# Onη,γf,g-Contractions in Extended b-Metric Spaces

Authors: Jayashree Patil; Basel Hardan; Ahmed A. Hamoud; Amol Bachhav; Homan Emadifar; Afshin Ghanizadeh; Seyyed Ahmad Edalatpanah; Hooshmand Azizi
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, we give a concept ofη,γf,g-contraction in the setting of expanded b–metric spaces and discuss the existence and uniqueness of a common fixed point. Introduced results generalize well-known fixed point theorems on contraction conditions and in the given spaces.


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## Body

## 1. Introduction and Preliminaries

The tremendous applications of fixed point theory had always inspired the growth of this domain. In 1922, Banach formulated his most simple but very natural result which is now popularly referred to as the Banach contraction principle. In the course of the last several decades, this principle has been extended and generalized in many directions with several applications in many branches. Employing simulation functions, Khojasteh et al. [1] initiated the idea of Z-contractions and utilized the same to cover the varied types of nonlinear contractions in the existing literature. Later, Argoubi et al. [2] and Roldán-López-de-Hierro et al. [3] independently sharpened the notion of simulation functions and also proved some coincidences and common fixed point results. Very recently, Lopez et al. [4] introduced the notion of R-contractions in order to extend several nonlinear contractions such as Z-contractions, manageable contractions, and Meir-Keeler contractions. Indeed, R-contractions are associated with R-functions that satisfy two independent conditions involving two sequences of nonnegative real numbers. Soon, inspired by R-contractions, Shahzad et al. [5] introduced the notion of η,γ-contractions which remains an extension of R,γ-contractions given in [6] by Roldán-López-de-Hierro and Shahzad wherein the authors proved very interesting results.Czerwik [7] established a successful generalization of the metric space concept by introducing the notion of b-metric space. Following this, a number of authors have introduced respective interesting theorems in b-metric, (see [8–14]). Newly, Kamran et al. [15] inspired by the concept of b-metric space, they introduced the concept of extended b space and also developed some fixed point theorems for self-mappings defined in such spaces. Their results extend/generalize many of the results already available in the literature. In this paper, we shall define a general contraction condition with the help of some auxiliary functions and investigate the existence and uniqueness of a fixed point for such mappings in the frame of b-metric space.Definition 1 (see [7]).
LetX be a nonempty set and let d:X×X⟶0,∞ satisfy the following for all u,v,w∈X.
(i)
du,v=0⇔u=v(ii)
du,v=dv,u(iii)
du,w≤bdu,v+dv,w, where b≥1
The pairX,d is called a b-metric space; when b=1, the b-metric space becomes a usual metric space.Example 2 (see [8, 9]).
LetX=ℝ2. Then, the functional d:X×X⟶0,∞ defined by
(1)du1,v1,u2,v2≔u1−u2+v1−v2,u1,v1,u2,v2∈0,1×0,1u1−u2+v1−v2,u1,v1,u2,v2∈1,∞×1,∞0,otherwise,is a b-metric space on X with b=2.Example 3 (see [16]).
The spaceLp0,1 (where 0<p<1) of all real functions ut,t∈0,1 such that ∫01utpdt<∞, together with the functional
(2)δu,v=∫01ut−vtpdt1/p,∀u,v∈Lp0,1,is a b-metric space with b=21/p.Example 4 (see [17]).
LetX=a,b,c and d:X×X⟶ℝ+ such that
(3)da,b=db,a=da,c=dc,a=b,db,c=dc,b=α≥c,da,a=db,b=dc,c=a.
Then,(4)du,v≤α2du,w+dw,v.
Therefore,X,d is a b-metric space for all u,v,w∈X. If α>c, then the ordinary triangle inequality does not hold, and X,d is not a metric space.Definition 5 (see [15]).
LetX be a nonempty set and ψ:X×X⟶1,∞, and let dψ:X×X⟶0,∞ satisfy:
(i)
dψu,v=0⇔u=v(ii)
dψu,v=dψv,u(iii)
dψu,w≤ψu,wdψu,v+dψv,w
The pairX,dψ is called an extended b-metric space. If ψu,v=b, for b≥1, then we reduce to Definition 1.Example 6 (see [17]).
LetX=a,b,c∪ℝ0+ and d:X×X⟶0,∞ be defined by
(i)
Ifu,v∈ℝ0+, then dψu,v=u−v2,(ii)
Ifu∈a,b,c and v∈ℝ0+, then dψu,v=dψv,u=1 and dψu,u=0(iii)
Ifu,v∈a,b,c, such that
(5)dψa,b=1,dψa,c=12anddψb,c=2,with dψu,v=dψv,u and dψu,u=0.Notice thatd is not a metric space since dψb,c>dψb,a+dψa,c. However, it is easy to see that d is an extended b-metric space for ψ:X×X⟶1,∞, where
(6)ψu,v≔43ifu,v∈a,b,c,2ifu,v∈ℝ0+,1ifu,vorv,u∈a,b,c×ℝ0+.Example 7 (see [15]).
LetX=Ca,b,ℝ be the space of all continuous real valued functions defined on a,b, let ψ:X×X⟶1,∞ where ψu,v=ut+vt+2, and note that X is a complete extended b-metric space by considering
(7)dψu,v=supt∈a,but−vt2.In this context, wonderful theorems established by the authors in extendedb-metric space, for examples, Fahed et al. and Swapna and Phaneendra [18, 19], got some new fixed point results in an extended b-metric space. Also, Ullah et al. [20] proved fixed point theorems in complex-valued extended b-metric spaces. In this bearing, Mitrović et al. [21] established new results in extended b-metric space, in follows that we recollect some fundamental notions, for example, convergence, the notion of the Cauchy sequence, and completeness in an extended b- metric space.Definition 8 (see [22]).
LetX,dψ be an extended b-metric space, and then
(i)
A sequenceun∈X is said to converge to u0∈X if, ∀ε>0, there exists N=Nε∈ℕ such that dψun,u0<ε,∀n≥N. We write limn⟶∞un=u0(ii)
A sequenceun in X is said to be Cauchy if, ∀ε>0, there exists N=Nε∈ℕ such that dψun,um<ε,∀n,m≥NDefinition 9 (see [15]).
An extendedb-metric space X,dψ is complete if every Cauchy sequence in X is convergent.Lemma 10 (see [22]).
LetX,dψ be a complete extended b-metric space. If dψ is continuous map, then every convergent sequence in X has a unique limit.Theorem 11 (see [15]).
SupposeX,dψ is an extended b-metric space such that dψ is a continuous mapping. Suppose T:X⟶X, it fulfills
(8)dψTu,Tv≤ηdψu,v,∀u,v∈X,where η∈0,1 is such that, for each u0∈X, we have limn,m⟶∞ψun,um<1/η. Here, Tnu0=un,n=1,2,⋯Then, T has exactly one fixed point u0, moreover ∀v∈X,Tnv⟶u0.For our objectives, we recall the definition of orbital admissible maps introduced by Popescu [23].Definition 12.
LetS be a self-map on X and η:X×X⟶0,∞. We say that S is an η-orbital admissible if for all u,v∈X, we have
(9)ηu,Su≥1⇒ηSu,S2u≥1.Remark 13 (see [23]).
Everyη-admissible mapping is an α-orbital admissible mapping.Definition 14 (see [24]).
For a nonempty setX, suppose T:X⟶X and η:X×X⟶0,∞ are mappings. One says that self-mapping T on X is η-admissible if for u,v∈X, one has
(10)ηu,v≥1⇒ηTu,Tv≥1.Definition 15 (see [17]).
LetΦ be the family of functions ϕ:0,∞⟶0,∞ satisfying the following conditions
(i)
ϑ is nondecreasing(ii)
ϑχ<χ,χ>0


## 2. Main results

Definition 16.
LetX,d be an extended b-metric space and let η:X×X⟶0,∞ and ψ:X×X⟶1,∞ such that
(11)ηu,vdψTu,Tv≤ψfu,v,where
(12)fu,v=supþψSu,Sv,þψTu,Su,þψSv,Tv,þψSu,TuþψSv,TvþψSu,Sv,þψSu,Tv+þψSv,Tu2supψSv,Tu,ψSu,Tv.Then,S and T are η,γf-contraction for all u,v∈X, where ψ∈Ψ.The headmost principal result of this paper is as follows:Theorem 17.
LetX,d be a complete extended b- metric space, and let S,T:X⟶X be an η,γf-contraction mappings. Let
(13)limn,m⟶∞supϑn+1rϑnrψun,um<1,for all u0∈X,r>0 where Tnun−n=Sun=un,n∈ℕ. Assume also that
(i)
S and T are η-orbital admissible(ii)
There existsw∈X such that ηSw,Tw≥1(iii)
S and T are continuous
Then,S and T possess a unique coincidence fixed point u0; that is, Su=Tu=u0.Proof.
By a supposition, for someu0∈X, we have un=Sun=Tnu0,∀n∈ℕ. Suppose that un0=Tun0−1=Sun0. Let that un≠un+1,∀n∈ℕ. Since T and S are η-admissible, we get
(14)ηu0,u1=ηSu0,Tu0≥1⇒ηSu1,Tu1=ηu1,u2≥1.
Repeatedly, we obtain for alln∈ℕ(15)ηun,un+1≥1.
On regard of (15) and (11), we get
(16)dψun,un+1=dψSun,Sun+1=dψTun−1,Tun≤ϑfun−1,un,where
(17)fun−1,un=supdψSun−1,Sun,dψTun−1,Sun−1,dψSun,Tun,dψSun−1,Tun−1dψSun,TundψSun−1,Sun,dψSun−1,Tun+dψSun,Tun−12supψSun,Tun−1,ψSun−1,Tun=supdψun−1,un,dψun,un−1,dψun,un+1,dψun−1,undψun,un+1dψun−1,un,dψun−1,un+1+dψun,un2supψun,un,ψun−1,un+1=supdψun,un−1,dψun,un+1,dψun−1,un+12supψun,un,ψun−1,un+1≤supdψun,un−1,dψun,un+1,dψun−1,un+dψun,un+12=supdψun,un−1,dψun,un+1.
Now, iffun−1,un=dψun,un+1 for some n∈ℕ, thus
(18)dψun,un+1≤ϑdψun,un+1<dψun,un+1,which is a contradiction. On the other hand, if fun−1,un=dψun,un−1, then for all n≥1, we have
(19)dψun,un+1≤ϑdψun,un−1<dψun,un−1.
Sequentially for alln∈ℕ, we get
(20)dψun,un+1≤ϑndψu0,u1.
Thus, there existsk≥0 such that
(21)dψun,un+1=k,whenn⟶∞.
Takingn⟶∞ to inequality (19), we obtain
(22)k≤ϑk⇒k=0.
Therefore, whenn⟶∞, we get
(23)dψun,un+1=0.
We will show thatun is a Cauchy sequence, as follows:
(24)dψun,un+m≤ψun,un+mdψun,un+1+dψun+1,un+m≤ψun,un+mdψun,un+1+ψun,un+mdψun+1,un+m.
Also,(25)dψun+1,un+m≤ψun+1,un+mdψun+1,un+2+dψun+2,un+m≤ψun+1,un+mdψun+1,un+2+ψun+1,un+mdψun+2,un+m.
And so, until the inequality (24) reaches to
(26)dψun,un+m≤ψun,un+mdψun,un+1+dψun+1,un+m≤ψu1,un+mψu2,un+m⋯ψun,un+mϑndψu0,u1+ψu1,un+mψu2,un+m⋯ψun,un+mψun+1,un+mϑn+1dψu0,u1+⋯+⋯+ψu1,un+mψu2,un+m⋯ψun+m−1,un+mϑn+m−1dψu0,u1=∑i=nn+m−1ϑidψu0,u1∏j=1iψuj,un+m=bn+m−1,
we conclude that(27)dψun,un+m<bn+m−1−bn.
The series(28)∑i=1∞ϑidψu0,u1∏j=1iψuj,un+m.
Supposeϑndψu0,u1∏j=1nψuj,un+m=Bn, then
(29)ϑn+1dψu0,u1∏j=1n+1ψuj,un+mϑndψu0,u1∏j=1nψuj,un+m=Bn+1Bn,ϑn+1dψu0,u1ϑndψu0,u1ψun+1,un+m=Bn+1Bn<1,by (13) where n,m⟶∞. Then, (28) converges by [25]. As result, in perspective of (27), we have
(30)dψun,un+m=0,asn,m⟶∞.
Then,un is a Cauchy sequence, and since X,d is a complete extended quasimetric space, there exists ω∈X such that
(31)dψun,ω=0,asn⟶∞.
By condition (ii), we obtain(32)limn⟶∞dψTun,Tω=limn⟶∞dψun+1,Tω=dψω,Tω=0,limn⟶∞dψSun,Sω=limn⟶∞dψun,Tω=dψω,Sω=0.
Hence, we deduce thatTω=Sω=ω. Furthermore, let ω∗∈X such that Tω∗=Sω∗=ω∗ where ω≠ω∗. So, by (12), we obtain
(33)dψSω,Sω∗=dψTω,Tω∗≤ϑfω,ω∗,where
(34)fω,ω∗=supdψSω,Sω∗,dψTω,Sω,dψSω∗,Tω∗,dψSω,TωdψSω∗,Tω∗dψSω,Sω∗,dψSω,Tω∗+dψSω∗,Tω2supψSω∗,Tω,ψSω,Tω∗=supdψω,ω∗,dψω,ω∗supψω∗,ω,ψω,ω∗=dψω,ω∗.
Then,(35)dψSω,Sω∗=dψTω,Tω∗=dψω,ω∗≤ϑdψω,ω∗.
Therefore,dψω,ω∗=0; thus, ω=ω∗. Hence, T and S possess a unique coincidence fixed point in X.To mitigation the continuity case on the given self-mappings, we will modify Definition16 as follows:Definition 18.
LetX,d be an extended quasimetric space, and we say that S and T are η,γg-contraction such that η:X×X⟶0,∞ and ψ:X×X⟶1,∞ if for all u,v∈X fulfilled
(36)ηu,vdψTu,Tv≤ϑgu,v,ψ∈Ψ,and
(37)gu,v=supdψSu,Sv2,dψSu,Tu+dψSu,Tv2,dψSu,Tv+dψSu,Tu2supψu,Tu,ψu,Su.By remove continuity of the given mappings, we get the following major result.Theorem 19.
LetX,d be a complete extended quasimetric space, and let S,T:X⟶X be η,γg-contraction mappings. Let (13) and conditions (i) and (ii) of Theorem 17 be satisfied. Assume also that
(iii)
Ifun is a sequence in X such that ηun,un+1≥1,∀n and un⟶u∈X as n⟶∞, then there exists a subsequence unk⊂un such that ηun,unk≥1,∀k
Then,S and T possess a coincidence fixed point u0, that is, Su=Tu=u0.Proof.
By inequality (11) and condition (iii) in Theorem 19, there exists unk⊂un such that ηunk,u≥1,∀k. Applying inequality (11), we obtain that
(38)dψunk+1,Tu=dψTunk,Tu≤ηunk,udψTunk,Tu≤ϑgunk,u.
And(39)dψunk,Su=dψSunk,Su≤ηunk,udψSunk,Su≤ϑgunk,u.
Also, we have(40)gunk,u=supdψSunk,Su2,dψSunk,Tunk+dψSunk,Tu2,dψSunk,Tunk+dψSunk,Tusupψunk,Tunk,ψunk,Sunk=supdψu,Su2,dψu,Tu2,dψu,TusupψSu,u,ψu,Tu.
Then,(41)gunk,u=supdψu,Su2,dψu,Tu2.
When(42)gunk,u=dψu,Su2,then neither dψu,Su>0 nor dψu,Su=0 for some u∈X when k⟶∞. If dψu,Tu>0, then by inequality (39), we obtain gunk,u>0. Thus, from Definition 18, we get
(43)ψgunk,u<gunk,u.
Also, by inequalities (38) and (42), we have
(44)dψSunk,Su≤ηunk,udψSunk,Su≤ϑgunk,u<gunk,u=dψu,Su2.
Thus,(45)2dψu,Su≤dψu,Su,ask⟶∞,
Which is an ambivalence. Therefore,dψu,Su=0 and u=Su. Likewise, we can get that u=Tu. Hence, u is a common fixed point for S and T in X, that is, u=Su=Tu.
Letu1,u2∈X be two common fixed points of S and T such that u1≠u2 then by (38) and (39), we get
(46)dψu1,u2=dψTu1,Tu2≤ηu1,u2dψTu1,Tu2≤ϑgu1,u2,where
(47)gu1,u2=supdψSu1,Su22,dψSu1,Tu1+dψSu1,Tu22,dψSu1,Tu1+dψSu1,Tu2supψu1,Tu1,ψu1,Su1=supdψu1,u22,dψu1,u2supψu1,u1=dψu1,u22.
Now, ifdψu1,u2>0, then ϑgu1,u2<gu1,u2, which implies by (46) that
(48)2dψu1,u2<dψu1,u2,but this is a contradiction. Hence, dψu1,u2=0, i.e., u1=u2. This proves the uniqueness of the common fixed point of given mappings.Example 20.
Suppose thatG1∪G2∪G3=X, where G1=−∞,0,G2=1/2,1/3,1/4,1/5 and G3=1,2. Consider the extended b-metric space on X as follows:
(49)dψ12,13=dψ14,15=310,dψ12,15=dψ13,14=210,dψ12,14=dψ15,13=610,dψ12,12=dψ13,13=⋯=0,dψu,v=u−v∈G3,else.It is apparent that the triangle inequality onG1 is not fulfilled. Actually,
(50)610=dψ12,14≥dψ12,13+dψ13,14=510.Observes that the condition (iii) in Definition5 is satisfied. Suppose that f,g:X⟶X are defined as
(51)Tu=0,ifu∈G1,14,ifu∈G2,12,ifu∈G3,Su=u,∀u∈X.Taking(52)ηu,v=1,ifu,v∈G2∪G30,ifelse,T and S are an η,γ-contractive mappings with γt=t/2,t∈0,∞. Furthermore, there exists u0∈X such that ηTu0,Su0≥1.Actually, we have foru=1/2∈X.
(53)ηT12,S12=η14,12=1.Now, suppose thatu,v∈X with ηu,v≥1. It implies that u,v∈G2∪G3.By Definition1, we have ηTu,Sv≥1, and then T and S are η-admissible mappings. Also, T and S are clearly not continuous mappings.Otherwise, ifun⊂X such that, ηun,un+1≥1, then un∈G2∪G3 for all n. Assume the sequence un is considered iteratively as fun−1=un for all n. Considering the arbitrary point u0∈X, it is located in either G2 or G3; so, we have two cases:In caseu0∈G2, thus un is constant sequence and un⟶1/4∈G2. Then, for all n, we have ηun,un+1≥1, which implies that ηun,1/4=ηun+1,1/4≥1.In caseu0∈G3, thus un is constant sequence and un⟶1/2∈G2. Then, for all n, we have ηun,un+1≥1, which implies that ηun,1/2=ηun+1,1/2≥1.Consequently,T and S fulfill the conditions of Theorem 19; hence, T and S have a unique common fixed point on X, which is u=1/4.


## 3. Conclusion

By replacingfu,v or gu,v with a proper one, we can conclude several results from the showed prime result in this paper on different sides. For example, we can obtain results in this frame of periodic contractions and partially ordered spaces.



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