# Theta Cone Metric Spaces and Some Fixed Point Theorems

Authors: Sahar Mohamed Ali Abou Bakr
Journal: Journal of Mathematics
 (2020)

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## Abstract

This paper introduces the concept of the theta cone metric, studies its various topological properties, and gives some examples of it. Furthermore, it proves some lemmas and then uses them to give further generalizations of some well-known fixed point theorems. Specifically, Theorem 2 of the paper is a generalization of Reich’s fixed point theorem.


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## Body

## 1. Introduction and Preliminaries

In 2007, Huang and Zhang [1] introduced cone metric spaces as a generalization of metric spaces. Let ℰ be a real Banach space, C⊂ℰ. C is called a cone in ℰ if(1)
C is nonempty, closed, and C≠Θ, where Θ is zero (neutral element) of ℰ(2)
αC+βC⊂C for all nonnegative real numbers α,β(3)
C∩−C=ΘIfintC is the set of all interior points of C, then a cone C in a normed space ℰ induces the following ordered relations [1, 2]:(1)u≺¯v⟺v−u∈C,u<v⟺v−u∈Candu≠v,u≠≺¯v⟺v−u∈intC.A sequencewnn∈N in ℰ is bounded above by w∈ℰ iff(2)wn≺¯≠w∀n∈ℕ.The coneC is called normal if there is a number M>0 such that(3)∀u,v∈ℰ,Θ≺¯u≺¯v⟹u≤Mv.The least positive number satisfying the above is called the normal constant ofC.Huang and Zhang [1] supposed that ℰ is a Banach space, C is a cone in ℰ with nonempty interior, and ≺¯ is partial ordering with respect to C. If X is a nonempty set, the distance dx,y of two elements x and y in the space X is defined to be a vector in the cone C of the ordered Banach space ℰ,≺¯.(1)
Θ≺¯≠dx,y∀x,y∈X.(2)
dx,y=Θ≺¯x=y.(3)
dx,y=dy,x∀x,y∈X.(4)
dx,z≺¯dx,y+dy,z∀x,y,z∈X.The tripleX,C,d is known as the cone metric space. They carefully studied convergence and completeness and then proved some fixed point theorems for the contractive type of mappings in this setting.In 2010, Haghi et al. [3] showed that some generalizations in fixed point theory are really consequences of Huang and Zhang results.In contrast, in 2012, Cakally et al. [4] obtained that any cone metric space X,d is equivalent to the usual metric space X,d⋆, where the real-valued metric function d⋆ is defined by a nonlinear scalarization function [5, 6].In 2013, Liu and Xu [2] introduced the concept of cone metric spaces with Banach algebras; they replaced Banach spaces ℰ by Banach algebras A, and they proved some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants (the constant k is a vector in a normal cone of the Banach algebra, and the essential conditions on the contraction constant k are neither order relations nor norm relations but spectrum radius). A well-formulated example shows that their main results concerning the fixed point theorems in the setting of cone metric spaces with Banach algebras are more useful than the standard results in cone metric spaces presented in the literature.In 2013, Khojasteh et al. [7] proposed the notion of the θ-metric as a proper generalization of a metric.Definition 1.
Letθ:0,∞×0,∞⟶0,∞ be a continuous mapping with respect to both variables. Let Imθ=θs,t:s≥0,t≥0. The mapping θ is called a B-action if and only if it satisfies the following conditions:(1)
θ0,0=0 and θu,v=θv,u for every u,v≥0.(2)(4)θu,v<θw,tif eitheru<wandv≤toru≤wandv<t.(3)
For everyu∈Imθ and every v∈0,u, there is w∈0,u such that θv,w=u.(4)
θu,0≤u for every u>0.
They functionally formulated the notion ofθ-metric spaces, and then, they gave the terminology of open and closed sets. Furthermore, they gave a detailed and comprehensive study of convergence and Cauchyness of sequences in this frame of work.
Now, we replace0,∞ by a cone in a normed space and introduce the following analogous generalization of the definition of the θ function.Definition 2.
Letℰ,≺¯ be an ordered normed space, where ≺¯ is the ordered relation induced by a cone C⊂ℰ. Let θ:C×C⟶C be a continuous mapping with respect to each variable. Let Imθ=t:t∈Csuch that∃u0,v0∈ℰ,θu0,v0=t. A mapping θ is called an ordered action on ℰ if and only if it satisfies the following conditions:(1)
θΘ,Θ=Θ and θu,v=θv,u for every u,v∈C.(2)(5)θu,v<θw,tif eitheru<w,v≺¯toru≺¯w,v<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.
Becausex−Θ∈C for every x∈C, one can write instead Θ≺¯x for every x∈C, [Θ<x for every x∈C/Θ].Example 1.
Letℝm be the vector space of all finite m sequences of real numbers with the usual operations of addition and scalar multiplication, for a=akk=1m∈ℝm. Denote ap=∑k=1makpp and a∞=maxk=1mak. If 1≤p, then ℝm,.p is the Banach space. If 0≤p<1, then ℝm,.p is the quasi-normed space.
Letℰ=Mnℝm be the vector space of all n-square matrices whose entries are elements in ℝm:(6)ℰ=u:u=uij1≤i,j≤n,uij=ukijk=1m∈ℝm,∀1≤i,j≤n.
Operations onℰ are defined as follows:(7)αu=αuij1≤i,j≤n=αukijk=1m1≤i,j≤n,u⊕v=uij1≤i,j≤n⊕vij1≤i,j≤n=ukij+vkijk=1m1≤i,j≤n.
The spaceℰ is a Banach space endowed with the following norms:(8)up=∑1≤i,j≤nukijk=1mp,oru∞=max1≤i≤n∑1≤j≤nukijk=1mp.
The zero element of the spaceℰ, is the matrix θ, and every entry of the matrix θ is the all of its entries are the zero element of ℝm, 0=0,0,…,0.
LetC=u:u=ukijk=1m1≤i,j≤n,1≤i,j≤n,andukij≥0for all1≤i,j≤n,and1≤k≤m, Each entry is a vector of the space ℝm, and the entries of these vectors are nonnegative real numbers. Then, C is a cone in ℰ. The cone C induces the ordered relation u=ukijk=1m1≤i,j≤n≺¯v=vkijk=1m1≤i,j≤n if and only if vkij≥ukij for every k∈1,2,…,m and every i,j, 1≤i,j≤n.
Now, letk be nonnegative real number such that 0≤k<1. Define θq:C×C⟶C,q∈1,2,3,4,5 by(9)θ1u,v=u⊕v=ukij+vkijk=1m1≤i,j≤n,θ2u,v=ku⊕v=kukij+vkijk=1m1≤i,j≤n,θ3u,v=kukij+vkij+ukij×vkijk=1m1≤i,j≤n,θ4u,v=ukij+vkij+ukij×vkijk=1m1≤i,j≤n,θ5u,v=ukij+vkij1+ukij×vkijk=1m1≤i,j≤n.
Then, the functionsθq,q=1,2,…,5, are all ordered actions on ℰ.
Furthermore, we replace0,∞ by a cone in a normed space and use ordered actions to introduce the concept of action function cone metric spaces.Definition 3.
Letℰ,≺¯ be an ordered normed space, where ≺¯ is the ordered relation induced by a cone C⊂ℰ, X be a nonempty set, and θ be an ordered action on ℰ. Then, the function dθ:X×X⟶C is called 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≺¯θdθx,z,dθz,y∀x,y,z∈X.
The tripleX,C,dθ is defined to be a θ-cone-metric space or equivalently action function cone metric space.Remark 1.
We mention that the class of metric spaces is included in the class ofθ-metric spaces if we consider θu,v=u+v,u,v∈0,∞. Also, we mention that the class of θ-metric spaces is included in the class of θ-cone-metric spaces if we take ℰ,≺¯=0,∞,≤.Example 2.
Letℰ, C, and θ1 be given as in Example 1X=Mnℝm be the space of all n×n matrices whose entries are elements of the space ℝm . Then, X is a nonempty set. The function dθ1, dθ1:X×X⟶C, defined by(10)dθ1x,y=xij−yijpδjkk=1m1≤i,j≤n=∑l=1kxlij−ylijpδjkpk=1m1≤i,j≤nis a θ1-cone metric on X, and X,C,dθ1 is the θ1-cone-metric space. In fact, conditions (1) and (2) are clear, and for any x,y,z∈X, we have(11)dθ1x,z+dθ1z,y=xij−zijpδjkk=1m1≤i,j≤n+zij−yijpδjkk=1m1≤i,j≤n=xij−zijpδjkk=1m+zij−yijpδjkk=1m1≤i,j≤n=xij−zijp+zij−yijpδjkk=1m1≤i,j≤n=∑l=1kxlij−zlijp+∑l=1kzlij−ylijpppδjkk=1m1≤i,j≤n.
Hence,dθ1x,y≺¯θ1dθ1x,z,dθ1z,y∀x,y,x∈X because(12)∑l=1kxlij−ylijpp≤∑l=1kxlij−zlijpp+∑l=1kzlij−ylijpp,∀1≤i,j≤n,1≤k≤m.Example 3.
Letℰ, C, and θ4 be given as in Example 1 and X=x,y,z be three elements. Then, the function dθ4, dθ4:X×X⟶C, is defined by(13)dθ4x,x=dθ4y,y=dθ4z,z=Θ,dθ4x,y=dθ4y,x=uij1≤i,j≤n,whereu11=2,0,0,…,uij=0,0,0,…,0∀1<i,j≤n,dθ4y,z=dθ4z,y=vij1≤i,j≤n,wherev11=3,0,0,…,vij=0,0,0,…,0∀1<i,j≤n,dθ4x,z=dθ4z,x=wij1≤i,j≤n,wherew11=6,0,0,…,wij=0,0,0,…,0∀1<i,j≤n.
Note thatdθ4 is not a metric on X because dθ4x,z>dθ4x,y+dθ4y,z, but it is a θ4-cone metric on X, and X,C,dθ4 is a θ4-cone-metric space.
In 2017, J. Fernandez et al. introducedF-cone-metric spaces over Banach algebra and gave some generalization of some previous fixed point theorems.
In 2017, Suzuki [8] introduced different generalized approaches for the strongest sequentially compatible topology on a ν-generalized metric space and studied its characterizations.
In 2019, Abou Bakr [9] gave some common fixed point theorems (and, in particular, fixed points) of the generalized contraction type of cyclic mappings defined on cone metric spaces on Banach algebra.
In 2020, Abou Bakr [10] gave a study on the common fixed point of joint generalized types of contraction mappings in quasi-metric spaces.


## 2. Main Results

We have the following two sections.


### 2.1. Convergence and Cauchyness inθ-Cone-Metric Spaces

Let us start with the following definition.Definition 4.
LetX,C,dθ be a θ-cone-metric space. An open ball Ndθx,u at center x∈X with a radius u∈Imθ (neighborhood of x at radius u) is defined as(14)Ndθx,u≕y:y∈X,dθx,y<u.
A subsetY⊂X is bounded if and only if there are x∈X and u∈Imθ such that Y⊂Ndθx,u. Limit and interior points are defined in the usual way. Additionally, bounded, open, and closed sets in X are also defined.Remark 2.
A sequencexnn∈ℕ in X,C,dθ converges to x whenever for each u∈Imθ with Θ<u, there is n0∈ℕ such that dθxn,x<u for all n≥n0. We instead write xn⟶n⟶∞dθx.
A sequencexnn∈ℕ in X,C,dθ is Cauchy whenever for each u∈C with Θ<u, there is n0∈ℕ such that dθxn,xm<u for all n,m≥n0.
A sequencexnn∈ℕ in X,C,dθ is bounded if and only if there are x∈X and u∈Imθ such that xnn∈ℕ⊂Ndθx,u. Equivalently, a sequence xnn∈ℕ in X,C,dθ is bounded if and only if there are x∈X and u∈Imθ such that dθxn,x<u for all n∈ℕ.
We have the following lemmas.Lemma 1.
Every neighborhood inX,C,dθ is an open subset.Proof.
Letx∈X be an arbitrary element and Ndθx,u be any neighborhood of x with any radius u∈C; we show that, for every y∈Ndθx,u, there is some w∈Imθ such that Ndθy,w⊂Ndθx,u. Set v=dθx,y, and we have u∈Imθ and Θ≺¯v≺¯u. Using the definition of θ, there is w, Θ≺¯w≺¯u, such that θv,w=u. We claim that Ndθy,w is the required neighborhood. In fact, if z∈Ndθy,w is an arbitrary element, then dθy,z<w. Now, using the definition of dθ and then the definition of θ, we have the following:(15)dθx,z≺¯θdθx,y,dθy,z<θv,w=u.
Hence,z∈Ndθx,u; this completes the proof.Lemma 2.
A sequencexnn∈ℕ in X,C,dθ converges to x if and only if dθxn,x⟶n⟶∞Θ converges in ℰ.Proof.
Suppose thatxn⟶n⟶∞dθx, and let ε>0. Then, choose u∈Imθ,Θ<u with Mu<ε, where M is the normal constant of C; for this u, there is n0∈ℕ such that dθxn,x<u for all n≥n0; hence, dθxn,x<Mu for all n≥n0, and accordingly, we have dθxn,x<ε for all n≥n0. This proves that dθxn,x⟶n⟶∞Θ.
Conversely, letdθxn,x⟶n⟶∞Θdθxn,x⟶n⟶∞0; we show that xn⟶n⟶∞dθx. Let u∈Imθ be arbitrary; since θ is continuous, there is a neighborhood of u with some radius ε>0, Nεu in ℰ and Nεu⊂C; for this ε, there is a natural number n0∈N such that dθxn,x<ε for every n≥n0, and since dθxn,x=u−u−dθxn,x, we see that u−u−dθxn,x<ε for every n≥n0; hence, u−dθxn,x∈Nεu, and consequently, u−dθxn,x∈C, that is, dθxn,x≺¯u. This proves that xn⟶n⟶∞dθx.Lemma 3.
The limit of any sequencexnn∈ℕ in X,C,dθ is unique.Proof.
Suppose thatxn⟶n⟶∞dθx and xn⟶n⟶∞dθy; we show that x=y. Using Lemma 2, we have dθxn,x⟶n⟶∞Θ and dθxn,y⟶n⟶∞Θ, and using the definition of dθ and the continuity of θ, we have(16)Θ≺¯dθx,y≺¯θdθxn,x,dθxn,y⟶n⟶∞θΘ,Θ=Θ.
Hence,dθx,y=Θ, that is, x=y.Lemma 4.
LetX,C,dθ be a θ-cone-metric space, xn⟶n⟶∞dθx, and yn⟶n⟶∞dθy. Then,(17)dθxn,yn⟶n⟶∞dθx,y.Proof.
Using Lemma4, we have dθxn,x⟶n⟶∞Θ and dθyn,y⟶n⟶∞Θ since dθx,yn≺¯θdθx,y,dθy,yn and θ is continuous; then, conditions (2) and (4) of θ imply the following:(18)dθx,yn≺¯θdθx,y,dθy,yn⟶n⟶∞θdθx,y,Θ≺¯dθx,y,and consequently,(19)dθxn,yn≼θdθyn,x,dθx,xnuse the continuity ofθ⟶n⟶∞θdθx,y,Θuse condition4ofθ≼dθx,y,Using the continuity of θ once more gives(20)dθx,y≺¯θdθx,xn,θdθxn,yn,dθyn,y⟶n⟶∞limn⟶∞θdθx,xn,θdθxn,yn,dθyn,y≺¯θlimn⟶∞dθx,xn,limn⟶∞θdθxn,yn,dθyn,y≺¯θlimn⟶∞dθx,xn,θlimn⟶∞dθxn,yn,limn⟶∞dθyn,y≺¯θΘ,θlimn⟶∞dθxn,yn,Θ≺¯θlimn⟶∞dθxn,yn,Θ≺¯limn⟶∞dθxn,yn.
Using inequalities (19) and (20) gives dθxn,yn⟶n⟶∞dθx,y and completes the proof. □Lemma 5.
A sequencexnn∈ℕ in X,C,dθ is Cauchy if and only if(21)dθxn,xm⟶n,m⟶∞Θ.Proof.
Suppose thatxnn∈ℕ is Cauchy, and let ε>0. Then, choose u∈Imθ,Θ<u with Mu<ε, where M is the normal constant of C; for this u, there is n0∈ℕ such that dθxn,xm<u for all n,m≥n0; hence, dθxn,xm<Mu for all n,m≥n0, and accordingly, we have dθxn,xm<ε for all n,m≥n0. This proves that dθxn,xm⟶n,m⟶∞Θ.
Conversely, letdθxn,xm⟶n,m⟶∞Θdθxn,xm⟶n,m⟶∞0; we show that xnn∈ℕ is Cauchy. Let u∈Imθ be arbitrary; since θ is continuous, there is a neighborhood of u with some radius ε>0, Nεu in ℰ such that Nεu⊂C; for this ε, there is a natural number n0∈N such that dθxn,xm<ε for every n,m≥n0, and since dθxn,xm=u−u−dθxn,xm, we see that u−u−dθxn,xm<ε for every n,m≥n0. Hence, u−dθxn,xm∈Nεu, and consequently, u−dθxn,xm∈C, that is, dθxn,xm≺¯u for every n,m≥n0. This proves thatxnn∈ℕ is Cauchy.Lemma 6.
Every convergent sequence in theθ-cone-metric space is Cauchy.Proof.
Letxnn∈ℕ in X,C,dθ converge to x, and let m,n∈ℕ with m≥n. Then, the continuity of θ insures the following:(22)dθxn,xm≺¯θdθx,xn,dθx,xm⟶n,m⟶∞θΘ,Θ=Θ.
The following lemmas are mainly used in the upcoming generalizations of fixed point theorems.Lemma 7.
LetX,C,dθ be a θ-cone-metric space and xnn∈N be in X such that dθxn,xn+1n∈ℕ converges to Θ. Then, xnn∈ℕ is a Cauchy sequence.Proof.
Suppose thatxnn∈ℕ is not Cauchy; then, there exist u∈Imθ,Θ<u, and sequences Jnn∈ℕ and knn∈ℕ of natural numbers such that, for any Jn>kn>n,(23)u≺¯dθxJn,xknanddθxJn−1,xkn<u∀n∈ℕ.
Using the definition and the continuity ofθ and the fact that any subsequence of dθxn,xn+1n∈ℕ converges to Θ, we have the following contradiction:(24)u≺¯dθxJn,xkn≺¯θdθxJn−1,xkn,dθxJn,xjn−1<θu,dθxJn,xjn−1<θu,limn⟶∞dθxJn,xjn−1=θu,Θ<u.Lemma 8.
LetX,C,dθ be a θ-cone-metric space and xnn∈N be in X such that(25)dθxn+2,xn+1≺¯t0dθxn+1,xn,n=0,1,2,…,for some real number t0, 0≤t0<1.Then, xnn∈N is Cauchy.Proof.
Taking into account the conditionsαC⊂C for every nonnegative real number α, C+C⊂C, and dθxn+1,xn≺¯t0dθxn,xn−1 imply(26)t0dθxn,xn−1−dθxn+1,xn∈C.
Hence,(27)t0t0dθxn,xn−1−dθxn+1,xn∈C,that is,(28)t02dθxn,xn−1−t0dθxn+1,xn∈C.
Additionally,dθxn+2,xn+1≺¯t0dθxn+1,xn gives(29)t0dθxn+1,xn−dθxn+2,xn+1∈C.
Inclusions (28) and (29) yield the following:(30)t02dθxn,xn−1−t0dθxn+1,xn+t0dθxn+1,xn−dθxn+2,xn+1∈C.
Thus,(31)t02dθxn,xn−1−dθxn+2,xn+1∈C,and hence,(32)dθxn+2,xn+1≺¯t0dθxn+1,xn≺¯t02dθxn,xn−1.
This implies successively(33)dθxn,xn+1≺¯t0dθxn−1,xn≺¯t02dθxn−2,xn−1⋯≺¯t0ndθx0,x1.
Since0≤t0<1, the sequence t0nn∈N is convergent to 0, and taking the limit as n⟶∞ of the two sides of (33) shows that t0ndθx1,x0⟶Θ as n⟶∞; hence, limn⟶∞dθxn,xn+1=Θ. Using Lemma 7 shows that xnn∈N is Cauchy and completes the proof. □


### 2.2. Fixed Point Theorem for Contraction Types of Mappings

We have the following definition.Definition 5.
Aθ-cone-metric space X,C,dθ is complete whenever every Cauchy sequence converges to an element belonging to it.
Banach’s contraction principle [11] is one of the pivotal results of nonlinear analysis and its applications, which establishes that every contraction mapping defined on a complete metric space has a unique fixed point. In this paper, we use the concept of θ-cone-metric spaces to generalize the Banach contraction principle as follows.Theorem 1.
LetX,C,dθ be a complete θ-cone-metric space on a normed space ℰ and S:X⟶X be a mapping that satisfies the following:(34)dθSx,Sy≺¯t0dθx,y,∀x,y∈X,for some real number t0, 0≤t0<1.Then, S has a unique fixed point.Proof.
Letx0∈X be a given element; then, the iterated sequence Snx0n∈ℕ, xn=Snx, satisfies the following:(35)dθxn+2,xn+1=dθSxn+1,Sxn≺¯t0dθxn+1,xn,n=0,1,2,….
Using Lemma8, the sequence of iterates is a Cauchy sequence. According to the completeness of X, there exists x∈X such that xn⟶n⟶∞dθx; we claim that x is a fixed point of S. In fact, we have(36)dθxn+1,Sx=dθSxn,Sx,n=0,1,2,…≺¯t0dθxn,x⟶n⟶∞Θ.
It means thatxn⟶n⟶∞dθSx; using Lemma 3 proves that Sx=x. Finally, we prove that x is the unique fixed point of S. Suppose, on the contrary, that x and y are two distinct fixed points of S; then, dθx,y≠Θ, and on the other side, 1−t0dθx,y∈C because 1−t0>0, that is, t0dθx,y<dθx,y. So, we get the following contradiction:(37)dθx,y=dθSx,Sy≺¯t0dθx,y<dθx,y.
One of beautiful generalizations of the Banach contraction principle was found by Reich [12] and further generalized by Hardy and Rogers [13]. In this paper, in case of the above introduced θ-cone-metric space setting, we give the following further generalization.Theorem 2.
LetX,C,dθ be a complete θ-cone-metric space on a normed space ℰ and S:X⟶X be a mapping that satisfies the following:(38)dθSx,Sy≺¯adθx,y+bdθx,Sx+cdθy,Sy,∀x,y∈X,for some real numbers a,b,c, 0<a+b+c<1.Then, S has a unique fixed point.Proof.
Letx0∈X be a given element. Then, the iterated sequence Snx0n∈ℕ, xn=Snx, satisfies the following:(39)dθxn+2,xn+1=dθSxn+1,Sxn,n=0,1,2,…≺¯adθxn+1,xn+bdθxn+1,Sxn+1+cdθxn,Sxn≺¯a+cdθxn+1,xn+bdθxn+1,xn+2,and consequently, a+cdθxn+1,xn−1−bdθxn+2,xn+1∈C; hence, 1/1−ba+cdθxn+1,xn−1−bdθxn+2,xn+1∈C, that is, a+c/1−bdθxn+1,xn-dθxn+2,xn+1∈C. Therefore, we have(40)dθxn+2,xn+1≺¯a+c1−bdθxn+1,xn,n=0,1,2,….
Sincea+c/1−b<1, dθxn+1,xn⟶n⟶∞dθΘ, and using Lemma 7 proves that the sequence of iterates is a Cauchy sequence. According to the completeness of X, there exists x∈X such that xn⟶n⟶∞dθx and limn⟶∞dθxn,x=Θ; we show that x is a fixed point of S. In fact, we have(41)dθxn+1,Sx=dθSxn,Sx,n=0,1,2,…≺¯adθxn,x+bdθx,Sx+cdθxn,Sxn≺¯adθxn,x+bdθx,Sx+cdθxn,xn+1⟶n⟶∞bdθx,Sx,(42)dθx,Sx≺¯θdθx,xn,dθxn,Sx≺¯θdθx,xn,θdθxn,xn+1,dθxn+1,Sx⟶n⟶∞θΘ,θΘ,limn⟶∞dθxn+1,Sx≺¯θΘ,limn⟶∞dθxn+1,Sx.≺¯limn⟶∞dθxn+1,Sx.
Using inequalities (41) and (42) proves that dθx,Sx≺¯bdθx,Sx; hence, b−1dθx,Sx∈C, and consequently, dθx,Sx=Θ because C∩−C=Θ, that is, Sx=x. Finally; we prove that x is the unique fixed point of S. Suppose, on the contrary, that x and y are two distinct fixed points of S. So, we get the following:(43)dθx,y=dθSx,Sy≺¯adθx,y+bdθx,Sx+cdθy,Sy=adθx,y.
Hence,adθx,y−dθx,y∈C and a−1dθx,y∈C, but a−1<0; this implies that dθx,y=Θ because C∩−C=Θ, that is, x=y.


## 2.1. Convergence and Cauchyness inθ-Cone-Metric Spaces

Let us start with the following definition.Definition 4.
LetX,C,dθ be a θ-cone-metric space. An open ball Ndθx,u at center x∈X with a radius u∈Imθ (neighborhood of x at radius u) is defined as(14)Ndθx,u≕y:y∈X,dθx,y<u.
A subsetY⊂X is bounded if and only if there are x∈X and u∈Imθ such that Y⊂Ndθx,u. Limit and interior points are defined in the usual way. Additionally, bounded, open, and closed sets in X are also defined.Remark 2.
A sequencexnn∈ℕ in X,C,dθ converges to x whenever for each u∈Imθ with Θ<u, there is n0∈ℕ such that dθxn,x<u for all n≥n0. We instead write xn⟶n⟶∞dθx.
A sequencexnn∈ℕ in X,C,dθ is Cauchy whenever for each u∈C with Θ<u, there is n0∈ℕ such that dθxn,xm<u for all n,m≥n0.
A sequencexnn∈ℕ in X,C,dθ is bounded if and only if there are x∈X and u∈Imθ such that xnn∈ℕ⊂Ndθx,u. Equivalently, a sequence xnn∈ℕ in X,C,dθ is bounded if and only if there are x∈X and u∈Imθ such that dθxn,x<u for all n∈ℕ.
We have the following lemmas.Lemma 1.
Every neighborhood inX,C,dθ is an open subset.Proof.
Letx∈X be an arbitrary element and Ndθx,u be any neighborhood of x with any radius u∈C; we show that, for every y∈Ndθx,u, there is some w∈Imθ such that Ndθy,w⊂Ndθx,u. Set v=dθx,y, and we have u∈Imθ and Θ≺¯v≺¯u. Using the definition of θ, there is w, Θ≺¯w≺¯u, such that θv,w=u. We claim that Ndθy,w is the required neighborhood. In fact, if z∈Ndθy,w is an arbitrary element, then dθy,z<w. Now, using the definition of dθ and then the definition of θ, we have the following:(15)dθx,z≺¯θdθx,y,dθy,z<θv,w=u.
Hence,z∈Ndθx,u; this completes the proof.Lemma 2.
A sequencexnn∈ℕ in X,C,dθ converges to x if and only if dθxn,x⟶n⟶∞Θ converges in ℰ.Proof.
Suppose thatxn⟶n⟶∞dθx, and let ε>0. Then, choose u∈Imθ,Θ<u with Mu<ε, where M is the normal constant of C; for this u, there is n0∈ℕ such that dθxn,x<u for all n≥n0; hence, dθxn,x<Mu for all n≥n0, and accordingly, we have dθxn,x<ε for all n≥n0. This proves that dθxn,x⟶n⟶∞Θ.
Conversely, letdθxn,x⟶n⟶∞Θdθxn,x⟶n⟶∞0; we show that xn⟶n⟶∞dθx. Let u∈Imθ be arbitrary; since θ is continuous, there is a neighborhood of u with some radius ε>0, Nεu in ℰ and Nεu⊂C; for this ε, there is a natural number n0∈N such that dθxn,x<ε for every n≥n0, and since dθxn,x=u−u−dθxn,x, we see that u−u−dθxn,x<ε for every n≥n0; hence, u−dθxn,x∈Nεu, and consequently, u−dθxn,x∈C, that is, dθxn,x≺¯u. This proves that xn⟶n⟶∞dθx.Lemma 3.
The limit of any sequencexnn∈ℕ in X,C,dθ is unique.Proof.
Suppose thatxn⟶n⟶∞dθx and xn⟶n⟶∞dθy; we show that x=y. Using Lemma 2, we have dθxn,x⟶n⟶∞Θ and dθxn,y⟶n⟶∞Θ, and using the definition of dθ and the continuity of θ, we have(16)Θ≺¯dθx,y≺¯θdθxn,x,dθxn,y⟶n⟶∞θΘ,Θ=Θ.
Hence,dθx,y=Θ, that is, x=y.Lemma 4.
LetX,C,dθ be a θ-cone-metric space, xn⟶n⟶∞dθx, and yn⟶n⟶∞dθy. Then,(17)dθxn,yn⟶n⟶∞dθx,y.Proof.
Using Lemma4, we have dθxn,x⟶n⟶∞Θ and dθyn,y⟶n⟶∞Θ since dθx,yn≺¯θdθx,y,dθy,yn and θ is continuous; then, conditions (2) and (4) of θ imply the following:(18)dθx,yn≺¯θdθx,y,dθy,yn⟶n⟶∞θdθx,y,Θ≺¯dθx,y,and consequently,(19)dθxn,yn≼θdθyn,x,dθx,xnuse the continuity ofθ⟶n⟶∞θdθx,y,Θuse condition4ofθ≼dθx,y,Using the continuity of θ once more gives(20)dθx,y≺¯θdθx,xn,θdθxn,yn,dθyn,y⟶n⟶∞limn⟶∞θdθx,xn,θdθxn,yn,dθyn,y≺¯θlimn⟶∞dθx,xn,limn⟶∞θdθxn,yn,dθyn,y≺¯θlimn⟶∞dθx,xn,θlimn⟶∞dθxn,yn,limn⟶∞dθyn,y≺¯θΘ,θlimn⟶∞dθxn,yn,Θ≺¯θlimn⟶∞dθxn,yn,Θ≺¯limn⟶∞dθxn,yn.
Using inequalities (19) and (20) gives dθxn,yn⟶n⟶∞dθx,y and completes the proof. □Lemma 5.
A sequencexnn∈ℕ in X,C,dθ is Cauchy if and only if(21)dθxn,xm⟶n,m⟶∞Θ.Proof.
Suppose thatxnn∈ℕ is Cauchy, and let ε>0. Then, choose u∈Imθ,Θ<u with Mu<ε, where M is the normal constant of C; for this u, there is n0∈ℕ such that dθxn,xm<u for all n,m≥n0; hence, dθxn,xm<Mu for all n,m≥n0, and accordingly, we have dθxn,xm<ε for all n,m≥n0. This proves that dθxn,xm⟶n,m⟶∞Θ.
Conversely, letdθxn,xm⟶n,m⟶∞Θdθxn,xm⟶n,m⟶∞0; we show that xnn∈ℕ is Cauchy. Let u∈Imθ be arbitrary; since θ is continuous, there is a neighborhood of u with some radius ε>0, Nεu in ℰ such that Nεu⊂C; for this ε, there is a natural number n0∈N such that dθxn,xm<ε for every n,m≥n0, and since dθxn,xm=u−u−dθxn,xm, we see that u−u−dθxn,xm<ε for every n,m≥n0. Hence, u−dθxn,xm∈Nεu, and consequently, u−dθxn,xm∈C, that is, dθxn,xm≺¯u for every n,m≥n0. This proves thatxnn∈ℕ is Cauchy.Lemma 6.
Every convergent sequence in theθ-cone-metric space is Cauchy.Proof.
Letxnn∈ℕ in X,C,dθ converge to x, and let m,n∈ℕ with m≥n. Then, the continuity of θ insures the following:(22)dθxn,xm≺¯θdθx,xn,dθx,xm⟶n,m⟶∞θΘ,Θ=Θ.
The following lemmas are mainly used in the upcoming generalizations of fixed point theorems.Lemma 7.
LetX,C,dθ be a θ-cone-metric space and xnn∈N be in X such that dθxn,xn+1n∈ℕ converges to Θ. Then, xnn∈ℕ is a Cauchy sequence.Proof.
Suppose thatxnn∈ℕ is not Cauchy; then, there exist u∈Imθ,Θ<u, and sequences Jnn∈ℕ and knn∈ℕ of natural numbers such that, for any Jn>kn>n,(23)u≺¯dθxJn,xknanddθxJn−1,xkn<u∀n∈ℕ.
Using the definition and the continuity ofθ and the fact that any subsequence of dθxn,xn+1n∈ℕ converges to Θ, we have the following contradiction:(24)u≺¯dθxJn,xkn≺¯θdθxJn−1,xkn,dθxJn,xjn−1<θu,dθxJn,xjn−1<θu,limn⟶∞dθxJn,xjn−1=θu,Θ<u.Lemma 8.
LetX,C,dθ be a θ-cone-metric space and xnn∈N be in X such that(25)dθxn+2,xn+1≺¯t0dθxn+1,xn,n=0,1,2,…,for some real number t0, 0≤t0<1.Then, xnn∈N is Cauchy.Proof.
Taking into account the conditionsαC⊂C for every nonnegative real number α, C+C⊂C, and dθxn+1,xn≺¯t0dθxn,xn−1 imply(26)t0dθxn,xn−1−dθxn+1,xn∈C.
Hence,(27)t0t0dθxn,xn−1−dθxn+1,xn∈C,that is,(28)t02dθxn,xn−1−t0dθxn+1,xn∈C.
Additionally,dθxn+2,xn+1≺¯t0dθxn+1,xn gives(29)t0dθxn+1,xn−dθxn+2,xn+1∈C.
Inclusions (28) and (29) yield the following:(30)t02dθxn,xn−1−t0dθxn+1,xn+t0dθxn+1,xn−dθxn+2,xn+1∈C.
Thus,(31)t02dθxn,xn−1−dθxn+2,xn+1∈C,and hence,(32)dθxn+2,xn+1≺¯t0dθxn+1,xn≺¯t02dθxn,xn−1.
This implies successively(33)dθxn,xn+1≺¯t0dθxn−1,xn≺¯t02dθxn−2,xn−1⋯≺¯t0ndθx0,x1.
Since0≤t0<1, the sequence t0nn∈N is convergent to 0, and taking the limit as n⟶∞ of the two sides of (33) shows that t0ndθx1,x0⟶Θ as n⟶∞; hence, limn⟶∞dθxn,xn+1=Θ. Using Lemma 7 shows that xnn∈N is Cauchy and completes the proof. □


## 2.2. Fixed Point Theorem for Contraction Types of Mappings

We have the following definition.Definition 5.
Aθ-cone-metric space X,C,dθ is complete whenever every Cauchy sequence converges to an element belonging to it.
Banach’s contraction principle [11] is one of the pivotal results of nonlinear analysis and its applications, which establishes that every contraction mapping defined on a complete metric space has a unique fixed point. In this paper, we use the concept of θ-cone-metric spaces to generalize the Banach contraction principle as follows.Theorem 1.
LetX,C,dθ be a complete θ-cone-metric space on a normed space ℰ and S:X⟶X be a mapping that satisfies the following:(34)dθSx,Sy≺¯t0dθx,y,∀x,y∈X,for some real number t0, 0≤t0<1.Then, S has a unique fixed point.Proof.
Letx0∈X be a given element; then, the iterated sequence Snx0n∈ℕ, xn=Snx, satisfies the following:(35)dθxn+2,xn+1=dθSxn+1,Sxn≺¯t0dθxn+1,xn,n=0,1,2,….
Using Lemma8, the sequence of iterates is a Cauchy sequence. According to the completeness of X, there exists x∈X such that xn⟶n⟶∞dθx; we claim that x is a fixed point of S. In fact, we have(36)dθxn+1,Sx=dθSxn,Sx,n=0,1,2,…≺¯t0dθxn,x⟶n⟶∞Θ.
It means thatxn⟶n⟶∞dθSx; using Lemma 3 proves that Sx=x. Finally, we prove that x is the unique fixed point of S. Suppose, on the contrary, that x and y are two distinct fixed points of S; then, dθx,y≠Θ, and on the other side, 1−t0dθx,y∈C because 1−t0>0, that is, t0dθx,y<dθx,y. So, we get the following contradiction:(37)dθx,y=dθSx,Sy≺¯t0dθx,y<dθx,y.
One of beautiful generalizations of the Banach contraction principle was found by Reich [12] and further generalized by Hardy and Rogers [13]. In this paper, in case of the above introduced θ-cone-metric space setting, we give the following further generalization.Theorem 2.
LetX,C,dθ be a complete θ-cone-metric space on a normed space ℰ and S:X⟶X be a mapping that satisfies the following:(38)dθSx,Sy≺¯adθx,y+bdθx,Sx+cdθy,Sy,∀x,y∈X,for some real numbers a,b,c, 0<a+b+c<1.Then, S has a unique fixed point.Proof.
Letx0∈X be a given element. Then, the iterated sequence Snx0n∈ℕ, xn=Snx, satisfies the following:(39)dθxn+2,xn+1=dθSxn+1,Sxn,n=0,1,2,…≺¯adθxn+1,xn+bdθxn+1,Sxn+1+cdθxn,Sxn≺¯a+cdθxn+1,xn+bdθxn+1,xn+2,and consequently, a+cdθxn+1,xn−1−bdθxn+2,xn+1∈C; hence, 1/1−ba+cdθxn+1,xn−1−bdθxn+2,xn+1∈C, that is, a+c/1−bdθxn+1,xn-dθxn+2,xn+1∈C. Therefore, we have(40)dθxn+2,xn+1≺¯a+c1−bdθxn+1,xn,n=0,1,2,….
Sincea+c/1−b<1, dθxn+1,xn⟶n⟶∞dθΘ, and using Lemma 7 proves that the sequence of iterates is a Cauchy sequence. According to the completeness of X, there exists x∈X such that xn⟶n⟶∞dθx and limn⟶∞dθxn,x=Θ; we show that x is a fixed point of S. In fact, we have(41)dθxn+1,Sx=dθSxn,Sx,n=0,1,2,…≺¯adθxn,x+bdθx,Sx+cdθxn,Sxn≺¯adθxn,x+bdθx,Sx+cdθxn,xn+1⟶n⟶∞bdθx,Sx,(42)dθx,Sx≺¯θdθx,xn,dθxn,Sx≺¯θdθx,xn,θdθxn,xn+1,dθxn+1,Sx⟶n⟶∞θΘ,θΘ,limn⟶∞dθxn+1,Sx≺¯θΘ,limn⟶∞dθxn+1,Sx.≺¯limn⟶∞dθxn+1,Sx.
Using inequalities (41) and (42) proves that dθx,Sx≺¯bdθx,Sx; hence, b−1dθx,Sx∈C, and consequently, dθx,Sx=Θ because C∩−C=Θ, that is, Sx=x. Finally; we prove that x is the unique fixed point of S. Suppose, on the contrary, that x and y are two distinct fixed points of S. So, we get the following:(43)dθx,y=dθSx,Sy≺¯adθx,y+bdθx,Sx+cdθy,Sy=adθx,y.
Hence,adθx,y−dθx,y∈C and a−1dθx,y∈C, but a−1<0; this implies that dθx,y=Θ because C∩−C=Θ, that is, x=y.


## 3. Conclusion

In this paper, we generalize the concept ofθ-metric space to the concept of the θ-cone-metric space as a generalization of a metric by replacing the triangle inequality with a more generalized inequality using some θ-action functions and replacing the usual ordered relation of real numbers by ordered relation induced by a cone in a normed space. We investigate the convergence and Cauchyness in such a θ-cone-metric space. Furthermore, we prove some lemmas and use them to give further generalizations of some well-known fixed point theorems in this context [14].



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