# Fixed Points of Generalizedα-Meir-Keeler Contraction Mappings in Sb-Metric Spaces

Authors: Thounaojam Stephen; Yumnam Rohen; Naeem Saleem; Mairembam Bina Devi; K. Anthony Singh
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this note, we define Meir-Keeler contraction inSb-metric spaces. Further, by adding the concept of α-admissible mappings, we define generalized αs-Meir-Keeler contraction and used it for examining the existence and uniqueness of fixed points. Various results are also given as a consequence of our results.


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## Body

## 1. Introduction and Preliminaries

The Banach contraction principle has been an important instrument for the study of a fixed point. It has been widely used in different areas like nonlinear analysis, applied mathematics, economics, and physics. Due to its importance, the result has been generalized in different ways. Meir and Keeler [1] introduce a generalization of the Banach contraction principle. According to them, self-mapping A in a metric space X,d is called Meir-Keeler contraction if for an ε>0 there exists δ>0 such that ε≤dθ,ϕ<ε+δε implies dAθ,Aϕ<ε for all θ,ϕ∈X. They also state and prove that if a self-mapping A in a complete metric space satisfies Meir-Keeler contraction, then there is a unique fixed point for the mapping A. There are a large number of works on Meir-Keeler contraction of which some of the recent works are mentioned here.Pourhadi et al. [2] introduced the concept of Meir-Keeler expansive mappings and obtained Krasnosel’skii-type fixed point theorem in Banach spaces. A new fixed point theorem was obtained by Du and Rassias [3] for a Meir-Keeler type condition as a generalization of the Banach contraction principle, Kannan’s fixed point theorem, Chatterjea’s fixed point theorem, etc., simultaneously.The idea ofSb-metric space [4–6] is defined by combining definitions of S-metric space [7] and b-metric space [8]. Samet et al. [9] introduced the concept of α-admissible mapping. This concept was further extended to G-metric space, S-metric space, Sb-metric space, etc. (for details, see [10–14]). There are various recent results on Meir-Keeler type and related topics which will be helpful to the readers for more information. Some of them can be seen in [15–21].In this article, we give the concept ofα-admissible and Meir-Keeler contraction in Sb-metric space. The new contraction will be known as generalized αs-Meir-Keeler contraction. By using generalized αs-Meir-Keeler contraction mappings, we study the existence and uniqueness of the fixed point in Sb-metric space.The following definitions and properties will be needed.Definition 1 (see [8]).
In a setX≠ϕ, suppose b≥1 is a real number and d:X×X⟶0,+∞ is a function satisfying
(1)
dθ,ϕ=0 if and only if θ=ϕ for all θ,ϕ∈X(2)
dθ,ϕ=dϕ,θ for all θ,ϕ∈X(3)
dθ,ϕ≤bdθ,ψ+dψ,ϕ for all θ,ϕ,ψ∈XThen,d is called b-metric on X and the pair X,d is called a b-metric space with coefficient b.Definition 2 (see [7]).
In a setX≠ϕ, suppose S:X×X×X⟶0,+∞ is a function satisfying
(1)
Sθ,ϕ,ψ=0 if and only if θ=ϕ=ψ for all θ,ϕ,ψ∈X(2)
Sθ,ϕ,ψ≤Sθ,θ,ω+Sϕ,ϕ,ω+Sψ,ψ,ω, for all θ,ϕ,ψ,ω∈XThen, the pairX,S is said to be an S-metric space.Definition 3 (see [5]).
In a setX≠ϕ, suppose b≥1 is a real number and S:X×X×X⟶0,+∞ is a function satisfying
(i)
Sθ,ϕ,ψ=0 if and only if θ=ϕ=ψ(ii)
Sθ,ϕ,ψ≤bSθ,θ,ω+Sϕ,ϕ,ω+Sψ,ψ,ω, for all θ,ϕ,ψ,ω in XHere,S is said to be a Sb-metric and X,S is said to be a Sb-metric space.Definition 4 (see [4]).
ASb-metric S satisfying Sθ,θ,ϕ=Sϕ,ϕ,θ for all θ,ϕ∈X is called a symmetric Sb-metric.Definition 5 (see [5]).
In aSb-metric space X,S, a sequence θn is called
(i)
convergent if and only ifSθn,θn,θ⟶0 as n⟶∞, where θ∈X and is expressed as limn⟶∞θn=θ(ii)
Cauchy if and only ifSθn,θm,θ⟶0 as n,m⟶∞, where θ∈X(iii)
completeSb-metric space if every Cauchy sequence θn is convergent and converging to θ in XWe recall some types ofα-admissible mappings in a metric space X,d.Definition 6 (see [9]).
LetA:X⟶X and α:X×X⟶[0,+∞ be functions. Here, A is said to be α-admissible if αθ,ϕ≥1 implies αAθ,Aϕ≥1 for all θ,ϕ∈X.Definition 7 (see [15]).
LetA,B:X⟶X and α:X×X⟶0,+∞ are functions. Here, the pair of mappings A,B is said to be anα-admissible if αθ,ϕ≥1 implies αAθ,Bϕ≥1 and αBθ,Aϕ≥1 for all θ,ϕ∈X.Definition 8 (see [16]).
LetA:X⟶X and α:X×X⟶[0,+∞ be functions. Here, A is known as triangular α-admissible, if
(i)
αθ,ϕ≥1, which implies αAθ,Aϕ≥1,θ,ϕ∈X(ii)
αθ,ϕ≥1, αϕ,ψ≥1, which implies αθ,ψ≥1,forallθ,ϕ,ψ∈XDefinition 9 (see [15]).
LetA,B:X⟶X and α:X×X⟶[0,+∞ be functions. Here, the pair A,B is said to be a triangular α-admissible, if
(i)
αθ,ϕ≥1, which implies αAθ,Bϕ≥1 and αBθ,Aϕ≥1,θ,ϕ∈X(ii)
αθ,ϕ≥1, αϕ,ψ≥1, which implies αθ,ψ≥1, for all θ,ϕ,ψ∈XWe extend the concept ofα-admissible mapping to be suitable for S-metric and Sb-metric spaces. Here, we consider X as S-metric space or Sb-metric space.Definition 10.
LetA:X⟶X and αs:X×X×X⟶[0,+∞ are functions, then A is called αs-admissible, if θ,ϕ,ψ∈X, αsθ,ϕ,ψ≥1 implies αsAθ,Aϕ,Aψ≥1.Example 11.
ConsiderX=0,+∞ and define A:X⟶X and αs:X×X×X⟶[0,+∞ by Aθ=4θ, for all θ,ϕ,ψ∈X, and
(1)αsθ,ϕ,ψ=eψ/θϕ,ifθ≥ϕ≥ψ,θ,ϕ≠0,0,ifθ<ϕ<ψ.
Then,A is an αs-admissible mapping.Definition 12.
LetA,B:X⟶X and αs:X×X×X⟶[0,+∞ be three functions. The pair A,B is called αs-admissible if θ,ϕ,ψ∈X such that αsθ,ϕ,ψ≥1, then we have αsAθ,Aϕ,Bψ≥1 and αsBθ,Bϕ,Aψ≥1.


## 2. Main Result

Here, we give various types of Meir-Keeler contractive mappings in order to extend various results of Gülyaz et al. [17] in Sb-metric space. Throughout this paper, assume X,S is a Sb-metric space, b≥1 is a real number, and A:X⟶X is a mapping.Definition 13.
Anαs-admissible mapping A in X,S is known as αs-Meir-Keeler contraction mapping of type I, if there esists δ>0 for all ε>0 such that
(2)ε≤Sθ,ϕ,ψ<ε+δimplies
(3)αsθ,ϕ,ψSAθ,Aϕ,Aψ<εbfor all θ,ϕ,ψ∈X.Definition 14.
Anαs-admissible mapping A in X,S is known as αs-Meir-Keeler contraction mapping of type II, if there exists δ>0 for all ε>0 such that
(4)ε≤Sθ,θ,ϕ<ε+δimplies
(5)αsθ,θ,ϕSAθ,Aθ,Aϕ<εbfor all θ,ϕ∈X.Remark 15.
(i)
IfA is an αs-Meir-Keeler contraction of type I, then
(6)αsθ,ϕ,ψSAθ,Aϕ,Aψ≤Sθ,ϕ,ψb,for all θ,ϕ,ψ∈X and equality is true, when θ=ϕ=ψ(ii)
IfA is an αs-Meir-Keeler contraction of type II, then
(7)αsθ,θ,ϕSAθ,Aθ,Aϕ≤Sθ,θ,ϕb,for all θ,ϕ∈X and equality is true, when θ=ϕNow, we introduce the following generalization of Meir-Keeler mappings.Definition 16.
Anαs-admissible mapping A in X,S is known as generalized αs-Meir-Keeler contraction mapping of type AI, if there exists δ>0 for all ε>0 such that
(8)ε≤Λθ,ϕ,ψ<ε+δimplies
(9)αsθ,ϕ,ψSAθ,Aϕ,Aψ<εb,where
(10)Λθ,ϕ,ψ=maxSθ,ϕ,ψ,Sθ,θ,Aθ,Sϕ,ϕ,Aϕ,Sψ,ψ,Aψfor all θ,ϕ,ψ∈X.Definition 17.
Anαs-admissible mapping A in X,S is known as generalized αs-Meir-Keeler contraction mapping of type AII, if there exists δ>0 for all ε>0 such that
(11)ε≤Λθ,θ,ϕ<ε+δimplies
(12)αsθ,θ,ϕSAθ,Aθ,Aϕ<εb,where
(13)Λθ,θ,ϕ=maxSθ,θ,ϕ,Sθ,θ,Aθ,Sϕ,ϕ,Aϕfor all θ,ϕ∈X.Definition 18.
Anαs-admissible mapping A in X,S is known as generalized αs-Meir-Keeler contraction mapping of type BI, if there exists δ>0 for all ε>0 such that
(14)ε≤Λθ,ϕ,ψ<ε+δimplies
(15)αsθ,ϕ,ψSAθ,Aϕ,Aψ<εb,where
(16)Λθ,ϕ,ψ=maxSθ,ϕ,ψ,Sθ,θ,Aθ,Sϕ,ϕ,Aϕ,Sψ,ψ,Aψ,14Sθ,θ,Aϕ+Sϕ,ϕ,Aψ+Sψ,ψ,Aθfor all θ,ϕ,ψ∈X.Definition 19.
Anαs-admissible mapping A in X,S is known as generalized αs-Meir-Keeler contraction mapping of type BII, if there exists δ>0 for all ε>0 such that
(17)ε≤Λθ,θ,ϕ<ε+δimplies
(18)αsθ,θ,ϕSAθ,Aθ,Aϕ<εb,where
(19)Λθ,θ,ϕ=maxSθ,θ,ϕ,Sθ,θ,Aθ,Sϕ,ϕ,Aϕ,14Sθ,θ,Aθ+Sθ,θ,Aϕ+Sϕ,ϕ,Aθfor all θ,ϕ∈X.Remark 20.
(i)
LetA:X⟶X be a generalized αs-Meir-Keeler contraction of type AI or BI. Then(20)αsθ,ϕ,ψSAθ,Aϕ,Aψ≤Λθ,ϕ,ψbfor all θ,ϕ,ψ∈X, where the equality holds only when θ=ϕ=ψ(ii)
LetA:X⟶X be a generalized αs-Meir-Keeler contraction of type AII or BII. Then(21)αsθ,θ,ϕSAθ,Aθ,Aϕ≤Λθ,θ,ϕb,for all θ,ϕ∈X, where the equality holds only when θ=ϕLemma 21.
LetX,S be a Sb-metric space and θn be a sequence satisfying
(i)
θm≠θn for all m≠n, m,n∈ℕ(ii)
Sθn,θn,θn+1≤1/bSθn−1,θn−1,θn, for all n∈ℕ
Then,θn is a Cauchy sequence in X,S.Proof.
In order to show that sequenceθn is Cauchy, we must prove that limn⟶∞Sθn,θn,θn+k=0 for any k∈ℕ.
From (ii), we have(22)Sθn,θn,θn+1≤1bnSθ0,θ0,θ1,foralln∈ℕ.
Applying limit asn⟶∞, we get
(23)0≤limn⟶∞Sθn,θn,θn+1≤1bnSθ0,θ0,θ1∴limn⟶∞Sθn,θn,θn+1=0.
Now,(24)Sθn,θn,θn+k≤2bSθn,θn,θn+1+b2Sθn+1,θn+1,θn+k≤2bSθn,θn,θn+1+2b3Sθn+1,θn+1,θn+2+b4Sθn+2,θn+2,θn+k≤2bSθn,θn,θn+1+b3Sθn+1,θn+1,θn+2+⋯+b2k−1+1Sθn+k−1,θn+k−1,θn+k≤2bSθ0,θ0,θ1bn+b3Sθ0,θ0,θ1bn+1+⋯+b2k−1+1Sθ0,θ0,θ1bn+k−1=2bn−11+b+⋯+bkSθ0,θ0,θ1=2bk−1bn−1b−1Sθ0,θ0,θ1∴limn⟶∞Sθn,θn,θn+k≤limn⟶∞2bk−1bn−1b−1Sθ0,θ0,θ1=0.
Thus,θn is a Cauchy sequence in Sb-metric space X,S.☐Theorem 22.
LetX,S be a complete Sb-metric space and αs:X×X×X⟶[0,+∞ be a mapping. Let A:X⟶X satisfy the following:
(i)
A is a generalized αs-Meir-Keeler contraction mapping of type AI(ii)
A is αs-admissible(iii)
There isθ0∈X so that αsθ0,θ0,Aθ0≥1(iv)
A is continuous
Then, there exists a fixed point ofA in X.Proof.
Supposeθ0∈X and αsθ0,θ0,Aθ0≥1. Define the sequence θn in X as
(25)θn+1=Aθn,foralln∈ℕ.
Supposeθn0=θn0+1 for some n0∈ℕ that is Sθn0,θn0,θn0+1=0 implies that θn0 is a fixed point of A. Thus, assume that θn≠θn+1 for all n≥0. From (ii), we have
(26)αsθ0,θ0,Aθ0=αsθ0,θ0,θ1≥1implies that
(27)αsAθ0,Aθ0,Aθ1=αsθ1,θ1,θ2≥1;continuing on the same lines, we have
(28)αsθn,θn,θn+1≥1,∀n∈ℕ.
Here, we need to show that sequenceθn satisfies the conditions of Lemma 21. If we put θ=ϕ=θn and ψ=θn+1 in (9), for all ε>0, there is δ>0 satisfying
(29)ε≤Λθn,θn,θn+1<ε+δimplies
(30)αsθn,θn,θn+1SAθn,Aθn,Aθn+1<εb,where
(31)Λθn,θn,θn+1=maxSθn,θn,θn+1,Sθn,θn,Aθn,Sθn+1,θn+1,Aθn+1.
From Remark20(ii), we have
(32)Sθn+1,θn+1,θn+2=SAθn,Aθn,Aθn+1≤αsθn,θn,θn+1SAθn,Aθn,Aθn+1≤Λθn,θn,θn+1b;due to the fact that θn≠θn+1, we see that equality does not hold, hence,
(33)Sθn+1,θn+1,θn+2<Λθn,θn,θn+1b.
IfΛθn,θn,θn+1=Sθn+1,θn+1,θn+2 for some n∈ℕ, then (11) implies
(34)Sθn+1,θn+1,θn+2<Sθn+1,θn+1,θn+2bwhich is not possible. Then, Λθn,θn,θn+1=Sθn,θn,θn+1 for all n∈ℕ, so that (11) yields
(35)Sθn+1,θn+1,θn+2<Sθn,θn,θn+1b,which shows that Lemma 21(ii) is true.☐Next, we consider the case forθn≠θm for all n≠m.If possible, letθn=θm for some m,n∈ℕ. We have Sθn,θn,θn+1≥0 for some n∈ℕ. In general, let m>n+1.We haveSθm,θm,θm+1=Sθn,θn,θn+1; by inequality (12), we have
(36)Sθn,θn,θn+1=Sθm,θm,θm+1<Sθm−1,θm−1,θmb<Sθm−2,θm−2,θm−1b2⋯<Sθn,θn,θn+1bm−nbecomes impossible. Thus, for some m≠n,λn=λm is not true, and hence, it must be θn≠θm for all n≠m. So, due to Lemma 21, θn is a Cauchy sequence in X,S. Thus, θn converges to u∈X, i.e.,
(37)limn⟶∞Sθn,θn,u=0.By the continuity ofA, we have
(38)limn⟶∞SAθn,Aθn,Au=limn⟶∞Sθn+1,θn+1,Au=0,so θn converges to Au. Since the limit is unique, Au=u.Theorem 23.
LetX,S be a complete Sb-metric space and αs:X×X×X⟶[0,+∞ be a mapping. Let A:X⟶X be a mapping such that
(v)
for a pair of fixed pointsθ,ϕ of A, αsθ,θ,ϕ≥1together with the four conditions of Theorem 22, then A has a unique fixed point in X.Proof.
The existence of a fixed point is proved in Theorem22. Now, for uniqueness, consider θ and ϕ as two different fixed points of A in X.
By (9), we have
(39)ε≤Λθ,θ,ϕ<ε+δimplies
(40)αsθ,θ,ϕSAθ,Aθ,Aϕ<εb,where
(41)Λθ,θ,ϕ=maxSθ,θ,ϕ,Sθ,θ,Aθ,Sϕ,ϕ,Aϕ=maxSθ,θ,ϕ,0,0=Sθ,θ,ϕ.
By (v),αsθ,θ,ϕ≥1, since Sθ,θ,ϕ>0, Remark 20(ii) becomes
(42)Sθ,θ,ϕ=SAθ,Aθ,Aϕ≤αsθ,θ,ϕSAθ,Aθ,Aϕ<Λθ,θ,ϕb=Sθ,θ,ϕb,which is a contradiction, hence, Sθ,θ,ϕ=0, i.e., θ=ϕ. Thus, the fixed point of A is unique.☐Definition 24.
InSb-metric space X,S, αs:X×X×X⟶[0,+∞ is a mapping. Then, Sb-metric space X,S is known as an α-regular if for any sequence θn, limn⟶∞Sθn,θn,θ=0 and αsθn,θn,θn+1≥1 for all n∈ℕ; we have αsθn,θn,θ≥1 for all n∈ℕ.Theorem 25.
In a completeSb-metric space X,S,b≥1 is a parameter and αs:X×X×X⟶[0,+∞ is an αs-admissible mapping. Let A:X⟶X be a generalized αs-Meir-Keeler contraction of type AI satisfying the following:
(i)
There isθ0∈X so that αsθ0,θ0,Aθ0≥1(ii)
TheSb-metric space X,S is an α-regular, then there exists a fixed point of A in X(iii)
For all pairs of fixed points,θ, ϕ∈X,αsθ,θ,ϕ≥1
Then,A has unique fixed point.Proof.
Supposeθ0∈X such that αsθ0,θ0,Aθ0≥1. Define a sequence θn∈X such that θn+1=Aθn for all n∈ℕ and converges to u∈X uniquely.
AsX,S is αs-regular, αsθn,θn,u≥1.
By (9), we have
(43)ε≤Λθn,θn,u<ε+δimplies
(44)αsθn,θn,uSAθn,Aθn,Au<εb,where
(45)Λθn,θn,u=maxSθn,θn,u,Sθn,θn,Aθn,Su,u,Au.
On the other hand, from Remark20(ii), we have
(46)Sθn+1,θn+1,Au=SAθn,Aθn,Au≤αsθn,θn,uSAθn,Aθn,Au<Λθn,θn,ub.
We have(47)limn⟶∞Sθn+1,θn+1,Au=Su,u,Au.
Also,(48)limn⟶∞Λθn,θn,u=limn⟶∞maxSθn,θn,u,Sθn,θn,Aθn,Su,u,Au=Su,u,Au.
Taking the limit asn⟶∞ in (46), we have
(49)Su,u,Au≤Su,u,Aub,which conclude that Su,u,Au=0.☐The uniqueness part is identical to Theorem23.Note: Theorems22, 23, and 25 will be true for generalized αs-Meir-Keeler contraction mapping of type BI and BII.Example 26.
LetX=0,∞ be endowed with Sb-metric
(50)Sx,y,z=y+z−2x,whereb=2.
DefineA:X⟶X by
(51)Ax=x28,x∈0,1,18+logx,x∈1,∞,αsx,y,z=1,x,y,z∈0,1,0,otherwise.
Clearly, mappingA is αs-admissible and continuous mapping. Let x,y∈0,1, without loss of generality, assume that x≤y, then
(52)SAx,Ax,Ay=Sx28,x28,y28=y28−x28.
Now, to calculate(53)Λx,y,z=maxSx,y,z,Sx,x,Ax,Sy,y,Ay,Sz,z,Az;in our case, if we take x=y, then after a simple calculation, we have
(54)Λx,x,y=maxSx,x,y,Sx,x,Ax,Sy,y,Ay=maxy−x,x−x28,y−y28.
Now, suppose that(55)ε<Λx,x,y=maxy−x,x−x28,y−y28<ε+δfor δ=3ε. Now, observe that maxx,y∈0,1y−x=1 and maxx,y∈0,1y+x=2, and assume that ε∈1/2,1, then we have
(56)y−xy+x8<28=14<ε2,which implies that
(57)SAx,Ax,Ay=y28−x28<ε2.Since αsx,y,z=1 for all x,y,z∈0,1; otherwise, αsx,y,z=0, and we have
(58)0=αsx,y,zSAx,Ay,Az<εb=ε2.
Hence,A satisfies the conditions of generalized αs-Meir-Keeler contraction mapping of type AI. Also, all the conditions of Theorem 22 are satisfied, and hence, x=0 is the unique fixed point of mapping A.


## 3. Consequences

Here, we consider some consequences of Theorems22, 23, and 25.Corollary 27.
LetX,S be complete Sb-metric space and A:X⟶X be an αs-admissible mapping satisfying the following:
(i)
For allε>0, there exists δ>0 such that(59)ε≤Nθ,ϕ,ψ<ε+δimplies
(60)αsθ,ϕ,ψSAθ,Aϕ,Aψ<εb,where
(61)Nθ,ϕ,ψ=maxSθ,ϕ,ψ,13Sθ,θ,Aθ+Sϕ,ϕ,Aϕ+Sψ,ψ,Aψfor all θ,ϕ,ψ∈X(ii)
There existsθ0∈X such that αsθ0,θ0,Aθ0≥1(iii)
A is continuous or Sb-metric space X,S is αs-regular
Then,A has a fixed point in X.
Also,(iv)
for every pair of fixed pointsθ,ϕ of A, if αsθ,θ,ϕ≥1
Then, the fixed point ofA is unique in X.Proof.
AsNθ,ϕ,ψ≤Λθ,ϕ,ψ for all θ,ϕ,ψ∈X, the proof is obvious from Theorems 22, 23, and 25.☐Corollary 28.
LetX,S be complete Sb-metric space and A:X⟶X be an αs-Meir-Keeler contraction of type I; that is, there exists δ>0 for every ε>0 such that
(62)ε≤Sθ,ϕ,ψ<ε+δimplies
(63)αsθ,ϕ,ψSAθ,Aϕ,Aψ<εbfor all θ,ϕ,ψ∈X.
IfA is continuous or Sb-metric space X,S is α-regular, then A has a fixed point. Further, with condition (v) in Theorem 23, the fixed point of A is unique.Proof.
The proof follows easily from the relationSθ,ϕ,ψ≤Λθ,ϕ,ψ for all θ,ϕ,ψ∈X.☐Takingαθ,ϕ,ψ=1 in Theorem 25, we get the following.Corollary 29.
LetX,S be a complete Sb-metric space and A:X⟶X be a continuous mapping. If there exists δ>0 for every ε>0 such that
(64)ε≤Λθ,ϕ,ψ<ε+δimplies
(65)SAθ,Aϕ,Aψ<εb,where
(66)Λθ,ϕ,ψ=maxSθ,ϕ,ψ,Sθ,θ,Aθ,Sϕ,ϕ,Aϕ,Sψ,ψ,Aψfor all θ,ϕ,ψ∈X. Then, the fixed point of A is unique.Corollary 30.
LetX,S be a complete Sb-metric space and A:X⟶X be a continuous mapping. If there exists δ>0 for every ε>0 such that
(67)ε≤Nθ,ϕ,ψ<ε+δimplies
(68)SAθ,Aϕ,Aψ<εb,where
(69)Nθ,ϕ,ψ=maxSθ,ϕ,ψ,13Sθ,θ,Aθ+Sϕ,ϕ,Aϕ+Sψ,ψ,Aψfor all θ,ϕ,ψ∈X. Then, A has a unique fixed point.
The Meir-Keeler contraction can be stated onSb-metric spaces as follows.Corollary 31.
LetX,S be a complete Sb-metric space and A:X⟶X be a continuous Meir-Keeler mapping. If there exists δ>0 for every ε>0 such that
(70)ε≤Sθ,ϕ,ψ<ε+δbecomes
(71)SAθ,Aϕ,Aψ<εbfor all θ,ϕ,ψ∈X. Then A has a unique fixed point.


## 4. Conclusion

In this article, we define Meir-Keeler contraction inSb-metric spaces using the concept of α-admissible mapping. Further, we define generalized αs-Meir-Keeler contraction. Using these definitions of contractive mappings, we prove theorems for the existence and uniqueness of fixed points. We show that obtained results are potential generalizations of various results in the literature.



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