Abstract
In this manuscript, we introduce the concept of complex-valued triple controlled metric spaces as an extension of rectangular metric type spaces. To validate our hypotheses and to show the usability of the Banach and Kannan fixed point results discussed herein, we present an application on Fredholm-type integral equations and an application on higher degree polynomial equations.
1. Introduction
Since the breakthrough of Banach [1] in 1922, where he was able to show that a contractive mapping on a complete metric space has a unique fixed point, the field of fixed point theory has become an important research focus in the field of mathematics; see [2–6]. Due to the fact that fixed point theory has many applications in many fields of science, many researchers have been working on generalizing his result by either generalizing the type of contractions [7–10] or by extending the metric space itself (-metric spaces [11, 12], controlled metric spaces [13], double controlled metric spaces [14], etc.). On the other hand, Azam et al. [15] defined complex-valued metric spaces and gave common fixed point results. Rao et al. [16] introduced the complex-valued -metric spaces in the year 2013. Going in the same direction, recently, Ullah et al. [17] presented complex-valued extended -metric spaces to extend the idea of extended -metric spaces.
In this manuscript, following the path of the work done in [18], we extend complex-valued rectangular extended -metric spaces [19] to complex-valued triple controlled metric spaces. The layout of our manuscript is as follows. In the second section, we present some backgrounds along with the definition of complex-valued triple controlled metric spaces. In the third section, we prove some fixed point results in such spaces. In the fourth section, we present an application for our findings. In closing, we present two open questions.
2. Preliminaries
In what follows, owing to Azam et al. [15], we recall several notations and definitions which will be used in the sequel.
Let be the set of all complex numbers and . The partial order on is defined as if and only if and . This implies that if one of the below conditions is fulfilled: (i)(ii), (iii)(, (iv),
Following [15], the authors in [17] developed the notion of complex-valued extended -metric spaces.
Definition 1 (see [17]). Let be a nonempty set and be a function. Then, is known as a complex-valued extended -metric space if the following are satisfied for all : (1) and if and only if (2)(3)
Then, the pair is known as a complex-valued extended -metric space.
As an extension of complex-valued extended -metric spaces, Ullah et al. in [19] introduced the concept of complex-valued rectangular extended -metric spaces.
Definition 2 (see [19]). Let be a nonempty set and and . We say that is a complex-valued rectangular extended -metric space if for all each of which is different from , we have (1) if and only if (2)(3)
The authors in [20] have recently introduced the idea of triple controlled metric type spaces as follows.
Definition 3 (see [20]). Let be a nonempty set. Given three functions and . We say that is a triple controlled metric type space if for all , we have (1) if and only if (2)(3)
Highly motivated by the abovementioned concepts, we now present the definition of complex-valued triple controlled metric spaces.
Definition 4. Let be a nonempty set. Given three functions and . We say that is a complex-valued triple controlled metric space if for all , each of which is different from , we have (1) if and only if (2)(3)
Throughout the rest of this paper, we will denote a complex-valued triple controlled metric space by (CV-TCMS). Next, we present the topology of (CV-TCMSs).
Definition 5. Let be a (CV-TCMS). (1)We say that a sequence is -convergent to some if (2)We say that a sequence is -Cauchy if and only if (3)We say that is -complete if for every -Cauchy sequence is -convergent(4)Let . An open ball of center and radius in the (CV-TCMS) is
Note that a CV rectangular metric space is a CV-TCMS. The converse is not true. Next, we present an example that confirms this statement.
Example 1. Let where and is the set of positive integers. We define by where . Now, define by . Given as and as .
Note that is a CV-TCMS. On the other hand, is not a CV rectangular metric space. Indeed,
In this paper, we prove the Banach and Kannan fixed point results in the setting of CV-TCMSs. Two related applications are also investigated.
3. Main Results
Theorem 1. Let be a -complete CV-TCMS. Let satisfy where . Assume that there exists such that the sequence defined by satisfies the following:
Then, has a unique fixed point in
Proof. First, we have . Then,
Now, let . We need to consider the following two cases.
Case 1. Let for some natural numbers and with . Without loss of generality, take . If ; then, by choosing and , we get , which implies that is a periodic point of . Hence, . Since , we get , so , that is, has a fixed point.
From now on, we consider the following case.
Case 2. Assume that for all natural numbers , we have . Let . To prove that is a -Cauchy sequence, we need to consider the following two subcases.
Subcase 1. If (where is a fixed natural number), then by the rectangle inequality of the CV-TCMS, we have
Now, given that we can easily deduce that
Since , the last right-hand side goes to zero at the limit (for any integer ). Therefore, is convergent.
Subcase 2. Let (where is a fixed integer). First, notice the following: which leads us to conclude that
Thus, by Subcase 1 and using the rectangular inequality of the complex-valued triple controlled metric, we have
Now, similar to Subcase 1, one can easily deduce that is a convergent sequence as (for any integer ). Hence, by Subcases 1 and 2, we conclude that is a -Cauchy sequence. Since is a -complete CV-TCMS, there is such that as .
Now, if there exists such that , then since we deal with Case 2, one writes for all . Also, for all . Next, assume that there exists with . Once again, we confirm that for all . Thus, without loss of generality, we may assume for all natural numbers . We have which implies
Therefore, in view of the assumptions in the theorem, as , we deduce that and that is as required.
In closing, assume there exist two fixed points of , say and where . Thus, which is a contradiction. Therefore, the fixed point of is unique.
Theorem 2. Let be a -complete CV-TCMS and be a self mapping on satisfying the following condition: for all , there exists such that and there exists in order that the sequence defined by satisfies the following:
Then, has a unique fixed point in
Proof. First of all, note that for all , we have
Consequently,
Since , one has . Set . One writes
Therefore,
Also, for all , we have
By (19), we deduce that as . Hence, is a -Cauchy sequence. Since is a -complete CV-TCMS, the sequence converges to some
By the argument of the proof of Theorem 1, assume that for all , we have . Thus,
As , we obtain
At the limit , we find that and that is as required. Now, assume that we have two fixed points of , say and . Therefore,
Hence, , as desired.
4. Applications
4.1. A Fredholm-Type Integral Equation
Consider the set . Given the following Fredholm-type integral equation where is a continuous function from into . Now, define
Note that is a complete CV-TCMS, where
Theorem 3. Assume that for all (1), for some (2) for all
Then, the above integral equation has a unique solution.
Proof. Let be defined by . Then,
Now, we have
Thus, . Since , one gets
Therefore, all the hypotheses of Theorem 1 are satisfied, and hence, equation (24) has a unique solution.
4.2. A Polynomial Equation of a Degree Greater or Equal to 3
The following is an application on higher degree polynomial equations.
Theorem 4. For any natural number and real , the following equation has a unique real solution.
Proof. It is not difficult to see that if , equation (30) does not have a solution. So, let and for all , let and and . Note that is a -complete CV-TCMS. Now, let
Notice that, since , we can deduce that . Thus,
Hence,
Moreover, it is easy to see that for all , we have
Note that all the conditions of Theorem 1 are satisfied. Thus, possesses a unique fixed point in , and equation (30) has a unique real solution.
5. Conclusion
Finally, we would like to leave the following questions.
Question 1. Let be a CV-TCMS and . Given a function . Suppose there exists such that, for all ,
Under what conditions does have a unique fixed point in ?
Question 2. Let be a CV-TCMS, and . Given a function . Suppose there exists such that, for all ,
Under what conditions does have a unique fixed point in
Data Availability
Data sharing is not applicable to this article as no data set were generated or analyzed during the current study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All the authors have equally contributed to the final manuscript.
Acknowledgments
The authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.