# On Fixed-Point Results of Generalized Contractions

Authors: Shazia Kanwal; Umair Hanif; Maha Eshaq Noorwali; Md. Ashraful Alam
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

The purpose of this research article is to introduce new contractive conditions and to examine the existence and uniqueness of fixed points of self-mappings in the context of b-metric spaces by applying these different contractive conditions. Furthermore, some examples are given to illustrate its validity and superiority. Our results generalize and extend several well-known results in metric and b-metric spaces.


---

## Body

## 1. Introduction and Preliminaries

Many issues in engineering and science explained by nonlinear equations may be tackled by confining them to analogous fixed-point case. An operator sumGx = 0 can be proved as fixed-point sum Fx = x, wherein F is a self-defining along with some relevant discipline. Fixed-point theory endows with some key modes to resolving problems ensuing from multiple offshoots of mathematical inspection such as split feasibility issues, supportive problems, equilibrium problems, and matching, as well as selective issues and such others. The theory of fixed points is the great vibrant and energetic zone of the study. This theory has previously been exposed as an excessive and major deployment for cramming nonlinear analysis. Specially, fixed-point procedures are being applied in a diversity of fields, for example, biology, chemistry, engineering, economics, physics, and game theory. Functional analysis is a very useful and important field of mathematics. Its results are supportive tools for other fields to solve many problems. Many researchers have put their efforts in obtaining these results; for further study, see [1–12] and literature. Because of Banach [4], the Banach-contraction theorem (1922) is indeed the most significant consequence in the theory of fixed points in metric spaces. This theorem promises the presence and distinctiveness of fixed points of self-mapping that satisfy the contraction condition on complete metric spaces as well as dispense a valuable approach for finding them. Since contractive condition deduces the uniform continuity of an operator f, so it was a natural question to raise the concern about existence of fixed point in the absence of continuity of f. In 1968, Kannan [12] answered this question by the introduction of Kannan contractive condition. One of the most famous generalizations of metric spaces was given by Bakhtin [3] in 1989. He presented the notion of b-metric space in which triangular inequality has been relaxed. In 1993, Czerwik [7] drew out the results in b-metric space. By accepting this idea, many researchers gave extensions of Banach’s principle in b-metric space. Boriceanu [5], Czerwik [7], Bota [13], and Pacurar [13] drew out the fixed-point theorems in b-metric space. Moreover, many authors examined and derived the existence of fixed point of a contraction function in the context of b-metric spaces; for detail, see [12, 14–17] and references therein. In this paper, we have established fixed-point results in the context of b-metric spaces for two different contractive conditions. Some direct consequences from our main results are also presented. In the support of these results, examples are created.Definition 1 (see [4]).
Consider a metric spaceΩ with metric d. A function G:Ω⟶Ω is known as Banach contraction on G if there exists a number α∈0,1 such that ∀, a,b∈Ω:(1)dGa,Gb≤αda,b.Definition 2 (see [9]).
ConsiderΩ,d a metric space and G:Ω⟶Ω is a function if ∃α∈0,1/2 such that, for all a1,a2∈Ω, we have(2)dGa1,Ga2≤αda1,Ga1+da2,Ga2.
Then,G is known as Kannan contraction.Definition 3 (see [3]).
LetΩ be any nonempty set and w≥1 be a real number. A function d:Ω×Ω⟶R+ is called b-metric if axioms given below are fulfilled for all μ,ν,ξ∈Ω:dμ,ν≥0anddν,μ=0iffμ=νdμ,ν≥0anddν,μ=0iffμ=νdμ,ξ≤wdμ,ν+dν,ξ
Then,Ω,d is called b-metric space.
If we takew=1, then b-metric space becomes ordinary metric space. Hence, set of all metric spaces is a subset of set of all b-metric spaces.Example 1.
LetΩ=R; then, dω1,ω2=ω1−ω22 is a b-metric space with s=2.(1)
Obviously,dω1,ω2=ω1−ω22 is real, finite, and nonnegative.(2)
Considerdω1,ω2=0⇔ω1−ω22=0⇔ω1−ω2=0⇔ω1=ω2 and dω1,ω2=ω1−ω22=ω2−ω12=dω2,ω1.So,dω1,ω2=dω2,ω1.(3)
To solve the triangular property in b-metric space, we will use the convexity of function, i.e.,“If1<p<∞, then convexity of function fa=apa>0 implies that a+b/2p≤ap/2+bp/bp, i.e.,(3)a+b2p≤ap2+bpbpa+bp≤2p−1ap+bp,holds,” and we have(4)dω1,ω3=ω1−ω32=ω1−ω2+ω2−ω32=ω1−ω2+ω2−ω32.
Using (3),(5)dω1,ω3≤2dω1,ω2+dω2,ω3.
Hence,dω1,ω2=ω1−ω22 is a b-metric with s=2.Example 2 (see [5]).
The setlp with 0<p<1, where lp=xn⊂ℝ:∑ι=1∞μιp<∞, together with the function d:lp×lp⟶0,∞, is (6)dμ,ν=∑ι=1∞μι−νιp1/p,where μ=μιandν=νι∈lp is b-metric space with w=21/p>1. Notice that the abovementioned result holds with 0<p<1.Definition 4 (see [5]).
LetΩ,d be b-metric space and zn be a sequence in Ω. Then,(1)
zn is called a convergent sequence if and only if there exists z∈Ω, such that for any r>0∃nr∈N such that, for all n≥nr, we get dzn,z<r. In this case, we write limn⟶∞zn=z.(2)
zn is said to be a Cauchy sequence if and only if for any r>0∃nr∈N such that, for each i,j≥nr, we get dzi,zj<r.(3)
Ω is called complete if every Cauchy sequence in Ω is convergent in Ω.
LetΩ be a nonempty set and R:Ω⟶Ω be a self-map. We say that x∈Ω is a fixed point of R if Rx=x.
LetΩ be any set and R:Ω⟶Ω be a self-map. For any given x∈Ω, we define Rnx inductively by R0x=x and Rn+1x=RRnx, and we recall Rnx, the nth iterative of x under R. For any x0∈Ω, the sequence xnn≥0⊂Ω is given by(7)xn=Rxn−1=Rnx0,n=1,2,…which is known as the sequence of successive approximations, where x0 is the initial value. This is also known as the Picard iteration starting at x0.


## 2. Fixed Points for Contractive Mappings

This section consists of our main results. We investigate the existence and uniqueness of fixed points of some new contractive conditions in the context of b-metric spaces. Moreover, results are supported with example.Theorem 1.
LetΩ,d be a complete b-metric space with coefficient ≥1. Let R be a function R:Ω⟶Ω such that(8)dRx,Ry≤adx,y+bmaxdx,Rx,dy,Ry,dx,Ry,dy,Rx,where a,b>0 such that a+2sb<1,∀x,y∈Ω and s≥1. Then, there is a unique fixed point of R.Proof.
Letɤ0∈Ω and ɤnn=1∞ be a sequence in Ω defined by the recursion:(9)ɤn=Rɤn−1=Rnɤ0,n=1,2,3,4,….
By (8) and (9),(10)dɤn,ɤn+1=dRɤn−1,Rɤn,(11)≤adɤn−1,ɤn+bmaxdɤn−1,Rɤn−1,dɤn,Rɤn,dɤn−1,Rɤn,dɤn,Rɤn−1,(12)=adɤn−1,ɤn+bmaxdɤn−1,ɤn,dɤn,ɤn+1,dɤn−1,ɤn+1,dɤn,ɤn.
So,(13)dɤn,ɤn+1≤adɤn−1,ɤn+bmaxdɤn−1,ɤn,dɤn,ɤn+1,dɤn−1,ɤn+1,dɤn,ɤn+1≤adɤn−1,ɤn+bM,where M=maxdɤn−1,ɤn,dɤn,ɤn+1,dɤn−1,ɤn+1,.Case 1.
IfM=dɤn−1,ɤn, thendɤn,ɤn+1≤adɤn−1,ɤn+bdɤn−1,ɤndɤn,ɤn+1≤a+bdɤn−1,ɤnLetk=a+b<1; then, dɤn,ɤn+1≤kdɤn−1,ɤn anddɤn,ɤn+1≤kndɤ0,ɤ1.Case 2.
IfM=dɤn,ɤn+1, then(14)dɤn,ɤn+1≤adɤn−1,ɤn+bdɤn,ɤn+1,(15)1−bdɤn,ɤn+1≤adɤn−1,ɤn,(16)dɤn,ɤn+1≤a1−bdɤn−1,ɤn.
Letk=a/1−b<1; then,(17)dɤn,ɤn+1≤kdɤn−1,ɤn,
(18)dɤn,ɤn+1≤kndɤ0,ɤ1.Case 3.
IfM=dɤn−1,ɤn+1, then(19)dɤn,ɤn+1≤adɤn−1,ɤn+bdɤn−1,ɤn+1,dɤn,ɤn+1≤adɤn−1,ɤn+bsdnn−1,ɤn+dɤn,ɤn+1,(20)dɤn,ɤn+1≤adɤn−1,ɤn+sbdɤn−1,ɤn+sbdɤn,ɤn+1,1−sbdɤn,ɤn+1≤a+sbdɤn−1,ɤn,(21)dɤn,ɤn+1≤a+sb1−sbdɤn−1,ɤn.
Letk=a+sb/1−sb<1; then,(22)dɤn,ɤn+1≤kdɤn−1,ɤn,(23)dɤn,ɤn+1≤kndɤ0,ɤ1.
Cases1–3 show that R is a contractive-type mapping.
Letm,n∈Nandm>n; then,(24)dɤn,ɤm≤sdɤn,ɤn+1+dɤn+1,ɤm,dɤn,ɤm≤sdɤn,ɤn+1+sdɤn+1,ɤm,dɤn,ɤm≤sdɤn,ɤn+1+sdɤn+1,ɤm,dɤn,ɤm≤sdɤn,ɤn+1+sdɤn+1,ɤm,dɤn,ɤm≤sdɤn,ɤn+1+ssdɤn+1,ɤn+2+dɤn+2,ɤm,dɤn,ɤm≤sdɤn,ɤn+1+s2dɤn+1,ɤn+2+s2dɤn+2,ɤm.
Continue in the following way:(25)dɤn,ɤm≤sdɤn,ɤn+1+s2dɤn+1,ɤn+2+s3dɤn+2,ɤn+3+⋯+sm−ndɤm−1,ɤm.
By Cases1–3,(26)dɤn,ɤm≤skndɤ0,ɤ1+s2kn+1dɤ0,ɤ1+s3kn+2dɤ0,ɤ1+⋯+sm−nkm−1dɤ0,ɤ1,=dɤ0,ɤ1skn+s2kn+1+s3kn+2+⋯+sm−nkm−1.
Sinceskn+s2kn+1+s3kn+2+⋯+sm−nkm−1 is geometric series with common ratio less than 1, so(27)dɤn,ɤm≤skn1−skn1−skdɤ0,ɤ1≤skn1−skdɤ0,ɤ1.
Asn⟶∞andk<1, skn/1−sk⟶0.
Hence,(28)dɤn,ɤm⟶0.
Hence,ɤn is a Cauchy sequence in Ω. Due to completeness of Ω, there exists ɤ∗∈Ω such that(29)ɤn⟶ɤ∗,asn⟶∞.
Now,(30)dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+dɤn+1,Rɤ∗,dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+sdɤn+1,Rɤ∗,dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+sdRɤn,Rɤ∗,dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+sadɤn,ɤ∗+bmaxdɤn,Rɤn,dɤ∗,Rɤ∗,dɤn,Rɤ∗dɤ∗,Rɤn,dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+sadɤn,ɤ∗+bMo,where Mo=maxdɤn,Rɤn,dɤ∗,Rɤ∗,dɤn,Rɤ∗,dɤ∗,Rɤn.Case 4.
IfMo=dɤn,Rɤn, then(31)dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+asdɤn,ɤ∗+bsdɤn,Rɤn,dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+asdɤn,ɤ∗+bsdɤn,ɤn+1.
Asn⟶∞andɤn⟶ɤ∗, so(32)dɤ∗,Rɤ∗⟶0.
Hence,(33)ɤ∗=Rɤ∗.
ɤ∗ is a fixed point of R.Case 5.
IfMo=dɤ∗,Rɤ∗, then(34)dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+asdɤn,ɤ∗+bsdɤ∗,Rɤ∗,1−bsdɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+asdsɤn,ɤ∗,dɤ∗,Rɤ∗≤s1−bsdɤ∗,ɤn+1+as1−bsdɤn,ɤ∗.
Asn⟶∞andɤn⟶ɤ∗, so(35)dɤ∗,Rɤ∗⟶0.
Hence,ɤ∗=Rɤ∗. ɤ∗ is a fixed point of R.Case 6.
IfMo=dωn,Rω∗, then(36)dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+asdɤn,ɤ∗+bsdɤn,Rɤ∗,dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+asdɤn,ɤ∗+bssdɤn,ɤ∗+dɤ∗,Rɤ∗,1−s2bdɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+as+s2bdɤn,ɤ∗,dɤ∗,Rɤ∗≤s1−s2bdɤ∗,ɤn+1+as+s2b/1−s2bdɤn,ɤ∗.
Asn⟶∞andɤn⟶ɤ∗, so(37)dɤ∗,Rɤ∗⟶0.
Hence,ɤ∗=Rɤ∗.
Hence,ɤ∗ is a fixed point of R.Case 7.
IfMo=dɤ∗,Rɤn, then(38)dn∗,Rɤ∗≤sdɤ∗,ɤn+1+asdɤn,ɤ∗+bsdɤ∗,Rɤn,dɤ∗,Rɤ∗≤sdɤ∗,ɤn+1+asdɤn,ɤ∗+bsdɤ∗,ɤn+1,dɤ∗,Rɤ∗≤s+bsdɤ∗,ɤn+1+asdɤn,ɤ∗.
Asn⟶∞andɤn⟶ɤ∗, so(39)dϒ∗,Rɤ∗⟶0.
Hence,ɤ∗=Rɤ∗. ɤ∗ is a fixed point of R.
Now, assume thatɤ' is another fixed point of R, i.e., Rɤ′=ɤ′. Consider(40)dɤ∗,ɤ′=dRɤ∗,Rɤ′,dɤ∗,ɤ′≤adɤ∗,ɤ′+bmaxdɤ∗,Rɤ∗,dɤ′,Rɤ′,dɤ∗,Rɤ′,dɤ′,Rɤ∗,dɤ∗,ɤ′≤adɤ∗,ɤ′ɤ′+bmaxdɤ∗,ɤ∗,dɤ′,ɤ′,dɤ∗,ɤ′,dɤ′,ɤ∗,dɤ∗,ɤ′≤adɤ∗,ɤ′+bmaxdɤ∗ɤ′,dɤ′,ɤ∗,dɤ∗,ɤ′≤adɤ∗,ɤ′+bdɤ∗,ɤ′,dɤ∗,ɤ′≤a+bdɤ∗,ɤ′,wherea,b>0. That is not possible; therefore, ɤ∗=ɤ′. Hence, ɤ∗ is unique fixed point of R.Example 3.
SupposeX=1,2,3; we define σ:X×X⟶ℝ as below:σ (1, 1) = σ (2, 2) = σ (3, 3) = 0σ (1, 2) = σ (2, 1) = σ (2, 3) = σ (3, 2) = 1σ (1, 3) = σ (3, 1) = z≥2
Then, (X, σ) is a b-metric space with coefficient z/2, where z≥2.
DefineR:X⟶X by Rx=x3−6x2+11x−4,for allx∈X.
Then, all conditions of the above theorems are satisfied fora=1/4,b=1/6,and2≤z<4.5. Hence, 2∈X is the unique fixed point of R.
In the following, some direct consequences of Theorem1 are as follows.Corollary 1 (Banach-contraction theorem in b-metric space).
LetΩ,d be a complete b-metric space with coefficient s≥1. Let R be a self-function on Ω such that(41)dRx,Ry≤⋯adx,y,where 0≤a<1,∀x,y∈Ω. Then, there exists a unique fixed point of R.Proof.
In Theorem1, by putting b = 0, the required result will be obtained.Corollary 2.
LetΩ,d be a complete metric space and a function R:Ω⟶Ω such that(42)dRx,Ry≤adx,y+bmaxdx,Rx,dy,Ry,dx,Ry,dy,Rx,where a,b>0 such that a+2b<1,∀x,y∈Ω. Then, there exists a unique fixed point of R.Proof.
Since every b-metric space is a metric space by takings = 1, so the proof is obvious.Corollary 3 (Banach-contraction theorem in metric space).
LetΩ,d be a complete metric space. Let R be a function R:Ω⟶Ω such that(43)dRx,Ry≤adx,y,where 0≤a<1,∀x,y∈Ω. Then, there exists a unique fixed point of R.Proof.
In Corollary2, by putting b = 0, the required will be obtained.Theorem 2.
LetΩ,d be a complete b-metric space. Let R be a mapping R:Ω⟶Ω such that(44)dRx,Ry≤amaxdx,Ry,dy,Rx,dx,y+bdx,Rx+dy,Ry,where a,b>0 such that 2as+2b<1,∀x,y∈Ωands≥1. Then, there exists a unique fixed point of R.Proof.
Letω0∈Ω and ωn be a sequence in Ω defined as(45)ωn=Rωn−1=Rnω0,n=1,2,3,…
By (44) and (45), we obtain that(46)dωn,ωn+1=dRωn−1,Rωn,dωn,ωn+1≤amaxdωn−1,Rωn,dωn,Rωn−1,dωn−1,ωn+bdωn−1,Rωn−1+dωn,Rωn,dωn,ωn+1≤amaxdωn−1,ωn+1,dωn,xn,dωn−1,ωn+bdωn−1,ωn+dωn,ωn+1,(47)dωn,ωn+1≤aM+bdωn−1,ωn+dωn,ωn+1,where(48)M=maxdωn−1,ωn+1,dωn,ωn,dωn−1,ωn,M=maxdωn−1,ωn+1,dωn−1,ωn.Case 8.
IfM=dωn−1,ωn+1, then (44) becomes(49)dωn,ωn+1≤adωn−1,ωn+1+bdωn−1,ωn+dωn,ωn+1,dωn,ωn+1≤asdωn−1,ωn+dωn,ωn+1+bdωn−1,ωn+bdωn,ωn+1,dωn,ωn+1≤asdωn−1,ωn+asdωn,ωn+1+bdωn−1,ωn+bdωn,ωn+1,1−b−asdωn,ωn+1≤as+bdωn−1,ωn,dωn,ωn+1≤as+b/1−b−asdωn−1,ωn,dωn,ωn+1≤kdωn−1,ωn,wherek=as+b/1−b−as<1.
Continuing in this way,(50)dωn,ωn+1≤k2dωn−2,ωn−1,
(51)dωn,ωn+1≤kndω0,ω1.Case 9.
IfM=dωn−1,ωn, then (47) becomes(52)dωn,ωn+1≤adωn−1,ωn+bdωn−1,ωn+dωn,ωn+1,dωn,ωn+1≤adωn−1,ωn+bdωn−1,ωn+bdωn,ωn+1,1−bdωn,ωn+1≤a+bdωn−1,ωn,dωn,ωn+1≤a+b1−bdωn−1,ωn,dωn,ωn+1≤kdωn−1,ωn,wherek=a+b/1−b<1.
Continuing in this way,(53)dωn,ωn+1≤k2dωn−2,ωn−1,
(54)dωn,ωn+1≤kndω0,ω1.
Cases8 and 9 show that R is a contractive-type mapping.
Now, letm and n be any two natural numbers and m > n; then,(55)dωn,ωm≤sdωn,ωn+1+dωn+1,ωm,dωn,ωm≤sdωn,ωn+1+sdωn+1,ωm,dωn,ωm≤sdωn,ωn+1+ssdωn+1,ωn+2+dωn+2,ωm,dωn,ωm≤sdωn,ωn+1+s2dωn+1,ωn+2+s2dωn+2,ωm.
Continuing in this way,(56)dωn,ωm≤sdωn,ωn+1+s2dωn+1,ωn+2+s3dωn+2,ωn+3+⋯+sm−ndωm−1,ωm,
By Cases1 and 2,(57)dωn,ωm≤skndωn,ωn+1+s2kn+1dωn+1,ωn+2+s3kn+2dωn+2,ωn+3+⋯+sm−nkm−1dωm−1,ωm,dωn,ωm≤dω0,ω1skn+s2kn+1+s2kn+2+⋯+sm−nkm−1.
Sinceskn+s2kn+1+s2kn+2+⋯+sm−nkm−1 is a geometric series with common ratio less than 1, so dωn,ωm≤dω0,ω1skn1−skn/1−sk≤dω0,ω1skn/1−sk.
Asn⟶∞,skn/1−sk⟶0,so(58)dωn,ωm⟶0.
Hence,ωn is a Cauchy sequence in Ω. Due to completeness of Ω, there exists ω∗inΩ such that ωn⟶ω∗. Now,(59)dω∗,Rω∗≤sdω∗,ωn+1+dωn+1,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sdωn+1,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sdRωn,Rω∗.
By (44),(60)dω∗,Rω∗≤sdω∗,ωn+1+samaxdωn,Rω∗,dω∗,Rωn,dωn,ω∗+sbdωn,Rωn+dω∗,Rω∗,(61)dω∗,Rω∗≤sdω∗,ωn+1+saM0+sbdωn,Rωn+dω∗,Rω∗,where M0=maxdωn,Rω∗,dω∗,Rωn,dωn,ω∗.Case 10.
IfM0=dωn,Rω∗, then (61) becomes(62)dω∗,Rω∗≤sdω∗,ωn+1+sadωn,Rω∗+sbdωn,Rωn+dω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sabdωn,ω∗+dω∗,Rω∗+sbdωn,Rωn+dω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sabdωn,ω∗+sabdω∗,Rω∗+sbdωn,Rωn+sbdω∗,Rω∗,1−sb−sabdω∗,Rω∗≤sdω∗,ωn+1+sabdωn,ω∗+sbdωn,ωn+1,dω∗,Rω∗≤s1−sb−sabdω∗,ωn+1+sab1−sb−sabdωn,ω∗+sb1−sb−sabdωn,ωn+1.
Asn⟶∞,dω∗,Rω∗⟶0. Hence,(63)ω∗=Rω∗.
ω∗ is a fixed point of R.Case 11.
IfM0=dω∗,Rωn, then (61) becomes(64)dω∗,Rω∗≤sdω∗,ωn+1+sadω∗,Rωn+sbdωn,Rωn+dω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sabdω∗,ωn+1+dωn+1,Rωn+sbdωn,Rωn+dω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sabdω∗,ωn+1+sabdωn+1,Rωn+sbdωn,Rωn+sbdω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sabdω∗,ωn+1+sabdωn+1,ωn+1+sbdωn,ωn+1+sbdω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sabdω∗,ωn+1+sbdωn,ωn+1+sbdω∗,Rω∗,1−sbdω∗,Rω∗≤s+sabdω∗,ωn+1+sbdωn,ωn+1,dω∗,Rω∗≤s+sab1−sbdω∗,ωn+1+sb1−sbdωn,ωn+1.
Asn⟶∞,dω∗,Rω∗⟶0. Hence,(65)ω∗=Rω∗.
ω∗ is a fixed point of R.Case 12.
IfM0=dωn,ω∗, then (61) becomes(66)dω∗,Rω∗≤sdω∗,ωn+1+sadωn,ω∗+sbdωn,Rωn+dω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sadωn,ω∗+sbdωn,Rωn+sbdω∗,Rω∗,dω∗,Rω∗≤sdω∗,ωn+1+sadωn,ω∗+sbdωn,ωn+1+sbdω∗,Rω∗,1−sbdω∗,Rω∗≤sdω∗,ωn+1+sadωn,ω∗+sbdωn,ωn+1,dω∗,Rω∗≤s1−sbdω∗,ωn+1+sa1−sbdωn,ω∗+sb1−sbdωn,ωn+1.
Asn⟶∞,dω∗,Rω∗⟶0. Hence,(67)ω∗=Rω∗,where ω∗ is a fixed point of R.
For uniqueness, assume thatω′ is another fixed point of R, i.e., Rω′=ω′. Consider(68)dω∗,ω′=dRω∗,Rω′.
By (44),(69)dω∗,ω′≤amaxdω∗,Rω′,dω′,Rω∗,dω∗,ω′+bdω∗,Rω∗+dω′,Rω′,dω∗,ω′≤amaxdω∗,ω′,dω′,ω∗,dω∗,ω′+bdω∗,ω∗+dω′,ω′,dω∗,ω′≤amaxdω∗,ω′,dω′,ω∗,dω∗,ω′+0,dω∗,ω′≤adω∗,ω′,which is a contradiction. Hence,(70)ω∗=ω′,where ω∗ is unique fixed point of R.Example 4.
SupposeX=1,2,3, and we define σ:X×X⟶ℝ as below:(71)σ1,1=σ2,2=σ3,3=0,σ1,2=σ2,1=σ2,3=σ3,2=1.
σ (1, 3) =  σ (3, 1) =  z≥2.
Then, (X, σ) is a b-metric space with coefficient z/2, where z≥2.
DefineT:X⟶X by Tx=x3−6x2+11x−3,for allx∈X.
Then, all conditions of the above theorems are satisfied fora=1/4,b=1/8,and2≤z<3. Hence, 3∈X is the unique fixed point of T.Corollary 4.
(Kannan-contraction theorem inb-metric space). Let Ω,d be a complete b-metric space. Let R be a mapping R:Ω⟶Ω such that(72)dRx,Ry≤bdx,Rx+dy,Ry,where 0<b<1/2,∀x,y∈Ω,ands≥1. Then, R has a unique fixed point.Proof.
By puttinga = 0 in Theorem 2, we get the required result.Corollary 5.
LetΩ,d be a complete metric space and R be a mapping R:Ω⟶Ω such that(73)dRx,Ry≤amaxdx,Ry,dy,Rx,dx,y+bdx,Rx+dy,Ry,where a,b>0 such that 2a+2b<1,∀x,y∈Ω. Then, there exists a unique fixed point of R.Corollary 6.
(Kannan-contraction theorem in metric space). LetΩ,d be a complete metric space and a self-mapping R:Ω⟶Ω such that(74)dRx,Ry≤bdx,Rx+dy,Ry.where 0<b<1/2 and ∀x,y∈Ω. Then, there exists a unique fixed point of R.Proof.
By puttinga = 0 in Corollary 5, we get the required result.


## 3. Conclusion

We have established and proved fixed-point results for different contractive conditions in b-metric space. To furnish, these results supportive examples are created. This generalization will be helpful for further investigation and applications. We conclude this paper by indicating, in the form of open questions, some directions for further investigation and work:(1)
Can the conditionsa + 2sb in Theorem 1 and 2as + 2b in Theorem 2 be relaxed?(2)
If the answer to 1 is yes, then what hypotheses ona and b are needed to guarantee the existence of fixed points?(3)
Can the concept of fixed point for these contractions be extended to more than one mapping?



---

*Source: 9167716-2022-02-25.xml*