On Complex-Valued Triple Controlled Metric Spaces and Applications

Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Sciences and Information Engineering, Asia University, Taichung, Taiwan Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Molotlegi St, GaRankuwa Zone 1, Ga-Rankuwa 0208, South Africa Institut de Mathématiques et de Sciences Physiques (IMSP/UAC), Laboratoire de Topologie Fondamentale, Computationnelle et Leurs Applications (Lab-ToFoCApp), BP 613 Porto-Novo, Benin Quantum Leap Africa (QLA), AIMS Rwanda Centre, Remera Sector KN 3, Kigali, Rwanda African Center for Advanced Studies (ACAS), P.O. Box 4477, Yaounde, Cameroon


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][3][4][5][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][8][9][10] or by extending the metric space itself (b-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 b-metric spaces in the year 2013. Going in the same direction, recently, Ullah et al. [17] presented complex-valued extended b-metric spaces to extend the idea of extended b-metric spaces.
In this manuscript, following the path of the work done in [18], we extend complex-valued rectangular extended b -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.

Preliminaries
In what follows, owing to Azam et al. [15], we recall several notations and definitions which will be used in the sequel.
Definition 1 (see [17]). Let X be a nonempty set and ξ : X × X ⟶ ½1,∞Þ be a function. Then, L e : X 2 ⟶ ℂ is known as a complex-valued extended b-metric space if the following are satisfied for all s, κ, u ∈ X: Then, the pair ðX, L e Þ is known as a complex-valued extended b-metric space.
As an extension of complex-valued extended b-metric spaces, Ullah et al. in [19] introduced the concept of complex-valued rectangular extended b-metric spaces.
Definition 2 (see [19]). Let X be a nonempty set and ξ : X 2 ⟶ ½1,∞Þ and L r : X 2 ⟶ ℂ. We say that ðX, L r Þ is a complex-valued rectangular extended b-metric space if for all a, b ∈ X each of which is different from κ, v ∈ X, we have The authors in [20] have recently introduced the idea of triple controlled metric type spaces as follows.
Definition 3 (see [20]). Let X be a nonempty set. Given three functions ξ, ρ, ς : X 2 ⟶ ½1,∞Þ and L T : X 2 ⟶ ½0,∞Þ. We say that ðX, L T Þ is a triple controlled metric type space if for all a, b, κ, v ∈ X, we have Highly motivated by the abovementioned concepts, we now present the definition of complex-valued triple controlled metric spaces.
Definition 4. Let X be a nonempty set. Given three functions ξ, ρ, ς : X 2 ⟶ ½1,∞Þ and L t : X 2 ⟶ ℂ. We say that ðX, L t Þ is a complex-valued triple controlled metric space if for all a, b ∈ X, each of which is different from κ, v ∈ X, we have 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). (1) We say that a sequence fa n g is L t -convergent to some a ∈ X if jL t ða n , aÞj ⟶ 0 as n ⟶ ∞ (2) We say that a sequence fa n g is L t -Cauchy if and only if lim n,m⟶∞ | L t ða n , a m Þ | = 0 An open ball of center x and radius η > 0 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.
Note that ðX, L t Þ is a CV-TCMS. On the other hand, ðX, L t Þ is not a CV rectangular metric space. Indeed, Journal of Function Spaces In this paper, we prove the Banach and Kannan fixed point results in the setting of CV-TCMSs. Two related applications are also investigated.

Main Results
Assume that there exists x 0 ∈ X such that the sequence fx n g defined by x n = T n x 0 satisfies the following: Then, T has a unique fixed point in X: Now, let L i = L t ðx n+i , x n+i+1 Þ. We need to consider the following two cases. Case 1. Let x n = x m for some natural numbers n and m with n ≠ m. Without loss of generality, take m > n. If T m−n ðx n Þ = x n ; then, by choosing y = x n and p = m − n, we get T p y = y, which implies that y is a periodic point of T . Hence, L t ðy, T yÞ = L t ðT p y, T p+1 yÞ≺δ p L t ðy, T yÞ. Since δ ∈ ð0, 1Þ, we get |L t ð y, T yÞ | = 0, so y = T y, that is, T has a fixed point.
From now on, we consider the following case.
Case 2. Assume that for all natural numbers n ≠ m, we have To prove that fx n g is a L t -Cauchy sequence, we need to consider the following two subcases.
Subcase 1. If m = n + 2p + 1 (where p ≥ 1 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 lim n⟶∞ δ n = 0, the last right-hand side goes to zero at the limit n ⟶ ∞ (for any integer p ≥ 1). Therefore, f|L t ðx n , x n+2p+1 Þ | g n is convergent.
Subcase 2. Let m = n + 2p (where p ≥ 1 is a fixed integer). First, notice the following: 3 Journal of Function Spaces 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 fjL t ðx n , x n+2p Þjg n is a convergent sequence as n ⟶ ∞ (for any integer p ≥ 1). Hence, by Subcases 1 and 2, we conclude that fx n g is a L t -Cauchy sequence. Since ðX, L t Þ is a L t -complete CV-TCMS, there is ν ∈ X such that fx n g ⟶ ν as n ⟶ ∞. Now, if there exists N ∈ ℕ such that x N = ν, then since we deal with Case 2, one writes x n = T n x 0 ≠ ν for all n > N. Also, x n = T n x 0 ≠ T ν for all n > N. Next, assume that there exists N ∈ ℕ with x N = T N x 0 = T ν. Once again, we confirm that x n = T n x 0 ∈fν, T νg for all n > N. Thus, without loss of generality, we may assume x n ∈fν, T νg for all natural numbers n. We have which implies Therefore, in view of the assumptions in the theorem, as n ⟶ ∞, we deduce that |L t ðν, T νÞ | = 0 and that is T ν = ν as required.
In closing, assume there exist two fixed points of T , say ν and μ where ν ≠ μ. Thus, which is a contradiction. Therefore, the fixed point of T is unique.

Theorem 2.
Let ðX, L t Þ be a L t -complete CV-TCMS and T be a self mapping on X satisfying the following condition: for all a, b ∈ X, there exists 0 < δ < 1/2 such that and there exists x 0 ∈ X in order that the sequence fx n g defined by x n = T n x 0 satisfies the following: Then, T has a unique fixed point in X: Proof. First of all, note that for all n ≥ 1, we have Consequently, Since 0 < δ < 1/2, one has 0 < ðδ/ð1 − δÞÞ < 1. Set μ = δ/ ð1 − δÞ. One writes Therefore, Also, for all n, m ≥ 1, we have By (19), we deduce that |L t ðx n , x m Þ | ⟶0 as n, m ⟶ ∞. Hence, fx n g is a L t -Cauchy sequence. Since ðX, L t Þ is a L t -complete CV-TCMS, the sequence fx n g converges to some ν ∈ X: By the argument of the proof of Theorem 1, assume that for all n ≥ 1, we have x n ∈fν, T νg. Thus, Journal of Function Spaces As n ⟶ ∞, we obtain At the limit n ⟶ ∞, we find that |L t ðν, T νÞ | = 0 and that is T ν = ν as required. Now, assume that we have two fixed points of T , say ν and s. Therefore, Hence, ν = s, as desired.

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 β ≥ 3 and real |α | ≤1, the following equation has a unique real solution.

Journal of Function Spaces
Notice that, since β ≥ 2, we can deduce that β 4 ≥ 6. Thus, Hence, Moreover, it is easy to see that for all α 0 ∈ X, we have Note that all the conditions of Theorem 1 are satisfied. Thus, T possesses a unique fixed point in X, and equation (30) has a unique real solution.

Conclusion
Finally, we would like to leave the following questions. Question 1. Let ðX, L t Þ be a CV-TCMS and T : X ⟶ X. Given a function ς : X 2 ⟶ ½1,∞Þ. Suppose there exists δ ∈ ð0, 1Þ such that, for all s, r ∈ X, Under what conditions does T have a unique fixed point in X? Question 2. Let ðX, L t Þ be a CV-TCMS, and T : X ⟶ X. Given a function ς : X 2 ⟶ ½1,∞Þ. Suppose there exists δ ∈ ð0, 1/2Þ such that, for all s, r ∈ X, L t T s, T r ð Þ≺δς s, r ð Þ L t s, T s ð Þ+ L t r, T r ð Þ ½ : Under what conditions does T have a unique fixed point in X?

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.