# Related Fixed Point Theorems via General Approach of Simulations Functions

Authors: Kastriot Zoto; Nabil Mlaiki; Hassen Aydi
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this work, we extend and complement some results in view of general and wider structures, such asb−metric spaces. By considering existing classes of Ζ−contractions and Ψ−simulating functions with a solid impact in database results of fixed point theory, we introduce a new general class of simulating functions, called as Ψ−s simulation functions, and also types of κψ−s− contractions in a more general framework. This approach covers, extends, and unifies several published works in the early and late literature.


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## Body

## 1. Introduction

Some of the significant generalizations of metric fixed point theory are related with the well-known Banach Contraction Principle [1] and classical contractions such as Boyd and Wong, Geraghty, Browder, and Ciric. In recent years, the theory of fixed points has attracted widespread attention and has been rapidly growing. It was massively studied by many researchers giving new results by using classes of implicit functions defining new and large contractive conditions. Recently, Khojasteh et al. [2] presented the notion of Ζ−contractions involving a new class of simulation functions that has been used and improved by many authors in various spaces, see [3–30]. Authors in [19] proposed new notion Ψ−simulation functions and established the type of Ζψ−contractions.Inspired by the above works, in this paper we introduce a new class of general type ofΨ−s simulation functions, defined in the setting of b−metric-like spaces. This class generalizes further and complements some results given in the framework of b−metric spaces.


## 2. Preliminaries

Definition 1 (see [6]).
LetX be a nonempty set and s≥1 be a given real number. A mapping d:X×X⟶0,+∞ is called a b−metric-like if for all x,y,z∈X, the following conditions are satisfied:(1)dx,y=0impliesx=y,dx,y=dy,x,dx,y≤sdx,z+dz,y.
The pairX,d is called a b−metric-like space.
In ab−metric-like space X,d, if x,y∈X, and dx,y=0, then x=y; however, the converse need not be true, and dx,xmay be positive for x∈X.Definition 2 (see [6]).
LetX,d be a b−metric-like space with parameter s≥1 and let xq be any sequence in X and x∈X. Then, we have the following:(a)
xq is said to be convergent to x if limq⟶+∞dxq,x=dx,x(b)
xq is said to be a Cauchy sequence in X,d if limq,p⟶+∞dxq,xp exists and is finite(c)
The pairX,d is called a complete b−metric-like space if, for every Cauchy sequence xq in X, there is x∈X such that limq,p⟶+∞dxq,xp=limq⟶+∞dxq,x=dx,xLemma 1 (see [6, 29, 30]).
Letxq and yq be two sequences in X,d that converge to x and y, respectively. Then, we have(2)s−2dx,y−s−1dx,x−dy,y≤lim infq⟶+∞dxq,yq≤lim supq⟶+∞dxq,yq≤sdx,x+s2dy,y+s2dx,y.In particular,dx,y=0⟹limq⟶+∞dxq,yq=0.Also, for eachz∈X, the above inequality becomes(3)s−1dx,z−dx,x≤lim infq⟶+∞dxq,z≤lim supq⟶+∞dxq,z≤sdx,z+sdx,x.In particular, if,dx,x=0, then(4)s−1dx,z≤lim infq⟶+∞dxq,z≤lim supq⟶+∞dxq,z≤sdx,z.Lemma 2 (see [23]).
Letxq be a sequence in theb-metric-like space X,d with parameter s≥1, such that(5)limq⟶+∞dxq,xq+1=0.Iflimq,p⟶+∞dxq,xp≠0, then there are ε>0 and two sequences of natural numbers pk, qk with qk>pk>k, (positive integers) such that(6)dxpk,xqk≥ε,dxpk,xqk−1<ε,εs2≤lim supk⟶∞dxpk−1,xqk−1≤εs,εs≤lim supk⟶∞dxqk−1,xpk≤ε,εs≤lim supk⟶∞dxpk−1,xqk≤εs2.Note: in the continuous section of the paper, we will useX∗,d,s (resp. X,d,s) to denote that the space with parameter s≥1 is complete (resp. noncomplete).


## 3. Main Results

LetX∗,d,s be a b-metric-like space and Ψ0,+∞ represent the collection of continuous functions ψ:0,+∞⟶0,+∞ with the following properties:(7)ψ1:ψis strictly increasing,ψ2:ψm=0,iffm=0.Definition 3.
A functionκ:0,+∞2⟶R is a Ψ−s simulation function if there are ψ∈Ψ and a coefficient λ≥1 so thatκ1: κt,v<ψv−ψsλt for all t,v>0κ2: If tn,vn are sequences in 0,∞ such that limn⟶+∞tn=limn⟶+∞vn=l>0, and tn≤vn, then limn⟶+∞supκtn,vn<0Remark 1.
If in the definition above we takes=1, then we obtain the definition of a Ψ− simulation function.If we takeψ as the identity function, then we get a definition of an s− simulation function.If we takes=1 and ψv=v, then we get the definition of a simulation function.
We denote byKψ−s the set of all Ψ−s simulation functions. In the following example, we give such a kind of functions.Example 1.
Letκ:0,+∞2⟶R be defined by(1)
κt,v=cψv−ψst for all t,v∈0,+∞, where c∈0,1(2)
κt,v=ψv−ϕv−ψsλt for all t,v∈0,+∞, where ϕ:0,+∞⟶0,+∞ is such that lim inft⟶vϕt>0 for all v>0(3)
κt,v=ϕv−ψsλt for all t,v∈0,+∞, where ψ;ϕ:0,+∞⟶0,+∞ are continuous and ψ is increasing such that ϕv<ψv for all v>0(4)
κt,v=Fψv,φv−ψsλt for all t,v∈0,+∞, where F:R+×R+⟶R is a C-class function where F is continuous such that Ft,v<t for all t>0
For a self-mappingf:X⟶X, we denote by Ax,y the following:(8)Ax,y=maxdx,y,dx,fx,dy,fy,14sdx,fy+dy,fx,for allx,y∈X.Theorem 1.
Letf:X⟶X be a self-map on a b−metric-like space X∗,d,s with parameter s≥1. Suppose that there is κ∈Kψ−s such that(9)κdfx,fy,Ax,y≥0,for all x,y∈X, where Ax,y, is defined as in (8), then the self-map f has a unique fixed point in X.Proof.
Letx0∈X be an arbitrary element. Define a sequence xq in X such that ∀q∈N∪0,xq+1=fxq.
Ifdxq,xq+1=0 for some q∈N∪0, that is, xq+1=xq and xq=xq+1=fxq; therefore, xq is a fixed point of f. Thus, suppose that dxq,xq+1>0 for all q∈N∪0.Considering the set Ax,y, we have(10)Axq−1,xq=maxdxq−1,xq,dxq−1,fxq−1,dxq,fxq,14sdxq−1,fxq+dxq,fxq−1.=maxdxq−1,xq,dxq−1,xq,dxq,xq+1,14sdxq−1,xq+1+dxq,xq..
Since(11)14sdxq−1,xq+1+dxq,xq≤14ssdxq−1,xq+dxq,xq+1+2sdxq−1,xq=143dxq−1,xq+dxq,xq+1≤maxdxq−1,xq,dxq,xq+1,we obtain using (10),(12)Axq−1,xq=maxdxq−1,xq,dxq,xq+1.
By the suppositiondxq,xq+1>0 and (12), we get Axq−1,xq>0. Assume that Axq−1,xq=dxq,xq+1.
Then, applying condition (9) and property κ1, we have for all q∈N(13)0≤κdxq,xq+1,Axq−1,xq=κdfxq−1,fxq,Axq−1,xq=κdxq,xq+1,dxq,xq+1<ψdxq,xq+1−ψsλdxq,xq+1.
That is, a contradiction. Therefore,(14)Axq−1,xq=dxq−1,xq.
From (9) and using (14), we obtain(15)0≤κdxq,xq+1,Axq−1,xq=κdfxq−1,fxq,Axq−1,xq=κdxq,xq+1,dxq−1,xq<ψdxq−1,xq−ψsλdxq,xq+1.
In view of property ofψ1, the above inequality gives dxq,xq+1<dxq−1,xq for all q∈N. Hence, dxq,xq+1 is a decreasing sequence of nonnegative reals, so there is l≥0 so that dxq,xq+1⟶l. Also, by (14),(16)limq⟶+∞dxq,xq+1=limq⟶+∞Axq−1,xq=l.
Suppose thatl>0, then limq⟶+∞dxq,xq+1=limq⟶+∞Axq−1,xq=l>0. By property κ2, we have(17)0≤lim supq⟶+∞κdxq,xq+1,Axq−1,xq<0,which is a contradiction. Therefore, l=0. Hence,(18)limq⟶+∞dxq,xq+1=limq⟶+∞Axq−1,xq=0.
Next, we show thatlimq,p⟶+∞dxq,xp=0. Suppose, to the contrary, that is, limq,p⟶+∞dxq,xp>0, then by Lemma 2, there are ε>0 and sequences pk and qk of positive integers with qk>pk>k such that(19)dxpk,xqk≥ε,dxpk,xqk−1<ε,εs2≤lim supk⟶∞dxpk−1,xqk−1≤εs,εs≤lim supk⟶∞dxqk−1,xpk≤ε,εs≤lim supk⟶∞dxpk−1,xqk≤εs2.
From the definition ofAx,y, we have(20)Axpk−1,xqk−1=maxdxpk−1,xqk−1,dxpk−1,fxpk−1,dxqk−1,fxqk−1,14sdxpk−1,fxqk−1+dxqk−1,fxpk−1=maxdxpk−1,xqk−1,dxpk−1,xpk,dxqk−1,xqk,14sdxpk−1,xqk+dxqk−1,xpk.
By the upper limitk⟶+∞ in (20) and keeping in mind (18–19), we obtain(21)lim supk⟶+∞Axpk−1,xqk−1=lim supk⟶∞maxdxpk−1,xqk−1,dxpk−1,xpk,dxqk−1,xqk,14sdxpk−1,xqk+dxqk−1,xpk≤maxεs,0,0,14sεs2+ε≤εs.
Also, from conditionκ1, we have(22)0≤κdxpk,xqk,Axpk−1,xqk−1=κdxpk,xqk,Axpk−1,xqk−1<ψAxpk−1,xqk−1−ψsλdxpk,xqk,which by property of ψ1 implies(23)sλdxpk,xqk<Axpk−1,xqk−1.
By taking upper limit on both sides of (23) in view of (19) and (21), it follows that(24)εsλ<εs,which contradicts ε>0. Thus, limq,p⟶+∞dxq,xp=0 and the sequence xq is Cauchy in X∗,d,s. So, there is ω∈X, such that(25)limq⟶+∞dxq,ω=dω,ω=limq,p⟶+∞dxq,xp=0.
For elementsω and xq, we consider(26)Axq,ω=maxdxq,ω,dxq,fxq,dω,fω,14sdxq,fω+dω,fxq=maxdxq,ω,dxq,xq+1,dω,fω,14sdxq,fω+dω,xq+1.
By Lemma1 together with (18) and (25), it follows by passing in the upper limit of (26):(27)limq⟶+∞supAxq,ω≤max0,0,dω,fω,sdω,fω4s=dω,fω.
Now, using theκ1 condition, we have(28)0≤κdxq+1,fω,Axq,ω=κdfxq,fω,Axq,ω<ψAxq,ω−ψsλdxq+1,fω,which implies(29)sλdxq+1,fω<Axq,ω.
Taking the limit superior in (29) and by Lemma 1 and inequality (27), we obtain(30)sλ−1dω,fω=sλ⋅1sdω,fω<dω,fω.
By (30), it follows that dω,fω=0, and so fω=ω.
Supposeω,y∈X are two different fixed points of f. By (14), we have dω,ω=0 (and also dy,y=0). Since ω≠ydω,y>0, one writes(31)Aω,y=maxdω,y,dω,ω,dy,y,14sdω,y+dy,ω=maxdω,y,0,0,12sdω,y=dω,y>0.
From condition (9) and property κ1, we have(32)0≤κdfω,fy,Aω,y=κdω,y,dω,y<ψdω,y−ψsλdω,y,which is a contradiction. Therefore, dω,y=0 and ω=y. Thus, there is a unique fixed point of f.Example 2.
LetX=0,1 with the b−metric-like dx,y=x+y2. Define f:X⟶X as fx=1/6xifx≠11/8ifx=1.
Also, we take the functionsϕx=x;ψx=2x and κt,v=ϕv−ψsλt, (where λ=2) for all t,v∈0,+∞, where ψ;ϕ:0,+∞⟶0,+∞ are continuous and ψ is increasing such that ϕv<ψv for all v>0.
The pairX,d is a b− metric-like space with coefficient s=2. We claim that the mapping f satisfies the contraction type condition (8):Case1.
Forx≠y≠1, we have(33)Ax,y=maxdx,y,dx,fxdy,fy14sdx,fy+dy,fx=maxx+y2,x+x62,y+y62,14sx+y62+y+x62.
Anddfx,fy=dx/6,y/6=x/6,y/62=1/36x+y2=1/36dx,y. Then,(34)κdfx,fy,Ax,y=ϕAx,y−ψs2dfx,fy=Ax,y−2s2dfx,fy=Ax,y−8dfx,fy=Ax,y−8136dx,y=Ax,y−29dx,y≥0.Case 2.
Forx=y=1, we note(35)A1,1=maxd1,1,d1,f1,d1,f1,14sd1,f1+d1,f1=maxd1,1,d1,f1=max1+12,1+182=4=d1,1.
Anddf1,f1=d1/8,1/8=1/8,+1/82=4/64=1/64d1,1<A1,1. Then,(36)κdf1,f1,A1,1=ϕA1,1−ψs2df1,f1=A1,1−2s2df1,f1=A1,1−8df1,f1=A1,1−8164d1,1=A1,1−29d1,1≥0.Case 3.
x<y=1 we note(37)dfx,f1=dx6,18=x6+182<136x+12=136dx,1⟹dfx,f1<136dx,1⟹8dfx,f1<8136dx,1⟹2s2dfx,f1<29dx,1<Ax,1.
Then,(38)κdfx,f1,Ax,1=ϕAx,1−ψs2dfx,f1=Ax,1−2s2dfx,f1≥0.
Here,0 is the unique fixed point of f.
Some applications of Theorem 1 are the following corollaries.Corollary 1.
Letf:X⟶X be a mapping on a b−metric-like space X∗,d,s. Suppose that there are ψ∈Ψ and λ≥1 such that(39)ψsλdfx,fy≤ψAx,y1+ψAx,y,for all x,y∈X, where Ax,y is defined as in (8). Then, the self-map f has a unique fixed point in X.Proof.
In Theorem1, take into account the function κt,v=ψv/1+ψv−ψsλt for all t,v∈0,+∞,Corollary 2.
Letf:X⟶Xbe a mapping on a b−metric-like spaceX∗,d,s. Suppose that there are ψ∈Ψ, φ:0,+∞⟶0,+∞ a lower semicontinuous function with φv=0iff v=0, and λ≥1 such that(40)ψsλdfx,fy≤ψAx,y1+φAx,y,for all x,y∈X, where Ax,y is defined as in (8). Then, the self-mapf admits a unique fixed point in X.Proof.
In Theorem1, take into account the function κt,v=ψv/1+φv−ψsλt for all t,v∈0,+∞.Corollary 3.
Letf:X⟶X be a mapping on a b−metric-like space X∗,d,s. Suppose that there are ψ∈Ψ, α∈0,1 and λ≥1 such that(41)ψsλdfx,fy≤αψAx,y,for all x,y∈X, where Ax,y is defined as in (8). Then, the self-map f has a unique fixed point in X.Proof.
In Theorem1, take into account the function κt,v=αψv−ψsλt for all t,v∈0,+∞, and α∈0,1.Corollary 4.
Letf:X⟶X be a mapping on a b−metric-like space X∗,d,s. Suppose that there are ψ∈Ψ, λ≥1, and ϕ:R+⟶R+ continuous with ϕv<ψv for v>0, such that(42)ψsλdfx,fy≤ϕAx,y,for all x,y∈X, where Ax,y is defined as in (8). Then, the self-mapf has a unique fixed point in X.Proof.
In Theorem1, take into account the function κt,v=ϕv−ψsλt for all t,v∈0,+∞.Corollary 5.
Letf:X⟶X be a mapping on a b−metric-like space X∗,d,s. Suppose that there are ψ∈Ψ, λ≥1, F:R+×R+⟶R a C-class function and φ:R+⟶R+ a continuous function, such that(43)ψsλdfx,fy≤FψAx,y,φAx,y,for all x,y∈X, where Ax,y is defined as in (8). Then, the self-map f has a unique fixed point in X.Proof.
In Theorem1, take into account the function κt,v=Fψv,φv−ψsλt for all t,v∈0,+∞, where F:R+×R+⟶R is a C-class function.Remark 2.
Corollary5 is much wider because condition (43) includes many other contractive conditions.Corollary 6.
Letf:X⟶X be a mapping on a b−metric-like space X∗,d,s. Suppose that there exist a function φ:0,+∞⟶0,+∞ with lim inft⟶vφt>0 for all v>0, and some constant λ≥1 such that(44)sλdfx,fy≤Ax,y−φAx,y,for all x,y∈X, where Ax,y is defined as in (8). Then, the self-map f has a unique fixed point in X.Proof.
In Theorem1, take into account the function κt,v=ψv−φv−ψsλt for all t,v∈0,+∞, and take ψv=v (it corresponds to Theorem 3.16 in [23]).
In the following result, we include two mappingsf and g in the set(45)Ex,y=maxdx,y,dy,gy,dx,fxdy,gy1+dx,y,dx,fxdy,gy1+dfx,gy.Theorem 2.
LetX∗,d,s be a b−metric-like space and f,g:X⟶X be two given mappings. Suppose that there exists κ∈Kψ−s such that(46)κdfx,gy,Ex,y≥0,for all x,y∈X, where Ex,y is denoted by (45); then, the mappings f and g have a unique common fixed point in X.Proof.
Letx0∈X be an arbitrary element. Define a sequence xq in X such that ∀q∈N∪0x2q+1=fx2q and x2q+2=gx2q+1.
Let for someq∈ℕ∪0x2q+1=x2q. Since(47)Ex2q,x2q+1=maxdx2q,x2q+1,dx2q+1,gx2q+1,dx2q,fx2qdx2q+1,gx2q+11+dx2q,x2q+1,dx2q,fx2qdx2q+1,gx2q+11+dfx2q,gx2q+1=maxdx2q,x2q+1,dx2q+1,x2q+2,dx2q,x2q+1dx2q+1,x2q+21+dx2q,x2q+1,dx2q,x2q+1dx2q+1,x2q+21+dx2q+1,x2q+2=max0,dx2q+1,x2q+2,0,0=dx2q+1,x2q+2.
Then, by (46) and κ1, we have(48)0≤κdfx2q,gx2q+1,Ex2q,x2q+1=κdx2q+1,x2q+2,dx2q+1,x2q+2<ψdx2q+1,x2q+2−ψsλdx2q+1,x2q+2.
By propertyψ1, we get dx2q+1,x2q+2=0, that is, x2q+1=x2q+2. We deduce that x2q=x2q+1=fx2q and gx2q=gfx2q=gx2q+1=x2q+2=x2q. Hence, x2q is a common fixed point of f and g.
Assume the general case thatdx2q,x2q+1>0 for all q∈N∪0, then(49)Ex2q,x2q−1=maxdx2q,x2q−1,dx2q−1,gx2q−1,dx2q,fx2qdx2q−1,gx2q−11+dx2q,x2q−1,dx2q,fx2qdx2q−1,gx2q−11+dfx2q,gx2q−1=maxdx2q,x2q−1,dx2q−1,x2q,dx2q,x2q+1dx2q−1,x2q1+dx2q,x2q−1,dx2q,x2q+1dx2q−1,x2q1+dx2q+1,x2q=maxdx2q,x2q−1,dx2q,x2q+1.
Ifdx2q−1,x2q≤dx2q,x2q+1 for some q∈Ν, then (49) implies(50)Ex2q−1,x2q=dx2q,x2q+1>0.
From (50), applying ψ1, (46), and κ1, we have(51)0≤κdx2q+1,x2q,Ex2q,x2q−1=κdfx2q,gx2q−1,Ex2q,x2q−1<ψEx2q,x2q−1−ψsλdfx2q,gx2q−1=ψdx2q+1,x2q−ψsλdx2q+1,x2q≤0.
That is a contradiction. So, we havedx2q,x2q+1<dx2q−1,x2q for all q∈N Hence, dx2q+1,x2q is a decreasing sequence of nonnegative reals, so there is l≥0 so that(52)limq⟶+∞dxq,xq+1=land alsolimq⟶+∞dxq,xq+1=limq⟶+∞Exq−1,xq=l.
Assume thatl>0; then, by applying κ2, we have(53)lim supq⟶+∞κdx2q+1,x2q,Ex2q,x2q−1≤0,a contradiction. Therefore,(54)limq⟶+∞dxq,xq+1=limq⟶+∞Exq−1,xq=l>0.
Now, we prove thatlimq,p⟶+∞dxq,xp=0. It is enough to prove that limq,p⟶+∞dx2q,x2p=0 On the contrary, assume that limq,p⟶+∞dx2q,x2p≠0. Then, from Lemma 2, there are ε>0 and two subsequences pk and qk of positive integers, with qk>pk>k, such that(55)ε≤limsupk⟶∞dx2qk,x2pk≤εs,εs≤lim supk⟶∞dx2pk,x2qk−1≤εs,εs2≤limsupk⟶∞dx2qk−1,x2pk+1≤εs2,εs≤lim supk⟶∞dx2pk+1,x2qk≤εs2.
From (45), we note(56)Ex2pk,x2qk−1=maxdx2pk,x2qk−1,dx2qk−1,gx2qk−1,dx2pk,fx2pkdx2qk−1,gx2qk−11+dx2pk,x2qk−1,dx2pk,fx2pkdx2qk−1,gx2pk−11+dfx2pk,gx2qk−1=maxdx2pk,x2qk−1,dx2qk−1,x2qk,dx2pk,x2pk+1dx2qk−1,x2qk1+dx2pk,x2qk−1,dx2pk,x2pk+1dx2qk−1,x2qk1+dx2pk+1,x2qk.
Hence, by (54)–(56), and Lemma 2, we have(57)limsupk⟶+∞Ex2pk,x2qk−1=lim supk⟶+∞maxdx2pk,x2qk−1,dx2qk−1,x2qk,dx2pk,x2pk+1dx2qk−1,x2qk1+dx2pk,x2qk−1,dx2pk,x2pk+1dx2qk−1,x2qk1+dx2pk+1,x2qk≤maxεs,0,0,0=εs.
By (46) and using properties ψ1, κ1, we have(58)0≤κdfx2pk,gx2qk−1,Ex2pk,x2qk−1=κdx2pk+1,x2qk,Ex2pk,x2qk−1<ψEx2pk,x2qk−1−ψsλdx2pk+1,x2qk,which leads to(59)sλdx2pk+1,x2qk<Ex2pk,x2qk−1.
Hence, by (55), (57), and (58) and taking the upper limit, we obtain(60)εsλ−1<εs,which implies that ε=0, a contradiction with ε>0. It remains that limq,p⟶+∞dxq,xp=0; therefore, xq is a Cauchy sequence in X. Since X∗,d,s is a complete b-metric-like space, there is ω∈X such that xq is convergent to ω, that is,(61)limq⟶+∞dxq,ω=limq,p⟶+∞dxq,xp=dω,ω=0.
Also, the subsequencesfx2q,gx2q+1 are convergent, so(62)limq⟶+∞dx2q+1,ω=limq⟶+∞dfx2q,ω=dω,ω=0,limq⟶+∞dx2q+2,ω=limq⟶+∞dgx2q+1,ω=dω,ω=0.
Consider(63)Ex2q,ω=maxdx2q,ω,dω,gω,dx2q,fx2qdω,gω1+dx2q,ω,dx2q,fx2qdω,gω1+dfx2q,gω=maxdx2q,ω,dω,gω,dx2q,x2q+1dω,gω1+dx2q,ω,dx2q,x2q+1dω,gω1+dx2q+1,gω.
Taking the limit superior in (63) and applying Lemma 1 and (62), it follows(64)lim supq⟶+∞Ex2q,ω≤max0,dω,gω,0,0,=dω,gω.
From condition (46),(65)0≤κdfx2q,gω,Ex2q,ω=κdx2q+1,gω,Ex2q,ω<ψEx2q,ω−ψsλdx2q+1,gω,which implies(66)sλdx2q+1,gω<Ex2q,ω.
Taking the upper limit asq⟶+∞ and using Lemma 1 and (64), we have
sλ−1dω,gω<dω,gω, that is, dω,gω=0 and ω is a fixed point of g. Similarly, we can get dfω,ω=0 and so ω is a common fixed point for mappings f and g. Suppose ω,δ∈X are two different common fixed points of f and g such that dω,δ>0. Then,(67)Eω,δ=maxdω,δ,dδ,gδ,dω,ωdδ,gδ1+dω,δ,dω,ωdδ,gδ1+dfω,gδ=maxdω,δ,dδ,δ,dω,ωdδ,δ1+dω,δ,dω,ωdδ,δ1+dω,δ=dω,δ>0.
Fromψ1, κ1, (67), and (46), we have(68)0≤κdfω,gδ,Eω,δ=κdω,δ,dω,δ<ψdω,δ−ψsλdω,δ≤0,which contradicts the supposition dω,δ>0. Hence, dω,δ=0 and the common fixed point is unique.Corollary 7.
Letf,g:X⟶X be two self-mappings given in a b−metric-like space X∗,d,s. Suppose that there exist ψ∈Ψ and α:0,∞⟶0,1 with limt⟶r+αt<1 for all r>0, such that(69)ψsλdfx,gy≤αEx,yψEx,y,for all x,y∈X, where Ex,y is defined as in (45).Then, the self-mappingsf and g have a unique common fixed point in X.Proof.
In Theorem2, take the Ψ−s simulation function κt,v=αvψv−ψsλt for all t,v∈0,+∞.Remark 3.
The above theorem reduces to a one mapping if we putg=f. Further corollaries can be stated for s=1, either by taking the function ψ as an identity function or by taking different functions κ∈Kψ−s as listed in Corollary 1–6.


## 4. Conclusion

In this work, we established common fixed point results for one and two mappings on ab−metric-like space which overcomes and unifies classical and previous results developed in papers [19–28]. The considered set of generalized contractive mappings contains the families of many contractions as a proper subset. We remark based on Example 2/4 which are functions of C−class used by many researchers and taken as a special case of Ψ−s simulation functions.By using additional set of functionsΨ, ϕ, coefficient λ, and parameter s, the rich class of Ψ−s simulation functions make it possible to collect, extend, and complement previously existing results related to a variety types of contractions.In terms ofΨ−s simulating functions, many classical and still recent contractions take a simple form as κdfx,fy,Ax,y≥0 not including other additional symbols and long formulas.This wide approach reflects a wide work and an unifying power for more general theorems made on the theory of fixed points.



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