# Some Fixed Point Theorems of Contractive Mappings inPbr-Cone Metric Spaces over Banach Algebras

Authors: Abba Auwalu; Evren Hinçal
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce the concept of aPbr-cone metric space over Banach algebras and prove some fixed point results under various contractive mappings in such a space. Some examples are given to elucidate the results. Our results extend and generalize many existing results in the literature.


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## Body

## 1. Introduction

In 2017, George et al. [1] introduced the concept of a rectangular cone b-metric space over Banach algebras as a generalization of metric space and many of its generalizations. They proved some fixed point results in such a space. Very recently, Fernandez et al. [2] introduced Pb-cone metric space over Banach algebras as a generalization of partial metric space and many of its generalizations. Motivated and inspired by these papers [1, 2], we introduce the concept of a Pbr-cone metric space over Banach algebras which generalized both rectangular cone b-metric space over Banach algebras and Pb-cone metric space over Banach algebras. Furthermore, we prove some fixed point results under various contractive mappings in such a space. Examples are also given to elucidate our results. Our results extend and generalize many results in [1, 3–10].


## 2. Preliminaries

We start with definitions and some basic facts needed in the sequel.Definition 1.
(see [11]). Let A be a real Banach algebra, i.e., A is a real Banach space in which an operation of multiplication is defined, for all y,z,x∈A and k∈ℝ, and the following are satisfied:(i)
yzx=yzx(ii)
yz+x=yz+yx and y+zx=yx+zx(iii)
kyz=kyz=ykz(iv)
yz≤yz
A Banach algebraA is called unital if there exists a unit e∈A such that ey=ye=y, for any y∈A.Definition 2.
(see [5]). A subset K of Banach algebra A is called a cone if(i)
K is nonempty, closed, and θ,e⊂A, where θ is the zero of A(ii)
αK+βK⊂K for all nonnegative real numbers α,β(iii)
K2=KK⊂K(iv)
K∩−K=θ
For a given coneK⊂A, we define a partial ordering ≺¯ with respect to K by y≺¯z if and only if z−y∈K. The notation y≪z will stand for z−y∈K°, where K° denotes the interior of K. K is called a solid cone if K°≠∅.Definition 3.
(see [6]). Let K be a solid cone in a Banach algebra A. A sequence yn⊂K is said to be a c-sequence if, for every c∈K°, there exists N∈ℕ such that yn≪c for all n>N.Lemma 1 (see [8]).
LetK be a solid cone in a Banach algebra A:(1)
Ifα,β∈A, γ∈K, and α≺¯β, then γα≺¯γβ(2)
Ifα≺¯βα, where α,β∈K and ρβ<1, then α=θ(3)
Ifα∈K and ρα<1, then ραq<1 for any fixed q∈ℕLemma 2 (see [11, 12]).
LetA be a unital Banach algebra and α∈A; then, limm⟶∞αm1/m exists and the spectral radius ρα satisfies(1)ρα=limm⟶∞αm1/m=infm≥1αm1/m.
Ifρα<β, then βe−α is invertible in A. Moreover,(2)βe−α−1=∑j=0∞αjβj+1,ρβe−α−1≤1β−ρα,where β is a complex constant.Lemma 3 (see [11]).
LetA be a unital Banach algebra and α,β∈A such that α commutes with β. Then,(3)ρα+β≤ρα+ρβ,ραβ≤ραρβ.Lemma 4 (see [13]).
LetK be a solid cone in a Banach algebra A, yn and zn be two c-sequences in K. If α,β∈K are two given vectors, then αyn+βzn is also a c-sequence in K.Lemma 5 (see [13]).
LetA be a unital Banach algebra. Let α∈A and ρα<1. Then, αn is a c-sequence in A.Lemma 6 (see [6]).
LetK be a solid cone in a Banach algebra A:(1)
Ifα,β,γ∈K and α≺¯β≪γ, then α≪γ(2)
Ifα∈A and θ≺¯α≪β for each β∈K°, then α=θ(3)
yn⊂K is a c-sequence provided that yn⟶θ as n⟶∞Definition 4.
(see [2]). Let Y be a nonempty set and A a Banach algebra. Suppose that, for all y,z,x∈Y, a mapping Pb:Y×Y⟶A satisfies(1)
y=z⇔Pby,y=Pby,z=Pbz,z(2)
θ≺¯Pby,y≺¯Pby,z(3)
Pby,z=Pbz,y(4)
Pby,z≺¯sPby,x+Pbx,z−Pbx,x
Then,Y,Pb is called a partial cone b-metric space over A with coefficient s≥1.Definition 5.
(see [1]). Let Y be a nonempty set and K a solid cone in a Banach algebra A. Suppose that, for all y,z∈Y and all distinct points x1,x2∈Y∖y,z, a mapping Prcb:Y×Y⟶A satisfies(1)
θ≺¯Prcby,z and Prcby,z=θ⇔y=z.(2)
Prcby,z=Prcbz,y.(3)
There existss∈K with e≺¯s such that(4)Prcby,z≺¯sPrcby,x1+Prcbx1,x2+Prcbx2,z.
Then,Prcb is called a rectangular cone b-metric on Y, and Y,Prcb is called a rectangular cone b-metric space over A with coefficient s.


## 3. Main Results

In this section, we introduce the concept of a partial rectangular coneb-metric space (Pbr-cone metric space) over Banach algebras and give some of its topological property. Furthermore, the notions of convergent sequence, θ-Cauchy sequence, and θ-completeness in the setting of this new space are defined. Moreover, some fixed point theorems under various contractive mappings are proved in such a space.Definition 6.
LetY be a nonempty set and K be a solid cone in a unital Banach algebra A. Suppose that, for all y,z∈Y and all distinct points x1,x2∈Y∖y,z, a mapping Pbr:Y×Y⟶A satisfies(P1)y=z⇔Pbry,y=Pbry,z=Pbrz,z.(P2)θ≺¯Pbry,y≺¯Pbry,z.(P3)Pbry,z=Pbrz,y.(P4) There existss∈K with e≺¯s such that(5)Pbry,z≺¯sPbry,x1+Pbrx1,x2+Pbrx2,z−Pbrx1,x1−Pbrx2,x2.
Then,Pbr is called a partial rectangular cone b-metric on Y, and Y,Pbr,A is called a partial rectangular cone b-metric space over A with coefficient s (in short PRCbMS-BA).Remark 1.
In any PRCbMS-BAY,Pbr,A, if Pbry,z=θ for all y,z∈Y, then y=z, but the converse may not be true. Also, every rectangular cone b-metric space over A is a Pbr-cone metric space over A with zero θ self distance, but there are Pbr-cone metric spaces over A which are not a rectangular cone b-metric space over A.Example 1.
LetA=Cℝ10,1 with the norm(6)y=y∞+y′∞,for ally∈A.
Define multiplication pointwisely onA. Then, A is a Banach algebra with unit et=1,∀t∈0,1. Let K=y∈A:y=yt≥0,t∈0,1. Then, K is a solid cone in A. Let Y=y1,y2,y3,y4 and, for all y,z∈Y, define a mapping Pbr:Y×Y⟶K by(7)Pbry,zt=θ,ify=z=y1,2t,ify,z∈y1,y2,y≠z,t,otherwise.
Then,Y,Pbr,A is a PRCbMS-BA with coefficient s=4/3 which is not a rectangular cone b-metric space over A because Pbry2,y2t≠θ and Pbry1,y2t=2t>t=Pbry1,y3t+Pbry3,y4t+Pbry4,y2t−Pbry3,y3t−Pbry4,y4t.Definition 7.
LetY,Pbr,A be a PRCbMS-BA and K be a solid cone in A. For each y∈Y and each c∈K°, let(8)BPbry,c=z∈Y:Pbry,z≪c+Pbry,y,ℬ=BPbry,c:y∈Yandc∈K°.
Then,(9)τP=U⊂Y:for ally∈Uthere existsBPbr∈ℬandy∈BPbr⊂U∪∅,is a topology on Y, BPbry,c is a Pbr-ball in Y,Pbr,A, ℬ is a subbase for the topology τP on Y, and U is a base generated by the subbase ℬ.Definition 8.
LetY,Pbr,A be a PRCbMS-BA, K be a solid cone in A, y∗∈Y, and yn be a sequence in Y. If, for every c∈K°, there exists N∈ℕ such that Pbryn,y∗≪c+Pbry∗,y∗ for all n>N, then yn is said to be convergent in Y and converges to y∗. This fact is denoted by yn⟶y∗ as n⟶∞ or limn⟶∞yn=y∗.Definition 9.
LetY,Pbr,A be a PCbMS-BA, K be a solid cone in A, and yn be a sequence in Y. Then, yn is called a θ-Cauchy sequence if Pbryn,ym is a c-sequence in A. That is, if, for every c∈K°, there exists N∈ℕ such that Pbryn,ym≪c for all n,m>N.Definition 10.
LetY,Pbr,A be a PRCbMS-BA, K be a solid cone in A, y∗∈Y, and yn be a sequence in Y. Then, Y,Pbr,A is called θ-complete if every θ-Cauchy sequence yn in Y converges to a point y∗∈Y. That is,(10)limn,m⟶∞Pbryn,ym=limn⟶∞Pbryn,y∗=Pbry∗,y∗=θ.Lemma 7.
LetY,Pbr,A be a PRCbMS-BA and yn be a sequence in Y. If yn converges to y∗∈Y, then(1)
Pbryn,y∗ is a c-sequence.(2)
For anym∈ℕ, Pbryn,yn+m is a c-sequence.Proof.
It follows from Definitions3, 6, and 8.
Firstly, we present a variant of the Banach contraction principle onPbr-cone metric space over Banach algebra A.Theorem 1.
LetY,Pbr,A be a θ-complete PRCbMS-BA with s∈K such that e≺¯s. Suppose F:Y⟶Y is a function satisfying(11)PbrFy,Fz≺¯αPbry,z,for ally,z∈Y,where α∈K such that α commutes with s and ρα<1. Then, F has a unique fixed point.Proof.
Lety0 be a point in Y. We define a sequence yn in Y by(12)yn=Fyn−1=Fny0,for alln≥1.
Ifyn=yn+1 for some n∈ℕ, then y∗=yn=Fyn is a fixed point of F, and the result is proved. Hence, we assume that yn≠yn+1for alln≥0. We will show that yn≠yn+qfor alln≥0 and q≥1. Suppose that yn=yn+qfor somen≥0,q≥1; then, yn+1=yn+q+1 and Fyn=Fyn+q. Then, (11) implies that(13)Pbryn,yn+1=Pbryn+q,yn+q+1≺¯αPbryn+q−1,yn+q≺¯⋯≺¯αqPbryn,yn+1.
Using Lemma1, we obtain that Pbryn,yn+1=θ, that is, yn=yn+1, which is a contradiction. Therefore, yn≠ym,for all distinctn,m∈ℕ. Hence, from (11) and (12), we have that(14)Pbryn,yn+1=PbrFyn−1,Fyn≺¯αPbryn−1,yn≺¯α2Pbryn−2,yn−1≺¯⋯∴Pbryn,yn+1≺¯αnPbry0,y1,for alln∈ℕ.
Similarly, for alln,m,q∈ℕ, we obtain that(15)Pbryn+q,ym+q=PbrFyn+q−1,Fym+q−1≺¯αPbryn+q−1,ym+q−1≺¯α2Pbryn+q−2,ym+q−2≺¯⋯∴Pbryn+q,ym+q≺¯αqPbryn,ym,for alln,m,q∈ℕ.
Observe thatρs exists because of Lemma 2, and since ρα<1, there exists q1∈ℕ such that ρsραq1<1 holds. Since α commutes with s, by Lemmas 2 and 3, we have that(16)ρsαq1≤ρsραq1<1ande−sαq1is invertible inA.
Hence, by condition (P4), for ally,z,x1,x2∈Y, we have(17)Pbry,z≺¯sPbry,x1+Pbrx1,x2+Pbrx2,z−Pbrx1,x1−Pbrx2,x2∴Pbry,z≺¯sPbry,x1+Pbrx1,x2+Pbrx2,z,for ally,z,x1,x2∈Y.
This, using (15) and (16), implies that(18)Pbryn,ym≺¯sPbryn,yn+q1+Pbryn+q1,ym+q1+Pbrym+q1,ym≺¯sαnPbry0,yq1+αq1Pbryn,ym+αmPbryq1,y0e−sαq1Pbryn,ym≺¯sαnPbry0,yq1+αmPbryq1,y0∴Pbryn,ym≺¯e−sαq1−1sαnPbry0,yq1+αmPbryq1,y0.
Using Lemmas4 and 5, we deduce that Pbryn,ym is a c-sequence in A. Therefore, yn is a θ-Cauchy sequence in Y. From the hypothesis, Y,Pbr,A is θ-complete; hence, there exists a point y∗∈Y such that yn converges to y∗. That is,(19)limn⟶∞Pbryn,y∗=limn,m⟶∞Pbryn,ym=Pbry∗,y∗=θ.
Next, we will show thaty∗ is the unique fixed point of F:(20)Pbry∗,Fy∗≺¯sPbry∗,yn+Pbryn,yn+1+Pbryn+1,Fy∗−Pbryn,yn−Pbryn+1,yn+1≺¯sPbry∗,yn+Pbryn,yn+1+PbrFyn,Fy∗≺¯sPbry∗,yn+Pbryn,yn+1+αPbryn,y∗≺¯se+αPbry∗,yn+Pbryn,yn+1.
By Lemmas6 and 7, we have Pbry∗,yn⟶θ as n⟶∞ and Pbryn,yn+1⟶θ as n⟶∞. Hence, we deduce that Pbry∗,Fy∗=θ. That is, y∗=Fy∗. So, y∗ is a fixed point of F. For uniqueness, we let z∗ be another fixed point of F. Then, it follows from (11) that(21)Pbry∗,z∗=PbrFy∗,Fz∗≺¯αPbry∗,z∗.
By Lemma1, we get that Pbry∗,z∗=θ, and hence, y∗=z∗.
Kindly, observe that Theorem1 extends and generalizes Theorem 3.5 in [1], Theorem 3.1 in [3], Theorem 2.1 in [4], Theorem 2.1 in [5], and Theorem 3.1 in [6].Example 2.
LetA=Cℝ10,1 with the norm(22)y=y∞+y′∞,for ally∈A.
Define multiplication pointwisely onA. Then, A is a Banach algebra with unit et=1,∀t∈0,1. Let K=y∈A:y=yt≥0,t∈0,1. Then, K is a solid cone in A. Let Y=0,1,2,3 and, for all y,z∈Y, define a mapping Pbr:Y×Y⟶K by(23)Pbry,zt=y2t,ify=z≠0,2y2+z2t,ify,z∉2,3,y≠z,y2+z2tify,z∈2,3,y≠z,12t,ify=z=0.
Then,Y,Pbr,A is a θ-complete PRCbMS-BA with coefficient s=2. Define a mapping F:Y⟶Y as follows:(24)Fy=0,ify∈0,1,1,ify∈2,3.
Hence, the mappingF satisfies all the conditions of Theorem 1 and y∗=0∈Y is the unique fixed point of F.
Secondly, we present a variant of the Reich contraction principle onPbr-cone metric space over Banach algebra A.Theorem 2.
LetY,Pbr,A be a θ-complete PRCbMS-BA with s∈K such that e≺¯s. Suppose F:Y⟶Y is a function satisfying(25)PbrFy,Fz≺¯αPbry,z+βPbry,Fy+γPbrz,Fz,for all y,z∈Y, where α,β,γ∈K commutes, ρα+ρβ+γ<1, and minρβ,ργ<1/ρs. Then, F has a unique fixed point.Proof.
Lety0 be a point in Y. We define a sequence yn in Y by(26)yn+1=Fyn=Fn+1y0,for alln≥0.
From (25) and (26), we have(27)Pbryn+1,yn=PbrFyn,Fyn−1≺¯αPbryn,yn−1+βPbryn,Fyn+γPbryn−1,Fyn−1∴e−βPbryn+1,yn≺¯α+γPbryn,yn−1.
On the contrary, we have(28)Pbryn+1,yn=PbrFyn,Fyn−1=PbrFyn−1,Fyn≺¯αPbryn−1,yn+βPbryn−1,Fyn−1+γPbryn,Fyn∴e−γPbryn+1,yn≺¯α+βPbryn,yn−1.
Adding up (27) and (28), we have(29)2e−λPbryn+1,yn≺¯2α+λPbryn,yn−1,where λ=β+γ∈K. Now, we observe that(30)2ρλ≤2ρα+2ρλ=2ρα+ρβ+γ<2.
This implies thatρλ<1<2; then, by Lemma 2, it follows that 2e−λ is invertible and 2e−λ−1=∑j=0∞λj/2j+1. From (29), we obtain(31)Pbryn+1,yn≺¯2e−λ−12α+λPbryn,yn−1≺¯kPbryn,yn−1,where k=2e−λ−12α+λ∈K. Hence,(32)Pbryn+1,yn≺¯kPbryn,yn−1≺¯k2Pbryn−1,yn−2≺¯⋯∴Pbryn+1,yn≺¯knPbry1,y0,for alln∈ℕ.
We claim thatρk<1. Indeed, since α commutes with λ=β+γ, it follows that(33)2e−λ−12α+λ=∑j=0∞λj2j+12α+λ=2∑j=0∞λj2j+1α+∑j=0∞λj+12j+1=2α∑j=0∞λj2j+1+λ∑j=0∞λj2j+1=2α+λ∑j=0∞λj2j+1=2α+λ2e−λ−1.
Therefore,2α+λ commutes with 2e−λ−1. Then, by Lemmas 2 and 3, we obtain(34)ρk=ρ2e−λ−12α+λ≤ρ2e−λ−1ρ2α+λ≤12−ρλ2ρα+ρλ<1sinceρα+ρλ<1.
Ifyn=yn+1 for some n∈ℕ, then y∗=yn=Fyn is a fixed point of F, and the result is proved. Hence, we assume that yn≠yn+1,for alln≥0. We will show that yn≠yn+q,for alln≥0 and q≥1. Suppose that yn=yn+q,for somen≥0,q≥1; then, yn+1=yn+q+1 and Fyn=Fyn+q. Then, (31) implies that(35)Pbryn+1,yn=Pbryn+q+1,yn+q≺¯kqPbryn+1,yn.
Using Lemma1, we obtain that Pbryn+1,yn=θ, that is, yn+1=yn, which is a contradiction. Therefore, yn≠ymfor all distinctn,m∈ℕ. Next, from (25), (26), and (32), we have(36)Pbryn,ym=PbrFyn−1,Fym−1≺¯αPbryn−1,ym−1+βPbryn−1,Fyn−1+γPbrym−1,Fym−1=αPbryn−1,ym−1+βPbryn−1,yn+γPbrym−1,ym≺¯αPbryn−1,ym−1+βkn−1Pbry0,y1+γkm−1Pbry0,y1∴Pbryn,ym≺¯qPbryn−1,ym−1+qn+qmPbry0,y1,where q∈α,β,γ,k such that ρq=maxρα,ρβ,ργ,ρk. Hence, from (36), we also obtain(37)Pbryn,ym≺¯qpPbryn−p,ym−p+pqn+qmPbry0,y1,for all p∈1,2,…,minn,m. Observe that ρs exists because of Lemma 2, and since ρq<1, there exists q0∈ℕ such that ρsρqq0<1 holds. Furthermore, since q commutes with s, by Lemmas 2 and 3, we have that(38)ρsqq0≤ρsρqq0<1ande−sqq0is invertible inA.
Therefore, from (37), we further obtain(39)Pbryn,yn+q0≺¯qnPbry0,yq0+nqn+qn+q0Pbry0,y1,(40)Pbrym+q0,ym≺¯qmPbryq0,y0+mqm+q0+qmPbry0,y1,(41)Pbryn+q0,ym+q0≺¯qq0Pbryn,ym+q0qn+q0+qm+q0Pbry0,y1.
Hence, from (P4), (38), (39), (40) and (41), we have(42)Pbryn,ym≺¯sPbryn,yn+q0+Pbryn+q0,ym+q0+Pbrym+q0,ym−Pbryn+q0,yn+q0−Pbrym+q0,ym+q0≺¯sqnPbry0,yq0+nqn+qn+q0Pbry0,y1+qq0Pbryn,ym+q0qn+q0+qm+q0Pbry0,y1+qmPbryq0,y0+mqm+q0+qmPbry0,y1e−sqq0Pbryn,ym≺¯sqn+qmPbry0,yq0+qnn+n+q0qq0+qmm+m+q0qq0Pbry0,y1∴Pbryn,ym≺¯e−sqq0−1sqn+qmPbry0,yq0+qnn+n+q0qq0+qmm+m+q0qq0Pbry0,y1.
Using Lemmas4 and 5, we deduce that Pbryn,ym is a c-sequence in A. Therefore, yn is a θ-Cauchy sequence in Y. From the hypothesis, Y,Pbr,A is θ-complete; hence, there exists a point y∗∈Y such that yn converges to y∗. That is,(43)limn⟶∞Pbryn,y∗=limn,m⟶∞Pbryn,ym=Pbry∗,y∗=θ.
Next, we will show thaty∗ is the unique fixed point of F:(44)Pbry∗,Fy∗≺¯sPbry∗,yn+Pbryn,yn+1+Pbryn+1,Fy∗−Pbryn,yn−Pbryn+1,yn+1≺¯sPbry∗,yn+Pbryn,yn+1+PbrFyn,Fy∗≺¯sPbry∗,yn+Pbryn,yn+1+αPbryn,y∗+βPbryn,Fyn+γPbry∗,Fy∗∴Pbry∗,Fy∗≺¯se+αPbry∗,yn+e+βPbryn,yn+1+γPbry∗,Fy∗.
On the contrary, we have(45)PbrFy∗,y∗≺¯sPbrFy∗,yn+1+Pbryn+1,yn+Pbryn,y∗−Pbryn+1,yn+1−Pbryn,yn≺¯sPbrFy∗,Fyn+Pbryn+1,yn+Pbryn,y∗≺¯sαPbry∗,yn+βPbry∗,Fy∗+γPbryn,Fyn+Pbryn+1,yn+Pbryn,y∗∴Pbry∗,Fy∗≺¯se+αPbry∗,yn+e+γPbryn,yn+1+βPbry∗,Fy∗.
By Lemmas6 and 7, we have Pbry∗,yn⟶θ as n⟶∞ and Pbryn,yn+1⟶θ as n⟶∞. Hence, from (44) and (45), we deduce that Pbry∗,Fy∗≺¯sγPbry∗,Fy∗ and Pbry∗,Fy∗≺¯sβPbry∗,Fy∗. Since minρβ,ργ<1/ρs, by Lemma 1, we have Pbry∗,Fy∗=θ so that y∗=Fy∗. That is, y∗ is a fixed point of F. For uniqueness, we let z∗ be another fixed point of F. Then, it follows from (25) that(46)Pbry∗,z∗=PbrFy∗,Fz∗≺¯αPbry∗,z∗+βPbry∗,Fy∗+γPbrz∗,Fz∗=αPbrz∗,y∗+βPbry∗,y∗+γPbrz∗,z∗≺¯αPbrz∗,y∗+βPbry∗,z∗+γPbrz∗,y∗∴Pbry∗,z∗≺¯α+β+γPbry∗,z∗.
By Lemmas1 and 3, we have that Pbry∗,z∗=θ, and hence, y∗=z∗, i.e., the fixed point of F is unique.
Note that Theorem2 extends and generalizes Theorem 3.1 in [7].
Thirdly, we present a variant of the quasi-contraction principle onPbr-cone metric space over Banach algebra A.Corollary 1.
LetY,Pbr,A be a θ-complete PRCbMS-BA with s∈K such that e≺¯s. Let F:Y⟶Y be a function satisfying(47)PbrFy,Fz≺¯αMy,z,for ally,z∈Y,where My,z∈Pbry,z,Pbry,Fy,Pbrz,Fz, α∈K commutes with s, and ρα<1. Then, F has a unique fixed point.Proof.
Since (47) implies (25), the proof follows from Theorem 2.
Note that Corollary1 generalizes Theorem 3.1 in [8–10].
Finally, we present a variant of the Kannan contraction principle onPbr-cone metric space over Banach algebra A.Corollary 2.
LetY,Pbr,A be a θ-complete PRCbMS-BA with s∈K such that e≺¯s. Suppose F:Y⟶Y is a function satisfying(48)PbrFy,Fz≺¯βPbry,Fy+Pbrz,Fz,for ally,z∈Y,where α∈K such that ρβ<1/2 and ρsβ<1. Then, F has a unique fixed point.Proof.
Putα=θ and β=γ in Theorem 2, and the result follows.
Note that Corollary2 generalizes Theorem 2.4 in [4], Theorem 2.3 in [5], and Theorem 3.3 in [6].


## 4. Conclusion

The concept of a partial rectangular coneb-metric space over Banach algebras was introduced, and some fixed point results under various contractive mappings were proved in such a space. Some examples were also given to elucidate the results. Our results extend and generalized many existing results in [1, 3–10].



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