35+γ3+34γ35+γ3−γ15+γ12≤345γ1−5γ225+345γ2−5γ325+345γ3−5γ1252+i345γ1−5γ225+345γ2−5γ325+345γ3−5γ1252=12534γ1−γ2+34γ2−γ3+34γ3−γ12+i34γ1−γ2+34γ2−γ3+34γ3−γ12.(113)Gγ1,γ2,γ3=34γ1−γ2+34γ2−γ3+34γ3−γ12+i34γ1−γ2+34γ2−γ3+34γ3−γ12.
Forγ1,γ2,γ3∈0,1, we discuss different cases with β1=1/10,β2=1/20, and b=2. Hence,(114)β1+β2=110+120<13,bβ2=2120<13,(115)β1+2bβ2=110+22120<23.Case 1: letγ1=0,γ2=0,γ3=0; then from (112) and (113), directly we get that GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3. Hence, (3.1) is satisfied with β1=1/10,β2=1/20, and b=2.Case 2: letγ1=0,γ2=0,γ3=1/2; then from (112) and (113), we find GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3 is satisfy with β1=1/10, i.e.,(116)125340−0+340−12+3412−02+i340−0+340−12+3412−02≤β1340−0+340−12+3412−02+i340−0+340−12+3412−02⇒9400+i9400≤9160+i9160.Hence, (3.1) is satisfied withβ1=1/10,β2=1/20, and b=2.Case 3: letγ1=1/2,γ2=1/3,γ3=1/4; then from (112) and (113), we find GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3 is satisfy with β1=1/10, i.e.,(117)1253412−13+3413−14+3414−122+i3412−13+3413−14+3414−122≤β13412−13+3413−14+3414−122+i3412−13+3413−14+3414−122⇒91600+i91600≤9640+i9640.Hence, (3.1) is satisfied withβ1=1/10,β2=1/20, and b=2.Case 4: letγ1=1/2,γ2=1/2,γ3=1; then from (112) and (113), we find GF1γ1,F2γ2,F3γ3≤β1Gγ1,γ2,γ3 is satisfy with β1=1/10, i.e.,(118)1253412−12+3412−1+341−122+i3412−12+3412−1+341−122≤β13412−12+3412−1+341−122+i3412−12+3412−1+341−122⇒9400+i9400≤9160+i9160.
Hence, (3.1) is satisfied withβ1=1/10,β2=1/20, and b=2. Thus, all the conditions of Theorem 2 are satisfied with noticing that the point 0∈Υ, which remains fixed under mappings F1,F2, and F3, is indeed unique.


## 4. Applications

In this section, we present an application of the three UTIEs to support our main work. LetΥ=Ck1,k2,Rn be the set of all real-valued continuous functions defined on k1,k2. Now, we state and prove a result based on the three UTIEs to uplift our work.Theorem 3.
LetΥ=Ck1,k2,Rn, where k1,k2⊆R and G:Υ×Υ×Υ⟶C are defined as(119)Gγ1,γ2,γ3=γ1q−γ2q+γ2q−γ3q+γ3q−γ1q21+k12eicotk1,for all γ1,γ2,γ3∈Υ and q∈k1,k2. Consider the UTIEs are(120)γ1q=∫k1k2Q1q,r,γ1rdr+ℏ1q,γ2q=∫k1k2Q2q,r,γ2rdr+ℏ2q,γ3q=∫k1k2Q3q,r,γ3rdr+ℏ3q,where r∈k1,k2. Let Q1,Q2,Q3:k1,k2×k1,k2×Rn⟶Rn are such that Dγ1,Eγ2,Fγ3∈Υ; for every γ1,γ2,γ3∈Υ, we have that(121)Dγ1q=∫k1k2Q1q,r,γ1rdr,Eγ2q=∫k1k2Q2q,r,γ2rdr,Fγ3q=∫k1k2Q3q,r,γ3rdr.
If there existsμ∈0,1,suchthatforallγ1,γ2,γ3∈Υ,(122)Dγ1q−Eγ2q+ℏ1q−ℏ2q+Eγ2q−Fγ3q+ℏ2q−ℏ3q+Fγ3q−Dγ1q+ℏ3q−ℏ1q21+k12eicotk1≤μMγ1,γ2,γ3,where(123)Mγ1,γ2,γ3=maxA1γ1,γ2,γ3q,A2γ1,γ2,γ3q,with(124)A1γ1,γ2,γ3q=γ1q−γ2q+γ2q−γ3q+γ3q−γ1q21+k12eicotk1,(125)A2γ1,γ2,γ3q=maxa1γ1,γ2,γ3q,a2γ1,γ2,γ3q,a3γ1,γ2,γ3q,a4γ1,γ2,γ3q,a5γ1,γ2,γ3q,a6γ1,γ2,γ3q,a7γ1,γ2,γ3q,a8γ1,γ2,γ3q,where(126)a1γ1,γ2,γ3q=γ1q−γ2q+Eγ2q+ℏ2q−γ2q+Eγ2q+ℏ2q−γ1q21+k12eicotk1,a2γ1,γ2,γ3q=2Dγ1q+ℏ1q−γ2q21+k12eicotk1,a3γ1,γ2,γ3q=2Dγ1q+ℏ1q−γ2q21+k12eicotk1,a4γ1,γ2,γ3q=2Eγ2q+ℏ2q−γ3q21+k12eicotk1,a5γ1,γ2,γ3q=2Eγ2q+ℏ2q−γ3q21+k12eicotk1,a6γ1,γ2,γ3q=4Dγ1q+ℏ1q−γ1q⋅Eγ2q+ℏ2q−γ2q21+k12eicotk121+2γ1q−γ2q21+k12eicotk1,a7γ1,γ2,γ3q=4Eγ2q+ℏ2q−γ2q⋅Fγ3q+ℏ3q−γ3q21+k12eicotk121+2Dγ1q+ℏ1q−γ3q21+k12eicotk1,a8γ1,γ2,γ3q=4Fγ3q+ℏ3q−γ3q⋅Dγ1q+ℏ1q−γ1q21+k12eicotk121+2Eγ2q+ℏ2q−Fγ3q−ℏ3q21+k12eicotk1.
Then, the three UTIEs, i.e., (120) have a unique common solution.Proof 1.
DefineF1,F2,F3:Υ⟶Υ as(127)F1γ1=F1γ1q=Dγ1q+ℏ1q=Dγ1+ℏ1,γ1q=γ1,F2γ2=F2γ2q=Eγ2q+ℏ2q=Eγ2+ℏ2,γ2q=γ2,F3γ3=F3γ3q=Fγ3q+ℏ3q=Fγ3+ℏ3,γ3q=γ3.
Then, we have the following main two cases:(1)
IfA1γ1,γ2,γ3q is the maximum term in A1γ1,γ2,γ3q,A2γ1,γ2,γ3q, then from (122), (123), and (127), we have that(128)GF1γ1,F2γ2,F3γ3≤μγ1−γ2+γ2−γ3+γ3−γ121+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy the hypothesis of Theorem 2 with μ=β1 and β2=0 in (31). Then, the given three UTIEs i.e., (4.1) have a unique common solution in Υ.(2)
IfA2γ1,γ2,γ3q is the maximum term in A1γ1,γ2,γ3q,A2γ1,γ2,γ3q, then from (123), we have that(129)Mγ1,γ2,γ3=A2γ1,γ2,γ3q.
Then, there are furthermore eight subcases arising:(i)
Ifa1γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a1γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(130)GF1γ1,F2γ2,F3γ3≤μγ1−γ2+Eγ2+ℏ2−γ2+Eγ2+ℏ2−γ121+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy the hypothesis of Theorem 2 with μ=β2 and β1=0 in (31). Then, the given three UTIEs, i.e., (4.1) have a unique common solution in Υ.(ii)
Ifa2γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a2γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(131)GF1γ1,F2γ2,F3γ3≤μ2Dγ1+ℏ1−γ221+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy the hypothesis of Theorem 2 with μ=β2 and β1=0 in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in Υ.(iii)
Ifa3γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a3γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(132)GF1γ1,F2γ2,F3γ3≤μ2Dγ1+ℏ1−γ221+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy the hypothesis of Theorem 2 with μ=β2 and β1=0 in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in Υ.(iv)
Ifa4γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a4γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(133)GF1γ1,F2γ2,F3γ3≤μ2Eγ2+ℏ2−γ321+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy the hypothesis of Theorem 2 with μ=β2 and β1=0 in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in Υ.(v)
Ifa5γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a5γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(134)GF1γ1,F2γ2,F3γ3≤μ2Eγ2+ℏ2−γ321+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy all the conditions of Theorem 2 with μ=β2 and β1=0 in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in Υ.(vi)
Ifa6γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a6γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(135)GF1γ1,F2γ2,F3γ3≤μ4Dγ1+ℏ1−γ1·Eγ2+ℏ2−γ221+k12eicotk121+2γ1−γ221+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy all the conditions of Theorem 2 with μ=β2 and β1=0 in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in Υ.(vii)
Ifa7γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a7γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(136)GF1γ1,F2γ2,F3γ3≤μ4Eγ2+ℏ2−γ2·Fγ3+ℏ3−γ321+k12eicotk121+2Dγ1+ℏ1−γ321+k12eicotk1,For allγ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy all the conditions of Theorem 2 with μ=β2 and β1=0 in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in Υ.(viii)
Ifa8γ1,γ2,γ3q is the maximum term in (125), then A2γ1,γ2,γ3q=a8γ1,γ2,γ3q; now, from (122), (127), and (129), we have that(137)GF1γ1,F2γ2,F3γ3≤μ4Fγ3+ℏ3−γ3·Dγ1+ℏ1−γ121+k12eicotk121+2Eγ2+ℏ2−Fγ3−ℏ321+k12eicotk1,for all γ1,γ2,γ3∈Υ. Thus, the maps F1,F2, and F3 satisfy all the conditions of Theorem 2 with μ=β2 and β1=0 in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in Υ.


## 5. Conclusions

We have established a generalized CFP-theorem in complex-valuedGb-metric spaces for three self-mappings. In this result, we have used a generalized rational contraction condition and proved a unique CFP-theorem. To justify our result, we presented an illustrative example in the said space by using three self-maps. Also, we present an application of integral equations to get the existing result for a common solution to support our work. By using this concept, one can prove different contractive-type FP and CFP results for many self-mappings in complex-valued Gb-metric spaces with different types of integral operators.



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