Abstract

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

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 of cone metric space showing that cone metric spaces provide larger categories of spaces for the fixed point theory. Recall that if is a normed space, is a cone in that generates the partial ordered relation , and is a cone metric space over , then a mapping is said to be contraction on if and only if there is a constant such that

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 - 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 - 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 be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all :where are nonnegative constants with . Then, has a unique coupled fixed point.
In 2010, Khamsi [14] gave the definition of metric type space (or -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 of the cone metric space generates a metric type and hence a metric type space . His remarkable notice enabled him to prove the existence of unique fixed point for Lipschitzian type mapping defined on complete cone metric but provided that the cone is normal.
In 2013, Luong et al. [15] followed another direction; they proved some coincidence and coupled fixed point of two compatible mappings: and , is continuous, has the mixed -monotone property, and satisfy some generalized contraction conditions, is partially ordered set endowed with a complete cone metric and some extra conditions on convergent monotone nondecreasing and convergent monotone nonincreasing sequences in .
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 be a nonempty set. Let be a function which satisfies the following:(1),if and only if(2)for every(3)for some positive real numberThe pair is called a metric type space, and it is called -metric space if .
We recall some standard notations and definitions in cone metric spaces.
A subset of a linear space is said to be a cone in if and only if(1) is nonempty closed and , where is the zero (neutral element) of (2) for all nonnegative real numbers (3)A cone in a normed space is said to be solid if and only if it has a nonempty interior, that is, the set of all interior points of is not empty set, .
If is a normed space, is a cone in ; then, generates the following ordered relations:A sequence in is bounded above by if and only ifand its bounded below by if and only ifA cone is called normal if there is a number (later proved to be greater than or equal 1) such that, for all ,The normal constant of the normal cone is defined to be the smallest constant satisfying (6).
A cone is called regular if every monotonically nonincreasing (no-decreasing) bounded above (bounded below) sequence has a limit in the norm sense of .

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 of any normal cone is such that , and for any real number , , there is a cone with normal constant .

Definition 2. Suppose that is a nonempty set, is a cone in a normed space , , (this does not depend on the cone itself, the cone may not be normal), and is a function; satisfies the following:(1)(2)(3)(4)Then, is defined to be a -cone metric space over . In particular, if , then is cone metric space.
A sequence in a cone (- cone) metric space (with solid) is Cauchy if and only if for every with there is such that for all .
A sequence in a cone (- cone) metric space (with solid) is convergent sequence if and only if there is such that, for every with , there is such that for all .
A cone (- cone) metric space is complete whenever every Cauchy sequence in converges to an element belonging to .
The following is a remarkable and an excitable notice given by Khamsi [14].

Remark 2. If is a cone metric space, where is normal cone with normal constant , then the composite function , is a -metric on . Indeed, for all , we have(1) and if and only if , if and only if , and if and only if (2)(3)Since , then ; hence, That is, the cone metric for normal cone generates a -metric space .
Next, we have the following.

Definition 3. (see [10]). Let be an ordered linear space, where is an ordered relation induced by some cone and be a continuous mapping with respect to each variable, and we denoteThen, is said to be an ordered-action mapping on if and only if it satisfies the following conditions:(1) and , for every (2)(3)For every and every , there is such that (4), for every In addition, the concept of -cone-metric space is as follows.

Definition 4. (see [10]). Let be an ordered normed space, where is an ordered relation induced by some cone and be an ordered-action mapping on . If is a nonempty set, then a function is said to be -cone-metric on if and only if satisfies the following conditions:(1)(2)(3), where is real number such that The triple is defined to be a --cone-metric space.

Remark 3. The constant does not depend on the cone in general whether it is normal cone or not.
If , then is -cone-metric space, meaning that the class of all -cone-metric spaces is included in the class of all --cone-metric spaces.
If , then is - cone-metric space, meaning that the class of all cone metric spaces is included in the class of all theta cone metric spaces.
A sequence in converges to whenever, for each with , there is such that for all . We instead write .
A sequence in is Cauchy whenever, for each with , there is such that for all .
A cone (- cone) metric space is complete whenever every Cauchy sequence in converges to an element belonging to .
On the other side, we have the following.

Definition 5. An element is said to be a coupled fixed point of the mapping if and only if and .
In this paper, we consider the corresponding definition for contraction type of mappings on complete -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 -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 all -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

Let be any real number, , and . Then, is a normal cone in the Banach space with normal constant , where for every , if , then , meaning that , for every and therefore , equivalently, .

The class of cone metric spaces is larger than the class of metric spaces. Indeed, we have the following.

From any metric space , we define infinitely many cone metric spaces in such a way that if is one of these cone metric spaces, then we have

Indeed, let be a metric space, be a given real number, and . Define

Clearly, is cone metric on and for every .

Specifically and in particular, the following is cone metric on and for every :

Every cone metric space is -cone metric space with and the converse is not true, meaning that the class of all -cone metric spaces is larger than the class of all cone metric spaces. Indeed, we have the following.

Let be the Banach space of all matrices, and the entries of each matrix in are elements of , the usual linear structure of addition and scalar multiplication:

Let . Then, is a normal cone in with normal constant .

Let us now give an example of -cone metric space which is not cone metric space.

Let , using the fact that the cone is normal with normal constant , and Khamsi notices that every cone metric on generates a metric on , .

Differently, we have the following. Let be a real number, , and , be defined by

Then, is -cone metric space with which is not a cone metric space. Indeed, let with , , and , and we have

Here, the constant is completely different from the normal constant of the cone which is in addition that the range of is a subset of the given cone not subset of positive real numbers . 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. Let be a complete - cone metric space. Suppose that the mapping satisfies the following contractive condition:where are nonnegative constants with . Then, has a unique coupled fixed point.

Proof. Select , in and set . Then, by (14), we haveSimilarly,Let . Then, we haveFor each , we haveIf , then is coupled fixed point of ; therefore, we continue by letting . Now, let , . Then,Similarly,Adding, we obtainThis shows thatUsing the two inequalities (18) and (22), we haveLet , that is, . Then, there is a neighborhood of with some radius say such that , and for this , there is a natural number such that for every ; hence, for every ; consequently, for every . This proves that for every . Therefore, ; hence, for every :Using (24), there is a sequence of neighborhoods such that , using (23) givesThis meansConsequently, for every , and we haveSince , , and , we similarly conclude thatThe two inequalities (28) and (29) prove that the two sequences and are Cauchy sequences in . Since is complete - cone metric space, there are two limits. Let , we have . Then,Take , and we haveSince is an arbitrary number, we have , and hence, , and similarly, , meaning that is a coupled fixed point of . Finally, we show that such a coupled fixed point is unique. Contrarily, suppose that is another coupled fixed point. Then, we haveAdding, we obtainSince , we see that ; hence, and meaning that and the proof of the theorem is completed.
The following corollary proves Theorem 1 of Sabetghadam et al.