# Fixed Point Results of Jaggi-Type Hybrid Contraction in Generalized Metric Space

Authors: Jamilu Abubakar Jiddah; Monairah Alansari; OM Kalthum S. K. Mohamed; Mohammed Shehu Shagari; Awad A. Bakery
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this manuscript, a new family of contractions called Jaggi-type hybridG−ϕ-contraction is introduced and some fixed point results in generalized metric space that are not deducible from their akin in metric space are obtained. The preeminence of this class of contractions is that its contractive inequality can be extended in a variety of manners, depending on the given parameters. Consequently, several corollaries that reduce our result to other well-known results in the literature are highlighted and analyzed. Substantial examples are constructed to validate the assumptions of our obtained theorems and to show their distinction from corresponding results. Additionally, one of our obtained corollaries is applied to set up unprecedented existence conditions for the solution of a family of integral equations.


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## Body

## 1. Introduction

The prominent Banach contraction in metric space has laid a solid foundation for fixed point theory in metric space. The applications of fixed point range across inequalities, approximation theory, optimization, and so on. Researchers in this area have introduced several new concepts in metric space and obtained a great deal of fixed point results for linear and nonlinear contractions. Recently, Karapınar and Fulga [1] introduced a new notion of hybrid contraction which is a unification of some existing linear and nonlinear contractions in metric space.On the other hand, Mustafa [2] pioneered an extension of a metric space by the name, generalized metric space (or more precisely, G-metric space), and proved some fixed point results for Banach-type contraction mappings. This new generalization was brought to spotlight by Mustafa and Sims [3]. Subsequently, Mustafa et al. [4] obtained some engrossing fixed point results for Lipschitzian-type mappings on G-metric space. However, Jleli and Samet [5] as well as Samet et al. [6] noted that most of the fixed point results in G-metric space are direct consequences of existence results in corresponding metric space. Jleli and Samet [5] further observed that if a G-metric is consolidated into a quasimetric, then the resultant fixed point results become the known fixed point results in the setting of quasimetric space. Motivated by the latter observation, many investigators (see for instance, [7, 8]) have established techniques of obtaining fixed point results in G-metric space that are not deducible from their ditto ones in metric space or quasimetric space.Following the existing literature, we realize that hybrid fixed point results inG-metric space are not adequately investigated. Hence, motivated by the ideas in [1, 7, 8], we introduce a new concept of Jaggi-type hybrid G−ϕ-contraction in G-metric space and prove some related fixed point theorems. An example is constructed to demonstrate that our result is valid, an improvement of existing results and the main ideas obtained herein do not reduce to any existence result in metric space. Some corollaries are presented to show that the concept proposed herein is a generalization and improvement of well-known fixed point results in G-metric space. Finally, one of our obtained corollaries is applied to establish novel existence conditions for solution of a class of integral equations.


## 2. Preliminaries

In this section, we will present some fundamental notations and results that will be deployed subsequently.Throughout, every setΦ is considered nonempty, ℕ is the set of natural numbers, and ℝ represents the set of real numbers and ℝ+ the set of nonnegative real numbers.Definition 1 (see [3]).
LetΦ be a nonempty set and let G:Φ×Φ×Φ⟶ℝ+ be a function satisfying
(G1) Gr,s,t=0 if r=s=t
(G2) 0<Gr,r,s for all r,s∈Φ with r≠s
(G3) Gr,r,s≤Gr,s,t, for all r,s,t∈Φ with t≠s
(G4) Gr,s,t=Gr,t,s=Gs,r,t=⋯ (symmetry in all variables)
(G5) Gr,s,t≤Gr,u,u+Gu,s,t, for all r,s,t,u∈Φ (rectangle inequality)Then, the functionG is called a generalized metric or, more precisely, a G-metric on Φ, and the pair Φ,G is called a G-metric space.Example 2 (see [4]).
LetΦ,d be a usual metric space; then, Φ,Gk and Φ,Gm are G-metric spaces, where
(1)Gkr,s,t=dr,s+ds,t+dr,t∀r,s,t∈Φ,Gmr,s,t=maxdr,s,ds,t,dr,t∀r,s,t∈Φ.Definition 3 (see [4]).
LetΦ,G be a G-metric space and let rn be a sequence of points of Φ. Then, rn is said to be G-convergent to r if limn,m⟶∞Gr,rn,rm=0; that is, for any ε>0, there exists n0∈ℕ such that Gr,rn,rm<ε, ∀n,m≥n0. We refer to r as the limit of the sequence rn.Proposition 4 (see [4]).
LetΦ,G be a G-metric space. Then, the following are equivalent:
(i)
rn is G-convergent to r(ii)
Gr,rn,rm⟶0, as n⟶∞(iii)
Grn,r,r⟶0, as n⟶∞(iv)
Grn,rn,r⟶0, as n⟶∞Definition 5 (see [4]).
LetΦ,G be a G-metric space. A sequence rn is called G-Cauchy if for any ε>0, we can find n0∈ℕ such that Grn,rm,rl<ε, ∀n,m,l≥n0, that is, Grn,rm,rl⟶0, as n,m,l⟶∞.Proposition 6 (see [4]).
IfΦ,G is a G-metric space, the following statements are equivalent:
(i)
The sequencern is G-Cauchy(ii)
For everyε>0, there exists n0∈ℕ such that Grn,rm,rm<ε, ∀n,m≥n0Definition 7 (see [4]).
LetΦ,G and Φ′,G′ be G-metric spaces and f:Φ,G⟶Φ′,G′ be a function. Then, f is G-continuous at a point u∈Φ if and only if for any ε>0, there exists δ>0 such that r,s∈Φ; and Gu,r,s<δ implies G′fu,fr,fs<ε. A function f is G-continuous on Φ if and only if it is G-continuous at all u∈Φ.Proposition 8 (see [4]).
LetΦ,G and Φ′,G′ be G-metric spaces. Then, a function f:Φ,G⟶Φ′,G′ is said to be G-continuous at a point r∈Φ if and only if it is G-sequentially continuous at r; that is, whenever rn is G-convergent to r, frn is G-convergent to fr.Definition 9 (see [4]).
AG-metric space Φ,G is called symmetric G-metric space if
(2)Gr,r,s=Gs,r,r∀r,s∈Φ.Proposition 10 (see [4]).
LetΦ,G be a G-metric space. Then, the function Gr,s,t is jointly continuous in all variables.Proposition 11 (see [4]).
EveryG-metric space Φ,G defines a metric space Φ,dG by
(3)dGr,s=Gr,s,s+Gs,r,r∀r,s∈Φ.
Note that for a symmetricG-metric space Φ,G,
(4)Φ,dG=2Gr,s,s∀r,s∈Φ.
On the other hand, ifΦ,G is not symmetric, then by the G-metric properties,
(5)32Gr,s,s≤dGr,s≤3Gr,s,s∀r,s∈Φ,
and that in general, these inequalities are sharp.Definition 12 (see [4]).
AG-metric space Φ,G is referred to as G-complete (or complete G-metric) if every G-Cauchy sequence in Φ,G is G-convergent in Φ,G.Proposition 13 (see [4]).
AG-metric space Φ,G is G-complete if and only if Φ,dG is a complete metric space.
Mustafa [2] proved the following result in the framework of G-metric space.Theorem 14 (see [2]).
LetΦ,G be a complete G-metric space, and let Γ:Φ⟶Φ be a mapping satisfying the following condition:
(6)GΓr,Γs,Γt≤kGr,s,t,
for allr,s,t∈Φ where 0≤k<1; then, Γ has a unique fixed point (say u, i.e., Γu=u), and Γ is G-continuous at u.Definition 15 (see [9]).
LetΨ be the set of all functions ϕ:ℝ+⟶ℝ+ satisfying
(i)
ϕ is monotone increasing, that is, t1≤t2 implies ϕt1≤ϕt2(ii)
the series∑n=0∞ϕnt is convergent for all t>0
Then,ϕ is called a c-comparison function.Remark 16.
Ifϕ∈Ψ, then ϕt<t for any t>0, ϕ0=0, and ϕ is continuous at 0.Karapınar and Fulga [1] gave the following definition of Jaggi-type hybrid contraction in metric space.Definition 17 (see [1]).
LetΦ,d be a complete metric space. A self-mapping Γ:Φ⟶Φ is called a Jaggi-type hybrid contraction; if there exists ϕ∈Φ such that
(7)dΓr,Γs≤ϕMr,s,for all distinct r,s∈Φ, where
(8)Mr,s=λ1dr,Γr·ds,Γsdr,sq+λ2dr,sq1/q,forq>0,r,s∈Φ,r≠s,dr,Γrλ1·ds,Γsλ2,forq=0,r,s∈Φ\FixΓ.
λ1,λ2≥0 with λ1+λ2=1 and FixΓ=r∈Φ:Γr=r.


## 3. Main Results

We begin this section by defining the notion of Jaggi-type hybridG−ϕ-contraction in G-metric space.Definition 18.
LetΦ,G be a G-metric space. A self-mapping Γ:Φ⟶Φ is called a Jaggi-type hybrid G−ϕ-contraction, if there exists ϕ∈Φ such that
(9)GΓr,Γs,Γ2s≤ϕMr,s,Γs,for all r,s∈Φ\FixΓ, where
(10)Mr,s,Γs=λ1Gr,Γr,Γ2r·Gs,Γs,Γ2sGr,s,Γsq+λ2Gr,s,Γsq1/q,forq>0,Gr,Γr,Γ2rλ1·Gs,Γs,Γ2sλ2,forq=0.
λ1,λ2≥0 with λ1+λ2=1 and FixΓ=r∈Φ:Γr=r.We now present the following results.Theorem 19.
LetΦ,G be a complete G-metric space and let Γ:Φ⟶Φ be a continuous Jaggi-type hybrid G−ϕ-contraction on Φ,G. Then, Γ has a fixed point in Φ (say c), and for any c0∈Φ, the sequence Γnc0n∈ℕ converges to c.Proof.
Letr0∈Φ be an arbitrary point and define a sequence rnn∈ℕ in Φ by rn=Γnr0. If there exists some n∈ℕ such that Γrn=rn+1=rn, then rn is a fixed point of Γ, and so the proof is complete. Assume now that rn≠rn−1 for any n∈ℕ. Since Γ is a Jaggi-type hybrid G−ϕ-contraction, then we have from (9) that
(11)Grn,rn+1,rn+2=GΓrn−1,Γrn,Γ2rn≤ϕMrn−1,rn,Γrn.We then consider the given cases of (10).Case 1.
Forq>0, we have
(12)Mrn−1,rn,Γrn=λ1Grn−1,Γrn−1,Γ2rn−1Grn,Γrn,Γ2rnGrn−1,rn,Γrnq+λ2Grn−1,rn,Γrnq1/q=λ1Grn−1,rn,rn+1Grn,rn+1,rn+2Grn−1,rn,rn+1q+λ2Grn−1,rn,rn+1q1/q=λ1Grn,rn+1,rn+2q+λ2Grn−1,rn,rn+1q1/q.Sinceϕ is nondecreasing, if we assume that
(13)Grn−1,rn,rn+1≤Grn,rn+1,rn+2,then (11) becomes
(14)Grn,rn+1,rn+2≤ϕλ1Grn,rn+1,rn+2q+λ2Grn−1,rn,rn+1q1/q≤ϕλ1Grn,rn+1,rn+2q+λ2Grn,rn+1,rn+2q1/q=ϕλ1+λ21/qGrn,rn+1,rn+2=ϕGrn,rn+1,rn+2<Grn,rn+1,rn+2,which is a contradiction. Therefore, for every n∈ℕ, we have
(15)Grn,rn+1,rn+2<Grn−1,rn,rn+1,so that (11) becomes
(16)Grn,rn+1,rn+2≤ϕλ1Grn,rn+1,rn+2q+λ2Grn−1,rn,rn+1q1/q≤ϕλ1+λ21/qGrn−1,rn,rn+1≤ϕGrn−1,rn,rn+1.Continuing inductively, we have(17)Grn,rn+1,rn+2≤ϕnGr0,r1,r2.Now, since(18)Grn,rn,rn+1≤Grn,rn+1,rn+2≤ϕnGr0,r1,r2,for all n∈ℕ with rn+1≠rn+2, then for any n,m∈ℕ with n<m and by rectangle inequality, we have
(19)Grn,rn,rm≤Grn,rn,rn+1+Grn+1,rn+1,rn+2+⋯+Grm−1,rm−1,rm≤ϕn+ϕn+1+ϕn+2+⋯+ϕm−1Gr0,r1,r2=∑i=nm−1ϕiGr0,r1,r2≤∑i=n∞ϕiGr0,r1,r2.Sinceϕ is a c-comparison function, then the series ∑i=0∞ϕiGr0,r1,r2 is convergent, and so denoting by Sp=∑i=0∞ϕiGr0,r1,r2, we have
(20)Grn,rn,rm≤Sm−1−Sn−1.Hence, asn,m⟶∞, we see that
(21)Grn,rn,rm⟶0.Thus,rn is a G-Cauchy sequence in Φ,G and so by the completeness of Φ,G, there exists c∈Φ such that rn is G-convergent to c, that is,
(22)limn⟶∞Grn,rn,c=0.We will now show thatc is a fixed point of Γ. By the assumption that Γ is continuous, we have
(23)limn⟶∞Gc,c,Γc=limn⟶∞Grn+1,rn+1,Γc=limn⟶∞GΓrn,Γrn,Γc=limn⟶∞GΓrn,Γrn,Γrn=0,so we getΓc=c, that is, c is a fixed point of Γ.Case 2.
Forq=0, we have
(24)Mrn−1,rn,Γrn=Grn−1,Γrn−1,Γ2rn−1λ1·Grn,Γrn,Γ2rnλ2=Grn−1,rn,rn+1λ1·Grn,rn+1,rn+2λ2.Now, ifGrn−1,rn,rn+1≤Grn,rn+1,rn+2, then (11) becomes
(25)Grn,rn+1,rn+2<Grn,rn+1,rn+2,which is a contradiction. Therefore,
(26)Grn,rn+1,rn+2<Grn−1,rn,rn+1.Hence, by (11) we have
(27)Grn,rn+1,rn+2<ϕGrn−1,rn,rn+1<ϕ2Grn−1,rn,rn+1<⋯<ϕnGr0,r1,r2.By similar argument as the case ofq>0, we can show that there exists a G-Cauchy sequence rn in Φ,G and a point c in Φ such that limn⟶∞rn=c. Similarly, under the assumption that Γ is continuous and by the uniqueness of limit, we have that Γc=c, that is, c is a fixed point of Γ.In the next result, we examine the existence of unique fixed point ofΓ under the restriction of continuity of some iterates of Γ.Theorem 20.
LetΦ,G be a complete G-metric space and let Γ:Φ⟶Φ be a Jaggi-type hybrid G−ϕ-contraction. If for some integer i>2, we have that Γi is continuous, then Γ has a unique fixed point in Φ.Proof.
In Theorem19, we have established that there exists a G-Cauchy sequence rnn∈ℕ in Φ,G with rn=Γrn−1 such that rn⟶c for some c in Φ. Let rnl be a subsequence of rnn∈ℕ where nl=l·i for all l∈ℕ, i>2 fixed. Notice that Γ0 is an identity self-mapping on Φ so that rnl=Γirnl−i. Hence, by the continuity of Γi, we have
(28)Gc,Γic,Γi+1c=liml⟶∞Gc,Γirnl−i,Γi+1rnl−i+1=liml⟶∞Gc,rnl,rnl=Gc,c,c=0,
that is,c is a fixed point of Γi.To see thatc is a fixed point of Γ, assume contrary that Γz≠z. Then in that case, Γi−j−1z≠Γi−jz for any j=0,1,⋯,i−1. Hence, by (9), we have
(29)GΓi−jc,Γi−j+1c,Γi−j+2c≤ϕMΓi−j−1c,Γi−jc,Γi−j+1c<MΓi−j−1c,Γi−jc,Γi−j+1c.Considering Case1, we obtain
(30)MΓi−j−1c,Γi−jc,Γi−j+1c=λ1GΓi−j−1c,ΓΓi−j−1c,Γ2Γi−j−1cGΓi−jc,ΓΓi−jc,Γ2Γi−jcGΓi−j−1c,Γi−jc,ΓΓi−jcq+λ2GΓi−j−1c,Γi−jc,ΓΓi−jcq1/q=λ1GΓi−j−1c,Γi−jc,Γi−j+1cGΓi−jc,Γi−j+1c,Γi−j+2cGΓi−j−1c,Γi−jc,Γi−j+1cq+λ2GΓi−j−1c,Γi−jc,Γi−j+1cq1/q=λ1GΓi−jc,Γi−j+1c,Γi−j+2cq+λ2GΓi−j−1c,Γi−jc,Γi−j+1cq1/q,so that (29) becomes
(31)GΓi−jc,Γi−j+1c,Γi−j+2cq1−λ1<λ2GΓi−j−1c,Γi−jc,Γi−j+1cq.Sinceλ1+λ2=1, then for every j=0,1,⋯,i−1, we have
(32)GΓi−jc,Γi−j+1c,Γi−j+2c<GΓi−j−1c,Γi−jc,Γi−j+1c.This clearly implies that for everyl=j,j+1,⋯,i−1,
(33)GΓi−jc,Γi−j+1c,Γi−j+2c<GΓi−j−l−1c,Γi−j−lc,Γi−j−l+1c.In particular, lettingj=0 and l=i−1, the above inequality becomes
(34)Gc,Γic,Γi+1c=GΓic,Γi+1c,Γi+2c<Gc,Γc,Γ2c,which is a contradiction. Hence,Γc=c.For Case2, we have
(35)MΓi−j−1c,Γi−jc,Γi−j+1c=GΓi−j−1c,ΓΓi−j−1c,Γ2Γi−j−1cλ1·GΓi−jc,ΓΓi−jc,Γ2Γi−jcλ2=GΓi−j−1c,Γi−jc,Γi−j+1cλ1·GΓi−jc,Γi−j+1c,Γi−j+2cλ2,so that (29) becomes
(36)GΓi−jc,Γi−j+1c,Γi−j+2c1−λ2<GΓi−j−1c,Γi−jc,Γi−j+1cλ1,implying that
(37)GΓi−jc,Γi−j+1c,Γi−j+2c<GΓi−j−1c,Γi−jc,Γi−j+1c.By similar argument as in Case1, we obtain a contradiction. Hence, Γc=c.Example 21.
LetΦ=−1,1 and let Γ:Φ⟶Φ be a self-mapping on Φ defined by
(38)Γr=r5,ifr∈−1,1,15,ifr∈−1,1,
for allr∈Φ. Define G:Φ×Φ×Φ⟶ℝ+ by
(39)Gr,s,Γs=r−s+r−Γs+s−Γs∀r,s∈Φ.
Then,Φ,G is a complete G-metric space and Γ is continuous for all r∈Φ. Define ϕ∈Ψ by ϕx=x/2 for all x≥0.
To see thatΓ is a Jaggi-type hybrid G−ϕ-contraction, notice that GΓr,Γs,Γ2s=0 for all r,s∈−1,1. Hence, inequality (9) holds for all r,s∈−1,1.Now, forr,s∈−1,1, if r=s=1, then GΓr,Γs,Γ2s=0 for all q≥0. If r=s=−1, then letting λ1=λ2=1/2 and q=1, we obtain
(40)GΓr,Γs,Γ2s=G−15,−15,15=45<1310=12135=12M−1,−1,−15=ϕMr,s,Γs.Also, forq=0, we have
(41)GΓr,Γs,Γ2s=45<12125=ϕMr,s,Γs.Ifr≠s, then letting λ1=2/10, λ2=4/5, and q=3, we obtain
(42)GΓr,Γs,Γ2s=G−15,15,15=G15,−15,15=45<85=12165=12M−1,1,15=12M1,−1,−15=ϕMr,s,Γs.Also, forq=0, we take λ1=λ2=1/2. Then,
(43)GΓr,Γs,Γ2s=G−15,15,15=G15,−15,15=45<4950=129850=12M−1,1,15=12M1,−1,−15=ϕMr,s,Γs.Hence, inequality (9) is satisfied for all r,s∈Φ. Therefore, Γ is a Jaggi-type hybrid G−ϕ-contraction. Consequently, all the assumptions of Theorem 19 are satisfied, and r=1/5 is the fixed point of Γ.We now demonstrate that our result is independent of the result of Karapınar and Fulga [1]. Let d:Φ×Φ⟶ℝ+ be defined by
(44)dr,s=r−s∀r,s∈Φ.Considerr,s∈−1,1 and take for Case 1, r≠s, λ1=3/4, λ2=1/4, and q=1. Then, inequality (9) becomes
(45)GΓr,Γs,Γ2s=G−15,15,15=G15,−15,15=45<4350=124325=12M−1,1,15=12M1,−1,−15=ϕMr,s,Γs,while inequality (7) due to Karapınar and Fulga [1] yields
(46)dΓr,Γs=d−15,15=d15,−15=25>37100=123750=12M−1,1=12M1,−1=ϕMr,s.Also, Karapınar and Fulga [1] declared in Definition (17) that r and s are distinct, since Mr,s is undefined for Case 1 if r=s. However, our result is valid for all r,s∈Φ\FixΓ.The above comparison is illustrated graphically for allr,s∈−1,1, using the following Figures 1 and 2.Figure 1
Illustration of contractive inequality (9) for all r,s∈−1,1.Figure 2
Illustration of contractive inequality (7) for all r,s∈−1,1.Therefore, Jaggi-type hybridG−ϕ-contraction is not Jaggi-type hybrid contraction defined by Karapınar and Fulga [1], and so Theorem 1 due to Karapınar and Fulga [1] is not applicable to this example.Corollary 22 (see Theorem14).
LetΦ,G be a complete G-metric space, and let Γ:Φ⟶Φ be a mapping satisfying the following condition:
(47)GΓr,Γs,Γt≤kGr,s,t,
for allr,s,t∈Φ where 0≤k<1; then, Γ has a unique fixed point (say u) and Γ is G-continuous at u.Proof.
Consider Definition (18) and let Γs=t, λ1=0, λ2=1, q>0, and ϕp=kp for all p≥0 and k∈0,1. Clearly, ϕ∈Ψ and Γ is a Jaggi-type hybrid G−ϕ-contraction. Hence, (9) coincides with (6) of Theorem 14 due to Mustafa [2]. Therefore, it is easy to see that we can find a unique point u in Φ such that Γu=u and Γ is G-continuous at u.Corollary 23 (see [10], Theorem 3.1).
LetΦ,G be a complete G-metric space. Suppose the mapping Γ:Φ⟶Φ satisfies
(48)GΓr,Γs,Γt≤ϕGr,s,t,
for allr,s,t∈Φ. Then, Γ has a unique fixed point (say u) and Γ is G-continuous at u.Proof.
Consider Definition18 and let Γs=t, λ1=0, λ2=1 and q>0. Then,
(49)Mr,s,t=Gr,s,t,
for allr,s,t∈Φ. Hence, inequality (9) becomes
(50)GΓr,Γs,Γt≤ϕGr,s,t,
for allr,s,t∈Φ and ϕ∈Ψ. This coincides with Theorem 3.1 due to Shatanawi [10] and so the proof follows in a similar manner.By specializing the parametersλii=1,2 and q, as well as letting ϕp=μp for all p≥0 and for μ∈0,1, the following result is also a direct consequence of Theorem 19.Corollary 24.
LetΦ,G be a complete G-metric space. If there exists μ∈0,1 such that for all r,s∈Φ, the mapping Γ:Φ⟶Φ satisfies
(51)GΓr,Γs,Γ2s≤μGr,s,Γs,
thenΓ has a fixed point in Φ.


## 4. Applications to Solution of Integral Equation

In this section, Corollary24 is applied to examine the existence criteria for a solution to a class of integral equations. Ideas in this section are motivated by [7, 11, 12].Consider the integral equation(52)ry=∫abLy,xfx,rxdx,y∈a,b.LetΦ=Ca,b,ℝ be the set of all continuous real-valued functions. Define G:Φ×Φ×Φ⟶ℝ+ by
(53)Gr,s,Γs=maxy∈a,bry−sy+maxy∈a,bry−Γsy+maxy∈a,bsyΓsy,(54)∀r,s∈Φ,y∈a,b.Then,Φ,G is a complete G-metric space.Define a functionΓ:Φ⟶Φ as follows:
(55)Γry=∫abLy,xfx,rxdx,y∈a,b.Then, a pointu∗ is said to be a fixed point of Γ if and only if u∗ is a solution to (52).Now, we study existence conditions of the integral equation (52) under the following hypotheses.Theorem 25.
Assume that the following conditions are satisfied:
(C1) L:a,b×a,b⟶ℝ+ and f:a,b×ℝ⟶ℝ are continuous
(C2) For all r,s∈Φ, x∈a,b, we have fx,rx−fx,sx≤rx−sx
(C3) maxy∈a,b∫abLy,xdx≤μ for some μ<1Then, the integral equation (52) has a solution u∗ in Φ.Proof.
Observe that for anyr,s∈Φ, using (55) and the above hypotheses, we obtain
(56)Γry−Γsy=∫abLy,xfx,rx−Ly,xfx,sxdx≤∫abLy,xfx,rx−fx,sxdx≤∫abLy,xrx−sxdx≤∫abLy,xmaxx∈a,brx−sxdx≤μmaxy∈a,bry−sy.Using this in (54), we have
(57)GΓr,Γs,Γ2s=maxy∈a,bΓr−Γs+maxy∈a,bΓr−Γ2s+maxy∈a,bΓs−Γ2s≤μmaxy∈a,br−s+μmaxy∈a,br−Γs+μmaxy∈a,bs−Γs=μmaxy∈a,br−s+maxy∈a,br−Γs+maxy∈a,bs−Γs=μGr,s,Γs.Hence, all the required hypotheses of Corollary24 are satisfied, implying that there exists a solution u∗ in Φ of the integral equation (52).Conversely, ifu∗ is a solution of (52), then u∗ is also a solution of (55) so that Γu∗=u∗, that is, u∗ is a fixed point of Γ.Remark 26.
(i)
We can deduce a number of corollaries by particularizing some of the parameters in Definition18(ii)
None of the results presented in this work can be expressed in the formGr,s,s or Gr,r,s. Hence, they cannot be obtained from their corresponding versions in metric space


## 5. Conclusion

A generalization of metric space was introduced by Mustafa and Sims [3], namely, G-metric space and several fixed point results were studied in that space. However, Jleli and Samet [5] as well as Samet et al. [6] established that most fixed point theorems obtained in G-metric space are direct consequences of their analogues in metric space. Contrary to the above observation, a new family of contractions called Jaggi-type hybrid G−ϕ-contraction is introduced in this manuscript and some fixed point theorems that cannot be deduced from their corresponding ones in metric space are proved. The main distinction of this class of contractions is that its contractive inequality is expressible in a number of ways with respect to multiple parameters. Consequently, some corollaries including recently announced results in the literature are highlighted and analyzed. Nontrivial comparative examples are constructed to validate the assumptions of our obtained theorems. Furthermore, one of our obtained corollaries is applied to set up novel existence conditions for solution of a class of integral equations.



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