Results on Complex Partial b-Metric Space with an Application
In this paper, we prove a fixed point theorem in complex partial -metric space under new contraction mapping. The proved results generalize and extend some of the well-known results in the literature. We also give some applications of our main results.
Introduced in 1989 by Bakhtin  and Czerwik , the concept of -metric spaces provided a framework to extend the results already known in the classical setting of metric spaces. About two decades later, more precisely in 2011, Azamet al.  came up with the notion of complex-valued metric spaces and provided some common fixed point theorems under some contractive conditions. Two years after, it was in  Rao et al. discussed for the first time the idea of complex-valued -metric spaces.
It was just very recently, in 2017, that Dhivya and Marudai  extended all the preceding results in the setting of complex partial metric spaces making use of a rational type contraction.
This was followed by Gunaseelan , who introduced the concepts of complex partial -metric spaces and discussed some results of fixed point theory for self-mappings in these new spaces.
Many authors have studied related interesting metric such as structures along with some applications. And, in this line, significant results have been obtained and can be read in [7–23]. In this paper, under new contraction condition, we prove a fixed point theorem in complex partial -metric space. Although there have been a significant amount of scientific contributions to the theory of partial -metric space, very few address that of complex valued, and even less, the applicability of complex partial -metrics in the resolution integral equations. This, however, in one the main contributions of the present work. We begin by recalling basic facts about complex partial -metric spaces.
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) and (ii) and (iii) and (iv) and
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 ). Let be a nonvoid set and let be a given real number. A function is called a complex-valued -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 -metric space.
Here, and denote the set of nonnegative complex numbers and the set of nonnegative real numbers, respectively. We now give the complex partial metric space.
Definition 2. (see ). 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 ). A complex partial -metric on a nonvoid set is a function such that, for all ,(i)(ii)(iii)(iv) a real number and is an independent of such that A complex partial -metric space is a pair such that is a nonvoid set and is the complex partial -metric on . The number is called the coefficient of .
Remark 1. (see ). In a complex partial -metric space if and , then , but the converse may not be true.
Every complex partial -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 -metric spaces.
Definition 4. (see ). Let be a complex partial -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 -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 -metric space and . A point is called an interior of set a 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 iff contains all its limit points.
Example 1. (see ). Let be endowed with complex partial -metric with .
In 2019, Gunaseelan  proved the following theorem.
Theorem 1. (see ). Let be a complete complex partial -metric space with coefficient and be a mapping satisfying the following condition:where . Then, has a unique fixed point and .
Inspired by Theorem 1, we prove a fixed point theorem on complex partial -metric space under new contraction mapping.
In Section 3, we first prove, under new contraction mapping, a fixed point theorem on complete complex partial -metric space. We also provide an example of the complete complex partial -metric space and clarify that, under certain conditions, it has a unique fixed point.
3. Main Results
Theorem 2. Let be a complete complex partial -metric space with constant and let be a self-mapping on . Suppose that there exist functions , , of into such that Each is upper semicontinuous from the right , For any distinct ,Then, has a unique fixed point.
Proof. Let us first prove that if fixed points of exists, then it is unique. Let be two distinct fixed points of , that is, . Therefore, . If , we have . If . By the definition of complex partial -metric space, we obtainFrom condition , we derivewhich implies thatwhich is impossible. Therefore, . Choose . SetLet . If there exists such that , the proof is complete. So, we presume that, for every , . Therefore, we haveFrom (2), we deriveand consequently,which implies thatFrom , we obtainConsequently, we deriveTherefore, from (10), we obtainwhich implies thatConsequently, and . Therefore, is a decreasing sequence of real numbers which is bounded from below. So, converges to some point . Similarly, converges to some point . Hence, converges to some point . If , then using condition , we obtainwhich implies that , which is impossible. Therefore,Next, we prove thatSuppose not, we assume that there exist and sequence and of natural numbers such thatTherefore, we deriveUsing (16), we deriveFrom (16), there exists such that , and using (18), we deriveTherefore, from , for every , we obtainHence, from , we deriveFrom (16)–(23), we obtainwhich is impossible. Hence, . By completeness of , there exists such thatWe shall prove that, for every ,orSuppose not, we assume that there exists such thatBy the definition of complex partial -metric space and (28), we derivewhich is impossible. Hence, (25) and (26) holds. From (25), we obtainFrom (12), we deriveUsing (25) and (26), we deriveSinceusing (25) and (32), we get . From (27), we obtainwhich means thatFrom (12) and (35), we derive thatUsing (16) and (25), we deriveSince