# Fixed Point of Almost Contraction inb-Metric Spaces

Authors: Maryam Iqbal; Afshan Batool; Ozgur Ege; Manuel de la Sen
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we introduce a generalized multivalued (α, L)-almost contraction in the b-metric space. Furthermore, we prove the existence and uniqueness of the fixed point for a specific mapping. The result presented in this paper extends some of the earlier results in the existing literature. Moreover, some examples are given to illuminate the usability of the obtained results.


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## Body

## 1. Introduction and Preliminaries

The idea ofb-metric was initiated from the works of Bourbaki [1] and Bakhtin [2]. A generalized form of the metric space is called a b-metric space. The concept of the b-metric space or metric type space was first introduced by Czerwik [3] as a generalization of the metric space. He provided an axiom which is weaker than the triangular inequality and formally defined a b-metric space with a view of generalizing the Banach contraction mapping principle. Later on, Fagin and Stockmeyer [4] worked on some kind of relaxation in the triangular inequality and called this new distance measure as nonlinear elastic matching (NEM). A similar type of relaxed triangle inequality was also used for trade measure [5] and to measure ice floes [6]. All these applications pushed us to introduce the concept of the b-metric space so that the results obtained for such rich spaces become more viable in different directions of applications. Since then, several authors proved fixed-point results of single-valued and multivalued operators in b-metric spaces [7–10]. Recently, Kamran et al. [11] worked on the b-metric space in which they discussed a generalization of the b-metric space called the extended b-metric space and established some fixed-point theorems for self-mappings defined on such spaces.Definition 1.
(see [3]). Let X be a nonempty set. A function d:X×X⟶0,∞ is said to be a b-metric if it satisfies the following conditions:(1)
0≤dx,y and dx,y=0 if and only if x=y,(2)
dx,y=dy,x, and(3)
dx,z≤sdx,y+dy,z for some s≥1,for allx,y,z∈X. The pair X,d is called a b-metric space with coefficient s.
We observe that every metric space is ab-metric space with s=1. The class of b-metric spaces is larger than the class of metric spaces, and the concept of the b-metric space coincides with the concept of the metric space. This is a weaker concept than that of a metric space. Conditions (1) and (2) are similar to the metric space, but (3) is a key feature of this concept. Therefore, it is important to study how to use (3) effectively. Each b-metric is not a continuous function. To show this statement, the following example was presented in [12].Example 1.
LetX,d be a metric space and σd:X×X⟶ℝ+ be defined by(1)σdx,y=dx,yp,for all x,y∈X, where p>1 is a fixed real number. Then, σd is a b-metric with s=2p−1. Indeed, conditions (1) and (2) in Definition 1 are satisfied, and thus, we only have to show that condition (3) holds for σd.
It is easy to see that if1<p<∞, then the convexity of the function fx=xp, where x≥0, implies(2)a+c2p≤12ap+cp,and hence,(3)a+cp≤2p−1ap+cp.
Therefore, for eachx, y, z∈X, we get(4)σdx,y=dx,yp≤dx,z+dz,yp≤2p−1dx,zp+z,yp=2p−1σdx,z+σdz,y.
So, condition (3) in Definition1 holds, and then σd is a b-metric space with coefficient s=2p−1.
Similarly, the concepts ofb-convergent sequence, b-Cauchy sequence, and complete b-metric space are defined accordingly.Definition 2.
(see [11]). Let X,d be a b-metric space.(1)
A sequencexn in X is called b-convergent if there exists x∈X such that dxn,x⟶0 as n⟶∞. In this case, we write limn⟶∞xn=x.(2)
A sequencexn in X is called b-Cauchy if dxn,xm⟶0 as n,m⟶∞.(3)
Ab-metric space X,d is said to be a complete b-metric space if every b-Cauchy sequence in X is b-convergent.


### 1.1. Hausdorff Metric

The concept of Hausdorff metric or Hausdorff distance was first introduced by Hausdorff in his bookGrundzuge der Mengenlehre [13]. The second name of Hausdorff distance is Pompeiu–Hausdorff distance. The Hausdorff distance has many applications in the computer field. The use of Hausdorff distance is to find a given template in an arbitrary target image in computer vision. The most important application of the Hausdorff metric in computer graphics is to measure the difference between two different representations of the same 3D object specifically when generating the level of detail for efficient display of complex 3D models.LetX,d be a b-metric space and CBX denote the family of all nonempty, closed, and bounded subsets of X.ForM,N⊂X and a∈X, the distance between a and N is defined as(5)da,N=infda,b:b∈N.The diameter ofM and N is defined as(6)δM,N=supda,b:a∈M,b∈N,HM,N=maxsupa∈Mda,N,supb∈Ndb,M,which is a Hausdorff metric on CBX induced by d. Then, CBX is a b-metric space under the Hausdorff distance H.IfX,d is a complete b-metric space, then CBX,H is a complete b-metric space, too. If M and N have the same closures, then H=0.


### 1.2.α,L-Almost Contraction Mappings

Berinde [14] extended the notion of almost contraction from single-valued mappings to multivalued mappings. There are many examples of contractive conditions which imply the almost contractiveness condition, for instance, Mizoguchi and Takahashi [15] used the concept of almost contraction in their work.Theorem 1 (see [16]).
LetX,d be a complete metric space and T:X⟶CBX be a generalized multivalued α,L-almost contraction, i.e., a mapping for which there exists a function α:0,∞⟶0,1 satisfying(7)limr⟶t+supαr<1,for every t∈0,∞, such that(8)HTx,Ty≤αdx,ydx,y+LN,where(9)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx,for all x,y∈X and Tx,Ty∈CBX. Then, T has a fixed point.Lemma 1 (see [4]).
LetX,d be a complete b-metric space with coefficient s and T:X⟶KX be a mapping, where KX is a nonempty compact subset of X. Assume that there exists a function α:0,∞⟶0,1 such that, for each t∈0,∞,(10)limr⟶t+supαr<1,HTx,Ty≤αdx,ydx,y,∀x,y∈X,x≠y.
Then,T has a fixed point.Lemma 2 (see [15]).
LetX,d be a complete b-metric space and T:X⟶CBX be a mapping. Assume that there exists a function α:0,∞⟶0,1 such that, for each t∈0,∞,(11)limr⟶t+supαr<1,HTx,Ty≤αdx,ydx,y,∀x,y∈X,x≠y.
Then,T has a fixed point.Lemma 3 (see [17]).
LetX,d be a b-metric space. Let A,B⊂X and q>1. Then, for every x∈A, there exists y∈B such that(12)dx,y≤qHA,B.


## 1.1. Hausdorff Metric

The concept of Hausdorff metric or Hausdorff distance was first introduced by Hausdorff in his bookGrundzuge der Mengenlehre [13]. The second name of Hausdorff distance is Pompeiu–Hausdorff distance. The Hausdorff distance has many applications in the computer field. The use of Hausdorff distance is to find a given template in an arbitrary target image in computer vision. The most important application of the Hausdorff metric in computer graphics is to measure the difference between two different representations of the same 3D object specifically when generating the level of detail for efficient display of complex 3D models.LetX,d be a b-metric space and CBX denote the family of all nonempty, closed, and bounded subsets of X.ForM,N⊂X and a∈X, the distance between a and N is defined as(5)da,N=infda,b:b∈N.The diameter ofM and N is defined as(6)δM,N=supda,b:a∈M,b∈N,HM,N=maxsupa∈Mda,N,supb∈Ndb,M,which is a Hausdorff metric on CBX induced by d. Then, CBX is a b-metric space under the Hausdorff distance H.IfX,d is a complete b-metric space, then CBX,H is a complete b-metric space, too. If M and N have the same closures, then H=0.


## 1.2.α,L-Almost Contraction Mappings

Berinde [14] extended the notion of almost contraction from single-valued mappings to multivalued mappings. There are many examples of contractive conditions which imply the almost contractiveness condition, for instance, Mizoguchi and Takahashi [15] used the concept of almost contraction in their work.Theorem 1 (see [16]).
LetX,d be a complete metric space and T:X⟶CBX be a generalized multivalued α,L-almost contraction, i.e., a mapping for which there exists a function α:0,∞⟶0,1 satisfying(7)limr⟶t+supαr<1,for every t∈0,∞, such that(8)HTx,Ty≤αdx,ydx,y+LN,where(9)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx,for all x,y∈X and Tx,Ty∈CBX. Then, T has a fixed point.Lemma 1 (see [4]).
LetX,d be a complete b-metric space with coefficient s and T:X⟶KX be a mapping, where KX is a nonempty compact subset of X. Assume that there exists a function α:0,∞⟶0,1 such that, for each t∈0,∞,(10)limr⟶t+supαr<1,HTx,Ty≤αdx,ydx,y,∀x,y∈X,x≠y.
Then,T has a fixed point.Lemma 2 (see [15]).
LetX,d be a complete b-metric space and T:X⟶CBX be a mapping. Assume that there exists a function α:0,∞⟶0,1 such that, for each t∈0,∞,(11)limr⟶t+supαr<1,HTx,Ty≤αdx,ydx,y,∀x,y∈X,x≠y.
Then,T has a fixed point.Lemma 3 (see [17]).
LetX,d be a b-metric space. Let A,B⊂X and q>1. Then, for every x∈A, there exists y∈B such that(12)dx,y≤qHA,B.


## 2. Generalized Multivalued Almost Contraction

In this section, we present and prove our main results on the existence of fixed points forα,L-almost contractions in the b-metric space. The list of following results is reestablished as the extension of the multivalued almost contraction in the b-metric space.Lemma 4.
LetX,d be a b-metric space and T:X⟶CX be a mapping. Then, for every x∈X with dx,Tx>0 and any b∈0,1, there exists y∈Tx (y≠x) such that(13)bsdx,y≤bdx,Tx.Lemma 5.
LetX,d be a b-metric space with s≥1 and T:X⟶CX be a mapping such that the following conditions hold:(1)
The functionf:X⟶ℝ,fx=dx,Tx, is lower semicontinuous.(2)
There existb∈0,1 and α:0,∞⟶0,1 such that, for all t∈0,∞,(14)limr⟶t+supαr<1.If for allx∈X, there exists y∈Ibx=y∈Tx:bdx,y≤dx,Tx satisfying(15)bsdy,Ty≤bαdx,ydx,y,then T has a fixed point.Lemma 6.
LetX,d be a complete b-metric space with coefficient s≥1, and assume that there exist a constant b∈0,1 and a set-valued α-contraction T:X⟶CBX, i.e., a mapping for which there exists a constant α∈0,∞ such that(16)bsHTx,Ty≤αdx,y,∀x,y∈X.
Then,T has a fixed point.


### 2.1.α,L-Almost Contraction Mappings in the b-Metric Space

LetX,d be a complete b-metric space with s≥1 and T:X⟶CBX be a generalized multivalued α,L-almost contraction, i.e., a mapping for which there exist a function α:0,∞⟶0,1 and a constant L>0 and b∈0,1 such that(17)bsHTx,Ty≤αdx,ydx,y+LN,where(18)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx,for all x,y∈X and Tx,Ty∈CBX.Theorem 2.
LetX,d be a b-metric space with s≥1 and T:X⟶CX satisfy the following conditions:(1)
The functionf:X⟶ℝ,fx=dx,Tx,x∈X, is lower semicontinuous.(2)
There existL≥0,b∈0,1 and α:0,∞⟶0,b such that, for all t∈0,∞,(19)limr⟶t+supαr<b.
If for allx∈X, there exists y∈Ibx=y∈Tx:bdx,y≤dx,Tx satisfying(20)bsdy,Ty≤αdx,ydx,y+LN,where(21)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx,for all x,y∈X, then T has a fixed point.Proof.
The main idea of the proof begins with the concept of the Banach contraction principle in which we take a Cauchy sequence in a completeb-metric space. Every Cauchy sequence is convergent in a complete b-metric space, and the converging point of that sequence is proved to be a fixed point of contraction.
If there existsx∈X such that dx,Tx=0, then x∈Tx, i.e., x is a fixed point of T. Since the range of T is closed, for each b∈0,1 and any x∈X with dx,Tx>0, there exists y∈Tx such that y∈Ibx, that is,(22)bsdx,y≤bdx,Tx.
So, we can assume that we havey∈Ibx,y≠x; otherwise, y=x∈Tx will be a fixed point of T, and the proof is done.
Letx1∈X be arbitrary but fixed with dx1,Tx1>0. By (16), there exists x2∈Tx1,x2≠x1, satisfying the inequality(23)bsdx1,x2≤bdx1,Tx1.
By (22),(24)bsdx2,Tx2≤αdx1,x2dx1,x2,(25)<bdx1,x2.
Sincedx2,Tx1=0, by (23) and (24), we have(26)bdx1,Tx1−bsdx2,Tx2≥bsdx1,x2−αdx1,x2dx1,x2=bs−αdx1,x2dx1,x2>0.
Now, forx2, there exists x3∈Tx2,x3≠x2, satisfying(27)bsdx2,x3≤bdx2,Tx2,(28)bsdx3,Tx3≤αdx2,x3dx2,x3<bdx2,x3.
By (27) and (28), we get(29)bdx2,Tx2−bsdx3,Tx3≥bsdx2,x3−αdx2,x3dx2,x3=bs−αdx2,x3dx2,x3>0.
Using (27) and (28), we obtain(30)dx2,x3≤bbsdx2,Tx2≤bbsαdx1,x2dx1,x2<dx1,x2.
By induction, forxn, n>1, obtained in the previous way, there exists xn+1∈Txn, xn≠xn+1, such that(31)bsdxn,xn+1≤bdxn,xn+1,(32)bsdxn+1,Txn+1≤αdxn,xn+1dxn,xn+1<bdxn,xn+1.
By the above pattern, we get(33)bdxn,Txn−bsdxn+1,Txn+1≥bsdxn,xn+1−αdxn,xn+1dxn,xn+1=bs−αdxn,xn+1dxn,xn+1>0,and we know that, in a complete b-metric space, every Cauchy sequence is convergent, and so,(34)dxn,xn+1<dxn,xn−1.
From (33) and (34), we can see that dxn,Txn and dxn,xn+1 are convergent and decreasing sequences, which are positive numbers. From assumption (2) in the theorem, it follows that there exists θ∈0,b such that(35)limn⟶∞supαdxn,xn+1=θ.
Then, for anyb0∈0,b, there exists n0∈ℕ such that(36)αdxn,xn+1<b0,∀n>n0.
By using (33) and by taking a=b−b0, we get(37)dxn,Txn−dxn+1,Txn+1≥adxn,xn+1,∀n>n0.
By (31) and (32) for any n>n0, we get(38)dxn+1,Txn+1≤αdxn,xn+1≤αdxn,xn+1bdxn,Txn≤αdxn,xn+1,…,αdx1,x2bndx1,Tx1=αdxn,xn+1,…,αdxn0+1,xn0+2bn−n0αdxn0,xn0+1,…,αdx1,x2bn0dx1,Tx1<b0bn−n0αdxn0,xn0+1,…,αdx1,x2bn0dx1,Tx1.
Sinceb0<b, we have(39)limn⟶∞b0bn−n0=0.
Thus,(40)limn⟶∞dxn,Txn=0.
Now, we have to show thatxn is a Cauchy sequence. For this, using condition (3) in Definition 1 and (37), for n,p∈ℕ, n,p>n0, we have(41)dxn,xn+p≤sdxn,xn+1+dxn+1,xn+p≤sdxn,xn+1+s2dxn+1,xn+2+s2dxn+2,xn+p⋮≤sdxn,xn+1+s2dxn+1,xn+2+,⋯,+spdxn+p−1,xn+p≤s1adxn,xn+1−dxn+1,xn+2+s21adxn+1,xn+2−dxn+2,xn+3+,…,+sp1adxn+p−2,xn+p−1−dxn+p−1,xn+p=sadxn,xn+1+ss−1adxn+1,xn+2+,⋯,+sn+p−1s−1adxn+p−1,xn+p⟶n,p⟶∞0.
Hence,xn is a Cauchy sequence. Since X,d is complete, xn is a convergent sequence in the b-metric space X,d. By using the definition of the convergent sequence, there exists x∗∈X such that(42)limn⟶∞xn=x∗.
As a result, we get the following:(43)0≤dx∗,Tx∗≤limn⟶∞infdxn,Txn=limn⟶∞dxn,Txn=0.
By the closeness property ofTx∗, x∗∈Tx∗, which is the definition of the fixed point, and so, x∗ is a fixed point.
To give the relation between our main result and works of Berinde, Nadler, and Mizoguchi [4, 15, 18–20], the following examples are provided.Example 2.
LetX=0,1,2,4,5,7,9 be a finite set, a b-metric d be defined on set X by dx,y=x−y2 with coefficient s=2, and CX=0,2,4,5,7,9 be the closed set of X. The mapping T:X⟶CX is piecewisely defined by(44)Tx=9−x,ifx∈2,4,7,9x+4,ifx∈0,1,5.
The values ofx and y are 0 and 2, whereas b=0.2. The function α is defined as αt=t/1+t.
By taking the notion of Theorem2 under consideration, Example 2 is elaborated. To choose the value of y,(45)y∈Ibx=y∈Tx:bdx,y≤dx,Tx.
Forx=0, Tx=4, and so,(46)0.2d0,2≤d0,4,0.24≤16,0.8≤16.
Thus,y=2. Now, by substituting values, we get(47)bSdy,Ty≤αdx,ydx,y+LN,where(48)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx.
After substitution, we get(49)1≤3.6.
Thus, we get the desired result.
The concept that “every metric space is ab-metric space with s=1” can be explained by the following example. The values of x, y, α, and b are also changed so that the concept of the theorem can be cleared in a more effective way. In the following example, the b-metric space is defined on an infinite set which is a generalization of the previous example.Example 3.
LetX=ℝ and a b-metric d on X be defined by dx,y=x−y. Note that dx,y is a b-metric with coefficient s=1. The closed set of X is taken to be ℝ itself. The mapping T:X⟶CX is defined as Tx=x/2 and αt=1+t/2. The values of x,y, and b are 4, 3, and 0.1, respectively. Since(50)x=4,Tx=42=2,y∈Ibx=y∈Tx:bdx,y≤dx,Tx,we have(51)0.1d4,3≤d4,2,0.1≤1.4142.
Thus,y=3. By substituting values, we get(52)bsdy,Ty≤αdx,ydx,y+LN,where(53)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx.
After substitution, we obtain(54)0.12≤1.
So, this is the desired result.


## 2.1.α,L-Almost Contraction Mappings in the b-Metric Space

LetX,d be a complete b-metric space with s≥1 and T:X⟶CBX be a generalized multivalued α,L-almost contraction, i.e., a mapping for which there exist a function α:0,∞⟶0,1 and a constant L>0 and b∈0,1 such that(17)bsHTx,Ty≤αdx,ydx,y+LN,where(18)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx,for all x,y∈X and Tx,Ty∈CBX.Theorem 2.
LetX,d be a b-metric space with s≥1 and T:X⟶CX satisfy the following conditions:(1)
The functionf:X⟶ℝ,fx=dx,Tx,x∈X, is lower semicontinuous.(2)
There existL≥0,b∈0,1 and α:0,∞⟶0,b such that, for all t∈0,∞,(19)limr⟶t+supαr<b.
If for allx∈X, there exists y∈Ibx=y∈Tx:bdx,y≤dx,Tx satisfying(20)bsdy,Ty≤αdx,ydx,y+LN,where(21)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx,for all x,y∈X, then T has a fixed point.Proof.
The main idea of the proof begins with the concept of the Banach contraction principle in which we take a Cauchy sequence in a completeb-metric space. Every Cauchy sequence is convergent in a complete b-metric space, and the converging point of that sequence is proved to be a fixed point of contraction.
If there existsx∈X such that dx,Tx=0, then x∈Tx, i.e., x is a fixed point of T. Since the range of T is closed, for each b∈0,1 and any x∈X with dx,Tx>0, there exists y∈Tx such that y∈Ibx, that is,(22)bsdx,y≤bdx,Tx.
So, we can assume that we havey∈Ibx,y≠x; otherwise, y=x∈Tx will be a fixed point of T, and the proof is done.
Letx1∈X be arbitrary but fixed with dx1,Tx1>0. By (16), there exists x2∈Tx1,x2≠x1, satisfying the inequality(23)bsdx1,x2≤bdx1,Tx1.
By (22),(24)bsdx2,Tx2≤αdx1,x2dx1,x2,(25)<bdx1,x2.
Sincedx2,Tx1=0, by (23) and (24), we have(26)bdx1,Tx1−bsdx2,Tx2≥bsdx1,x2−αdx1,x2dx1,x2=bs−αdx1,x2dx1,x2>0.
Now, forx2, there exists x3∈Tx2,x3≠x2, satisfying(27)bsdx2,x3≤bdx2,Tx2,(28)bsdx3,Tx3≤αdx2,x3dx2,x3<bdx2,x3.
By (27) and (28), we get(29)bdx2,Tx2−bsdx3,Tx3≥bsdx2,x3−αdx2,x3dx2,x3=bs−αdx2,x3dx2,x3>0.
Using (27) and (28), we obtain(30)dx2,x3≤bbsdx2,Tx2≤bbsαdx1,x2dx1,x2<dx1,x2.
By induction, forxn, n>1, obtained in the previous way, there exists xn+1∈Txn, xn≠xn+1, such that(31)bsdxn,xn+1≤bdxn,xn+1,(32)bsdxn+1,Txn+1≤αdxn,xn+1dxn,xn+1<bdxn,xn+1.
By the above pattern, we get(33)bdxn,Txn−bsdxn+1,Txn+1≥bsdxn,xn+1−αdxn,xn+1dxn,xn+1=bs−αdxn,xn+1dxn,xn+1>0,and we know that, in a complete b-metric space, every Cauchy sequence is convergent, and so,(34)dxn,xn+1<dxn,xn−1.
From (33) and (34), we can see that dxn,Txn and dxn,xn+1 are convergent and decreasing sequences, which are positive numbers. From assumption (2) in the theorem, it follows that there exists θ∈0,b such that(35)limn⟶∞supαdxn,xn+1=θ.
Then, for anyb0∈0,b, there exists n0∈ℕ such that(36)αdxn,xn+1<b0,∀n>n0.
By using (33) and by taking a=b−b0, we get(37)dxn,Txn−dxn+1,Txn+1≥adxn,xn+1,∀n>n0.
By (31) and (32) for any n>n0, we get(38)dxn+1,Txn+1≤αdxn,xn+1≤αdxn,xn+1bdxn,Txn≤αdxn,xn+1,…,αdx1,x2bndx1,Tx1=αdxn,xn+1,…,αdxn0+1,xn0+2bn−n0αdxn0,xn0+1,…,αdx1,x2bn0dx1,Tx1<b0bn−n0αdxn0,xn0+1,…,αdx1,x2bn0dx1,Tx1.
Sinceb0<b, we have(39)limn⟶∞b0bn−n0=0.
Thus,(40)limn⟶∞dxn,Txn=0.
Now, we have to show thatxn is a Cauchy sequence. For this, using condition (3) in Definition 1 and (37), for n,p∈ℕ, n,p>n0, we have(41)dxn,xn+p≤sdxn,xn+1+dxn+1,xn+p≤sdxn,xn+1+s2dxn+1,xn+2+s2dxn+2,xn+p⋮≤sdxn,xn+1+s2dxn+1,xn+2+,⋯,+spdxn+p−1,xn+p≤s1adxn,xn+1−dxn+1,xn+2+s21adxn+1,xn+2−dxn+2,xn+3+,…,+sp1adxn+p−2,xn+p−1−dxn+p−1,xn+p=sadxn,xn+1+ss−1adxn+1,xn+2+,⋯,+sn+p−1s−1adxn+p−1,xn+p⟶n,p⟶∞0.
Hence,xn is a Cauchy sequence. Since X,d is complete, xn is a convergent sequence in the b-metric space X,d. By using the definition of the convergent sequence, there exists x∗∈X such that(42)limn⟶∞xn=x∗.
As a result, we get the following:(43)0≤dx∗,Tx∗≤limn⟶∞infdxn,Txn=limn⟶∞dxn,Txn=0.
By the closeness property ofTx∗, x∗∈Tx∗, which is the definition of the fixed point, and so, x∗ is a fixed point.
To give the relation between our main result and works of Berinde, Nadler, and Mizoguchi [4, 15, 18–20], the following examples are provided.Example 2.
LetX=0,1,2,4,5,7,9 be a finite set, a b-metric d be defined on set X by dx,y=x−y2 with coefficient s=2, and CX=0,2,4,5,7,9 be the closed set of X. The mapping T:X⟶CX is piecewisely defined by(44)Tx=9−x,ifx∈2,4,7,9x+4,ifx∈0,1,5.
The values ofx and y are 0 and 2, whereas b=0.2. The function α is defined as αt=t/1+t.
By taking the notion of Theorem2 under consideration, Example 2 is elaborated. To choose the value of y,(45)y∈Ibx=y∈Tx:bdx,y≤dx,Tx.
Forx=0, Tx=4, and so,(46)0.2d0,2≤d0,4,0.24≤16,0.8≤16.
Thus,y=2. Now, by substituting values, we get(47)bSdy,Ty≤αdx,ydx,y+LN,where(48)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx.
After substitution, we get(49)1≤3.6.
Thus, we get the desired result.
The concept that “every metric space is ab-metric space with s=1” can be explained by the following example. The values of x, y, α, and b are also changed so that the concept of the theorem can be cleared in a more effective way. In the following example, the b-metric space is defined on an infinite set which is a generalization of the previous example.Example 3.
LetX=ℝ and a b-metric d on X be defined by dx,y=x−y. Note that dx,y is a b-metric with coefficient s=1. The closed set of X is taken to be ℝ itself. The mapping T:X⟶CX is defined as Tx=x/2 and αt=1+t/2. The values of x,y, and b are 4, 3, and 0.1, respectively. Since(50)x=4,Tx=42=2,y∈Ibx=y∈Tx:bdx,y≤dx,Tx,we have(51)0.1d4,3≤d4,2,0.1≤1.4142.
Thus,y=3. By substituting values, we get(52)bsdy,Ty≤αdx,ydx,y+LN,where(53)N=mindx,Tx,dy,Ty,dx,Ty,dy,Tx.
After substitution, we obtain(54)0.12≤1.
So, this is the desired result.



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*Source: 3218134-2020-10-28.xml*