Solving Integral Equations by Common Fixed Point Theorems on Complex Partial b-Metric Spaces

Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur 603203, Kanchipuram, Chennai, Tamil Nadu, India Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, 31000 Oran, Algeria Department of Mathematics, Sri Sankara Arts and Science College (Autonomous), Affiliated to Madras University, Enathur, 631 561, Kanchipuram, Tamil Nadu, India Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt


Introduction
Introduced in 1989 by Bakhtin [1] and Czerwick [2], the concept of b-metric spaces provided a framework to extend the results already known in the classical setting of metric spaces. The concept of complex valued metric spaces was introduced in 2011 by Azam et al. [3] and given some common fixed point theorems under the condition of contraction. Rao et al. [4] introduced the definition of complex valued b -metric spaces in 2013 and provided a scheme to expand the results, as well as proved a common fixed point theorem under contraction. In 2017, Dhivya and Marudai [5] introduced the concept of complex partial metric space and suggested a plan to expand the results, as well as proved common fixed point theorems under the rational expression contraction condition. Gunaseelan [6,7] presented the concept of complex partial b-metric space in 2019, as well as proved the fixed point theorem under the contractive condition. Many researchers have studied some intriguing concepts and applications and have shown significant results [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In this paper, we prove some common fixed point theorems for rational contraction mapping on complex partial b-metric space.

Preliminaries
Let C be the set of complex numbers and 1, 2, 3 ∈ C. Define a partial order ≤ on C as follows: 1 ≤ 2 if and only if Rð1Þ ≤ Rð2Þ and I ð1Þ ≤ I ð2Þ.
(iv) ∃ a real number s ≥ 1 and s is an independent of ℵ, ϑ, Z such that φ cb ðℵ, ϑÞ ≤ s½φ cb ðℵ, ZÞ + φ cb ðZ, ϑÞ − φ cb ðZ, ZÞðtriangularityÞ A complex partial b-metric space is a pair ðW, φ cb Þ such that W is a nonvoid set and φ cb is the complex partial b-metric on W. The number s is called the coefficient of ðW, φ cb Þ.
Definition 2 [6]. Let ðW, φ cb Þ be a complex partial b-metric space with coefficient s. Let fℵ n g be any sequence in W and ℵ ∈ W. Then, (1) The sequence fℵ n g is said to converge to ℵ, if lim n⟶∞ φ cb ðℵ n , ℵÞ = φ cb ðℵ, ℵÞ (2) The sequence fℵ n g is said to be Cauchy sequence in ðW, φ cb Þ if lim n,m⟶∞ φ cb ðℵ n , ℵ m Þ exists and is finite (3) ðW, φ cb Þ is said to be a complete complex partial b-metric space if for every Cauchy sequence fℵ n g in W, there exists ℵ ∈ W such that lim n,m⟶∞ φ cb ðℵ n , ℵ m Þ = lim n⟶∞ φ cb ðℵ n , ℵÞ = φ cb ðℵ, ℵÞ. In 2019, Gunaseelan [6] proved some fixed point theorems on complex partial b-metric space as follows.
Inspired by the study made by Gunaseelan [6], here, we prove some common fixed point theorems for rational contraction mapping on complex partial b-metric space with an application.

Main Results
In this section, we will give our main result of this paper, where some common fixed point theorems for rational contraction mapping on complex partial b-metric space are given.
for all ℵ, ϑ ∈ W, where a 1 , a 2 are nonnegative reals with a 1 + sa 2 < 1. Then, S and T have a unique common fixed point in W.
Proof. Let ℵ 0 be arbitrary point in W, and define a sequence fℵ n g in W such that Next, show that the sequence fℵ n g is Cauchy. By using (3), we get so that Journal of Function Spaces By the hypothesis of theorem, we get Hence, Similarly, since a 1 + sa 2 < 1. Therefore, with δ = a 1 + sa 2 < 1 and for all n ≥ 0, consequently, we have That is, For any m > n, m, n ∈ ℕ, we have From (10), we get Hence, and hence, Thus, fℵ n g is a Cauchy sequence in W. Since W is complete, there exists some u ∈ W such that ℵ n ⟶ u as n ⟶ ∞ and Assume on the contrary that there exists z ∈ W such that By using the triangular inequality and (2), we obtain 3 Journal of Function Spaces which implies that As n ⟶ ∞ in (18), we obtain that jzj = jφ cb ðu, SuÞj ≤ 0, a contradiction with (16). Therefore jzj = 0. Hence, Su = u. Similarly, we obtain Tu = u.
Assume that u * is another common fixed point of S and T. Then, so that Hence, u = u * , which proves the uniqueness. This completes the proof of the theorem. ☐ Theorem 6. Let ðW, φ cb Þ be a complete complex partial b-metric space with the coefficient s ≥ 1 and S, T : W ⟶ W be mappings satisfying for all ℵ, ϑ ∈ W, where a 1 , a 2 , and a 3 are nonnegative reals with a 1 + 2sa 2 + 2a 3 < 1. Then, S and T have a unique common fixed point in W.
Proof. Let ℵ 0 be arbitrary point in W, and define a sequence fℵ n g in W such that Next, show that the sequence fℵ n g is Cauchy. By using (22), we get so that By the notion of complex partial b-metric space, we get Hence, Similarly, Set δ = ða 1 + sa 2 + a 3 Þ/ð1 − sa 2 − a 3 Þ. Since a 1 + 2sa 2 + 2 a 3 < 1 and for all n ≥ 0, consequently, we have That is,

Journal of Function Spaces
For any m > n, m, n ∈ ℕ, we have From (29), we get Hence,