Solving a Fredholm integral equation via coupled ﬁxed point on bicomplex partial metric space

: In this paper, we obtain some coupled ﬁxed point theorems on a bicomplex partial metric space. An example and an application to support our result are presented.


Introduction
Segre [1] made a pioneering attempt in the development of special algebra. He conceptualized the commutative generalization of complex numbers, bicomplex numbers, tricomplex numbers, etc. as elements of an infinite set of algebras. Subsequently, in the 1930s, researchers contributed in this area [2][3][4]. The next fifty years failed to witness any advancement in this field. Later, Price [5] developed the bicomplex algebra and function theory. Recent works in this subject [6,7] find some significant applications in different fields of mathematical sciences as well as other branches of science and technology. An impressive body of work has been developed by a number of researchers. Among these works, an important work on elementary functions of bicomplex numbers has been done by Luna-Elizaarrarás et al. [8]. Choi et al. [9] proved some common fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces. Jebril [10] proved some common fixed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces. In 2017, Dhivya and Marudai [11] introduced the concept of a complex partial metric space, suggested a plan to expand the results and proved some common fixed point theorems under a rational expression contraction condition. In 2019, Mani and Mishra [12] proved coupled fixed point theorems on a complex partial metric space using different types of contractive conditions. In 2021, Gunaseelan et al. [13] proved common fixed point theorems on a complex partial metric space. In 2021, Beg et al. [14] proved fixed point theorems on a bicomplex valued metric space. In 2021, Zhaohui et al. [15] proved common fixed theorems on a bicomplex partial metric space. In this paper, we prove coupled fixed point theorems on a bicomplex partial metric space. An example is provided to verify the effectiveness and applicability of our main results. An application of these results to Fredholm integral equations and nonlinear integral equations is given.
There are four idempotent elements in C 2 : They are 0, 1, e 1 = 1+i 1 i 2 2 , e 2 = 1−i 1 i 2 2 of which e 1 and e 2 are nontrivial, such that e 1 + e 2 = 1 and e 1 e 2 = 0. Every bicomplex number z 1 + i 2 z 2 can be uniquely expressed as the combination of e 1 and e 2 , namely This representation of ς is known as the idempotent representation of a bicomplex number, and the complex coefficients ς 1 = (z 1 − i 1 z 2 ) and ς 2 = (z 1 + i 1 z 2 ) are known as the idempotent components of the bicomplex number ς. An element ς = z 1 + i 2 z 2 ∈ C 2 is said to be invertible if there exists another element η in C 2 such that ςη = 1, and η is said to be inverse (multiplicative) of ς. Consequently, ς is said to be the inverse(multiplicative) of η. An element which has an inverse in C 2 is said to be a non-singular element of C 2 , and an element which does not have an inverse in C 2 is said to be a singular element of C 2 . An element ς = z 1 + i 2 z 2 ∈ C 2 is non-singular if and only if ||z 2 1 + z 2 2 || 0 and singular if and only if ||z 2 1 + z 2 2 || = 0. When it exists, the inverse of ς is as follows.
Zero is the only element in C 0 which does not have a multiplicative inverse, and in C 1 , 0 = 0 + i 1 0 is the only element which does not have a multiplicative inverse. We denote the set of singular elements of C 0 and C 1 by O 0 and O 1 , respectively. However, there is more than one element in C 2 which does not have a multiplicative inverse: for example, e 1 and e 2 . We denote this set by O 2 , and clearly A bicomplex number ς = ϑ 1 + ϑ 2 i 1 + ϑ 3 i 2 + ϑ 4 i 1 i 2 ∈ C 2 is said to be degenerated (or singular) if the matrix ϑ 1 ϑ 2 ϑ 3 ϑ 4 is degenerated (or singular). The norm ||.|| of an element in C 2 is the positive real valued function ||.|| : C 2 → C + 0 defined by The linear space C 2 with respect to a defined norm is a normed linear space, and C 2 is complete. Therefore, C 2 is a Banach space. If ς, η ∈ C 2 , then ||ςη|| ≤ √ 2||ς||||η|| holds instead of ||ςη|| ≤ ||ς||||η||, and therefore C 2 is not a Banach algebra. For any two bicomplex numbers ς, η ∈ C 2 , we can verify the following: 2||ς||||η||, and the equality holds only when at least one of ς and η is degenerated, The partial order relation i 2 on C 2 is defined as follows. Let C 2 be the set of bicomplex numbers and ς = z 1 + i 2 z 2 , η = ω 1 + i 2 ω 2 ∈ C 2 . Then, ς i 2 η if and only if z 1 ω 1 and z 2 ω 2 , i.e., ς i 2 η if one of the following conditions is satisfied: In particular, we can write ς i 2 η if ς i 2 η and ς η, i.e., one of 2, 3 and 4 is satisfied, and we will write ς ≺ i 2 η if only 4 is satisfied. Now, let us recall some basic concepts and notations, which will be used in the sequel.
A bicomplex partial metric space is a pair (U, ρ bcpms ) such that U is a non-void set and ρ bcpms is a bicomplex partial metric on U.
Definition 2.8. [15] Let (U, ρ bcpms ) be a bicomplex partial metric space. Let {ϕ τ } be any sequence in U. Then, 1. If every Cauchy sequence in U is convergent in U, then (U, ρ bcpms ) is said to be a complete bicomplex partial metric space.
Inspired by Theorem 2.11, here we prove coupled fixed point theorems on a bicomplex partial metric space with an application.
Thus, we have g = ν. Similarly, we get h = µ. Therefore S has a unique coupled fixed point.
Thus, S has a unique coupled fixed point.
Consider the mapping S : U × U → U defined by

Applications to integral equations
As an application of Theorem 3.3, we find an existence and uniqueness result for a type of the following system of nonlinear integral equations: κ(µ, p)[G 1 (p, ϕ(p)) + G 2 (p, ζ(p))]dp + δ(µ).
Then, the integral equation (4.1) has a unique solution in U.