Theta Cone Metric Spaces and Some Fixed Point Theorems

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.

1. Introduction and Preliminaries

In 2007, Huang and Zhang [1] introduced cone metric spaces as a generalization of metric spaces. Let be a real Banach space, . is called a cone in if(1) is nonempty, closed, and , where is zero (neutral element) of (2) for all nonnegative real numbers (3)

If is the set of all interior points of , then a cone in a normed space induces the following ordered relations [1, 2]:

A sequence in is bounded above by iff

The cone is called normal if there is a number such that

The least positive number satisfying the above is called the normal constant of .

Huang and Zhang [1] supposed that is a Banach space, is a cone in with nonempty interior, and is partial ordering with respect to . If is a nonempty set, the distance of two elements and in the space is defined to be a vector in the cone of the ordered Banach space .(1).(2).(3).(4).

The triple 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 is equivalent to the usual metric space , where the real-valued metric function 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 by Banach algebras , and they proved some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants (the constant is a vector in a normal cone of the Banach algebra, and the essential conditions on the contraction constant 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 -metric as a proper generalization of a metric.

Definition 1. Let be a continuous mapping with respect to both variables. Let . The mapping is called a -action 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 .They functionally formulated the notion of -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 replace by a cone in a normed space and introduce the following analogous generalization of the definition of the function.

Definition 2. Let be an ordered normed space, where is the ordered relation induced by a cone . Let be a continuous mapping with respect to each variable. Let . A mapping is called an ordered action 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 .Because for every , one can write instead for every , [ for every ].

Example 1. Let be the vector space of all finite sequences of real numbers with the usual operations of addition and scalar multiplication, for . Denote and . If , then is the Banach space. If , then is the quasi-normed space.
Let be the vector space of all -square matrices whose entries are elements in :Operations on are defined as follows:The space is a Banach space endowed with the following norms:The zero element of the space , is the matrix θ, and every entry of the matrix θ is the all of its entries are the zero element of , .
Let , Each entry is a vector of the space , and the entries of these vectors are nonnegative real numbers. Then, is a cone in . The cone induces the ordered relation if and only if for every and every , .
Now, let be nonnegative real number such that . Define byThen, the functions , are all ordered actions on .
Furthermore, we replace by a cone in a normed space and use ordered actions to introduce the concept of action function cone metric spaces.

Definition 3. Let be an ordered normed space, where is the ordered relation induced by a cone , be a nonempty set, and be an ordered action on . Then, the function is called -cone-metric on if and only if satisfies the following conditions:(1).(2).(3).The triple is defined to be a -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 of -metric spaces if we consider . Also, we mention that the class of -metric spaces is included in the class of -cone-metric spaces if we take .

Example 2. Let , , and be given as in Example 1 be the space of all matrices whose entries are elements of the space . Then, is a nonempty set. The function , , defined byis a -cone metric on , and is the -cone-metric space. In fact, conditions (1) and (2) are clear, and for any , we haveHence, because

Example 3. Let , , and be given as in Example 1 and be three elements. Then, the function , , is defined byNote that is not a metric on because , but it is a -cone metric on , and is a -cone-metric space.
In 2017, J. Fernandez et al. introduced -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 -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 in -Cone-Metric Spaces

Let us start with the following definition.

Definition 4. Let be a -cone-metric space. An open ball at center with a radius (neighborhood of at radius ) is defined asA subset is bounded if and only if there are and such that . Limit and interior points are defined in the usual way. Additionally, bounded, open, and closed sets in are also defined.

Remark 2. 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 sequence in is bounded if and only if there are and such that . Equivalently, a sequence in is bounded if and only if there are and such that for all .
We have the following lemmas.

Lemma 1. Every neighborhood in is an open subset.

Proof. Let be an arbitrary element and be any neighborhood of with any radius ; we show that, for every , there is some such that . Set , and we have and . Using the definition of , there is , , such that . We claim that is the required neighborhood. In fact, if is an arbitrary element, then . Now, using the definition of and then the definition of , we have the following:Hence, ; this completes the proof.

Lemma 2. A sequence in converges to if and only if converges in .

Proof. Suppose that , and let . Then, choose with , where is the normal constant of ; for this , there is such that for all ; hence, for all , and accordingly, we have for all . This proves that .
Conversely, let ; we show that . Let be arbitrary; since is continuous, there is a neighborhood of with some radius , in and ; for this , there is a natural number such that for every , and since , we see that for every ; hence, , and consequently, , that is, . This proves that .

Lemma 3. The limit of any sequence in is unique.

Proof. Suppose that and ; we show that . Using Lemma 2, we have and , and using the definition of and the continuity of , we haveHence, , that is, .

Lemma 4. Let be a -cone-metric space, , and . Then,

Proof. Using Lemma 4, we have and since and is continuous; then, conditions (2) and (4) of imply the following:and consequently,Using the continuity of once more givesUsing inequalities (19) and (20) gives and completes the proof. □

Lemma 5. A sequence in is Cauchy if and only if

Proof. Suppose that is Cauchy, and let . Then, choose with , where is the normal constant of ; for this , there is such that for all ; hence, for all , and accordingly, we have for all . This proves that .
Conversely, let ; we show that is Cauchy. Let be arbitrary; since is continuous, there is a neighborhood of with some radius , in such that ; for this , there is a natural number such that for every , and since , we see that for every . Hence, , and consequently, , that is, for every . This proves that is Cauchy.

Lemma 6. Every convergent sequence in the -cone-metric space is Cauchy.

Proof. Let in converge to , and let with . Then, the continuity of insures the following:The following lemmas are mainly used in the upcoming generalizations of fixed point theorems.

Lemma 7. Let be a -cone-metric space and be in such that converges to . Then, is a Cauchy sequence.

Proof. Suppose that is not Cauchy; then, there exist , and sequences and of natural numbers such that, for any ,Using the definition and the continuity of and the fact that any subsequence of converges to , we have the following contradiction:

Lemma 8. Let be a -cone-metric space and be in such thatfor some real number , .Then, is Cauchy.

Proof. Taking into account the conditions for every nonnegative real number , , and implyHence,that is,Additionally, givesInclusions (28) and (29) yield the following:Thus,and hence,This implies successivelySince , the sequence is convergent to 0, and taking the limit as of the two sides of (33) shows that as ; hence, . Using Lemma 7 shows that is Cauchy and completes the proof. □

2.2. Fixed Point Theorem for Contraction Types of Mappings

We have the following definition.

Definition 5. A -cone-metric space 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 -cone-metric spaces to generalize the Banach contraction principle as follows.

Theorem 1. Let be a complete -cone-metric space on a normed space and be a mapping that satisfies the following:for some real number , .Then, has a unique fixed point.

Proof. Let be a given element; then, the iterated sequence , , satisfies the following:Using Lemma 8, the sequence of iterates is a Cauchy sequence. According to the completeness of , there exists such that ; we claim that is a fixed point of . In fact, we haveIt means that ; using Lemma 3 proves that . Finally, we prove that is the unique fixed point of . Suppose, on the contrary, that and are two distinct fixed points of ; then, , and on the other side, because , that is, . So, we get the following contradiction: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 -cone-metric space setting, we give the following further generalization.

Theorem 2. Let be a complete -cone-metric space on a normed space and be a mapping that satisfies the following:for some real numbers , .Then, has a unique fixed point.

Proof. Let be a given element. Then, the iterated sequence , , satisfies the following:and consequently, ; hence, , that is, . Therefore, we haveSince , , and using Lemma 7 proves that the sequence of iterates is a Cauchy sequence. According to the completeness of , there exists such that and ; we show that is a fixed point of . In fact, we haveUsing inequalities (41) and (42) proves that ; hence, , and consequently, because , that is, . Finally; we prove that is the unique fixed point of . Suppose, on the contrary, that and are two distinct fixed points of . So, we get the following:Hence, and , but ; this implies that because , that is, .

3. Conclusion

In this paper, we generalize the concept of -metric space to the concept of the -cone-metric space as a generalization of a metric by replacing the triangle inequality with a more generalized inequality using some -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 -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].

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The sole author contributed to the article. The author read and approved the final manuscript.