Application to Coupled Fixed-Point Theorems on Complex Partial b-Metric Space

Abstract

In this paper, we prove coupled fixed-point theorems in complex partial b-metric space. The proved results generalize and extend some of the well-known results in the literature. We also give some applications of our main results.

1. Introduction

The notion of b-metric space was introduced by Backhtin [1] in 1989, and Czerwik [2] extended the results of metric spaces. The notion of complex valued metric spaces was introduced by Azam et al. [3] in 2011 who also proved some common fixed-point theorems under the contraction condition. In 2013, Rao et al. [4] introduced the concept of complex valued b-metric space which is more general than the well-known complex valued metric space and also proved common fixed-point theorem under the contraction condition. In 2017, Dhivya and Marudai [5] introduced the notion of complex partial metric space and also proved common fixed-point theorems under the contraction condition of rational expression. In 2019, Gunaseelan [6] introduced the notion of complex partial b-metric space and also proved fixed-point theorem under the contractive condition. Some interesting concepts and applications have been studied by many authors, and important results have been obtained in [7–12]. In this paper, we prove coupled fixed-point theorems in complex partial b-metric space.

In the next section, we give basic definitions, examples, and primary results for the better understanding of our major results presented in this research paper.

2. Preliminaries

Let be the set of complex numbers and . Define a partial order on as follows: if and only if , .

Consequently, one can infer that if one of the following conditions is satisfied:(i), (ii), (iii), (iv),

In particular, we write if and one of (i), (ii), and (iii) is satisfied and we write if only (iii) is satisfied. Notice that(a)If , then (b)If and , then (c)If , and , then for all

Definition 1 (see [4]). Let be a nonvoid set and let be a given real number. A function is called a complex valued b-metric on if for all the following conditions are satisfied:(i) and if and only if (ii)(iii)The pair is called a complex valued b-metric space.
Here, and denote the set of non-negative complex numbers and the set of non-negative real numbers, respectively. We now give the complex partial metric space.

Definition 2 (see [5]). A complex partial metric on a nonvoid set is a function such that for all :(i)(ii)(iii) if and only if (iv)A complex partial metric space is a pair such that is a nonvoid set and is the complex partial metric on .

Definition 3 (see [6]). A complex partial b-metric on a nonvoid set is a function such that for all :(i)(ii)(iii)(iv) a real number such that A complex partial b-metric space is a pair such that is a nonvoid set and is the complex partial b-metric on . The number is called the coefficient of .

Remark 1 (see [6]). In a complex partial b-metric space , if and , then , but the converse may not be true.

Remark 2 (see [6]). It is clear that every complex partial metric space is a complex partial b-metric space with coefficient and every complex valued b-metric is a complex partial b-metric space with the same coefficient and zero self-distance. However, the converse of this fact need not hold.

Example 1 (see [6]). Let and be defined by . Then, is a complex partial b-metric space with coefficient , but it is neither a complex valued b-metric nor a complex partial metric.

Proposition 1 (see [6]). Let be a nonvoid set such that is a complex partial metric and is a complex valued b-metric with coefficient on . Then, the function defined by is a complex partial b-metric on , that is, is a complex partial b-metric space.

Proposition 2 (see [6]). Let be a complex partial metric space, ; then, is a complex partial b-metric space with coefficient , where is defined by .

Every complex partial b-metric on a nonvoid set generates a topology on whose base is the family of open -balls where and . Now, we define Cauchy sequence and convergent sequence in complex partial b-metric spaces.

Definition 4 (see [6]). Let be a complex partial b-metric space with coefficient . Let be any sequence in and . Then,(i)The sequence is said to be convergent with respect to and converges to , if .(ii)The sequence is said to be Cauchy sequence in if exists and is finite.(iii) is said to be a complete complex partial b-metric space if for every Cauchy sequence in there exists such that .(iv)A mapping is said to be continuous at if for every , there exists such that .Let be a complex partial b-metric space and . A point is called an interior of set , if there exists such that . A subset is called open, if each point of is an interior point of . A point is said to be a limit point of , for every , . A subset is called closed if contains all its limit points.

Example 2 (see [6]). Let , be any constant and define by .
Then, is a complex partial b-metric space with arbitrary coefficient . Now, define a sequence in by for all . Note that, if , we have .Therefore, for all . Thus, the limit of convergent sequence in complex partial b-metric space need not be unique.

Example 3 (see [6]). Let be endowed with complex partial b-metric with .
It is easy to verify that is a complex partial b-metric space and note that self-distance need not be zero, for example, . Now, the complex valued b-metric not induced by is as follows: ; without loss of generality, suppose that ; then, . Therefore, .
Thus, we have the following proposition.

Proposition 3 (see [6]). There exists a complex partial b-metric which does not define a complex b-metric , where .

Definition 5 (see [13]). Let be a partially ordered set and . We say that has the mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in , that is, for any ,

Definition 6. Let be a complex partial b-metric space. An element , is called a coupled fixed point of the mapping if and .

Example 4. Let be endowed with complex partial b-metric defined by . Consider the mapping with . Here, is the coupled fixed point of .
In 2019, Gunaseelan and Mishra [14] proved the following theorem.

Theorem 1 (see [14]). Let be a complete complex partial 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.

Inspired by Theorem 1, we prove coupled fixed-point theorem on partially ordered complex partial b-metric space using mixed monotone property.

In the next section, we firstly prove that a continuous mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space has a coupled fixed point under certain conditions. Secondly, we give result of a coupled fixed point for a mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space by losing the property of continuity. Then, we find the condition under which a continuous mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space has a unique coupled fixed point under certain conditions. We also give an example of continuous mapping having the mixed monotone property on a partially ordered complete complex partial b-metric space and show that it has unique coupled fixed point under said conditions.

3. Main Results

Let be a partially ordered set and be a complex partial b-metric space on . Further, we endow the product space with the following partial order:

We begin with the following theorem that establishes the existence of a fixed-point theorem for a function on the product .

Theorem 2. Let be a partially ordered complete complex partial b-metric space with the coefficient . Let be a continuous mapping having the mixed monotone property on . Assume that there exists a with

If there exists such thatthen has a coupled fixed point.

Proof. Since (say) and (say), letting and , we denoteDue to the mixed monotone property of ,Further, for , we letWe can easily verify thatIf and for some , then and , and hence is a coupled fixed point of . Suppose, further, thatNow, we claim that, for ,Indeed, for , using , , we getwhich implies thatSimilarly,which implies thatAdding (12) and (16), we haveIn a similar way, proceeding by induction, if we assume that (12) holds, we get thatHence, by induction, (12) is proved. SetThen, the sequence is decreasing andBy assumption (9), for . Then, for each , we havewhich implies thatTherefore,Therefore, and are Cauchy sequences in . Since is complete complex partial b-metric space, there exists such thatand and . Finally, we claim that is a coupled fixed point of . Indeed, from and , using (24) and the continuity of , it immediately follows that and .
In the next theorem, we will substitute the continuity hypothesis on by an additional property satisfied by the space .

Theorem 3. Let be a partially ordered complete complex partial b-metric space with the coefficient . Let be a mapping having the mixed monotone property on . Assume that there exists with

Finally, assume that has the following properties:(i)If a nondecreasing sequence in converges to , then for all .(ii)If a nonincreasing sequence in converges to , then for all .

Then, has a coupled fixed point.

Proof. Following the proof of Theorem 2, we only have to show is a coupled fixed point of . We haveSince the nondecreasing sequence converges to and the nonincreasing sequence converges to , by (i)–(iii), we haveNow, from the contractive condition (25), we haveThen, from (26), we getwhich implies thatTaking limit as , we haveTherefore, . Similarly, we can prove that . Hence, is a coupled fixed point of .

Theorem 4. Assume that

Adding (32) to the hypotheses of Theorem 2, we obtain the uniqueness of the coupled fixed point of .

Proof. From Theorem 2, we know that there exists a coupled fixed point of , which is obtained as and . Suppose that is another coupled fixed point, i.e.,Let us claim thatWe discuss two cases. Case 1: is comparable with with respect to the ordering in . Let, e.g., and . Then, we can apply the contractive condition (4) to obtain which implies that which implies that Adding (36) and (38), we get Since , (34) holds. Case 2: is not comparable with . In this case, there exists that is comparable both to and . Then, for all , is comparable both to and . We havewhich implies thatSince , (34) holds.
We deduce that in all cases, (34) holds. This implies that and the uniqueness of the coupled fixed point of is proved.

Theorem 5. In addition to the hypotheses of Theorem 2 (resp. Theorem 3), suppose that in are comparable. Then, .

Proof. Suppose that . We claim thatFrom the mixed monotone property of , we haveAssume that for some . Now,Hence, (42) holds.
Now, using (42) and the contractive condition, we getwhich implies thatPassing to the limit as , we getSince , this implies that , i.e., .

Example 5. Let be equipped with the partial order defined byand with the functional defined by for all . Clearly, is a partially ordered complete complex partial b-metric space with . Define the mapping byObviously, the mapping has the mixed monotone property and is continuous. Let be such that and . We have considered the following cases. Case 1: . Since , we have : Case 2: , , and : Case 3: , , and : Case 4: and :Thus, satisfies all assumptions of Theorem 4 and it has a unique coupled fixed point (which is ).
Next, we present a result for the existence of a unique solution for a particular system of integral equations.

3.1. Applications to Integral Equations

We study the existence of solutions for the following system of integral equations:where .

We assume that , , satisfy the following conditions:(i) and for all .(ii)There exist , such that(iii).

Theorem 6. Consider integral equations (54) and (55) with , and . Under assumptions (i)–(iii), equations (54) and (55) have a unique solution.

Proof. Consider the natural order relation on ; that is, for ,It is well known that is a complete complex partial b-metric space with respect toSuppose that is a monotone nondecreasing sequence in that converges to a point . Then, for every , the sequence of real numbersconverges to . Therefore, for all , , . Hence, , for all . Similarly, it can be verified that, if for all , is a limit of a monotone nondecreasing sequence in , then for all , and hence for all .
Also, is a partially ordered set under the following order relation in :For any , and , for each , are in and are the upper and lower bounds of , respectively. Therefore, for every , there exists that is comparable to and . Define byWe now claim that has the mixed monotone property. If , thenSimilarly, if , thenThus, is monotone nondecreasing in and is monotone nonincreasing in . Also, for , that is, , , it follows thatNow, all the hypotheses of Theorem 4 are satisfied. Therefore, has a unique coupled fixed point.

4. Conclusion

In 2019, Gunaseelan and Mishra [14] proved coupled fixed-point theorem on complex partial metric space. In this paper, we proved coupled fixed-point theorems on complex partial b-metric space using partially ordered set and mixed monotone property. An illustrative application in partially ordered complex partial b-metric space is given.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally in preparation of this paper. All authors read and approved the final manuscript.

Acknowledgments

This study was supported by the Higher Education Commission of Pakistan and the Basque Government (grant no. IT1207-19).