Coupled Fixed-Point Theorems in Theta-Cone-Metric Spaces

Abstract

This paper gives further generalizations of some well-known coupled fixed-point theorems. Specifically, Theorem 3 of the paper is the generalization of the Baskar–Lackshmikantham coupled fixed-point theorem, and Theorem 5 is the generalization of the Sahar Mohamed Ali Abou Bakr fixed-point theorem, where the underlying space is complete -cone-metric space.

1. Introduction and Preliminaries

Since 1922, the pioneering fixed-point principle of Banach [1] showed exclusive interest of researchers because it has many applications, including variational linear inequalities and optimization, and applications in differential equations, in the field of approximation theory, and in minimum norm problems.

Since then, several types of contraction mappings have been introduced and many research papers have been written to generalize this Banach contraction principle.

In 1987, Guo and Lakshmikantham [2] introduced one of the most interesting concepts of coupled fixed point.

Definition 1. An element is said to be a coupled fixed point of the mapping if and only if and .

In 2006, Bhaskar and Lakshmikantham [3] introduced the concept of the mixed monotone property as follows.

Definition 2. Let be a partially ordered set and be a mapping from to . Then,(1) is said to be monotone nondecreasing in if and only if, for any ,(2) is said to be monotone nonincreasing in if and only if, for any ,(3) is said to have a mixed monotone property if and only if is both monotone nondecreasing in and monotone nonincreasing in

Definition 3. An element is said to be a lower-anti-upper coupled point of the mapping if and only ifA mapping is said to have a lower-upper property if and only if has at least one lower-anti-upper coupled point.

Definition 4. Let be a partially ordered normed space. Then,(1) is said to be a sequentially lower ordered space if it fulfills the condition: If is a nondecreasing sequence in such that converges strongly to , then for all (2) is said to be a sequentially upper-ordered space if it fulfills the condition: If is a nonincreasing sequence in such that converges strongly to , then for all (3) is said to be a sequentially lower-upper ordered space if it is both a lower- and upper-ordered space

In 2006, Bhaskar and Lakshmikantham [3] proved the existence of coupled fixed points for mixed monotone mappings with weak contractivity assumption in a partial-ordered Banach space as follows.

Theorem 1. (see [3]). Let be a sequentially both lower- and upper-ordered Banach space and be a mapping with mixed monotone and lower-upper properties. If there is a real number such thatthen has coupled fixed points in .

In 2013, Mohamed Ali [4] introduced novel contraction type of mappings and proved the following fixed-point theorem.

Theorem 2. (see [4]). Let be a Banach space and be a mapping from into , and we suppose there are three constants and such that

Then, there is a unique point such that .

There are many interesting coupled fixed-point theorems concerning some other type of contraction mappings, see [5–10].

Recently, more advanced approaches for studying coupled fixed points have been presented by the authors in [11–13].

In 2007, Huang and Zhang [14] introduced the concept of cone-metric spaces as follows: First, a subset of the real Banach 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)

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

If is a nonempty set, the distance between any two elements is defined to be a vector in the cone , and the space is said to be a cone-metric space if and only if satisfied the three axioms of metric but using the ordered relation induced by for the triangle inequality instead. They studied the topological characterizations of such a defined space, and then, they applied their concept to have more generalizations of some previous fixed-point theorems for contractive type of mappings.

A mapping is said to be a contraction if and only if there is a constant such that

In 2019, Mohamed Ali Abou Bakr [15] proved the existence of a unique common fixed point of generalized joint cyclic Banach algebra contractions and Banach algebra Kannan type of mappings on cone quasimetric spaces.

In 2013, Khojasteh et al. [10] introduced the notion of -action function, , the concept of -metric, and then, they studied the topological structures of -metric spaces in detail. Their work led to a step-forward generalization of metric spaces.

In 2020, Mohamed Ali Abou Bakr [16] replaced by a cone in a normed space and used the ordered relation induced by this cone to introduce the following analogous generalization of -action function.

Definition 5. Let be an ordered normed 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

Because for every , one can write instead for every , ( for every ).

In addition, Mohamed Ali Abou Bakr [16] gave further replacement, replaced the set of nonnegative real numbers by a cone in a normed space, and used -ordered actions to introduce the concept of -cone-metric space as follows.

Definition 6. (see [16]). Let be an ordered normed space, where is the ordered relation induced by some cone , and be an ordered action on . If is a nonempty set, then the function is said to be a -cone-metric on if and only if satisfies the following conditions:(1)(2)(3)The double is defined to be a -cone-metric space.

The author has further given some topological characterizations of this space and then generalized some previous fixed-point theorems in this setting.

Remark 1. If , then we have a cone-metric space.

In this paper, we extend and generalize the coupled fixed-point theorem of Baskar–Lackshmikantham (1.5) to a more general one (2.1), where the underlying space is a complete -cone-metric space. On the other side, if is a continuous mapping in the second argument and there are three constants and such thatthen we proved that has a unique fixed point in the sense that there is a unique point such that .

We also claim that some results of [6–10, 17] can be proved in the case of -cone-metric spaces.

2. Main Results

Let be a partially ordered -cone-metric space. Then, the following relation defines a partial-ordered relation on :

We have the following coupled fixed-point theorem.

Theorem 3. Let be a partially ordered, sequentially lower-upper ordered complete -cone-metric space and be a mapping having mixed monotone and lower-upper properties on . We assume that there exists with

Then, has coupled fixed points in .

Proof. Since has a lower-upper property, then there exist such thatWe denote and and then give notations for the elements of the following inductively constructed sequences:Using the mixed monotonicity property of insures that each step leads to the next step in each of the following:The mixed monotonicity property, the contractiveness of , and the inductive process prove the following for every :Consequently, we haveHence, we claim that both and are Cauchy sequences in . Indeed, if one of them, say , is not Cauchy, then there exist , and sequences of natural numbers and such that, for any ,Since any subsequence of is convergent to , the properties of imply the following contradiction:Similarly, the sequence is also Cauchy. Since is a complete -cone-metric space, there exist such thatNow, we are going to show that is a coupled fixed point of . Since the sequence is nondecreasing with , then , and since the sequence is nonincreasing with , then for every , and accordingly, we haveTaking the limit as with the help of equation (20), we find thatHence, ; therefore, . Similarly, .

If the partial-ordered relation on is defined asthen the following theorem is similarly proved.

Theorem 4. Let be a partially ordered, sequentially lower-upper ordered complete -cone-metric space and be a mapping having mixed monotone property, and we suppose that there are such that . If there exists withthen has coupled fixed points in .

Corollary 1. Let be a sequentially both lower- and upper-ordered Banach space and be a mapping with mixed monotone and lower-upper properties. If there is a real number such thatthen has coupled fixed point in .

Proof. We just notice that any Banach space is a -cone-metric space , where is the Banach space of real numbers with the absolute value metric and with the usual ordered relation of real numbers, , , and the metric is the metric induced by the norm on , .

Remark 2. Corollary 1 is Baskar–Lackshmikantham coupled fixed-point Theorem 1. This proves that Theorem 3 is a quite good generalization of the Baskar–Lackshmikantham coupled fixed-point theorem.

On the other side, we have the following results:

Lemma 1. Let be a -cone-metric space and be a mapping, . It is supposed that there are constants and such that

If and are arbitrary elements in , then the sequence defined iteratively bywhich satisfies the following:where . Moreover, the sequence is a Cauchy sequence.

Proof. Using the contractiveness property of the given mapping givesHence,and repeating the last step times with the term proves the inequalities given in (29). To prove that the sequence (27) is Cauchy, we take the limit of both sides of (29) as gives and suppose that is not Cauchy; then, there exist , and sequences of natural numbers and such that, for any ,Since any subsequence of is convergent to , the continuity and the properties of imply the following contradiction:

Theorem 5. Let be a complete -cone-metric space and be a continuous mapping in the second argument, and we suppose there are three constants and such thatand then, has a unique fixed point in the sense that there is a unique point such that .

Proof. Since is complete, the Cauchy sequence given in Lemma 1 is converging to some element in . We show that is fixed point of . Using the properties of and the continuity of , we see thatSince , we get ; consequently, . Now, let and be two arbitrarily distinct elements in with and , and we haveThus, , that is, . Similarly, we get ; therefore, is a coupled fixed point of . On the other hand, we have the following contradiction:Since , we have ; consequently, .

We conclude the following.

Corollary 2. Let be a Banach space and be a mapping from into , and we suppose that there are three constants and such that

Then, there is a unique point such that .

Proof. It can be proved in a similar way of Corollary 1 with the same notice.

Remark 3. Corollary 2 is the fixed-point theorem of Mohamed Ali Abou Bakr; accordingly, Theorem 5 is a generalization of fixed-point Theorem 2 in the setting of a complete -cone-metric space.

3. Conclusions

This paper gives further generalizations of some well-known coupled fixed-point theorems. Specifically, Theorem 3 generalizes the Baskar–Lackshmikantham coupled fixed-point theorem [3], and Theorem 5 generalizes the Sahar Mohamed Ali Abou Bakr fixed-point theorem [4]; the underlying space is a complete -cone-metric space, and we claim that some results of [6–10] can be proved in the case of -cone-metric spaces.

Data Availability

No data were used to support this study.

Disclosure

This research was performed as part of the employment of Dr. Sahar Mohamed Ali Abou Bakr at Ain Shams University, Faculty of Science, Department of Mathematics, Cairo, Egypt.

Conflicts of Interest

The author has no conflicts of interest.

Authors’ Contributions

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