# Coincidence Fixed-Point Theorems forp-Hybrid Contraction Mappings in Gb-Metric Space with Application

Authors: Lucas Wangwe
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

By combining the notions ofG-metric space and b-metric space, in this paper, we present coincidence fixed-point theorems for p-hybrid mappings in Gb-metric spaces. An example is given to demonstrate the novelty of our main results. Henceforth, the illustrative applications are given by using nonlinear fractional differential equations.


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## Body

## 1. Introduction

In 1922, Banach [1] initiated the study of constructive theory in metric space. The constructive theory is used for nonlinear functional analysis, approximation theory, optimization theory (saddle function), variation inequalities, game theory (Nash equilibrium), and economics (Black-Scholes theorem). In addition, it is used in many practical and research problems in various fields beyond mathematics. It can reduce to fixed-point problems, which include biology, chemistry, physics, computer science, economics, engineering, global analysis statistics, and operations research.In 1969, Nadler [2] proved the multivalued version of Banach’s contraction principle [1]. Naimpally et al. [3] generalized Goebel’s [4] result in a hybrid contraction mapping. The method of hybrid fixed points can be used to derive another classical fixed-point theorem result. The concept of hybrid pair of mapping is very consequential for the theory of fixed point, and it has an important role in game theory, optimization theory, and differential equations.Definition 1 (see [3]).
LetP:W⟶CLW and Q:W⟶W be such that PW is a complete subspace of W and QW⊂PW. Further, assume there exists θ, 0<θ<1 such that for every κ,μ∈W,
(1)HPκ,Pμ≤θdQκ,Qμ.
Then,P and Q have a coincidence; that is, there exists ν∈W such that Qν∈Pν.Later, Chauhan et al. [5] proved the results in unified common fixed-point theorems for a hybrid pair of mappings via an implicit relation involving altering distance function. Imdad et al. [6, 7] generalized the hybrid fixed-point theorems in symmetric spaces via common limit range property and joint common limit range property in metric spaces. Nashine et al. [8] gave the proof using (JCLR) property for hybrid fixed-point theorems via quasi F-contractions. Wangwe and Kumar [9, 10] proved the common fixed-point theorem for a hybrid pair of mappings in weak partial b-metric spaces and G-metric space with some applications.Bakhtin [11] and Czerwik [12] generalized the concept of metric space to b-metric space due to some problems, especially the convergence issue of measurable functions to a measure that led to a generalization of a metric’s notion. Czerwik [12] established b-metric spaces by introducing a parameter s≥1 in the triangle inequality as a coefficient and generalized Banach’s contraction principle to these spaces. Later, Czerwik [13] proved the multivalued results in b-metric spaces. These findings motivated several potential researchers to perform and analyze contraction condition variants using single- and multivalued maps in b-metric space. One can refer to [14–18] and the references therein.The concepts ofG-metric space were initiated by Mustafa and Sims [19, 20] due to the shortcoming of the fundamental topological structure on D-metric spaces. Also, they replaced the tetrahedral inequality with an inequality involving the repetition of indices. Further, several researchers generalized the results for single-valued mapping and multivalued mappings in G-metric spaces. For more results, we refer the reader in [21–25] and the references contained. Furthermore, Aghajani et al. [26] using both concepts of b-metrics and G-metrics initiated the results on Gb-metric spaces. Since then, several results followed for single- and multivalued mappings for various abstract spaces. For more literature, we refer the reader to [27–32] and the references contained.This paper is aimed at proving a coincidence fixed-point theorem forp-hybrid contraction mappings in Gb-metric space with some application to the fractional differential equation. In particular, we modify and extend the works due to Karapinar et al. [33, 34], Wangwe and Kumar [10], and Aghajani et al. [26]. The results proved to have a novelty in the study of fixed-point theory.


## 2. Preliminaries

This part introduces some preliminary results of definitions and theorems, which will help develop the main result.Bakhtin [11] and Czerwik [12] established a new metric on a nonempty set, known as a b-metric.Definition 2 (see [12]).
LetW be a nonempty set and s≥1 be a given real number. Suppose that a function db:W×W⟶0,∞ satisfies
(B1)dbκ,μ=0 iff κ=μ
(B2)dbκ,μ=dbμ,κ
(B3)dbκ,μ≤sdbκ,ν+dbν,κ, for all κ,μ,ν∈W
Then,db is said to be b-metric and W,db,s is a b-metric space. s≥1 is defined as a parameter of W,db,s.We give some examples which satisfyb-metric space axioms.Example 1 (see [11, 12]).
Consider the setW=0,1 endowed with the function db:W×W⟶0,∞ defined by dbκ,μ=κ−μ2 for all κ,μ∈W. Thus, W,db,2 is a b-metric space for s=2.Mustafa and Sims [20] gave the following axioms for G-metric space.Definition 3 (see [20]).
LetW represent a nonempty set with G:W×W×W⟶ℝ+ as a function which satisfies the following axioms:
(G1)Gκ,μ,ν=0 for κ=μ=ν
(G2)Gκ,κ,μ>0,∀κ,μ∈W if ν=μ
(G3)Gκ,κ,μ≤Gκ,μ,ν∀κ,μ∈W if ν=μ
(G4)Gκ,μ,ν=Gκ,ν,μ=Gμ,ν,κ=⋯
(G5)Gκ,μ,ν≤Gκ,a,a+Ga,μ,ν, ∀κ,μ,ν,a∈W
Then,G is called a metric and W,G is a G-metric space.We give an example from [20].Example 2 (see [20]).
LetW=ℝ represent a set of real numbers. Define G:ℝ3⟶ℝ+ as follows:
(2)Gκ,μ,ν=κ−μ+μ−ν+κ−ν,for all κ,μ,ν∈W.
Recall that ifGκ,μ,ν=0, then κ=μ=ν.Example 3 (see [20]).
LetW=ℝ. Then, G-metric G is defined by
(3)Gκ,μ,ν=13κ−μ+μ−ν+κ−ν,for all κ,μ,ν∈W.On the other hand, Aghajani et al. combining the concepts from Bakhtin [11], Czerwik [12] and Mustafa and Sims [20], they established a new generalized space known as Gb-metric space.Definition 4 (see [26]).
LetW be a nonempty set and s≥1. Suppose that Gb:W×W×W⟶0,∞ is a function satisfying the following conditions:
Gb1Gbκ,μ,ν=0 if κ=μ=ν
Gb20<Gκ,κ,μ, for all κ,μ∈W with ν=μ
Gb3Gκ,κ,μ≤Gκ,μ,ν for all κ,μ∈W with ν=μ
Gb4Gκ,μ,ν=Gκ,ν,μ=Gμ,ν,κ=⋯ (symmetry in all three variables)
Gb5Gκ,μ,ν≤sGκ,a,a+Ga,μ,ν, for all κ,μ,ν,a∈W (rectangle inequality)The distance metricGb is called a Gb-metric, and W,Gb is called a Gb-metric space. The real number s≥1 is called the coefficient of W,Gb.LetW=ℝ be the set of real numbers; then, the mapping Gb:W×W×W⟶0,∞ is defined by
(4)Gbκ,μ,ν=19κ−μ+μ−ν+κ−ν2,for all κ,μ,ν∈ℝ, is a Gb-metric.Example 4 (see [26]).
LetW,Gb be a Gb-metric space. Consider
(5)Gbκ,μ,ν=Gbκ,μ,νp,for all κ,μ,ν∈W and p>1. Therefore, Gb is a Gb-metric with s=2p−1.Example 5 (see [26]).
LetW,Gb be a Gb-metric space. Then, for κ0∈W,r>0, the Gb-ball with center κ0 and radius r is
(6)BGκ0,r=μ∈WGκ0,μ,μ<r.For example,κ=ℝ, and consider a Gb-metric G defined by
(7)Gbκ,μ,ν=19κ−μ+μ−ν+κ−ν2,for all κ,μ,ν∈ℝ. Then,
(8)BG3,4=μ∈WG3,μ,μ<4=μ∈W19κ−μ+μ−ν+κ−ν2<4=μ∈W193−μ+μ−ν+3−ν2<4=μ∈W193−μ+3−ν2<4=μ∈W493−μ2<4=μ∈W3−μ2<9=0,6.Aghajani et al. [26] gave the proposition below that satisfies Gb-metric axioms.Proposition 5 (see [26]).
LetW,Gb be a Gb-metric space, ∀κ,μ,ν,a∈W; we have the following:
(i)
IfGbκ,μ,ν≤Gbκ,κ,μ+Gbκ,κ,ν, then κ=μ=ν(ii)
Gbκ,μ,ν≤sGbκ,κ,μ+Gbκ,κ,ν(iii)
Gbκ,μ,μ≤2sGbμ,κ,κ(iv)
Gbκ,μ,ν≤sGbκ,a,ν+Gba,μ,νMotivated by Aghajani et al. [26], we recall some properties in Gb-metric spaces as follows.Definition 6 (see [26]).
LetW,Gb be a Gb-metric space. If κi∈W then, we have the following:
(i)
Gb is convergence to a point κ∈W if, for each ε>0, there exists a positive integer n0 such that, for all i,j≥n0, Gbκi,κj,κ<ε(ii)
Gb is a Cauchy sequence if, for ε>0, ∃ a positive integer n0 such that, for all i,j,k≥n0, Gbκi,κj,κk<εProposition 7 (see [26, 27]).
LetW,Gb be a Gb-metric space. Then, the function is given by
(9)dGbκ,μ=Gbκ,μ,μ+Gbμ,κ,κ,for all κ,μ∈W. Define a b-metric on W. It is called a b-metric induced by the Gb-metric G.Proposition 8 (see [27]).
LetW,Gb be a Gb-metric space. The properties below are similar:
(i)
κi is Gb-convergence to κ(ii)
limi⟶∞Gbκi,κi,κ=0(iii)
limi⟶∞Gbκi,κ,κ=0(iv)
limi⟶∞Gbκi,κj,κk=0Proposition 9 (see [27]).
LetW,Gb be a Gb-metric space. Therefore, the properties below hold:
(i)
κi is a Gb-Cauchy sequence(ii)
∀ε>0, there exists n0∈ℕ such that, ∀i,j∈n0, Gκi,κj,κk=0Definition 10 (see [26]).
AGb-metric space W is called Gb-complete if every Gb Cauchy sequence is a Gb-convergent in W.Further, Makran et al. [29] extended the works due to Aghajani et al. [26] and Kaewcharoen and Kaewkhao [22] by introducing the multivalued versions in Gb-metric spaces as follows:LetW be a Gb-metric space. We shall denote CBGbW as a nonempty, closed, and bounded subsets of W. Let HGb.,.,. represent the Haursdorff-Gb-metric on CBGbW, and A,B,C∈CBGbW define
(10)HGbA,B,C=maxsupκ∈AGbκ,B,C,supκ∈BGbκ,C,A,supκ∈CGbκ,A,B,where
(11)Gbκ,B,C=dGbκ,B+dGbB,C+dGbκ,C,(12)dGbκ,B=infdGbκ,μ,μ∈B,(13)dGbA,B=infdGba,b,a∈A,b∈B,(14)dGbκ,μ,C=infGbκ,μ,ν,ν∈C.Lemma 11 (see [29]).
LetW,Gb be a Gb-metric space with s≥1 and A,B∈CBGbW. Then, for each a∈A, we have
(15)Gba,B,B≤HGbA,B,B.Lemma 12 (see [29]).
LetW,Gb be a Gb-metric space with s≥1. If A,B∈CBGbW and κ∈A, then for each ε>0, there exists μ∈B such that
(16)Gbκ,μ,μ≤HGbA,B,B+σ.Lemma 13 (see [35]).
LetW,Gb be a Gb-metric space with s≥1, and suppose that κi,μi and νi are Gb-convergent to κ, μ, and ν, respectively. Then, we have
(17)1s3Gbκ,μ,ν≤limi⟶∞infGbκi,μi,νi≤limi⟶∞supGbκi,μi,νi≤s3Gbκ,μ,ν.In particular, ifκ=μ=ν, then we have limi⟶∞Gbκi,μi,νi=0.Definition 14 (see [22]).
LetW be a nonempty set. Assume Q:W⟶W and P:W⟶2W are two mappings. If w=Qκ∈Pκ for some κ∈W, then κ is a coincidence point of two mapping Q,P. Then, the coincidence point of Q and P is w. On the other hand, the mappings Q and P are said to be weakly compatible if Qκ∈Pκ for some κ∈W consequently QPκ⊆PQκ.Proposition 15 (see [22]).
LetW be a nonempty set. Assume Q:W⟶W and P:W⟶2W are said to be weakly compatible mappings. If u=Qκ∈Pκ is a unique coincidence of Q and P, u is said to be a unique common fixed point of Q and P.Definition 16 (see [36]).
LetQ,P be the two self-mappings on an ordered metric space W,d,° with QW⊆PW. For every κ0∈W, consider the sequence κi∈W defined by Qκi=Pκi+1, ∀i∈ℕ0. A sequence Qκi is a Q-P sequence starting at κ0.Wangwe and Kumar [10] gave the following definition and theorem.Definition 17 (see [10]).
LetX,G be a G-metric space and let f,T be two hybrid mapping on this space for p≥0 and ki≥0,i=1,2,3,4, such that ∑i=14ki=1. We define the following expression:
(18)MGpζ,η,η=k1Gfζ,fη,fηp+k2Gfζ,Tζ,Tζp+k3GTη,fη,fηp+k4Gfζ,Tη,Tη+GTζ,fη,fη2p1/p,p≥0,ζ,η∈X,Gfζ,fζ,fηK1Gfζ,Tζ,TζK2GTη,fη,fηK3GTζ,fη,fη+Gfζ,Tη,Tη2K4,p=0,ζ,η∈X.Theorem 18 (see [10]).
LetX,G be a G-metric space, and suppose f,T:X⟶CBX is a p-hybrid mapping with almost altering distance ψ∈Ψ satisfying the following conditions:
(a)
f and T are weakly compatible(b)
f and T satisfy CLRf property(c)
Tx⊆fX(d)
TX is a G-complete subspace of X(e)
HGpTζ,Tη,Tη≤ψMGpζ,η,η, for all ζ,η∈X and p≥0
Therefore,f and T admit a unique common fixed point in X.


## 3. Main Results

We commence this section by extending Definition17 to Gb-metric space setting.Definition 19.
LetW,Gb be a Gb-metric space, Q,P be a pair of hybrid mapping, and s≥1 with p≥0 and zi≥0,i=1,2,3,4, such that ∑i=14zi=1. Then, we define the following expression:
(19)NGbpκ,μ,ν=z1GbQκ,Qμ,Qνp+z2GbQκ,Pκ,Pκp+z3GbPμ,Qμ,Qνp+z4GbQκ,Pμ,Pν+GbPκ,Qμ,Qνsp,for p≥0,κ,μ,ν∈W.Now, we are equipped to prove the following theorem.Theorem 20.
LetW,Gb be a complete Gb-metric space, and suppose Q:W⟶W and P:W⟶CBGbW are p-hybrid contraction mapping on W and s≥1 satisfying the following conditions:
(i)
∃QW⊆PW, since W,Gb is complete(ii)
Q and P are weakly compatible(iii)
PW is a Gb-complete subspace of W(iv)
Fori, Qκi and Pκi converge to a common fixed point(v)
∃ a constant θ=z1+z2+z4/1−z3+z4∈0,1, s≥1, and θ>0 such that ∀κ,μ,ν∈W; we have
(20)sHGbpPκ,Pμ,Pν≤θNGbpκ,μ,ν
Then,Q and P pose a unique coincidence fixed point.Proof.
Assume thatQW⊆PW and PW is a Gb-complete subspace of W. We can construct a P-Q-sequence such that Pκi∈W with initial point κ0 satisfying
(21)Qκi,Pκi−1,Pκi−1,Qκi+1,Pκi,Pκi,Qκi+2,Pκi+1,Pκi+1⋯,∀i∈ℕ0, such that Pκi, Qκi∈PW.
Letκ0 be an arbitrary element in W. If Qκ0=Pκ0, then κ0 is a coincidence point of Q and P. Therefore, our proof is completed. Otherwise, for Qκ0≠Pκ0, it implies that QW⊂PW. Now, we choose κ1∈W such that Qκ1=Pκ0. Again, we can choose κ2∈W such that Qκ2=Pκ1. By repeating the above procedure and applying Definition 16, we formulate a sequence κn⊂W, such that
(22)Qκi=Pκi−1,Qκi+1=Pκi.
Equivalently,(23)Qκi,Qκi+1=Pκi−1,Pκi.
Using Lemmas11 and 12, we obtain that there exists
(24)GbQκi,Qκi+1,Qκi+1≤sHGbPκi−1,Pκi,Pκi.
Consequently, we have(25)GbQκi,Qκi+1,Qκi+1≤sHGbPκi−1,Pκi,Pκi,<HGbPκi−1,Pκi,Pκi.
Applyκ=κi−1, μ=κi, and ν=κi in (20); we get
(26)GbQκi,Qκi+1,Qκi+1p≤sHGbpPκi−1,Pκi,Pκi,≤θNGbpκi−1,κi,κi,where
(27)NGbpκi−1,κi,κi=z1GbQκi−1,Qκi,Qκip+z2GbQκi−1,Pκi−1,Pκi−1p+z3GbPκi,Qκi,Qκip+z4GbQκi−1,Pκi,Pκi+GPκi−1,Qκi,Qκisp,(28)NGbpκi−1,κi,κi=z1GbQκi−1,Qκi,Qκip+z2GbQκi−1,Qκi,Qκip+z3GbQκi+1,Qκi,Qκip+z4GbQκi−1,Qκi+1,Qκi+1+GbQκi,Qκi,Qκisp.
By (Gb5), we have
(29)GbQκi−1,Qκi+1,Qκi+1≤sGbQκi−1,Qκi,Qκi+GbQκi,Qκi+1,Qκi+1.
Applying (29) in (28), we obtain
(30)NGbpκi−1,κi,κi=z1GbQκi−1,Qκi,Qκip+z2GbQκi−1,Qκi,Qκip+z3GbQκi+1,Qκi,Qκip+z4sGbQκi−1,Qκi,Qκi+GbQκi,Qκi+1,Qκi+1+GbQκi,Qκi,Qκisp,(31)NGbpκi−1,κi,κi=z1+z2+z4GbQκi−1,Qκi,Qκip+z3+z4GbQκi,Qκi+1,Qκi+1p.
Using (9) and (31) in (26), we get
(32)GbQκi,Qκi+1,Qκi+1p<z1+z2+z4GbQκi−1,Qκi,Qκip+z3+z4GbQκi,Qκi+1,Qκi+1p.1−z3+z4GbQκi,Qκi+1,Qκi+1p≤z1+z2+z4GbQκi−1,Qκi,QκipGbQκi,Qκi+1,Qκi+1p≤z1+z2+z41−z3+z4GbQκi−1,Qκi,Qκip.
Letθ=z1+z2+z4/1−z3+z4<1; we have
(33)GbQκi,Qκi+1,Qκi+1p≤θGbQκi−1,Qκi,Qκip.
By repeating the above procedure, we construct a sequenceQκi⊂W such that Qκi∉Pκi,Qκi+1∈Pκi.
(34)GbQκi,Qκi+1,Qκi+1p≤θiGbQκi−1,Qκi,Qκip.
In order to simplify the above equation, letGbi=GbQκi,Qκi+1,Qκi+1p. Thus, by (34), we have
(35)∑i=0∞Gbi≤∑i=1∞θiGbi−1,for all i∈ℕ.
By taking limits asi⟶∞ in (35), we obtain
(36)limi⟶∞Gbi=0.
Therefore,∑i=0∞Gbi=∑i=0∞Gbi−1 converges.
Using (34) and Gb4-symmetric properties, for all i,j∈ℕ0 with j>i, we obtain
(37)GbQκi,Qκi+j,Qκi+jp≤sGbQκi,Qκi+1,Qκi+1p+s2GbQκi+1,Qκi+2,Qκi+2p+s3GbQκi+1,Qκi+2,Qκi+2p+⋯+sj−2GbQκi+j−2,Qκi+j−1,Qκi+j−1p+sj−1GbQκi+j−1,Qκi+j,Qκi+jp.
On the other hand, using (35), we obtain
(38)GbQκi,Qκi+j,Qκi+jp≤sGbi+s2Gbi+1+s3Gbi+2⋯+sj−2Gbi+j−1+sj−1Gbi+j−2,≤si∑i=1∞θiGbi−1,⟶0asi⟶∞.where Gbi=GbQκi,Qκi+j,Qκi+jp=0. This proves that the sequence Qκi satisfies the Gb-Cauchy sequence conditions on complete subspace QW. Henceforth, Qκi is a Cauchy sequence.
LetQ,P be closed and weakly compatible mappings. From Definition 14, we have
(39)QPκi=PQκi,Qκi=Pκi,PQκi=PPκi=Pκi+1,QPκi=QQκi=Qκi+1.
Now, we find thatκ∈W such that Qκ=μ. We will show that Qκ∈Pκ. For each i∈ℕ, using (20), it follows that
(40)GbQκi+1,Pκ,Pκp≤sHGbpPκi,Pκ,Pκ≤θNGbpκi,κ,κ,where
(41)NGbpκi,κ,κ=z1GbQκi,Qκ,Qκp+z2GbQκi,Pκi,Pκip+z3GbPκ,Qκ,Qκp+z4GbQκi,Pκ,Pκ+GPκi,Qκ,Qκsp.
Taking the limit asi⟶∞ in (40) and (41) with θ<1, we obtain
(42)GbQκ,Pκ,Pκp≤z1GbQκ,Qκ,Qκp+z2GbQκ,Pκ,Pκp+z3GbPκ,Qκ,Qκp+z4GbQκ,Pκ,Pκ+GPκ,Qκ,Qκsp,≤z1GbQκ,Qκ,Qκp+z2GbQx,Pκ,Pκp+z3GbPκ,Qκ,Qκp+2s−1pz4GbQκ,Pκ,Pκp,GbQκ,Pκ,Pκp≤z2+z3+2s−1pz4GbQκ,Pκ,Pκp.1−z2+z3+2s−1pz4GbQκ,Pκ,Pκp≤0,Qκ,Pκ,Pκp≤0.
This shows thatQκ∈Pκ. That is, Q and P have a point of coincidence.
Next, we prove the uniqueness of the point of coincidence ofQ and P. Let Qκ∈Pκ and Qμ∈Pμ. Assume that GbQκ,Pμ,Pμp>0. Using (20), we get
(43)GbQκ,Pμ,Pμp≤sHGbpPκ,Pμ,Pμ≤θNGbpκ,μ,μ,where
(44)NGbpκ,μ,μ=z1GbQκ,Qμ,Qyp+z2GbQκ,Pκ,Pκp+z3GbPμ,Qμ,Qμp+z4GbQμ,Pμ,Pμ+GPκ,Qμ,Qμsp.
FromGb4 the symmetric properties, using (43) and (44) with θ<1, we obtain
(45)GbQκ,Pμ,Pμp≤z1GbQκ,Qμ,Qμp+z2GbQκ,Pκ,Pκp+z3GbPμ,Qμ,Qμp+z4GbQκ,Pμ,Pμ+GPκ,Qμ,Qμsp,GbQκ,Pμ,Pμp≤z1+2s−1pz4GbQκ,Pμ,Pμp,1−z1+2s−1pz4GbQκ,Pμ,Pμp≤0,Qκ,Pμ,Pμp≤0.
We can conclude thatQκ=Pμ. Suppose that Q and P are weakly compatible. By applying Proposition 15 and Definition 14, we obtain that Q and P have a unique coincidence fixed point, which is a contradiction. Thus, μ is a coincidence point of Q and P, for κ=μ. The proof is completed.Inspired by the idea of Theorem20, we can deduce the corollary as follows:Corollary 21.
LetW,Gb be a Gb-metric space, and let Q:W⟶W and P:W⟶CBGbW be a p-hybrid mapping which satisfies the following hypotheses:
(i)
QW⊂W,Gb(ii)
∃ a continuous function θ<1 and s≥1 such that
(46)sHGbpPκ,Pμ,Pν≤θMGbpκ,μ,ν,where
(47)MGbpκ,μ,ν=GbQκ,Qμ,Qν,GbQκ,Pκ,QνGbQν,Pν,Qν,GbQμ,Qμ,Qν
Then,Q and P have the unique coincidence fixed point.Proof.
We prove the above corollary by following similar steps of Theorem20. Therefore, the proof is completed.Next, we demonstrate with an example for Theorem20.Example 6.
LetW=0,∞ be endowed with the usual ordering on ℝ and Gb-metric on W be given by Gbκ,μ,ν=1/9κ−μ+μ−ν+κ−ν2, where s=2p−1 and p≥1 with z1=0.5,z2=0.1, z3=0.2, and z4=0.1.ConsiderW to be Gb-complete. Define a self-map as Q:W⟶W by
(48)Qκ=κ3,∀κ∈W,and P:W⟶CBGbW by
(49)Pκ=0,κn,∀κ∈W.Therefore,(i)
PW is a Gb-complete subspace in W(ii)
forθ∈0,1 in (20), we have
(50)sHGbpPκ,Pμ,Pν≤θNGbpκ,μ,ν,where
(51)NGbpκ,μ,ν=z1GbQκ,Qμ,Qνp+z2GbQκ,Pκ,Pκp+z3GbPμ,Qμ,Qνp+z4GbQκ,Pμ,Pν+GbPκ,Qμ,Qνsp,for all κ,μ,ν∈WApplying (i), we prove thatPW is a Gb-complete subspace in W. By Proposition 7, we have
(52)Gbκ,μ,μ≤19κ−μ+μ−μ+κ−μ2=49κ−μ2,(53)Gbμ,κ,κ≤19μ−κ+κ−κ+μ−κ2=49μ−κ2.By (52) and ((53)) in (9), we obtain
(54)dGbκ,μ≤49κ−μ2+49μ−κ2=89κ−μ2.From (ii), assume thatκ,μ,ν∈W. If κ=μ=ν=0, then Pκ=Pμ=Pν=0 and sHGbPκ,Pμ,Pν=0. The proof is completed. Otherwise, we suppose that the value of κ,μ,ν are not all zero.Forκ≤μ≤ν, we get
(55)HGbpPκ,Pμ,Pν=HGbp0,κn,0,μn,0,νn.By (10), (55) is equivalent to
(56)=maxsup0≤a≤κnGba,0,μn,0,νnpsup0≤b≤μnGbb,0,κn,0,νnpsup0≤c≤νnGbc,0,κn,0,μnp.Sinceκ≤μ≤ν, then 0,κn⊆0,μn⊆0,νn, using (11) yields
(57)dGb0,κn,0,μnp=0,dGb0,μn,0,νnp=0,dGb0,κn,0,νnp=0.Now, for each0≤a≤κn and dGbκ,μ=8/9κ−μ2 in (11) and (12), we have
(58)Gba,0,μn,0,νnp=dGba,0,μn+dGb0,μn,0,μn+dGba,0,νnp≤89a−μn2+0+89a−νn2p=89a−μn2+89a−νn2p=89pκn−μn2+89κn−νn2p.Next, for each0≤b≤μn and dGbκ,μ=8/9κ−μ2 in (11) and (12), we get
(59)Gbb,0,κn,0,νnp=dGbb,0,κn+dGb0,κn,0,νn+dGbb,0,νnp≤89b−κn2+0+89b−νn2p=89b−κn2+89b−νn2p=89pμn−κn2+κn−νn2p.Using the same approach, for each0≤νn, ((11)) and ((12)) give
(60)Gbc,0,κn,0,μnp=dGbc,0,κn+dGbκn,0,μn+dGbc,0,μnp≤89c−κn2+0+89c−μn2p=89c−κn2+89c−μn2p=89pνn−κn2+νn−μn2p.Consequently, using (58), (59), and (60) in (55), we obtain
(61)HGbpPκ,Pμ,Pν≤maxsup0≤a≤κn89pκn−μn2+κn−μn2psup0≤b≤μn89pμn−κn2+κn−νn2psup0≤c≤νn89pνn−κn2+νn−μn2p=89pνn−κn2+νn−μn2p.Further, we calculate the followingGb-metrics. Equations ((11)) and ((12)) yield
(62)GbQκ,Qμ,Qνp=Gbκ3,μ3,ν3p=19pκ3−μ3+μ3−ν3+κ3−ν32p.In the same manner, we can calculate the followingGb-metrics.
(63)GbQκ,Pκ,Pκp=Gbκ3,0,κn,0,κnp=169pκ3−κn2p,GbPμ,Qμ,Qν=Gb0,μn,μ3,ν3p=89pμn−μ32+μ3−ν32+μn−ν32p,GbQκ,Pμ,Pνp=Gbκ3,0,μn,0,νnp=89pκ3−μn2+κ3−νn2p,GbPκ,Qμ,Qνp=Gb0,κn,μ3,ν3p=89pκn−μ32+μ3−ν32+κn−ν32p.Applying the above equality in (20), we obtain
(64)s89pνn−κn2+νn−μn2p≤θz119pκ3−μ3+μ3−ν3+κ3−ν32p+z2169pκ3−κn2p+z389pμn−μ32+μ3−ν32+μn−ν32p89pκ3−μn2+κ3−νn2p+89pκn−μ32+μ3−ν32++z4κn−ν32ps.By using the above inequality, forGb1/4,1/3,1/2,n=1,p=2, s=2p−1=s=2, and z1=0.5,z2=0.1, z3=0.2, and z4=0.1, it follows that
(65)2×0.006439567≤θ0.5×0.000160182+0.1×0.065873215+0.2×0.097691847+0.1×0.020276367,0.012879134≤θ0.000080091+0.006587321+0.019538369+0,0202763670.012879134≤0.046482148θ,for θ=z1+z2+z4/1−z3+z4≤1.This shows that allκ,μ,ν∈W and the hypothesis given in Theorem 20 are correct. Therefore, p-hybrid mappings have an amazing coincidence fixed point. So, κ=0 is a unique coincidence fixed point of Q and P. Clearly, W is P,Q-closed, and κ0=0,Q0,P0,P0∈W.


## 4. An Application to Nonlinear Fractional Boundary Valued Problem inGb-Metric Space

The nonlinear fractional differential equation is used as convolution mapping. Convolution and associated functions are found in many sciences, engineering, and mathematics applications, such as the following:(i)
In physics: the system linear system with a “superposition principle,” a convolution operation, makes an appearance(ii)
Used in acoustics: Doppler effect of the sound which is a convolution(iii)
In image processing: in digital image processing, convolutional filtering plays a vital role in many essential algorithms in edge detection and related processes(iv)
In optics: an out-of-focus photograph is a convolution of the sharp image with a lens function(v)
In radiotherapy: Most parts of all modern codes of calculation apply the convolution-superposition algorithm in the treatment of planning systems(vi)
In electric transmission lines(vii)
In a control system: to increase the speed of response, decrease the relative stability, and decrease or eliminate the steady-state error(viii)
Application of fractional-order circuit models for modelling human tissue, plant physiology, and respiratory system(ix)
Used in the tautochrone problem: a cycloidFor further literature, we refer the reader to [37–39] and the references contained.The Caputo derivative is used for modelling phenomena that account for interaction within the past and problems with nonlocal properties. In this case, one can think of the equation as having memory; the groundwater equation within confined, unconfined, leaky aquifers; and other diffusion problems. In addition, fractional differential arises in a dynamic process, rheology, fluid flows, viscoelasticity, chemical physics, electrical network, and numerous other shorts of science and engineering.Inspired by [40, 41], we demonstrate the result of Theorem 20 using a nonlinear fractional boundary valued problem by transforming it into a system of integral equations.Now, we investigate the Caputo derivative with the fractional order of the nonlinear fractional differential equation.Definition 22 (see [41]).
For continuous functionf:0,∞⟶ℝ, the Caputo derivative of functional order q is defined as
(66)CDtqft=1Γn−q∫0tt−sn−q−1fnsds,n−1<q<n,n=q+1,where q denotes the integer part of the real number q.Definition 23 (see [41]).
The Riemann-Liouville of fractional-orderq for continuous function ft is defined as
(67)Dqft=1Γn−qddtn∫0tt−sn−q−1fsds,n=q+1,provided that the right-hand side is point-wise defined on 0,∞.Definition 24 (see [41]).
The Riemann-Liouville of the fractional integral of orderq for continuous function ft is defined as
(68)Iqft=1Γq∫0tt−sq−1fsds,q>0,provided that such integral exists.The Caputo fractional differential equation and the nonlinear fractional equation are used as convolution mapping, which has several applications in science, engineering, and mathematics, i.e., optics and radiotherapy.We consider the following Caputo nonlinear boundary value fractional differential equation, defined as follows:(69)CDqκt=ft,κt,t∈0,1,1<α≤2,κ0=0,αIpκη=κ1,where CDq denotes the Caputo fractional derivative of order q and f:0,1⟶W is a continuous function and α≠Γp+2/ηp+1, 1<q≤2.LetW=C0,1 be the set of real continuous functions defined on 0,1, for κ,μ,ν∈W, and define the Gb-metric Gb:W×W×W⟶ℝ+ given by
(70)Gbκ,μ,ν=19supt∈0,1κ−μ+supt∈0,1μ−ν+supt∈0,1κ−ν2,∀κ,μ,ν∈W. Then, W,Gb is a complete Gb-metric space with s=2. For μ=ν, the above inequality becomes
(71)Gbκ,μ,μ=49supt∈0,1κ−μ2.The nonlinear fractional differential equation (69) can be transformed to integral equation as follows:
(72)κt=1Γq∫0tt−sq−1fs,κsds−Γp+2tΓqΓp+2−αηp+1∫011−sα−1fs,κsds+αpp+1tΓqΓp+2−αηp+1∫0η∫0sη−sp−1s−zq−1fz,κzdzds.Now, we investigate the theorem below.Theorem 25.
Consider the hypotheses below:(i)
f∈CI×W,W is continuous(ii)
∃ an increasing function f:0,1×ℝ⟶ℝ+, such that
(73)ft,κs−ft,μs≤θsNGbpκ,μ,ν,where NGbpκ,μ,ν=4/9κs−μs and
(74)NGbpκ,μ,ν=z1GbQκ,Qμ,Qνp+z2GbQκ,Pκ,Pκp+z3GbPμ,Qμ,Qνp+z4GbQκ,Pμ,Pν+GbPκ,Qμ,Qνsp,for p≥0,κ,μ,ν∈W(iii)
There existsθ/s∈0,1 such that
(75)tqqΓq+Γp+2tqΓqΓp+2−αηp+1+αpp+1tΓqΓp+2−αηp+12≤θs
Then, equation (69) has a coincidence solution as a fixed point κ∈CI,W.Proof.
Let us definePQ:C0,1⟶C0,1 by
(76)PQκt=1Γq∫0tt−sq−1fs,κsds−Γp+2tΓqΓp+2−αηp+1∫011−sq−1fs,κsds+αpp+1tΓqΓp+2−αηp+1∫0η∫0sη−sp−1s−zq−1fz,κzdzds,for t∈0,1; then, PQ is a compatible and continuous mapping, such that
(77)Qκt=∫0qs−zα−1fz,κzdz.
This implies thatQ∈PQ and Q possess a coincidence fixed point κ∗∈PQ. To prove the existence of the coincidence point of PQ, we prove that PQ is continuous and a contraction. To see that PQ is continuous, assume that PQκ≠PQμ, for all κ,μ∈0,1.
Forκ,μ,ν∈C0,1, μ=ν with κ<μ, we claim that HGbpPκ,Pμ,Pμ≰θ/sNGbpκ,μ,μ.
By hypothesis (ii), we have(78)HGbpPκ,Pμ,Pμ≤49supt∈0,1κ−μ2,PQκ−PQμ=1Γq∫0tt−sq−1fs,κsds−Γp+2tΓqΓp+2−αηp+1∫011−sq−1fs,κsds+αpp+1tΓqΓp+2−αηp+1∫0η∫0sη−sp−1s−zq−1fz,κzdzds−1Γq∫0tt−sq−1fs,μsds+Γp+2tΓqΓp+2−αηp+1∫011−sq−1fs,μsds−αpp+1tΓqΓp+2−αηp+1∫0η∫0sη−sp−1s−zq−1fz,μzdzds2≤1Γq∫0tt−sq−1fs,κs−fs,μsds+Γp+2tΓqΓp+2−αηp+1∫011−sq−1fs,κs−fs,μsds,+αpp+1tΓqΓp+2−αηp+1∫0η∫0sη−sp−1s−zq−1fz,κz−fz,μzdzds2≤491Γq∫0tt−sq−1κs−μsds+Γp+2tΓqΓp+2−αηp+1∫011−sq−1κs−μsds+αpp+1tΓqΓp+2−αηp+1∫0η∫0sη−sp−1s−zq−1κz−μzdzds2=49tqqΓqκs−μs∞+Γp+2tqΓqΓp+2−αηp+1κs−μs∞+αpp+1tΓqΓp+2−αηp+1κs−μs∞2≤tqqΓq+Γp+2tqΓqΓp+2−αηp+1+αpp+1tΓqΓp+2−αηp+1249κs−μs∞2≤4θ9sκs−μs∞2=4θ9ssupt∈0,1κ−μ2=θsNGbpκ,μ,ν.
This implies that(79)HGbpPκ,Pμ,Pμ=PQκ−PQμ≤θsNGbpκ,μ,ν,equivalent to
(80)sHGbpPκ,Pμ,Pμ≤θNGbpκ,μ,ν,which is a contradiction. Therefore, PQ is a p-hybrid contraction mapping on W.
Hence,κ is a coincidence fixed point of P and Q and also a solution to integral equation (72) and a solution of the nonlinear fractional differential equation (69). Therefore, we can conclude that all the hypotheses given in Theorem 20 and Theorem 25 are satisfied. Hence, the proof is completed.



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