Research Article Results on Complex Partial b-Metric Space with an Application

In this paper, we prove a ﬁxed point theorem in complex partial b -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.


Introduction
Introduced in 1989 by Bakhtin [1] and Czerwik [2], the concept of b-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. [3] 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 [4] Rao et al. discussed for the first time the idea of complex-valued b-metric spaces.
It was just very recently, in 2017, that Dhivya and Marudai [5] extended all the preceding results in the setting of complex partial metric spaces making use of a rational type contraction.
is was followed by Gunaseelan [6], who introduced the concepts of complex partial b-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][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In this paper, under new contraction condition, we prove a fixed point theorem in complex partial b-metric space. Although there have been a significant amount of scientific contributions to the theory of partial b-metric space, very few address that of complex valued, and even less, the applicability of complex partial b-metrics in the resolution integral equations. is, however, in one the main contributions of the present work. We begin by recalling basic facts about complex partial b-metric spaces.
Definition 2 (see [5]). A complex partial metric on a nonvoid set Θ is a function ξ c : Θ × Θ ⟶ C + such that, for all α, β, c ∈ Θ, A complex partial metric space is a pair (Θ, ξ c ) such that Θ is a nonvoid set and ξ c is the complex partial metric on Θ.
Definition 3 (see [6]). A complex partial b-metric on a nonvoid set Θ is a function ⋎ cb : Θ × Θ ⟶ C + such that, for all α, β, c ∈ Θ, A complex partial b-metric space is a pair (Θ, ⋎ cb ) such that Θ is a nonvoid set and ⋎ cb is the complex partial b-metric on Θ.
e number s is called the coefficient of (Θ, ⋎ cb ).
Every complex partial b-metric ⋎ cb on a nonvoid set Θ generates a topology τ cb on Θ whose base is the family of . Now, we define Cauchy sequence and convergent sequence in complex partial b-metric spaces.
Definition 4 (see [6]). Let (Θ, ⋎ cb ) be a complex partial b-metric space with coefficient s. Let α n be any sequence in Θ and α ∈ Θ. en, (i) e sequence α n is said to be convergent with respect to τ cb and converges to A subset E⊆Θ is called closed iff E contains all its limit points.
Inspired by eorem 1, we prove a fixed point theorem on complex partial b-metric space under new contraction mapping.
In Section 3, we first prove, under new contraction mapping, a fixed point theorem on complete complex partial b-metric space. We also provide an example of the complete complex partial b-metric space and clarify that, under certain conditions, it has a unique fixed point.

Main Results
(2) en, Ξ has a unique fixed point.
Proof. It is clear that (Ξ, ⋎ cb ) is a complete complex partial b-metric space with coefficient s � 2 2 and 2 is a unique fixed point of Ξ. Let j < r: Mathematical Problems in Engineering