# Coupled Fixed Point Theorems for Some Type of Contraction Mappings inb-Cone and b-Theta Cone Metric Spaces

Authors: Sahar Mohamed Ali Abou Bakr
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper proves the existence of a unique coupled fixed point of some type of contraction mappings defined on a completeb-cone and b-theta cone metric spaces; consequently, it extends and generalizes many previous coupled fixed point theorems.


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## Body

## 1. Introduction

In 2007, Huang and Zhang [1] introduced cone metric spaces as generalization of metric spaces by considering vector-valued metrics with values in an ordered real Banach space and hence generalized the concept of metric spaces and its completeness. They proved the existence of a unique fixed point for a contraction self-map T of cone metric space X,d showing that cone metric spaces provide larger categories of spaces for the fixed point theory. Recall that if A is a normed space, C is a cone in A that generates the partial ordered relation ⪯, and X,d is a cone metric space over A, then a mapping T:X⟶X is said to be contraction on X if and only if there is a constant α∈0,1 such that(1)dTx,Ty⪯αdx,y,∀x,y∈X.In 2008, Rezapour and Hamlbarani [2] improved some of the results in [1] by omitting the normality assumption of the cone induced the partial relation.After that, more topological characterization of cone metric spaces linked with some fixed point theorems have been studied by many other authors (see [2–5] and the references therein).In 2013, 2014, 2016, 2017, and then in 2020, Azam et al., citeAkbar, Xu and Radenovi [6], Huang and Radenovi [7], Sharma [8], and Sahar [9], respectively, considered cyclic and cone metric spaces over Banach algebra and b- cone metric spaces as a generalization of cone metric spaces, and they gave some further generalizations of some fixed points and proved fixed point theorems of contractive mapping in b- cone metric spaces, some of these results are proved without using the normality condition of a cone and some proved for generalized contraction multivalued mappings.In 2020, Sahar [10] introduced the concept of theta cone metric spaces which gave larger categories of metric spaces and generalized some previous fixed points’ theorems in the setting of this concept.On the other side, in 1987, the concept of coupled fixed point was initiated by Gue and Lakshmikantham [11], in partially ordered metric spaces; after that, in 2006, Bhaskar and Lakshmikantham [12] proved existence of coupled fixed points for mappings having the mixed monotone property.In 2009, Sabetghadam et al. [13] proved some coupled fixed point theorems for mappings satisfying different contractive conditions on complete cone metric spaces. Specifically, they proved the following.Theorem 1.
(see [13]). Let X,Cd be a complete cone metric space. Suppose that the mapping F:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(2)dFx,y,Fu,v⪯kdx,u+ldy,v,where kandl are nonnegative constants with k+l<1. Then, F has a unique coupled fixed point.
In 2010, Khamsi [14] gave the definition of metric type space (or b-metric space) and used this approach to proved the existence of a unique fixed point for Lipschitzian type mapping defined on a complete metric type space, and then, he noticed that, in case of normal cone, the cone metric q of the cone metric space X,C,q generates a metric type D:X×X⟶ℝ+ and hence a metric type space X,D. His remarkable notice enabled him to prove the existence of unique fixed point for Lipschitzian type mapping defined on complete cone metric X,C,q but provided that the cone C is normal.
In 2013, Luong et al. [15] followed another direction; they proved some coincidence and coupled fixed point of two compatible mappings: F:X×X⟶X and g:X⟶X, F is continuous, F has the mixed g-monotone property, F and g satisfy some generalized contraction conditions, X is partially ordered set endowed with a complete cone metric and some extra conditions on convergent monotone nondecreasing and convergent monotone nonincreasing sequences in X.
In 2021, Sahar [16] generalized coupled fixed point result for some generalized contraction type of mappings of Bhaskar and Lakshmikantham [12] and of Sahar [17, 18] in theta cone metric spaces.
We have the following notations and basic definitions.


## 2. Preliminaries and Basic Definitions

We start with the following.Definition 1.
(see [14]). Let X be a nonempty set. Let D:X×X⟶ℝ+ be a function which satisfies the following:(1)
Dx,y≥0,Dx,y=0if and only ifx=y(2)
Dx,y=Dy,xfor everyx,y∈X(3)
Dx,y≤KDx,w+Dw,yfor some positive real numberK>0
The pairX,D is called a metric type space, and it is called b-metric space if K≥1.
We recall some standard notations and definitions in cone metric spaces.
A subsetC of a linear space A is said to be a cone in A if and only if(1)
C is nonempty closed and C≠θ, where θ is the zero (neutral element) of A(2)
λC+μC⊂C for all nonnegative real numbers λ,μ(3)
C∩−C=θ
A coneC in a normed space A,⋅ is said to be solid if and only if it has a nonempty interior, that is, the set of all interior points of C is not empty set, IntC≠∅.
IfA,⋅ is a normed space, C is a cone in A; then, C generates the following ordered relations:(3)u⪯v⟺v−u∈C,u<v⟺v−u∈Candu≠v,u≪v⟺v−u∈IntC.
A sequenceznn∈N in A is bounded above by z∈A if and only if(4)zn⪯z,∀n∈ℕ,and its bounded below by z∈A if and only if(5)z⪯zn,∀n∈ℕ.
A coneC is called normal if there is a number M>0 (later proved to be greater than or equal 1) such that, for all x,y∈A,(6)ifθ≤x≤y,thenx≤My.
The normal constant of the normal coneC is defined to be the smallest constant M satisfying (6).
A coneC is called regular if every monotonically nonincreasing (no-decreasing) bounded above (bounded below) sequence has a limit in the norm sense of A.Remark 1.
(see [3]). Every regular cone is normal, there are normal cones which are not regular, there are cones which are not normal, the normal constant M of any normal cone is such that M≥1, and for any real number k, k≥1, there is a cone with normal constant M=k.Definition 2.
Suppose thatX is a nonempty set, C is a cone in a normed space A, r∈ℝ+, r≥1 (this r does not depend on the cone C itself, the cone may not be normal), and q is a function; q:X×X⟶C satisfies the following:(1)
θ≪qu,v,∀u,v∈X(2)
qu,v=θ⇔u=v(3)
qu,v=qv,u,∀u,v∈X(4)
qu,v⪯rqu,w+qw,v,∀u,v,w∈X
Then,X,C,q is defined to be a b-cone metric space over C. In particular, if r=1, then X,C,q is cone metric space.
A sequencevnn∈ℕ in a cone (b- cone) metric space X,C,q (with C solid) is Cauchy if and only if for every z∈A with θ≪z there is n0∈ℕ such that qvn,vm≪z for all n,m≥n0.
A sequencevnn∈ℕ in a cone (b- cone) metric space X,C,q (with C solid) is convergent sequence if and only if there is v such that, for every z∈A with θ≪z, there is n0∈ℕ such that qvn,v≪z for all n≥n0.
A cone (b- cone) metric space X,C,q is complete whenever every Cauchy sequence in X,C,q converges to an element belonging to X,C,q.
The following is a remarkable and an excitable notice given by Khamsi [14].Remark 2.
IfX,C,q is a cone metric space, where C is normal cone with normal constant M≥1, then the composite function Dx,y=⋅oqx,y=qx,y, D:X×X⟶ℝ+ is a b-metric on X. Indeed, for all x,y,w∈X, we have(1)
Dx,y≥0 and Dx,y=0 if and only if qx,y=0, if and only if qx,y=θ, and if and only if x=y(2)
Dx,y=qx,y=qy,x=Dy,x(3)
Sinceqx,y≤qx,w+qw,y, then qx,y≤Mqx,w+qw,y; hence, Dx,y=qx,y≤Mqx,w+qw,y≤Mqx,w+qw,y=MDx,w+Dw,y
That is, the cone metricX,C,q for normal cone C generates a b-metric space X,D.
Next, we have the following.Definition 3.
(see [10]). Let be an ordered linear space, where ⪯ is an ordered relation induced by some cone C⊂A and Θ:C×C⟶C be a continuous mapping with respect to each variable, and we denote(7)ImΘ=t:t∈Csuch that∃u0,v0∈A,Θu0,v0=t.
Then,Θ is said to be an ordered-action mapping on A if and only if it satisfies the following conditions:(1)
Θθ,θ=θ and Θu,v=Θv,u, for every u,v∈C(2)
Θu,v<Θw,tif eitheru<wandv⪯toru⪯wandv<t(3)
For everyu∈ImΘ and every θ⪯v⪯u, there is θ⪯w⪯u such that Θv,w=u(4)
Θu,θ⪯u, for every u∈C/θ
In addition, the concept ofΘ-cone-metric space is as follows.Definition 4.
(see [10]). Let A,⪯ be an ordered normed space, where ⪯ is an ordered relation induced by some cone C⊂A and Θ be an ordered-action mapping on A. If X is a nonempty set, then a function dΘ:X×X⟶C is said to be Θ-cone-metric on X if and only if dΘ satisfies the following conditions:(1)
dΘx,y=θ⇔x=y(2)
dΘx,y=dΘy,x,∀x,y∈X(3)
dΘx,y⪯rΘdΘx,z,dΘz,y,∀x,y,z∈X, where r is real number such that r≥1
The tripleX,C,dΘ is defined to be a b-Θ-cone-metric space.Remark 3.
The constantr does not depend on the cone C in general whether it is normal cone or not.
Ifr=1, then X,C,dΘ is Θ-cone-metric space, meaning that the class of all Θ-cone-metric spaces is included in the class of all b-Θ-cone-metric spaces.
IfΘu,v=u+v, then X,C,dΘ is b- cone-metric space, meaning that the class of all cone metric spaces is included in the class of all theta cone metric spaces.
A sequenceunn∈ℕ in X,C,dΘ converges to u whenever, for each z∈Imθ with θ<z, there is n0∈ℕ such that dΘun,u<z for all n≥n0. We instead write un⟶n⟶∞dΘu.
A sequenceunn∈ℕ in X,C,dΘ is Cauchy whenever, for each z∈Imθ with θ<z, there is n0∈ℕ such that dΘun,um<z for all n,m≥n0.
AΘ cone (b- cone) metric space X,C,dΘ is complete whenever every Cauchy sequence in X,C,dΘ converges to an element belonging to X,C,dΘ.
On the other side, we have the following.Definition 5.
An elementx,y∈X×X is said to be a coupled fixed point of the mapping F:X×X⟶X if and only if Fx,y=x and Fy,x=y.
In this paper, we consider the corresponding definition for contraction type of mappings on completeb-cone metric spaces and generalize the coupled fixed point theorem of Sabetghadam et al. (Theorem 1) in this setting. On the other side, we consider the concept of b-theta-cone metrics and also prove the existence of unique coupled fixed point of some contraction type of mappings that gives another generalization of some previous coupled fixed point theorems.
Since the class of allb-cone metric spaces is including the class of cone metric spaces as supported with the example below, the results of this paper are real generalizations of some previous results. Moreover and in particular, some results of [15] can be extended to the case of theta cone metric spaces and do not affect the validity of this paper.


## 3. Main Results

Letp be any real number, p>0, and Cp≔λnn∈ℕ:λnn∈ℕ∈lp,λn≥0∀n∈ℕ. Then, Cp is a normal cone in the Banach space lp with normal constant M=1, where λnn∈ℕp=∑n∈ℕλnpp for every λnn∈ℕ∈lp, if λnn∈ℕ⪯μnn∈ℕ, then μn−λnn∈ℕ∈Cp, meaning that μn−λn≥0, λn≤μn for every n∈ℕ and therefore ∑n∈ℕλnpp≤∑n∈ℕμnpp, equivalently, λnn∈ℕp≤μnn∈ℕp.The class of cone metric spaces is larger than the class of metric spaces. Indeed, we have the following.From any metric spaceX,d, we define infinitely many cone metric spaces in such a way that if X,C,d∗ is one of these cone metric spaces, then we have(8)du,v=d∗u,v,∀u,v∈X.Indeed, letX,d be a metric space, p be a given real number, and λ=λnn∈ℕ∈lp. Define(9)d∗u,v=du,vλnλpn∈ℕ,∀u,v∈X.Clearly,d∗:X×X⟶Cp is cone metric on X and du,v=d∗u,vp for every u,v∈X.Specifically and in particular, the following is cone metric onX and du,v=d∗u,vp for every u,v∈X:(10)d∗u,v=du,v2n/pn∈ℕ,∀u,v∈X.Every cone metric space isb-cone metric space with r=1 and the converse is not true, meaning that the class of all b-cone metric spaces is larger than the class of all cone metric spaces. Indeed, we have the following.LetMmlp be the Banach space of all m×m matrices, and the entries of each matrix in Mmlp are elements of lp, the usual linear structure of addition and scalar multiplication:(11)A=aij1≤i,j≤m∈Mmlp⟹aij=anijn∈ℕ∈lp,∀1≤i,j≤m,A∞=aij1≤i,j≤m∞=maxi=1m∑j=1maijp.LetC∗≔aij1≤i,j≤m:aij∈Cp,∀1≤i,j≤m. Then, C∗ is a normal cone in A=Mmlp with normal constant M=1.Let us now give an example ofb-cone metric space which is not cone metric space.LetX=Mmlp, using the fact that the cone is normal with normal constant M=1, and Khamsi notices that every cone metric p on X generates a metric Dx,y=px,y on X, D:X×X⟶ℝ+.Differently, we have the following. Letp be a real number, 0<p<1, and q, q:X×X⟶C∗ be defined by(12)qA,B=qaij1≤i,j≤m,bij1≤i,j≤m≔aij−bijpδjkk∈ℕ1≤i,j≤m.Then,X,C∗,q is b-cone metric space with r=21/p which is not a cone metric space. Indeed, let A,B,C∈X with A=aij1≤i,j≤m, B=bij1≤i,j≤m, and C=cij1≤i,j≤m, and we have(13)qA,B=qaij1≤i,j≤m,bij1≤i,j≤m=aij−bijpδjkk∈ℕ1≤i,j≤m=aij−cij+cij−bijpδjkk∈ℕ1≤i,j≤m≺¯21/paij−cijp+cij−bijpδjkk∈ℕ1≤i,j≤m=21/paij−cijpδjkk∈ℕ+cij−bijpδjkk∈ℕ1≤i,j≤m=21/paij−cijpδjkk∈ℕ1≤i,j≤m+cij−bijpδjkk∈ℕ1≤i,j≤m=21/pqA,C+qC,B.Here, the constantr=21/p is completely different from the normal constant of the cone which is M=1 in addition that the range of q is a subset of the given cone not subset of positive real numbers ℝ+. That is why the results of this paper do not interfere, but it provides generalized results of [14].We have the following results, where cones are not necessarily normal but solid.Theorem 2.
LetX,C,q be a complete b- cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition:(14)qTx,y,Tu,v⪯kqx,u+lqy,v,∀x,y,u,v∈X,where kandl are nonnegative constants with rk+l<1. Then, T has a unique coupled fixed point.Proof.
Selectu0, v0 in X and set u1=Tu0,v0,v1=Tv0,u0,…,un+1=Tun,vn,vn+1=Tvn,un. Then, by (14), we have(15)qun+1,un=qTun,vn,Tun−1,vn−1⪯kqun,un−1+lqvn,vn−1.
Similarly,(16)qvn+1,vn=qTvn,un,Tvn−1,un−1⪯kqvn,vn−1+lqun,un−1.
Letzn=qvn,vn−1+qun,un−1. Then, we have(17)zn+1=qun+1,un+qvn+1,vn⪯kqun,un−1+lqvn,vn−1+kqvn,vn−1+lqun,un−1=k+lqun,un−1+qvn,vn−1=k+lzn.
For eachn∈ℕ, we have(18)θ⪯zn⪯k+lzn−1⪯k+l2zn−2⪯,…,⪯k+lnz0.
Ifz0=θ, then x0,y0 is coupled fixed point of T; therefore, we continue by letting θ<z0. Now, let n,m∈ℕ, n≤m. Then,(19)qun,um⪯rqun,un+1+qun+1,um⪯rqun,un+1+rqun+1,un+2+qun+2,um⪯rqun,un+1+rqun+1,un+2+rqun+2,un+3+qun+3,um=rqun,un+1+r2qun+1,un+2+r3qun+2,un+3+⋯⋯+rm−nqum−1,um.
Similarly,(20)qvn,vm⪯rqvn,vn+1+r2qvn+1,vn+2+r3qvn+2,vn+3+⋯⋯+rm−nqvm−1,vm.
Adding, we obtain(21)qun,um+qvn,vm⪯rqun,un+1+qvn,vn+1++r2qun+1,un+2+qvn+1,vn+2++r3qun+2,un+3+qvn+2,vn+3+⋯+rm−nqum−1,um+qvm−1,vm.
This shows that(22)qun,um+qvn,vm⪯rzn+1+r2zn+2+r3zn+3+⋯+rm−nzm.
Using the two inequalities (18) and (22), we have(23)qun,um+qvn,vm⪯rk+ln+1z0+r2k+ln+2z0+⋯+rm−nk+ln+m−nz0=k+lnrk+l+r2k+l2+r3k+l3+⋯+rm−nk+lm−nz0=k+lnrk+l+rk+l2+rk+l3+⋯+rk+lm−nz0⪯k+lnrk+l1−rk+lz0.
Letθ≪z, that is, z∈IntC. Then, there is a neighborhood of z with some radius δ>0 say z+Nδθ such that z+Nδθ⊂C, and for this δ>0, there is a natural number n0 such that k+ln<δ1−rk+l/z0rk+l for every n≥n0; hence, z0rk+l/1−rk+lk+ln<δ for every n≥n0; consequently, ‖rk+l/1−rk+l‖k+lnz0<δ for every n≥n0. This proves that ±rk+l/1−rk+lk+lnz0∈Nδθ for every n≥n0. Therefore, z±rk+l/1−rk+lk+lnz0∈z+Nδθ⊂C; hence, z±rk+l/1−rk+lk+lnz0∈IntC for every n≥n0:(24)rk+l1−rk+lk+lnz0≪z,∀n≥n0.
Using (24), there is a sequence of neighborhoods Nδnθn≥n0 such that z−rk+l/1−rk+lk+lnz0+Nδnθ⊂C, using (23) gives(25)k+lnrk+l1−rk+lz0−qun,um+qvn,vm+z−rk+l1−rk+lk+lnz0+Nδnθ⊂C+C⊂C,∀n≥n0.
This means(26)z−qun,um+qvn,vm+Nδnθ⊂C,∀n≥n0.
Consequently,z−qun,um+qvn,vm∈IntC for every n≥n0, and we have(27)qun,um+qvn,vm≪z,∀n≥n0.
Sincen≤m, qun,um⪯qun,um+qvn,vm, and qvn,vm⪯qun,um+qvn,vm, we similarly conclude that(28)qun,um≪z,∀n,m≥n0,(29)qvn,vm≪z,∀n,m≥n0.
The two inequalities (28) and (29) prove that the two sequences unn∈ℕ and vnn∈ℕ are Cauchy sequences in X,C,q. Since X,C,q is complete b- cone metric space, there are two limits. Let f≥1, we have θ≪1/rk+l+1fz. Then,(30)∀θ≪z∃n1∈ℕsuch thatqun,w0≪1rk+l+1fz,∀n≥n1,∀θ≪z∃n2∈ℕsuch thatqvn,z0≪1rk+l+1fz,∀n≥n2.
TakeN0=maxn1,n2, and we have(31)qTw0,z0,w0⪯rqTw0,z0,uN0+1+quN0+1,w0=rqTw0,z0,TuN0,vN0+quN0+1,w0=rkqw0,uN0+lqz0,vN0+rquN0+1,w0=rk1rk+l+1fz+l1rk+l+1fz+r1rk+l+1fz=rk+l+11rk+l+1fz=1fz.
Sincef≥1 is an arbitrary number, we have qTw0,z0,w0=θ, and hence, Tw0,z0=w0, and similarly, Tz0,w0=z0, meaning that w0,z0 is a coupled fixed point of T. Finally, we show that such a coupled fixed point is unique. Contrarily, suppose that u1,v1 is another coupled fixed point. Then, we have(32)qw0,u1=qTw0,z0,Tu1,v1⪯kqw0,u1+lqz0,v1,qz0,v1=qTz0,w0,Tv1,u1⪯kqz0,v1+lqw0,u1.
Adding, we obtain(33)qw0,u1+qz0,v1⪯k+lqw0,u1+qz0,v1.
Sincek+l<1, we see that qw0,u1+qz0,v1=θ; hence, qw0,u1=θ and qz0,v1=θ meaning that w0,z0=u1,v1 and the proof of the theorem is completed.
The following corollary proves Theorem1 of Sabetghadam et al.Corollary 1.
LetX,C,q be a complete cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(34)qTx,y,Tu,v⪯kqx,u+lqy,v,where kandl are nonnegative constants with k+l<1. Then, T has a unique coupled fixed point.Proof.
Using Theorem2 with r=1 completes the proof.
We also have the following.Corollary 2.
LetX,C,q be a complete cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(35)qTx,y,Tu,v⪯tqx,u+qy,v,where t is a nonnegative constant with t<1/2. Then, T has a unique coupled fixed point.Proof.
Using Corollary1 with l=k<1/2, we obtain the following.Corollary 3.
LetX,C,q be a complete b- cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(36)qTx,y,Tu,v⪯tqx,u+qy,v,where t is nonnegative constant with rt<1/2. Then, T has a unique coupled fixed point.Proof.
Using Theorem2 with k=l completes the proof.Theorem 3.
LetX,C,q be a complete b-cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(37)qTx,y,Tu,v⪯aqx,u+bqTx,y,x+cqTu,v,u,∀x,y,u,v∈X,where a,b,c∈ℝ are nonnegative constants with ra+b+c<1. Then, T has a unique coupled fixed point.Proof.
Construct the two sequencesunn∈ℕ and vnn∈ℕ as in Theorem 2. Then, by (37), we have(38)qun+1,un=qTun,vn,Tun−1,vn−1⪯aqun,un−1+bqTun,vn,un+cqTun−1,vn−1,un−1⪯aqun,un−1+bqun+1,un+cqun,un−1⪯a+cqun,un−1+bqun+1,un.
Consequently,(39)qun+1,un⪯a+c1−bqun,un−1.
Similarly,(40)qvn+1,vn⪯a+c1−bqvn,vn−1.
We havera+c/1−b<1; then, the two sequences unn∈ℕ and vnn∈ℕ are Cauchy sequences, and the rest of the proof is similar to the proof of Theorem 2.Corollary 4.
LetX,C,q be a complete b- cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(41)qTx,y,Tu,v⪯tqTx,y,x+qTu,v,u,∀x,y,u,v∈X,where t∈ℝ is nonnegative constants with rt<1/2. Then, T has a unique coupled fixed point.
The following is a coupled fixed point theorem of contraction type of mappings inb-Θ-cone metric spaces.Theorem 4.
LetX,C,dΘ be a complete b-Θ-cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(42)dΘTx,y,Tu,v⪯kdΘx,u+ldΘy,v,where kandl are nonnegative constants with k+l<1. Then, T has a unique coupled fixed point.Proof.
Construct the two sequencesunn∈ℕ and vnn∈ℕ in X and the sequence znn∈ℕ in C as in Theorem 2. Then, by (42), we have(43)dΘun+1,un⪯kdΘun,un−1+ldΘvn,vn−1≺¯k+ldΘun,un−1+dΘvn,vn−1,dΘvn+1,vn⪯kdΘvn,vn−1+ldΘun,un−1⪯k+ldΘun,un−1+dΘvn,vn−1,θ⪯zn⪯k+lzn−1⪯k+l2zn−2⪯⋯⪯k+lnz0.
The last inequalities give(44)dΘun+1,un⪯k+lzn⪯k+ln+1z0,(45)dΘvn+1,vn⪯k+lzn⪯k+ln+1z0.
Ifz0=θ, then x0,y0 is coupled fixed of T; therefore, we continue by letting θ<z0. Now, let θ≪z, z∈IntC. Then, there is a neighborhood of z with some radius δ>0 say z+Nδθ such that z+Nδθ⊂C, and for this δ>0, there is a natural number n0 such that k+ln+1<δ1/z0 for every n≥n0; hence, z0k+ln+1<δ for every n≥n0; consequently, k+ln+1z0<δ for every n≥n0. This proves that ±k+ln+1z0∈Nδθ for every n≥n0. Therefore,(46)k+ln+1z0≪z,∀n≥n0.
Using (46) with both (44) and (45), we have(47)dΘun+1,un≪z,∀n≥n0,(48)dΘvn+1,vn≪z,∀n≥n0.
Now, we are going to show that the two sequencesunn∈ℕ and vnn∈ℕ are Cauchy sequences; for this reason, suppose that one of them is not Cauchy, say unn∈ℕ. Then, there exists z¯∈ImΘ, θ<z¯ and sequences of natural numbers inn∈N and jnn∈N such that for any in>jn>n,(49)z¯⪯dΘuin,ujn,dΘuin−1,ujn<z¯.
Letf∈ℝ+ be such that rΘfz¯,z¯<z¯; using (47), for such z¯, there is n0∈ℕ such that dΘun+1,un≪fz¯ for every n≥n0; in particular, we have dΘuin,uin−1≪fz¯ for every in≥n0; hence, we have the following contradiction:(50)z¯⪯dΘuin,ujn⪯rΘdΘuin,uin−1,dΘuin−1,ujn<rΘdΘuin,uin−1,z¯⪯rΘfz¯,z¯<z¯.
Hence, the two sequences are Cauchy sequences. SinceX,C,dΘ is complete, there are two limits for both of them say u0 and v0, respectively:(51)un⟶n⟶∞dΘu0,vn⟶n⟶∞dΘv0.
We show thatu0,v0 is coupled fixed point of T. Using the properties of Θ, we see that(52)dΘTu0,v0,u0⪯rΘdΘTu0,v0,un,dΘun,u0⪯rΘdΘTu0,v0,Tun−1,vn−1,dΘun,u0⪯rΘkdΘu0,un−1+ldΘv0,vn−1,dΘun,u0⟶n⟶∞rΘkθ+lθ,θ=rΘθ,θ=rθ=θ.
Consequently,dΘTu0,v0,u0=θ, meaning that Tu0,v0=u0; similarly, Tv0,u0=v0. The uniqueness of such a coupled fixed point is as before; clearly, the coupled fixed point is unique.Corollary 5.
LetX,C,dΘ be a complete theta cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(53)dΘTx,y,Tu,v⪯kdΘx,u+ldΘy,v,where kandl are nonnegative constants with k+l<1. Then, T has a unique coupled fixed point.
As in Theorem3, we have the following.Corollary 6.
LetX,C,dΘ be a complete Θ-cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(54)dΘTx,y,Tu,v⪯adΘx,u+bdΘTx,y,x+cdΘTu,v,u,∀x,y,u,v∈X,where a,b,c∈ℝ are nonnegative constants with a+b+c<1. Then, T has a unique coupled fixed point.Corollary 7.
LetX,C,dΘ be a complete b- Θ-cone metric space. Suppose that the mapping T:X×X⟶X satisfies the following contractive condition for all x,y,u,v∈X:(55)qTx,y,Tu,v⪯tdΘTx,y,x+dΘTu,v,u,∀x,y,u,v∈X,where t∈ℝ is nonnegative constants with t<1/2. Then, T has a unique coupled fixed point.


## 4. Conclusion

The results of this paper prove the existence of a unique coupled fixed point of some well-known type contraction of mappings but are defined on completeb-cone and b-theta cone metric spaces; consequently, it extends and generalizes some previous coupled fixed point theorems.



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*Source: 5569674-2021-05-03.xml*