# Alpha-F-Convex Contraction Mappings in b-Metric Space and a Related Fixed Point Theorem

Authors: Kitila Wirtu Geleta; Kidane Koyas Tola; Solomon Gebregiorgis Teweldemedhin
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we establish fixed point theorems forα-F-convex contraction mappings in b-metric space and prove the existence and uniqueness of fixed points for such mappings. Our result extends and generalizes comparable results in the existing literature. Finally, we provide an example in support of our main finding.


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## Body

## 1. Introduction

The theoretical framework of metric fixed point theory has been an active research field, and the contraction mapping principle is one the most important theorems in functional analysis. The contraction mapping principle introduced by Banach [1] has a wide range of applications in fixed point theory. The family of contractive mappings in different spaces is of great interest and has already been studied in the literature for a long time. Istratescu [2] introduced a class of convex contraction mappings in metric spaces and generalized the well-known Banach contraction principle. Afterwards, some works have appeared on the generalization of such classes of mappings in the setting of various spaces [3–12]. We focus on the research works of Eke et al. [5] for our purpose. They introduced the convexity condition to some classes of contraction mappings such as the Chatterjea contractive mapping and the Hardy and Rogers contractive mapping and proved fixed points of these maps in complete metric spaces.Inspired and motivated by the works of Eke et al. [5], in this paper we extend and generalize their main theorem in the setting of b-metric spaces.


## 2. Preliminaries

Throughout this research, we use the following notations.(1)
R=−∞,∞(2)
R+=0,∞(3)
ℕ=1,2,3,⋯(4)
Φ=F:R+⟶R be a mapping satisfying the following conditions:
(a)
F is strictly increasing, that is, for all x,y∈R+, if x<y, then Fx<Fy;(b)
For each sequenceαn of positive numbers, limn⟶∞αn=0, if and only if limn⟶∞Fαn=−∞;(c)
There existsk∈0,1 such that limα⟶0+αkFα=0.Definition 1 (see [13]).
LetT:X⟶X be a self-mapping on a nonempty set X and α:X×X⟶R+ be a function. We say that T is an α-admissible mapping if αx,y≥1⟹αTx,Ty≥1 for all x,y∈X.Definition 2 (see [11]).
Anα-admissible mapping T is said to be α∗-admissible, if for each x,y∈FixT≠∅, we have αx,y≥1. If FixT=∅, we say that T is vacuously α∗-admissible.Definition 3 (see [14]).
LetX,d be a metric space. T:X⟶X is said to be orbitally continuous on X if z∈X, such that limn⟶∞Tnx=z for some x∈X, then limn⟶∞TTnx=Tz.Definition 4 (see [15, 16]).
LetX be a nonempty set and s≥1 be a given real number. A function d:X×X⟶R+ is said to be a b-metric if and only if for all x,y,z∈X, the following conditions are satisfied:
(1)
dx,y=0, if and only if x=y(2)
dx,y=dy,x(3)
dx,y≤sdx,z+dz,y
The pairX,d is called a b-metric space.
It should be noted that the class ofb-metric spaces is effectively larger than that of metric spaces, since a b-metric is a metric when s=1. But, in general, the converse is not true.Example 5 (see [16]).
LetX=−1,0,1 and d:X×X⟶R+ be given by dx,y=dy,x for all x,y∈X, dx,x=0, d−1,0=3, and d−1,1=d0,1=d1,0=1; then, X,d is a b-metric on X with s=3/2, but it is not a metric space since for x=−1, y=0, and z=1, we have d−1,0=3≮2=d−1,1+d1,0.Hence, the triangle inequality for a metric is not satisfied.Definition 6 (see [17]).
LetX,d be a b-metric space, and let T:X⟶X be a given mapping. We say that T is continuous at x0∈X if and only if for every sequence xn in X, we have xn⟶x0, as n⟶∞; then, Txn⟶Tx0, as n⟶∞. If T is continuous at each point of x0∈X, then we say that T is continuous on X.In general, ab-metric is not necessarily continuous.Example 7 (see [18]).
LetX=ℕU∞.
Define a mappingd:X×X⟶R+ as follows:
(1)dm,n=0,ifm=n,1m−1n,ifoneofmandnisevenandtheotherevenor∞,5,ifoneofmandnisoddandtheotheroddor∞,2,ifothers.
It satisfies all the conditions of ab-metric space with the value of the parameter s=5/2. Hence, X,d is a b-metric space.
Now, choosexn=2n for each n∈ℕ. Then, we have
(2)dxn,∞=d2n,∞=12n⟶0,asn⟶∞.
That is,xn⟶∞, as n⟶∞. But dxn,1=2↛5=d∞,1asn⟶∞. Hence, it is not continuous.Definition 8 (see [17]).
LetX be a b-metric space and xn be a sequence in X; thus, we say that
(1)
xnb-converges to x∈X if dxn,x⟶0, as n⟶∞(2)
xn is a b-Cauchy sequence if dxn,xm⟶0, as n,m⟶∞(3)
X,d is b-complete if every b-Cauchy sequence in X is b-convergentDefinition 9 (see [2]).
A mappingT:X⟶X defined on a metric space X is called a two-sided convex contraction mapping if there exist positive numbers a1,a2,b1,b2∈0,1, such that the following inequality holds:
(3)dT2x,T2y≤a1dx,Tx+a2dTx,T2x+b1dy,Ty+b2dTy,T2y,for all x,y∈X and a1+a2+b1+b2<1.Theorem 10 (see [2]).
LetX,d be a complete metric space and T:X⟶X be any two-sided convex contraction mapping. Then T has a unique fixed point.Definition 11 (see [19]).
LetX,d be a metric space and T:X⟶X be a self-map; then, T is said to be an F-contraction mapping on X,d if there exists a F∈Φ and τ>0 such that
(4)dTx,Ty>0⟹τ+FdTx,Ty≤Fdx,y,for all x,y∈X.Remark 12.
If the mapT satisfies (4), then T is contractive, i.e.,
(5)FdTx,Ty≤Fdx,y−τ<Fdx,y⟹dTx,Ty<dx,y,for all x,y∈X such that Tx≠Ty.T is also a continuous mapping.Definition 13 (see [11]).
LetX,d be a metric space. A mapping T:X⟶X is said to be an α-F-convex contraction on X,d if there exist two functions α:X×X⟶R+ and F∈Φ, and a constant τ>0 such that
(6)dT2x,T2y>0⟹τ+Fαx,ydT2x,T2y≤FadTx,Ty+bdx,y,for all x,y∈X, where a,b≥0 and a+b<1.Theorem 14 (see [11]).
LetX,d be a complete metric space and T:X⟶X be an α-F-convex contraction satisfying the following conditions:
(1)
T is α-admissible(2)
There existsx0∈X such that αx0,Tx0≥1(3)
T is continuous or orbitally continuous on X
Then,T has a fixed point in X.Definition 15 (see [5]).
A mappingT:X⟶X defined on a metric space X is called the Chatterjea two-sided convex contraction mapping if there exist positive numbers a1,a2,b1,b2∈0,1 such that
(7)dT2x,T2y≤a1dx,Ty+a2dTy,T2y+b1dy,Tx+b2dTx,T2x,for all x,y∈X and a1+a2+b1+b2<1.Theorem 16 (see [5]).
LetX,d be a complete metric space and T be a self-mapping of X satisfying the Chatterjea two-sided convex contraction condition. Suppose also that T is orbitally continuous. Then, T has a unique fixed point in X. For any x0∈X, the Picard iteration xnn=0∞ given by xn+1=Tnx0, n≥0 converges to the fixed point of T.Definition 17 (see [5]).
LetX,d be a metric space. A mapping T:X⟶X is called the Hardy and Rogers convex contraction mapping of type 2 if there exist positive numbers a1,a2,b1,b2,c1,c2,e1,e2,f1,f2∈0,1 such that the following inequality holds:
(8)dT2x,T2y≤a1dx,y+a2dTx,Ty+b1dx,Tx+b2dTx,T2x+c1dy,Ty+c2dTy,T2y+e1dx,Ty+e2dTy,T2y+f1dy,Tx+f2dTx,T2x,for all x,y∈X and a1+a2+b1+b2+c1+c2+e1+e2+f1+f2<1.Theorem 18 (see [5]).
LetX,d be a complete metric space and T be a self-mapping satisfying the Hardy and Rogers convex contraction condition. Suppose T is orbitally continuous. Then, T has a unique fixed point in X. For any x0∈X, the Picard iteration xnn=0∞ given by xn+1=Tnx0, n≥0 converges to the fixed point of T.


## 3. Main Results

In this section, we prove a fixed point result forα-F-convex contraction mappings in the setting of b-metric spaces, but first we introduce our definition as follows.Definition 19.
LetX,d be a b-metric space with parameter s≥1, T:X⟶X, α:X×X⟶R+, and F∈Φ. Then, T is called a Hardy and Rogers α-F-convex contraction mapping if there exists ai,bi,ci,ei,fi∈0,1 with a1+a2+b1+b2+c1+c2+e1+e2+f1+f2<1/s and for all x,y∈X and τ>0,
(9)dT2x,T2y>0⟹τ+Fαx,ydT2x,T2x≤Fa1dx,y+a2dTx,Ty+b1dx,Tx+b2dTx,T2x+c1dy,Ty+c2dTy,T2y+e1dx,Ty+e2dTy,T2y+f1dy,Tx+f2dTx,T2x.Theorem 20.
LetX,d be a complete b-metric space with s≥1.
LetT:X⟶X be a Hardy and Rogers α-F-convex contraction mapping satisfying the following conditions:
(a)
T is α-admissible(b)
There existsx0∈X such that αx0,Tx0≥1(c)
T is orbitally continuous on X
Then,T has a fixed point in X. Further, if T is α∗-admissible, then T has a unique fixed point z∈X. Furthermore, for any x0∈X if xn+1=Tn+1x0≠Txn for all n≥0, then limn⟶∞Tnx0=z.Proof.
By (b) there exists a pointx0∈X such that αx0,Tx0≥1 and define a sequence xn by
(10)x1=Tx0,x2=Tx1,x3=Tx2,⋯,xn+1=Txn,⋯for all n=0,1,2,⋯.
Ifxn=xn+1 for some n, then xn=xn+1=Txn, and hence, xn is a fixed point of T.
Now, assumexn≠xn+1 for all n=0,1,2,⋯. Then dxn,xn+1>0 for all n=0,1,2,⋯.
SinceT is α-admissible, we have
(11)αx0,Tx0≥1⟹αx1,x2=αTx0,T2x0≥1.
Therefore, one can obtain inductively that(12)αxn,xn+1=αTnx0,Tn+1x0≥1,for all n=0,1,2,⋯.
Substitutingx=x0 and y=Tx0 in (9), we have
(13)dT2x0,T3x0>0⟹τ+Fαx0,Tx0dT2x0,T3x0≤Fa1dx0,Tx0+a2dTx0,T2x0+b1dx0,Tx0+b2dTx0,T2x0+c1dTx0,T2x0+c2dT2x0,T3x0+e1dx0,T2x0+e2dT2x0,T3x0+f1dTx0,Tx0+f2dTx0,T2x0=Fa1dx0,Tx0+a2dTx0,T2x0+b1dx0,Tx0+b2dx0,T2x0+c1dTx0,T2x0+c2dT2x0,T3x0+e1dx0,T2x0+e2dT2x0,T3x0+f2dTx0,T2x0≤Fa1dx0,Tx0+a2dTx0,T2x0+b1dx0,Tx0+b2dx0,T2x0+c1dTx0,T2x0+c2dT2x0,T3x0+e1sdx0,T2x0+e2dT2x0,T3x0+f2dTx0,T2x0=Fa1dx0,Tx0+a2dTx0,T2x0+b1dx0,Tx0+b2dx0,T2x0+c1dTx0,T2x0+c2dT2x0,T3x0+e1dx0,T2x0+e2dT2x0,T3x0+f2dTx0,T2x0=Fa1+b1+se1sdx0,Tx0+a1+b1+e1+f2dTx0,T2x0+c2+e2dT2x0,T3x0≤Fa1+b1+2se1+a2+b2+f2maxdx0,Tx0,dTx0,T2x0+c2+e2dT2x0,T3x0=Fa1+b1+c1+2se1s+a2+b2+f2v+c2+e2dT2x0,T3x0,where v=maxdx0,Tx0,dTx0,T2x0.
SinceF is strictly increasing, and τ>0, (13) becomes
(14)dT2x0,T3x0<a1+b1+2se1s+a2+b2+f2v+c2+e2dT2x0,T3x0.
Collecting like terms and rearranging in (14), we get
(15)1−c2−e2dT2x0,T3x0<a1+b1+c1+2se1+a2+b2+f2v<dT2x0,T3x0<a1+b1+c1+2e1s+a2+b2+f21−c2−e2v≤λv,where λ=a1+b1+c1+2e1s+a2+b2+f2/1−c2−e2.
Substitutingx=Tx0 and y=T2x0, in (9) and using (12) we have
(16)T3x0,T4x0>0⟹τ+FαTx0,T2x0dT3x0,T4x0≤Fa1dTx0,T2x0+a2dT2x0,T3x0+b1dTx0,T2x0+b2dT2x0,T3x0+c1dT2x0,T3x0+c2dT3x0,T4x0+e1dTx0,T3x0+e2dT3x0,T4x0+f1dT2x0,T2x0+f2dT2x0,T3x0=Fa1dTx0,T2x0+a2dT2x0,T3x0+b1dTx0,T2x0+b2dT2x0,T3x0+c1dT2x0,T3x0+c2dT3x0,T4x0+e1dTx0,T3x0+eedT3x0,T4x0+f2dT2x0,T3x0≤Fa1dTx0,T2x0+a2dT2x0,T3x0+b1dTx0,T2x0+b2dT2x0,T3x0+c1dT2x0,T3x0+c2dT3x0,T4x0+e1sdTx0,T2x0+sdT2x0,T3x0+e2dT3x0,T4x0+f2dT2x0,T3x0=Fa1dTx0,T2x0+a2dT2x0,T3x0+b1dTx0,T2x0+b2dT2x0,T3x0+c1dT2x0,T3x0+c2dT3x0,T4x0+e1sdTx0,T2x0+e1sdT2x0,T3x0+e2dT3x0,T4x0+f2dT2x0,T3x0=Fa1+b1+e1sdTx0,T2x0+a2+b2+C1+e1s+f2dT2x0,T3x0+c2+e2dT3x0,T4x0≤Fa1+b1+e1sv+a2+b2+c1+e1s+f2υ+c2+e2dT3x0,T4x0.
SinceF is strictly increasing, and τ>0, we get
(17)dT3x0,T4x0<a1+b1+2se1+a2+b2+f2v+c2+e2dT3x0,T4x0.
Collecting like terms and rearranging in (17), we get
(18)1−c2−e2dT3x0,T4x0<a1+b1+c1+2se1+a2+b2+f2vdT3x0,T4x0<a1+b1+c1+2e1s+a2+b2+f21−c2−e2υ≤λv.
Similarly, we can show thatdT4x0,T5x0≤λ2v and dT5x0,T6x0≤λ2v.
Continuing this process inductively, we get(19)dTmx0,Tm+1x0<λl⟶0,asl⟶∞.
wherem=2l when m is even and m=2l+1 when m is odd for l≥1.
Now, we show thatxn is a b-Cauchy sequence in X.
Hence, we consider two cases, i.e., whenm is even or m=2l and m is odd or m=2l+1 for l≥1. Let m,n>0 with n>m.Case 1.
m=2l and l≥1.(20)dTmx0,Tnx0=dT2lx0,Tnx0≤sdT2lx0,T2l+1x0+dT2l+1x0,Tnx0≤sdT2lx0,T2l+1x0+s2dT2l+1x0,T2l+2x0+s3dT2l+2x0,T2l+3x0+s4dT2l+3x0,T2l+4x0+s5dT2l+4x0,T2l+5x0+⋯+sn−1dTn−1x0,Tnx0≤sλlv+s2λlv+s3λl+1v+s4λl+1v+s5λl+2v+s6λl+2v+⋯=sλl1+s2λ+s4λ2+s6λ3+⋯v+s2λl1+s2λ+s4λ2+s6λ3+⋯v=s+s2λl1+s2λ+s4λ2+s6λ3+⋯v≤s+s2λl11−s2λvdTmx0,Tnx0≤s+s2λl11−s2λv⟶0,asl⟶∞.Case 2.
m=2l+1 and l≥1.
(21)dTmx0,Tnx0=dT2l+1x0,Tnx0≤sdT2l+1x0,T2l+2x0+dT2l+2x0,Tnx0≤sdT2l+1x0,T2l+2x0+s2dT2l+2x0,T2l+3x0+s3dT2l+3x0,T2l+4x0+s4dT2l+4x0,T2l+5x0+s5dT2l+5x0,T2l+6x0+⋯+sn−1dTn−1x0,Tnx0≤sλlv+s2λl+1v+s3λl+1v+s4λl+2v+s5λl+2v+s6λl+3v+⋯=sλl1+s2λ+s4λ2+s6λ3+⋯v+s2λl1+s2λ+s4λ2+s6λ3+⋯v=s+s2λl1+s2λ+s4λ2+s6λ3+⋯v≤s+s2λl11−s2λvdTmx0,Tnx0≤s+s2λl11−s2λv⟶0asl⟶∞.
That is(22)limn⟶∞dTmx0,Tnx0=0.
This shows thatxn is a b-Cauchy sequence in X.
SinceX is b-complete, there exists z∈X such that
(23)limn⟶∞xn=limn⟶∞Tnx0=z.
Now, we prove thatz is a fixed point of T.
SinceT is orbitally continuous, we obtain
(24)z=limn⟶∞TTnx0=Tz.
This shows thatz is a fixed point of T.
Now, we show the uniqueness of the fixed point.
We suppose thatT is α∗-admissible. Since FixT≠∅, let z,z∗∈FixT. By α∗-admissible of T, we have αz,z∗≥1.
Using (9), we have
(25)Fdz,z∗=FdT2z,T2z∗=Fαz,z∗dT2z,T2z∗≤Fa1dz,z∗+a2dTz,Tz∗+b1dz,Tz+b2dTz,T2z+c1dz∗,Tz∗+c2dTz∗,T2z∗+e1dz,Tz∗+e2dTz∗,T2z∗+f1dz∗,Tz+f2dTz,T2z−τ≤Fa1dz,z∗+a2dz,z∗+e1dz,z∗+f1dz∗,z−τ.
Sinceτ>0 and F is strictly increasing, (25) becomes
(26)dz,z∗<a1dz,z∗+a2dz,z∗+e1dz,z∗+f1dz∗,z<a1+a2+e1+f1dz,z∗.
dz,z∗<dz,z∗ is a contradiction, which in turn gives z∗=z.
Hence,T has a unique fixed point in X.Now, we give an example in support of Theorem20.Example 21.
LetX=0,1 and d:X×X⟶R+ be given by
(27)dx,y=x−y2,for x,y∈X. Then, X,d is a complete b-metric space with s=2. Define a mapping T:X⟶X by
(28)Tx=x24+18,for all x∈X with αx,y=1, and for all x,y∈X.Then,T is α-admissible. Let F∈Φ be Fγ=lnγ, γ>0.Since we have(29)Tx−Ty=12x2−y2≤x−y,for all x∈X, we have
(30)αx,yT2x−T2y2=T2x−T2y2=12564x4−4y4+4x2−4y22=164x4−y4+x2−y22≤164x4−y42+x2−y22≤12Tx−Ty2+14x−y2≤114dx,y+116dTx,Ty+130dx,Tx+125dTx,T2x+114dy,Ty+116dTy,T2y+130dx,Ty+140dTy,T2y+125dy,Tx+140dTx,T2x≤19514200maxx−y2,Tx−Ty2,x−Tx2,Tx−T2x2,y−Ty2,Ty−T2y2,x−Ty2,Ty−T2y2,y−Tx2,Tx−T2x2≤e−τdx,y,dTx,Ty,dx,Tx,dTx,T2x,y,Ty,dTy,T2y,dx,Ty,dTy,T2y,dy,Tx,dTx,T2x,where −τ=ln1951/4200.Taking a natural logarithm on both sides of (30), we obtain
(31)τ+Fαx,ydT2x,T2y≤Fa1dx,y+a2dTx,Ty+b1dx,Tx+b2dTx,T2x+c1dy,Ty+c2dTy,T2y+e1dx,Ty+e2dTy,T2y+f1dy,Tx+f2dTx,T2x,where a1=c1=1/14, a2=c2=1/16, b1=e1=1/30, b2=f1=1/25, e2=f2=1/40.This shows thatT is the Hardy and Rogers α-F-convex contraction mapping.We define a sequencexn by
(32)xn=nn+1−12,then xn⟶1−1/2, as n⟶∞.Therefore,(33)xn+1=Txn=14nn+1−122+18⟶1−12,asn⟶∞.Thus, all conditions of Theorem20 are satisfied, and x=1−1/2 is the unique fixed point of T in X.Remark 22.
If we takes=1 and F is the identity function in Theorem 20, then we get the extended and generalized form of Theorem 18 of Eke et al. [5].Remark 23.
Ife1=e2=f1=f2=0 and we take s=1 in Theorem 20, then we get Theorem 18 of Singh [11].


## 4. Conclusion

Eke et al. [5] established fixed point theorems for convex contractive mappings in complete metric spaces and proved the existence and uniqueness of fixed points. In this paper, we establish fixed point theorems for α-F-convex contraction mappings and prove the existence and uniqueness of fixed points in the setting of complete b-metric spaces. Our result extends and generalizes comparable results in the existing literature. We also provided an example in support of our main results.



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