EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6314 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Rational-Type Contractive Mappings in Bi-Complex Valued Control Metric Spaces and Applications Muhammad Sarwar1,2,∗, Nahid Fatima2, Syed Khayyam Shah3, Asad Khan1, Kamaleldin Abodayeh2 1 Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan 2 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia 3 Department of Sustainable Environment and Energy Systems (SEES), Middle East Technical University, Northern Cyprus Campus, 99738 Kalkanli, Guzelyurt, Mersin 10, Turkey Abstract. This manuscript studies some unique and common fixed point results in the context of bi-complex valued control metric space(BVCMS) using rational-type inequalities. The presented work explains the idea of BVCMS and then shows the necessary criteria for a pair of contractive type mappings in this space to have common fixed points. To show how applicable our results are, we also give an example. Finally, the existence of solutions of a system of fractional differential equations has been studied using the obtained results. 2020 Mathematics Subject Classifications: 47H10, 54H25, 34A08 Key Words and Phrases: Bi-complex valued controlled metric spaces, Fixed Point, Rational type contractions, Fractional Differential Equations 1. Introduction and Preliminaries The notion of conventional differential equations may be extended to non-integer or- ders using fractional differential equations (FDEs), an excellent mathematical tool. Unlike fractional calculus, which introduces the idea of fractional derivatives that may be derived for non-integer orders, conventional differential equations only deal with integer-order derivatives. Numerous scientific fields, including physics, engineering, economics, biology, and more, have begun to pay close attention to the study of fractional differential equa- tions. This is because FDEs offer a more precise and adaptable method for simulating ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6314 Email addresses: sarwarswati@gmail.com (M. Sarwar), nfatima@psu.edu.sa (N. Fatima), shah.syed@metu.edu.tr (S. K. Shah), asad.ah.ak@gmail.com (A. Khan), kamal@psu.edu.sa (K. Abodayeh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 2 of 28 complicated processes that display non-local and memory-dependent behavior. Multiple applications of fractional differential equations (FDEs) may be found in physics, engineer- ing, economics, and biology. They faithfully represent complex systems with anomalous diffusion, long-range interactions, and memory dependence. Control systems, the behavior of viscoelastic materials, asset pricing, image processing, and biomedicine are all improved by FDEs. In the research of FDEs, [1–4] have outstanding data. Furthermore, fixed point theorems offer helpful methods for demonstrating that solu- tions to specific fractional differential equations exist, supporting the mathematical study and real-world use of these equations in various scientific and technical fields. Since the proof of the well-known Banach contraction theorem, the metric fixed point theorem has appeared. Since then, there have been numerous discoveries relating to maps satisfying various contractive requirements and different metric spaces. In the context of non-linear analysis the theory of fixed point gained extraordinary im- portance. After the famous Banach presented the very first result in metric fixed point theory [5], many researchers in this direction put forward the generalized structure of the contraction principle. One of the recent generalizations in this sequel was the introduction of b-metric space by Bakhtin [6]. In generalizing the metrix space the triangular inequal- ity was introduced in a different manner by introducing s ≥ 1, a constant multiple. This generalization leads to some important developments notably, extended b-metric spaces, which was developed by Kamran et al. [7]. Likewise, Mlaiki et al.[8] presented controlled metric spaces in 2018. Moreover, researcher expanded many results in this space such as [9, 10] In this direction, the concept of complex-valued metric spaces was first presented by Azam et al. [11], and they also developed some fixed point results for pairs of mappings that fulfill the contraction requirement for rational expressions. Furthermore, Segre [12] established a base for bi-complex numbers and supported a commutative replacement for the skew field of quaternions. These numbers more strongly and explicitly generalized and extended the complex numbers to quaternions. Bi-complex valued metric spaces (BCVMS) were first proposed by Choi et al. in 2017 [13], who connected the two ideas, bi-complex numbers and complex-valued metric spaces. They developed common fixed- point outcomes for weakly compatible mappings. A contractive type common fixed point for two maps in bicomplex-valued metric spaces was established by [14]. Later, several researchers used this notion to describe their results; see [15–22]. Guechi [23] first discussed the idea of optimum control for Hilfer fractional equations and demonstrated fixed-point outcomes. Similarly, inspired by the above work, G.Mani and S.Haque at [24] investigate the existence of a unique solution of the fractional differential equation in a bi-complex, controlled metric space. In the realm of fixed point theory, the exploration of results within bi-complex valued metric spaces holds substantial significance due to the enriched algebraic and topolog- ical structures these spaces exhibit in comparison to classical and even complex-valued metric spaces. Bi-complex numbers, which generalize complex numbers by incorporating two imaginary units, provide a more flexible and comprehensive framework for analyzing various mathematical models. This generalization is particularly beneficial in domains M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 3 of 28 involving multidimensional or hyper-complex systems, such as quantum mechanics, signal processing, and dynamic systems analysis. The inherent complexity of bi-complex valued metric spaces allows researchers to in- vestigate more intricate contraction mappings and convergence behaviors that may not be adequately represented in traditional settings. This adaptability not only broadens the applicability of fixed point results but also facilitates the development of novel theorems and techniques tailored for more abstract and challenging mathematical problems. Moreover, fixed point results in bi-complex settings serve to unify and generalize exist- ing results in real, complex, and complex-valued metric spaces. As a result, they contribute to a deeper understanding and a more universally applicable theoretical foundation within the scope of fixed point theory. The current manuscript illustrates some unique common Fixed point results on (BCVMS). Then, we provide an application to identify the unique common solution for the fractional differential equation (FDE) system. { ϵDβℵ(y) + ϑ(y,Λ(y)) = 0, ϵDβΩ(y) + υ(y, χ(y)) = 0, 1 < ϵ ≤ 2,y ∈ [0, 1]. λ(0) = ω(0) = ℓ,λ(1) = ω(1) = ȷ, where ℓ and ȷ are constant. Where ϵDβ represent the order of β as the Caputo fractional derivatives and Λ.Ω : [0, 1]× [0,+∞) → [0,+∞). In the subsequent sections, we will review fundamental definitions and notations de- rived from existing literature, which will be employed throughout the remainder of this work. Throughout the manuscript, we represent the sets of real, complex, and bi-complex numbers by C0, C1, and C2, respectively. Segre [12] provided the following list of complex numbers. γ = ℘1 + ℘2i1, where ℘1, ℘2 ∈ C0, i 2 1 = −1. C1 is represented as follows: C1 = {γ : γ = ℘1 + ℘2i1, ℘1, ℘2 ∈ C0}. Let γ ∈ C1, then |γ| = (℘2 1 + ℘2 2) 1/2. All entries in C1 with a real-valued positive norm function ∥ · ∥ : C1 → C+ 0 is defined by ∥γ∥ = (℘2 1 + ℘2 2) 1/2. Segre [12] described the bi-complex number (BCN) as: χ = ℘1 + ℘2i1 + ℘3i2 + ℘4i1i2, where ℘1, ℘2, ℘3, ℘4 ∈ C0, and the independent units i1, i2 satisfy i21 = i22 = −1 and i1i2 = i2i1. We represent the BCN set C2 as: C2 = {χ : χ = ℘1 + ℘2i1 + ℘3i2 + ℘4i1i2, ℘1, ℘2, ℘3, ℘4 ∈ C0}, M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 4 of 28 that is, C2 = {χ : χ = γ1 + i2γ2, γ1, γ2 ∈ C1}, where γ1 = ℘1 + ℘2i1 ∈ C1 and γ2 = ℘3 + ℘4i1 ∈ C1. If χ = γ1 + i2γ2 and ν = ω1 + i2ω2 are any two BCNs, then their sum is χ± ν = (γ1 + i2γ2)± (ω1 + i2ω2) = γ1 ± ω1 + i2(γ2 ± ω2) and the product is χ.ν = (γ1 + i2γ2)(ω1 + i2ω2) = (γ1ω1 − γ2ω2) + i2(γ1ω2 + γ2ω1). In C2, there exist four idempotent elements, they are 0, 1, ε1 = 1+i1i2 2 , ε2 = 1−i1i2 2 of which ε1 and ε2 are non-trivial, such that ε1 + ε2 = 1 and ε1ε2 = 0. Every BCN γ1 + i2γ2 may be written in a specific way as a combination of ε1 and ε2. Namely, χ = γ1 + i2γ2 = (γ1 − i1γ2)ε1 + (γ1 + i1γ2)ε2. The complex components χ1 = (γ1 − i1γ2) and χ2 = (γ1 + i1γ2) are referred to as the idempotent components of the BCN χ, and this representation of χ is known as the idempotent representation of a BCN. Each element in C2 with a positive real-valued norm function ∥ · ∥ : C2 → C+ 0 is defined by ∥χ∥ = ∥γ1 + i2γ2∥ = { ∥γ1∥2 + ∥γ2∥2 }1/2 = [ |γ1 − i1γ2|2 + |γ1 + i1γ2|2 2 ]1/2 = (℘2 1 + ℘2 2 + ℘2 3 + θ24) 1/2 where χ = ℘1 + ℘2i1 + ℘3i2 + ℘4i1i2 = γ1 + i2γ2 ∈ C2. The linear space C2 with respect to a defined norm is a normed linear space, and C2 is complete. Therefore, C2 is a Banach space. If χ, ν ∈ C2, then ∥χν∥ ≤ √ 2∥χ∥∥ν∥ holds instead of ∥fν∥ ≤ ∥χ∥∥ν∥, and therefore C2 is not a Banach algebra. For any two BCNs χ, ν ∈ C2, then (i) χ ⪯ ν ⇔ ∥χ∥ ≤ ∥ν∥; (ii) ∥χ+ ν∥ ≤ ∥χ∥+ ∥ν∥; (iii) ∥℘χ∥ = |℘|∥χ∥, where ℘ is in C0; (iv) ∥χν∥ ≤ √ 2∥χ∥∥ν∥, and ∥χν∥ = √ 2∥χ∥∥ν∥ holds if just one of χ or ν is degenerated; (v) ∥χ−1∥ = ∥χ∥−1, if χ is degenerated with χ ≻ 0; M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 5 of 28 (vi) ∥χν ∥ = ∥χ∥ ∥ν∥ , if ν is a degenerated BCN. The relation ⪯ (partial order) is defined on C2 as given below. Let C2 be a set of BCNs and χ = γ1 + i2γ2 and ν = ω1 + i2ω2 ∈ C2. Then, χ ⪯ ν if and only if γ1 ⪯ ω1 and γ2 ⪯ ω2, i.e., χ ⪯ ν, if one of the following conditions are fulfilled: (i) γ1 = ω1, γ2 = ω2; (ii) γ1 ⪯ ω1, γ2 = ω2; (iii) γ1 = ω1, γ2 ⪯ ω2; (iv) γ1 ⪯ ω1, γ2 ⪯ ω2. It is obvious that we can write χ ⋨ ν if χ ⪯ ν and χ ̸= ν, i.e., if 2, 3, or 4 are fulfilled, and we will write χ ⪯ ν if only 4 is satisfied. Definition 1. [8] Let S ̸= ∅ and ϑ : S × S → [1,+∞). The functional mc : S × S → [0,+∞) is called controlled-type metric (CM) if: (CM1) mc(ς, α) = 0 ⇐⇒ ς = α, (CM2) mc(ς, α) = mc(α, ς), (CM3) mc(ς, β) ≤ ϑ(ς, α)mc(ς, α) + ϑ(α, β)mc(α, β), for all ς, α, β ∈ S. Then, the doublet (S,mc) is called a CM space. Example 1. [8] Choose S = {1, 2, . . . , }. Take mc : S × S −→ [0,+∞) such that mc(ς, α) =  0 if and only if ς = α 1 ς if ς = 2n and α = 2n+ 1, 1 α if ς = 2n+ 1 and α = 2n, 1 otherwise. Consider γ : S × S −→ [1,+∞) as γ(ς, α) =  ς if ς = 2n and α = 2n+ 1, α if ς = 2n+ 1 and α = 2n, 1 otherwise. It is clear that condition (CM1) and (CM2) are satisfied. Now, we have to investigate condition (CM3) case 1. If β = ς or β = α, (d3) is satisfied. case 2. If β ̸= ς and β ̸= α, (CM3) is true when ς = α. Now, we may assume that ς ̸= α. Then, we have ς ̸= α ̸= β. It is clear that (CM3) holds in all of the following possible subcases: M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 6 of 28 (i) α is odd, and ς, β are even. (ii) α, β are odd, and ς is even. (iii) α is even, and ς, β are odd. (iv) ς, α, β are even. (v) β is odd, and ς, α are even. (vi) β is even, and ς, α are odd. (vii) ς, α, β are odd. Thus, mc is a controlled metric type. Definition 2. [25] Let S ̸= ∅ and ϑ : S × S → [1,+∞). The functional db : S × S → C2 is termed the briefly bicomplex valued controlled-type metric (BCVMS) if: (BVCM1) db(ς, α) ≾ 0, (BVCM2) db(ς, α) = 0 ⇐⇒ ς = α, (BVCM3) db(ς, α) = db(α, ς), (BVCM4) db(ς, β) ≾ ϑ(ς, α)db(ς, α) + ϑ(α, β)db(α, β), for all ς, α, β ∈ S. Then, the pair (S, db) is termed as a BVCM space. Example 2. [24] Let S = [0, 1] and define the function db : S × S → C2 by db(σ, v) = |σ − v|2 + i2|σ − v|2. Then, (S, db) is a complete bi-complex b-metric space with ϑ(σ, v) = 2. Remark 1. [24] Every bicomplex-valued b-metric space is a BVCM space. Example 3. [25] Let S = {1, 2, 3} and b : S × S → C2 be defined as follows: db(1, 1) = db(2, 2) = db(3, 3) = 0, db(2, 1) = db(1, 2) = 4 + 4i2, db(3, 2) = db(2, 3) = 1 + 2i2, db(3, 1) = db(1, 3) = 1− i2. Also, let ϑ : S × S → [1,+∞) be defined as follows: ϑ(1, 1) = ϑ(2, 2) = ϑ(3, 3) = 3, ϑ(1, 2) = ϑ(2, 1) = 2, ϑ(2, 3) = ϑ(3, 2) = 4, ϑ(1, 3) = ϑ(3, 1) = 1. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 7 of 28 It is obvious that the conditions (BVCM1) and (BVCM3) fulfilled. Now, case 1. If ς = β then the condition (BVCM3) fulfilled. case 2. If ς = 1 and β = 3 (same as β = 1 and ς = 3) and α = 2, db(ς, β) = |db(1, 3)| = |1− i2| ≾ |12 + 16i2| = |2(4 + 4i2) + 4(1 + 2i2)| ≾ 2 |4 + 4i2|+ 4 |1 + 2i2| = ϑ(1, 2)db(1, 2) + ϑ(2, 3)db(2, 3) = ϑ(ς, α)db(ς, α) + ϑ(α, β)db(α, β). case 3. If ς = 1 and β = 2 (same as β = 1 and ς = 2) and α = 3, db(ς, β) = |db(1, 2)| = |4 + 4i2| ≾ |5 + 7i2| = |1(1− i2) + 4(1 + 2i2)| ≾ |1− i2|+ 4|1 + 2i2| = ϑ(1, 3)db(1, 3) + ϑ(3, 2)db(3, 2) = ϑ(ς, α)db(ς, α) + ϑ(α, β)db(α, β). case 4. If ς = 2 and β = 3 (same as β = 3 and ς = 2) and α = 1, db(ς, β) = |db(2, 3)| = |1 + 2i2| ≾ |9 + 7i2| = |2(4 + 4i2) + 1(1− i2)| ≾ |4 + 4i2|+ 1|1− i2| = ϑ(2, 1)db(2, 1) + ϑ(1, 3)db(1, 3) = ϑ(ς, α)db(ς, α) + ϑ(α, β)db(α, β). Then, (S, db) is a (BCVMS). Definition 3. [25] Let (S, db) be a (BCVMS) with a sequence {κj} in S and κ ∈ S. Then, [i.] (i) A sequence {κj} in S is convergent to κ ∈ S if for all 0 ≾ α ∈ C2, there exists a natural number N such that db(κj,κ) ≾ α for each j ≥ N. Then, limj→+∞ κj = κ or κj → κ as j → +∞. (ii) If, for each 0 ≾ α where α ∈ C2, there exists a natural number N such that db(κj,κj+m) ≾ α for each m ∈ N and j > N. Then, {κj} is referred to as a Cauchy sequence in (S, db). (iii) If each Cauchy sequence is convergent in S, the (BCVMS) (S,£bvc) is said to be complete. Theorem 1. [25] Suppose that (S, db) is (BCVMS) which is complete and ψ : S → S is a map, therefore db(ψς, ψα) ≾ ωdb(ς, α), for all ς, α ∈ S, where 0 < ω < 1. For ς0 ∈ S, we denote ςm = ψmς0. Suppose that max m≥1 lim i→+∞ ϑ(ςi+1, ςi+2)ϑ(ςi+1, ςm) ϑ(ςi, ςi+1) < 1 ω . In addition, for each ς ∈ S, lim η→+∞ ϑ(ςη, ς) and lim η→+∞ ϑ(ς, ςη)∃ and is finite. Then, ψ has a UFP. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 8 of 28 Theorem 2. [25] Suppose that (S, db) is (BCVMS) which is complete and ψ : S → S is a map, therefore db(ψς, ψI) ≾i2 κ(db(ψς, ς) + db(ψα, α)) . for all κ, σ ∈ S, where 0 ≤ ω < 1 2 . For ς0 ∈ S, we denote ςm = ψmς0. Suppose that max m≥1 lim i→+∞ ϑ(ςi+1, ςi+2)ϑ(ςi, ςm) ϑ(ςi, ςi+1) < 1 ω . where ω = κ 1−κ In addition, for each ς ∈ S, lim η→+∞ ϑ(ςη, ς) and lim η→+∞ ϑ(ς, ςη)exists and is finite. Then, ψ has a UFP. 2. MAIN RESULTS In this section, we provide the proof of the unique and common fixed point theorem in bi-complex valued controlled metric space . On the basis of the theorems, we also offer examples and applications. The following is the first theorem. Theorem 3. Let (S, db) be a (BCVMS) which is complete and ψ : S −→ S be such that there are µ, ν, γ ∈ (0, 1) with ω = µ+ν 1−γ < 1, such that db (ψκ, ψσ) ≾ µdb(κ, σ) + νdb (κ, ψκ) + γdb (σ, ψσ) , (1) for all κ, σ ∈ S, where 0 ≤ ω < 1. For ς0 ∈ S, we assume that ςm = ψmς0. Let max m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi,κm) ϑ(κi,κi+1) < 1 ω . (2) Suppose that, lim η→+∞ ϑ(κη,κ) and lim η→+∞ ϑ(κ,κη) exist and are finite, and γ limη→+∞ ϑ(κη,κ) < 1 for every κ ∈ S, then ψ have a UFP. Proof. The examined sequence (κn) verifies κn+1 = Ψ(κn) for all n ∈ N. Clearly, if there exists n0 ∈ N for which κn0+1 = κn0, then Ψ(κn0) = κn0, and the proof is complete. Thus, we assume that κn+1 = κn for every n ∈ N. Thus by (1), we have db(κn,κn+1) ≾ db(Ψ(κn−1),Ψ(κn)) ≾ µdb(κn−1,κn)+νdb(κn−1,Ψ(κn−1))+γdb(κn,Ψ(κn)), which implies that db(κn,κn+1) ≾ µdb(κn−1,κn) + νdb(κn−1,κn) + γdb(κn,κn+1), M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 9 of 28 db(κn,κn+1)− γdb(κn,κn+1) ≾ µdb(κn−1,κn) + νdb(κn−1,κn), (1− γ)db(κn,κn+1) ≾ (µ+ ν)db(κn−1,κn), db(κn,κn+1) ≾ (µ+ ν) (1− γ) db(κn−1,κn), db(κn,κn+1) ≾ (µ+ ν) (1− γ) db(κn−1,κn) = ωdb(κn−1,κn). Thus, we have db(κn,κn+1) ≾ ωdb(κn−1,κn) ≾ ω2db(κn−2,κn−1) ≾ · · · ≾ ωndb(κ0,κ1). For all n,m ∈ N (n < m), we have db(κn,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + ϑ(κn+1,κm)db(κn+1,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + ϑ(κn+1,κm)ϑ(κn+1,κn+2) × db(κn+1,κn+2) + ϑ(κn+1,κm)ϑ(κn+2,κm)db(κn+2,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + ϑ(κn+1,κm)ϑ(κn+1,κn+2) × db(κn+1,κn+2) + ϑ(κn+1,κm)ϑ(κn+2,κm)ϑ(κn+2,κn+3) × db(κn+3,κm) ≾ . . . ≾ ϑ(κn,κn+1)db(κn,κn+1) + m−2∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm)  × ϑ(κi,κi+1)db(κi,κi+1) + m−1∏ i=n+1 ϑ(κi,κm)db(κm−1,κm). (3) This implies that db(κn,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + m−2∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm)  × ϑ(κi,κi+1)db(κi,κi+1) + [ m−1∏ i=n+1 ϑ(κj ,κm) ] ϑ(κm−1,κm)db(κm−1,κm) ≾ ϑ(κn,κn+1)ω ndb(κn,κn+1) + m−2∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm) ϑ(κi,κi+1)ω idb(κ0,κ1) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 10 of 28 + [ m−1∏ i=n+1 ϑ(κj ,κm) ] ϑ(κm−1,κm)ωm−1db(κ0,κ1) = ϑ(κn,κn+1)ω ndb(κ0,κ1) + m−1∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm) ϑ(κi,κi+1)ω idb(κ0,κ1). (4) Let Υℓ = ℓ∑ i=0 [ i∏ j=0 ϑ(κj ,κm) ] ϑ(κi,κi+1)ω idb(κ0,κ1). (5) Consider Λi = [ i∏ j=0 ϑ(κj ,κm) ] ϑ(κi,κi+1)ω idb(κ0,κ1), (6) we have Λi+1 Λi = ϑ(κi+1,κm) ϑ(κi+1,κi+2) ϑ(κi,κi+1) ω. (7) We make sure that the series ∑ i Λi converges in the context of the condition (2) and ratio test. Therefore, there is limn→+∞Υℓ. Thus the Υℓ is cauchy as a result. Now, using (4), we get db(κn,κm) ≾ db(κ0,κ1)[ω nϑ(κi,κi+1) + (Υm−1 −Υn)]. (8) Above, we used ϑ(κ, σ) ≥ 1. Letting n,m→ +∞ in (8) we obtain lim n,m→+∞ db(κn,κm) = 0. (9) Thus, the sequence {κn} is a Cauchy in BCVMS (S, db).For some κ⋆ ∈ S so that lim n→+∞ db(κn,κ⋆) = 0, (10) that is κn → κ⋆ as n→ +∞. We shall now demonstrate that κ⋆ is a fixed point of S. By applying condition (iii) and using (1), we obtain db(κ⋆, ψκ⋆) ≾ ϑ(κ⋆,κn+1)db(κ⋆,κn+1) + ϑ(κn+1, ψκ⋆)db(κn+1, ψκ⋆) = ϑ(κ⋆,κn+1)db(κ⋆,κn+1) + ϑ(κn+1, ψκ⋆)db(ψκn, ψκ⋆) ≾ ϑ(κ⋆,κn+1)db(κ⋆,κn+1) + ϑ(κn+1, ψκ⋆) [µdb(κn,κ⋆) +νdb(κn, ψκn) + γdb(κ⋆, ψκ⋆)] = ϑ(κ⋆,κn+1)db(κ⋆,κn+1) + ϑ(κn+1, ψκ⋆) [µdb(κn,κ⋆) +νdb(κn,κn+1) + γdb(κ⋆, ψκ⋆)] (11) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 11 of 28 Employing the limit, n→ +∞ and utilizing (3),(4) and the fact that limη→+∞ ϑ(κη,κ) and limη→+∞ ϑ(κ,κη) exist and are finite, we would have db(κ⋆, ψκ⋆) ≾ [ γ lim n→+∞ ϑ(κ⋆,κψκ⋆) ] db(κ⋆,κψκ⋆). (12) Suppose that κ⋆ ̸= ψκ⋆, having in mind that [ γ limn→+∞ ϑ(κ⋆,κψκ⋆) ] < 1, so 0 ≺i2 db(κ⋆, ψκ⋆) ≾ [ γ lim n→+∞ ϑ(κ⋆,κψκ⋆) ] db(κ⋆,κψκ⋆) ≺i2 db(κ⋆,κψκ⋆). (13) It is a contradiction. This yields that κ⋆ = ψκ⋆. Uniqueness: Next, we need to justify that κ⋆ is a unique fixed point of ψ. Suppose that there is one more fixed point κ• that is κ• = ψκ• it follows that; db(κ⋆,κ•) = db(ψκ⋆, ψκ•) ≾ µdb(κ⋆,κ•) + νdb(κ⋆,κ⋆) + γdb(κ•,κ•)] db(κ⋆,κ•) ≾ µdb(κ⋆,κ•). Since µ ∈ (0, 1), so we have db(κ⋆,κ•). Therefore, we have κ⋆ = κ• and thus κ⋆ is a unique fixed point of ψ. Theorem 4. Let (S, ϑ, db) be (BCVMS) which is complete and Φ,Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: (i) µ(Φκ) ≤ µ(κ) and ν(Φκ) ≤ ν(κ); (ii) µ(Ψκ) ≤ µ(κ) and ν(Ψκ) ≤ ν(κ); (iii) (µ+ ν)(κ) < 1; (iv) db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) (14) for all κ, σ ∈ S. For κ0 ∈ S, we set µ(κ0) 1−ν(κ0) = ω. Suppose that: sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 ω (15) where κ2n+1 = Φκ2n and κ2n+2 = Ψκ2n+1 for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which exist and are finite. Then, Φ and Ψ have a UCFP. Proof. Suppose µ0 ∈ S. We find {κn} in S by κ2n+1 = Φκ2n and κ2n+2 = Ψκ2n+1 for each n ≥ 0. From hypothesis and (14) we get: db(κ2n+1,κ2n+2) = db(Φκ2n,Ψκ2n+1) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 12 of 28 ≾ µ(κ2n)db(κ2n,κ2n+1) + ν(κ2n) db(κ2n,Φκ2n)db(κ2n+1,Ψκ2n+1) 1 + db(κ2n,κ2n+1) = µ(κ2n)db(κ2n,κ2n+1) + ν(κ2n) db(κ2n,κ2n+1)db(κ2n+1,κ2n+2) 1 + db(κ2n,κ2n+1) ≾ µ(κ2n)db(κ2n,κ2n+1) + ν(κ2n)db(κ2n+1,κ2n+2) = µ(Ψκ2n−1)db(κ2n,κ2n+1) + ν(Ψκ2n−1)db(κ2n+1,κ2n+2) ≾ µ(κ2n−1)db(κ2n,κ2n+1) + ν(κ2n−1)db(κ2n+1,κ2n+2) = µ(Φκ2n−2)db(κ2n,κ2n+1) + ν(Φκ2n−2)db(κ2n+1,κ2n+2) ≾ µ(κ2n−2)db(κ2n,κ2n+1) + ν(κ2n−2)db(κ2n+1,κ2n+2) · · · ≾ µ(κ0)db(κ2n,κ2n+1) + ν(κ0)db(κ2n+1,κ2n+2). Which implies that, db(κ2n+1,κ2n+2) ≾ ( µ(κ0) 1− ν(κ0) ) db(κ2n,κ2n+1). Similarly, db(κ2n+2,κ2n+3) = db(Ψκ2n+1,Φκ2n+2) = db(Φκ2n+2,Ψκ2n+1) ≾ µ(κ2n+2)db(κ2n+2,κ2n+1) +ν(κ2n+2) db(κ2n+2,Φκ2n+2)db(κ2n+1,Ψκ2n+1) 1 + db(κ2n+2,κ2n+1) = µ(κ2n+2)db(κ2n+2,κ2n+1) +ν(κ2n+2) db(κ2n+2,κ2n+3)db(κ2n+1,κ2n+2) 1 + db(κ2n+2,κ2n+1) = µ(κ2n+2)db(κ2n+2,κ2n+1) + ν(κ2n+2)db(κ2n+3,κ2n+2) = µ(Ψκ2n+1)db(κ2n+2,κ2n+1) + ν(Ψκ2n+1)db(κ2n+3,κ2n+2) µ(Ψκ2n+1)db(κ2n+2,κ2n+1) + ν(Ψκ2n+1)db(κ2n+3,κ2n+2) ≾ µ(κ2n+1)db(κ2n+2,κ2n+1) + ν(κ2n+1)db(κ2n+3,κ2n+2) = µ(Φκ2n)db(κ2n+2,κ2n+1) + ν(Φκ2n)db(κ2n+3,κ2n+2) ≾ µ(κ2n)db(κ2n+2,κ2n+3) + ν(κ2n)db(κ2n+3,κ2n+2) . . . ≾ µ(κ0)db(κ2n+2,κ2n+1) + ν(κ0)db(κ2n+3,κ2n+2). M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 13 of 28 Which implies that db(κ2n+3,κ2n+2) ≾ ( µ(κ0) 1− ν(κ0) ) db(κ2n+2,κ2n+1) = ωdb(κ2n+2,κ2n+1. To continue the process in this direction, we obtain, db(κn,κn+1) ≾ ωdb(κn−1,κn) ≾ ω2db(κn−2,κn−1) ≾ · · ·ωndb(κ0,κ1). Thus, db(κn,κn+1) ≾ ωndb(κ0,κ0). (16) For all n,m ∈ N (n < m), we have db(κn,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + ϑ(κn+1,κm)db(κn+1,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + ϑ(κn+1,κm)ϑ(κn+1,κn+2) × db(κn + 1,κn+2) + ϑ(κn+1,κm)ϑ(κn+2,κm)db(κn+2,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + ϑ(κn+1,κm)ϑ(κn+1,κn+2) × db(κn+1,κn+2) + ϑ(κn+1,κm)ϑ(κn+2,κm)ϑ(κn+2,κn+3) × db(κn+2,κn+3) + ϑ(κn+1,κm)ϑ(κn+2,κm) × ϑ(κn+3,κm)db(κn+3,κm) ≾ . . . ≾ ϑ(κn,κn+1)db(κn,κn+1) + m−2∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm)  × ϑ(κi,κi+1)db(κi,κi+1) + m−1∏ i=n+1 ϑ(κi,κm)db(κm−1,κm) (17) This implies that db(κn,κm) ≾ ϑ(κn,κn+1)db(κn,κn+1) + m−2∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm)  × ϑ(κi,κi+1)db(κi,κi+1) + [ m−1∏ i=n+1 ϑ(κj ,κm) ] × ϑ(κm−1,κm)db(κm−1,κm) ≾ ϑ(κn,κn+1)ω ndb(κn,κn+1) + m−2∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm)  × ϑ(κi,κi+1)ω idb(κ0,κ1) + [ m−1∏ i=n+1 ϑ(κj ,κm) ] M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 14 of 28 × ϑ(κm−1,κm)ωm−1db(κ0,κ1) = ϑ(κn,κn+1)ω ndb(κ0,κ1) + m−1∑ i=n+1  i∏ j=n+1 ϑ(κj ,κm)  × ϑ(κi,κi+1)ω idb(κ0,κ1) (18) Let Υℓ = ℓ∑ i=0 [ i∏ j=0 ϑ(κj ,κm) ] ϑ(κi,κi+1)ω idb(κ0,κ1). (19) Consider Λi = [ i∏ j=0 ϑ(κj ,κm) ] ϑ(κi,κi+1)ω idb(κ0,κ1), (20) we have Λi+1 Λi = ϑ(κi+1,κm) ϑ(κi+1,κi+2) ϑ(κi,κi+1) ω. (21) We make sure that the series ∑ i Λi converges in the context of the condition (15) and ratio test. Therefore, there is limn→+∞Υℓ. Thus the sequence Υℓ is cauchy as a result. Now, using (18), we get db(κn,κm) ≾ db(κ0,κ1)[ω nϑ(κi,κi+1) + (Υm−1 −Υn)]. (22) Above, we used ϑ(κ, σ) ≥ 1. Letting n,m→ +∞ in (22) we obtain lim n,m→+∞ db(κn,κm) = 0. (23) Thus, the sequence {κn} is a Cauchy in (BCVMS) (S, db, ϑ). Thus for all κ⋆ ∈ S such that lim n→+∞ db(κn,κ⋆) = 0, (24) that is κn → κ⋆ as n→ +∞. Now, we’ll show that κ⋆ is a fixed point of S. By using (14) and condition (iii), we get db(κ⋆,Φκ⋆) ≾ ϑ(κ⋆,κ2n+2)db(κ⋆,κ2n+2) + ϑ(κ2n+2,Φκ⋆)db(κ2n+2,Φκ⋆) = ϑ(κ⋆,κ2n+2)db(κ⋆,κ2n+2) + ϑ(κ2n+2,Φκ⋆)db(Ψκ2n+1,Φκ⋆) = ϑ(κ⋆,κ2n+2)db(κ⋆,κ2n+2) + ϑ(κ2n+2,Φκ⋆)db(Φκ⋆, ψκ2n+1) = ϑ(κ⋆,κ2n+2)db(κ⋆,κ2n+2) + ϑ(κ2n+2,Φκ⋆) [ µ(κ⋆)db(κ⋆,κ2n+1) + ν(κ⋆) db(κ⋆,Φκ⋆)db(κ2n+1,κ2n+2) 1 + db(κ⋆,κ2n+2) ] (25) Letting n → +∞ and by using (24), there arise contradiction to db(κ⋆,Φκ⋆) ≺i2 0. Thus, db(κ⋆,Φκ⋆) = 0. This implies that κ⋆ = Φκ⋆. Similarly one can show that κ⋆ = M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 15 of 28 Ψκ⋆. Therefore, κ⋆ is common fixed point of Φ and Ψ. Uniqueness: Now we have to show that κ⋆ is a unique fixed point of Ψ and Φ. Assume that there exists another common fixed point κ• that is κ• = Ψκ• = Φκ•. It follows that: db(κ⋆,κ•) = db(Φκ⋆,Ψκ•) ≾ µ(κ⋆)db(κ⋆,κ•) + ν(κ⋆) db(κ⋆,Φκ⋆)db(κ•,Ψκ•) 1 + db(κ⋆,κ•) db(κ⋆,κ•) ≾ µ(κ⋆)db(κ⋆,κ•). Since µ ∈ [0, 1), so we have db(κ⋆,κ•). Therefore, we have κ⋆ = κ• and thus κ⋆ is a unique common fixed point ofΦ and Ψ. Corollary 1. Let (S, ϑ, db) be a complete controlled metric space and Φ,Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: db(Φκ,Ψν) ≾ µdb(κ, σ) + ν db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) for all κ, σ ∈ S. For κ0 ∈ S, we set µ(κ) 1−ν(κ) = ω. Suppose that, sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 ω where κ2n+1 = Φκ2n and κ2n+2 = Ψκ2n+1 for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which are finite and exist. Then, Φ and Ψ have a UCFP. Proof. The proof is trivial by setting µ(κ) = µ and ν(κ) = ν in the Theorem (4). Corollary 2. Let (S, ϑ, db)be a (BCVMS) which is complete and Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: (i) µ(Ψκ) ≤ µ(κ) and ν(Ψκ) ≤ ν(κ); (ii) (µ+ ν)(κ) < 1; (iii) db(Ψκ,Ψν) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψκ)db(σ,Ψσ) 1 + db(κ, σ) for all κ, σ ∈ S. For κ0 ∈ S, we set µ(κ0) 1−ν(κ0) = ω. Suppose that, sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 ω M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 16 of 28 where κn+1 = Ψκn and for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which exist and are finite. Then,Ψ have a UFP. Proof. The proof is trivial by setting Ψ = Φ in the Theorem (4) this result can be obtained. Corollary 3. Let (S, ϑ, db)be a (BCVMS) which is complete and Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: (i) µ(Ψκ) ≤ µ(κ) ; (ii) db(Ψκ,Ψν) ≾ µ(κ)db(κ, σ) for all κ, σ ∈ S. For κ0 ∈ S. Suppose that: sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 µ(κ0) where κn+1 = Ψκn and for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which exist and are finite. Then,Ψ have a UFP. Proof. This result can be obtained by putting ν(κ) = 0 in the Corollary (2). Corollary 4. Let (S, ϑ, db)be a (BCVMS) which is complete and Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: db(Ψκ,Ψν) ≾ µdb(κ, σ) + ν db(κ,Ψκ)db(σ,Ψσ) 1 + db(κ, σ) for all κ, σ ∈ S. For κ0 ∈ S, we set µ(κ0) 1−ν(κ0) = ω. Suppose that, sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 ω where κn+1 = Ψκn and for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which exist and are finite. Then,Ψ have a UFP. Proof. By setting µ(κ) = µ and ν(κ) = ν in the Corollary (2) this result can be obtained. Corollary 5. Let (S, ϑ, db)be a (BCVMS) which is complete and Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 17 of 28 db(Ψκ,Ψν) ≾ µdb(κ, σ) for all κ, σ ∈ S. For κ0 ∈ S. Suppose that, sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 µ where κn+1 = Ψκn and for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which exist and are finite. Then,Ψ have a UFP. Proof. This result can be obtained by putting µ(κ) = µ in Corollary (3). Remark 2. Corollary (4) and (5) are the outcomes of paper [24]. Theorem 5. Let (S, ϑ, db)be a (BCVMS) which is complete and Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: (i) µ(Ψnκ) ≤ µ(κ) and ν(Ψnκ) ≤ ν(κ); (ii) (µ+ ν)(κ) < 1; (iii) db(Ψ nκ,Ψnσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) (26) for all κ, σ ∈ S. For κ0 ∈ S, we set µ(κ0) 1−ν(κ0) = ω. Suppose that, sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 ω where κn+1 = Ψκn and for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which exist and are finite. Then,Ψ have a UFP. Proof. By the Corollary (2), Ψn possess a unique fixed point κ⋆. It can be deduced from, Ψn(Ψκ⋆) = Ψ(Ψnκ⋆) = Ψκ⋆ that Ψκ⋆ is a fixed point of Ψn. Thus Ψκ⋆ = κ⋆ by the uniqueness of a fixed point of κn and then κ⋆ is also fixed point of Ψ. As a result, since the fixed point is unique so it is the fixed point of both Ψ and Ψn. Corollary 6. Let (S, ϑ, db)be a (BCVMS) which is complete and Ψ : S → S. If there exist µ, ν : S → [0, 1) such that: M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 18 of 28 db(Ψ nκ,Ψnσ) ≾ κdb(κ, σ) + νκ db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) for all κ, σ ∈ S. For κ0 ∈ S, we set κ 1−κ = ω. Suppose that, sup m≥1 lim i→+∞ ϑ(κi+1,κi+2)ϑ(κi+1,κm) ϑ(κi,κi+1) < 1 ω where κn+1 = Ψκn and for each n ≥ 0. Assume further, that for every κ ∈ S, we have limn→+∞ ϑ(κn,κ) and limn→+∞ ϑ(κ,κn), which exist and are finite. Then,Ψ have a UFP. Proof. This result can be obtained by setting µ(κ) = µ and ν(κ) = ν in Theorem (5). Example 4. Let db be a (BCVMS) defined on S = {0, 1, 2} such that, db(0, 1) = db(1, 2) = 1 + i2, db(0, 2) = 4 + 4i2 . Where ϑ : S × S → [1,+∞) such that ϑ(0, 0) = ϑ(1, 1) = ϑ(2, 2) = ϑ(1, 2) = 1, ϑ(0, 2) = 2, ϑ(0, 1) = 3 2 . Let ψ : S → S as ψ(0) = 2 and ψ(1) = ψ(2) = 1. Consider, µ = 1 11 and ν = γ = 3 11 , and κ0 = 0 , then κ1 = 2 and κn = 1 for all n ≥ 2. Clearly, (2) is satisfied. Now, we take different cases to check that (1) is also satisfied. Case I: If κ = σ = 0, κ = σ = 1, κ = σ = 2. Then clearly our result can be obtained. Case II: If κ = 0 and σ = 1 then we obtain, db(ψκ, ψσ) = db(ψ(0), ψ(1)) = 1 + i2, db(κ, σ) = db(0, 1) = 1 + i2 db(κ, ψκ) = db(0, ψ(0)) = 4 + 4i2, db(σ, ψσ) = db(1, ψ(1)) = 0 + 0i2. db (ψκ, ψσ) ≾ µdb(κ, σ) + νdb (κ, ψκ) + γdb (σ, ψσ) , The above distances contraction condition of Theorem (3) by using the partial order for BCNs, Thus the result is obvious for Case(II). M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 19 of 28 Case III: If κ = 0 and σ = 2 then we obtain, db(ψκ, ψσ) = db(ψ(0), ψ(2)) = 1 + i2, db(κ, σ) = db(0, 2) = 4 + 4i2, db(κ, ψκ) = db(0, ψ(0)) = 4 + 4i2, db(σ, ψσ) = db(2, ψ(2)) = 1 + i2. db (ψκ, ψσ) ≾ µdb(κ, σ) + νdb (κ, ψκ) + γdb (σ, ψσ) , The above distances contraction condition of Theorem (3) by using the partial order for BCNs, Thus the result is obvious for Case(III). Case IV: If κ = 1 and σ = 2 then we obtain, db(ψκ, ψσ) = db(ψ(1), ψ(2)) = 0 + 0i2, db(κ, σ) = db(1, 2) = 1 + i2, db(κ, ψκ) = db(1, ψ(1)) = 0 + 0i2, db(σ, ψσ) = db(2, ψ(2)) = 1 + i2. db (ψκ, ψσ) ≾ µdb(κ, σ) + νdb (κ, ψκ) + γdb (σ, ψσ) , The above distances satisfies contraction condition of Theorem (3) by using the partial order for BCNs, Thus the result is also obvious for Case(IV). Therefore, all the necessities of the Theorem (3) are true for all the cases so ψ has a unique fixed point. Example 5. Let db be a (BCVMS) defined on S = {0, 1, 2} such that, db(0, 1) = 60 + 60i2, db(1, 2) = 1 + i2 and db(0, 2) = 90 + 90i2 . Where ϑ : S × S → [1,+∞) such that, ϑ(0, 0) = ϑ(1, 1) = ϑ(2, 2) = ϑ(1, 2) = 1, ϑ(0, 2) = 2, ϑ(0, 1) = 3 2 . Let Φ,Ψ : S → S as Ψ(0) = Φ(0) = 1 and Ψ(1) = Φ(1) = Ψ(2) = Φ(2) = 2. Consider, µ(κ) = (κ−3)2 30 and ν(κ) = (κ−3)2 20 , and κ0 = 0 , then κ1 = 2 and κn = 1 for all n ≥ 2. Clearly, condition(i),(ii),(iii) and (15) are satisfied. Now, we take different cases to check that (14) is also satisfied. Case I: M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 20 of 28 If κ = σ = 0, κ = σ = 1, κ = σ = 2. Then clearly our result can be obtained. Case II: If κ = 0 and σ = 1 then we obtain db(Φκ,Ψσ) = db(Φ(0),Ψ(1)) = 1 + i2, db(κ, σ) = db(0, 1) = 60 + 60i2, db(σ,Ψσ) = db(1,Ψ(1)) = 1 + i2, db(κ, ϕκ) = db(0,Φ(0)) = 60 + 60i2. db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) The above distances satisfies the contraction for Theorem (4) by using type partial order for BCNs, Thus the result is obvious for Case(II). Case III: If κ = 0 and σ = 2 then we obtain db(Φκ,Ψσ) = db(Φ(0),Ψ(2)) = 1 + i2, db(κ, σ) = db(0, 2) = 90 + 90i2, db(κ,Φκ) = db(0,Φ(0)) = 60 + 60i2, db(σ, ψσ) = db(2, ψ(2)) = 0 + 0i2. db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) The above distances satisfies the contraction for Theorem (4) by using type partial order for BCNs, Thus the result is obvious for Case(III). Case IV: If κ = 1 and σ = 2 then we obtain, db(Φκ,Ψσ) = db(Φ(1),Ψ(2)) = 0 + 0i2, db(κ, σ) = db(1, 2) = 1 + i2, db(κ,Φκ) = db(1,Φ(1)) = 1 + i2, db(σ,Ψσ) = db(2,Ψ(2)) = 0 + 0i2. db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) The above distances satisfies the contraction for Theorem (4) by using type partial order for BCNs, Thus the result is obvious for Case(IV). Case V: If κ = 1 and σ = 0 then we obtain, db(Φκ,Ψσ) = db(Φ(1),Ψ(0)) = 1 + 1i2, db(κ, σ) = db(1, 0) = 60 + 60i2, db(κ,Φκ) = db(1,Φ(1)) = 1 + i2, db(σ,Ψσ) = db(0,Ψ(0)) = 60 + 60i2. db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 21 of 28 The above distances satisfies the contraction for Theorem (4) by using type partial order for BCNs, Thus the result is obvious for Case(V). Case VI: If κ = 2 and σ = 0 then we obtain, db(Φκ,Ψσ) = db(Ψ(2),Φ(0)) = 1 + i2, db(κ, σ) = db(2, 0) = 90 + 90i2, db(κ,Φκ) = db(2,Φ(2)) = 0 + 0i2, db(σ,Ψσ) = db(0,Ψ(0)) = 1 + i2. db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) The above distances satisfies the contraction for Theorem (4) by using type partial order for BCNs, Thus the result is obvious for Case(VI). Case VII: If κ = 2 and σ = 1 then we obtain, db(Φκ,Ψσ) = db(Ψ(2),Φ(1)) = 0 + 0i2, db(κ, σ) = db(2, 1) = 1 + i2, db(κ,Φκ) = db(2,Φ(2)) = 0 + 0i2, db(σ,Ψσ) = db(2,Ψ(1)) = 0 + 0i2. db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) The above distances satisfies the contraction for Theorem (4) by using type partial order for BCNs, Thus the result is obvious for Case(VII). Therefore all the axioms of Theorem (4) are fulfilled for all the cases, So Φ and Ψ have a unique common fixed point. Example 6. Let db be a BCCM defined on S = {0, 1, 2} such that; db(0, 1) = 60 + 60i2, db(1, 2) = 1 + i2 and db(0, 2) = 90 + 90i2. Where ϑ : S × S → [1,+∞) such that ϑ(0, 0) = ϑ(1, 1) = ϑ(2, 2) = ϑ(1, 2) = 1, ϑ(0, 2) = 2, ϑ(0, 1) = 3 2 . Let Ψ : S → S as Ψn(0) = 1 and Ψn(1) = Ψn(2) = 2. Consider, µ(κ) = (κ−3)2 30 and ν(κ) = (κ−3)2 20 . and κ0 = 0 , then κ1 = 2 and κn = 1 for all n ≥ 2. Clearly, condition(i),(ii),(iii) and (5) are satisfied. Now, we take different cases to check that (26) is also satisfied. Case I: If κ = σ = 0, κ = σ = 1, κ = σ = 2. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 22 of 28 Then clearly our result can be obtained. Case II: If κ = 0 and σ = 1 then we obtain, db(Ψ nκ,Ψnσ) = db(Ψ n(0),Ψn(1)) = 1 + i2, db(κ, σ) = db(0, 1) = 60 + 60i2, db(σ,Ψ nσ) = db(1,Ψ n(1)) = 1 + i2, db(κ,Ψnκ) = db(0,Ψ n(0)) = 60 + 60i2. db(Ψ nκ,Ψnσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) The above distances satisfies contraction condition for Theorem (5) by using type partial order for BCNs, Thus the result is obvious for Case(II). Case III: If κ = 0 and σ = 2 then we obtain db(Ψ nκ,Ψnσ) = db(Ψ n(0),Ψn(2)) = 1 + i2, db(κ, σ) = db(0, 2) = 90 + 90i2, db(κ,Ψnκ) = db(0,Ψ n(0)) = 60 + 60i2, db(σ,Ψ nσ) = db(2,Ψ n(2)) = 0 + 0i2. db(Ψ nκ,Ψnσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) The above distances satisfies contraction condition for Theorem (5) by using type partial order for BCNs, Thus the result is obvious for Case(III). Case IV: If κ = 1 and σ = 2 then we obtain db(Ψ nκ,Ψnσ) = db(Ψ n(1),Ψn(2)) = 0 + 0i2, db(κ, σ) = db(1, 2) = 1 + i2, db(κ,Ψnκ) = db(1,Ψ n(1)) = 1 + i2, db(σ,Ψ nσ) = db(2,Ψ n(2)) = 0 + 0i2. db(Ψ nκ,Ψnσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) The above distances satisfies contraction condition for Theorem (5) by using type partial order for BCNs, Thus the result is obvious for Case(IV). Case V: If κ = 1 and σ = 0 then we obtain db(Ψ nκ,Ψnσ) = db(Ψ n(1),Ψn(0)) = 1 + 1i2, db(κ, σ) = db(1, 0) = 60 + 60i2, db(κ,Ψnκ) = db(1,Ψ n(1)) = 1 + i2, db(σ,Ψ nσ) = db(0,Ψ n(0)) = 60 + 60i2. db(Ψ nκ,Ψnσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 23 of 28 The above distances satisfies contraction condition for Theorem (5) by using type partial order for BCNs, Thus the result is obvious for Case(V). Case VI: If κ = 2 and σ = 0 then we obtain db(Ψ nκ,Ψnσ) = db(Ψ n(2),Ψn(0)) = 1 + i2, db(κ, σ) = db(2, 0) = 90 + 90i2, db(κ,Ψnκ) = db(2,Ψ n(2)) = 0 + 0i2, db(σ,Ψ nσ) = db(0,Ψ n(0)) = 1 + i2. db(Ψ nκ,Ψnσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) The above distances satisfies contraction condition for Theorem (5) by using type partial order for BCNs, Thus the result is obvious for Case(VI). Case VII: If κ = 2 and σ = 1 then we obtain db(Ψ nκ,Ψnσ) = db(Ψ n(2),Ψn(1)) = 0 + 0i2, db(κ, σ) = db(2, 1) = 1 + i2, db(κ,Ψnκ) = db(2,Ψ n(2)) = 0 + 0i2, db(σ,Ψ nσ) = db(2,Ψ n(1)) = 0 + 0i2. db(Ψ nκ,Ψnσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Ψnκ)db(σ,Ψnσ) 1 + db(κ, σ) The above distances satisfies contraction condition for Theorem (5) by using type partial order for BCNs, Thus the result is obvious for Case(VII). Therefore all the axioms of Theorem (5) are fulfilled for all the cases, Thus Ψ has a UFP. 3. Application In this section we solve the system of fractional differential equation with the help of Theorem(4){ ϵDβℵ(y) + ϑ(y,Λ(y)) = 0, ϵDβΩ(y) + υ(y, χ(y)) = 0, 1 < ϵ ≤ 2,y ∈ [0, 1]. λ(0) = ω(0) = ℓ,λ(1) = ω(1) = ȷ, where ℓ and ȷ are constant. (27) Where ϵDβ represent the order of β as the Caputo fractional derivatives and Λ.Ω : [0, 1]× [0,+∞) → [0,+∞). By using the result of [26] we have , Iβ[Dβℵ(y)] = ℵ(y) + C0 + yC1 + y2C2 + ...+ yn− + Cn−1y n−1, where n = [β] + 1 and C ∈ R+ and Iβ is the integral operator of fractional order.The system of integral equations then provides the solution to (27);{ ℵ(y) = ℓ+ y(ȷ− ℓ) + ∫ 1 0 ℑ(y, s)∆(s,κ(s))ds, ω(y) = ℓ+ y(ȷ− ℓ) + ∫ 1 0 ℑ(y, s)Θ(s, σ(s))ds (28) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 24 of 28 where ℑ(y, s) = 1 ℶ(β) { y(1− s)β−1 − (y − s)β−1, 0 ≤ s ≤ y ≤ 1, y(1− s)β−1, 0 ≤ s ≤ y ≤ 1. In the above system (28), let us denote Γ(y) = ℓ+ y(ȷ− ℓ) , Rκ(y) = ∫ 1 0 ℑ(y, s)∆(s,κ(s))ds, Sσ(y) = ∫ 1 0 ℑ(y, s)Θ(s, σ(s))ds. Consider C([0, 1],R) = S is a space described by [0, 1], and db : S×S → C2 is a (BCVMS), such that; db(κ, σ) = sup m∈[0,1] | κ(m)− σ(m) |2 +i2 sup m∈[0,1] | κ(m)− σ(m) |2, for all κ, σ ∈ S. Let ϑ : S × S → [1,+∞) be defined by ϑ(κ, σ) = 2, for all κ, σ ∈ S. Then, (S, db) is (BCVMS). Theorem 6. Consider a system of non-linear fractional differential equations (27). As- sume that the following claims are verified: If for all y ∈ [0, 1] there exist µ, ν : S → [0, 1) such that: (i) µ(Rκ + Γ(y)) ≤ µ(κ) and ν(Rκ + Γ(y)) ≤ ν(κ); (ii) µ(Sσ + Γ(y)) ≤ µ(σ) and ν(Sσ + Γ(y)) ≤ ν(σ); (iii) (µ+ ν)(κ) < 1 ; (iv) ∥ Rκ(y)− Sσ(y) ∥2≾ µ(κ)𭟋1(κ, σ)(y) + ν(κ)𭟋2(κ, σ)(y) (v) supy∈[0,1] ∫ 1 0 ℑ(y, s)ds < 1. For all κ, σ ∈ S, where: 𭟋1(κ, σ)(y) =∥ κ(y)− σ(y) ∥2, and 𭟋2(κ, σ)(y) = ∥ Rκ(y) + Γ(y)− κ(y) ∥2∥ Sσ(y) + Γ(y)− σ(y) ∥2 1+ ∥ κ(y)− σ(y) ∥ . Then the system of FDE (27) has a unique common solution. Proof. Let us define Φ,Ψ : S → S by Φκ = Rκ + Γ, and Ψσ = Sσ + Γ. Then db(Φκ,Ψσ) = sup y∈[0,1] ( ∥ Rκ(y)− Sσ(y) + Γ(y)− Γ(y) ∥2 ) (1 + i2) = sup y∈[0,1] ( ∥ Rκ(y)− Sσ(y) ∥2 ) (1 + i2), M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 25 of 28 db(κ, σ) = sup y∈[0,1] ( ∥ κ(y)− σ(y) ∥2 ) (1 + i2), db(κ,Φκ) = sup y∈[0,1] ( ∥ Rκ(y) + Γ(y)− κ(y) ∥2 ) (1 + i2), and db(σ,Φσ) = sup y∈[0,1] ( ∥ Sσ(y) + Γ(y)− σ(y) ∥2 ) (1 + i2). Now by (14) of Theorem (4) we obtain, ∥ Rκ(y)− Sσ(y) ∥2 ≾ µ(κ) ∥ κ(y)− σ(y) ∥2 + ν(κ) ∥ Rκ(y) + Γ(y)− κ(y) ∥2∥ Sσ(y) + Γ(y)− σ(y) ∥2 (1 + i2) 1+ ∥ κ(y)− σ(y) ∥2 (1 + i2) ≾ µ(κ) ∥ κ(y)− σ(y) ∥2 + ν(κ) ∥ Rκ(y) + Γ(y)− κ(y) ∥2∥ Sσ(y) + Γ(y)− σ(y) ∥2 1+ ∥ κ(y)− σ(y) ∥2 = µ(κ)𭟋1(κ, σ)(y) + ν(κ)𭟋2(κ, σ)(y). Thus, it implies that: (i) µ(Φκ) ≤ µ(κ) and ν(Φκ) ≤ ν(κ); (ii) µ(Ψκ) ≤ µ(κ) and ν(Ψκ) ≤ ν(κ); (iii) (µ+ ν)(κ) < 1; (iv) db(Φκ,Ψσ) ≾ µ(κ)db(κ, σ) + ν(κ) db(κ,Φκ)db(σ,Ψσ) 1 + db(κ, σ) , (29) for all κ, σ ∈ S. Therefore, by the Theorem (4), we obtain that Φ and Ψ have a unique common fixed point. Thus we claim that the system of FDE (27) have a unique common solution. 4. Conclusion Fractional differential equations (FDEs) have emerged as powerful tools for modeling a variety of complex, real-world processes encountered in physics, engineering, biology, and finance. Their ability to incorporate non-integer order derivatives makes them particularly effective in capturing memory effects and hereditary properties inherent in many dynamic systems. Analyzing such equations often involves transforming them into equivalent inte- gral formulations, which in turn require robust mathematical frameworks to examine the M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 26 of 28 existence, uniqueness, and stability of their solutions. Fixed point theory, especially in the context of generalized metric spaces and non-traditional contraction principles, provides a foundational approach in this analytical endeavor. In particular, rational-type contractive conditions have proven to be especially useful in establishing strong convergence results and solution behaviors. In this study, we investigated unique and common fixed point results within the set- ting of bi-complex valued control metric spaces using rational-type inequalities. The use of bi-complex numbers, which extend complex analysis through the introduction of two imag- inary units, provides a richer and more flexible algebraic and topological structure. This enhanced framework enables the examination of more generalized and complex contractive mappings that cannot be adequately addressed within conventional real or complex-valued metric spaces. The theoretical results obtained not only contribute meaningfully to the broader field of fixed point theory but also have direct applications in the analysis and solution of fractional differential equations. Overall, this work opens up new avenues for the study of intricate mathematical models and offers innovative tools for addressing the analytical challenges presented by fractional systems in higher-dimensional and abstract settings. Acknowledgements The authors M. Sarwar, N. Fatima and K. Abodayeh would like to thank Prince Sultan University for APC and for the support of this work through TAS research lab. Author’s contributions All authors contribute equally to the writing of this manuscript. All authors reads and approved the final version. References [1] Anatolii A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo. Theory and Applications of Fractional Differential Equations, volume 204. Elsevier, 2006. [2] Igor Podlubny. Fractional differential equations, volume 198 of. Mathematics in Science and Engineering, 198:7–35, 1999. [3] Vangipuram Lakshmikantham, Srinivasa Leela, and J Vasundhara Devi. Theory of fractional dynamic systems. (No Title), 2009. [4] Kenneth S Miller and Bertram Ross. An introduction to the fractional calculus and fractional differential equations. (No Title), 1993. [5] Stefan Banach. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta mathematicae, 3(1):133–181, 1922. [6] I.A. Bakhtin. The contraction mapping principle in quasimetric spaces. Functional Analysis, 30:26–37, 1989. In Russian. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 27 of 28 [7] Tayyab Kamran, Maria Samreen, and Qurat UL Ain. A generalization of b-metric space and some fixed point theorems. Mathematics, 5(2):19, 2017. [8] Nabil Mlaiki, Hassen Aydi, Nizar Souayah, and Thabet Abdeljawad. Controlled metric type spaces and the related contraction principle. Mathematics, 6(10):194, 2018. [9] Bhawna Soni and Abha Tenguria. Expansion mapping in controlled metric space and extended b-metric space. 2025. [10] Thabet Abdeljawad, Nabil Mlaiki, Hassen Aydi, and Nizar Souayah. Double con- trolled metric type spaces and some fixed point results. Mathematics, 6(12):320, 2018. [11] Akbar Azam, Brian Fisher, and M Khan. Common fixed point theorems in complex valued metric spaces. Numerical Functional Analysis and Optimization, 32(3):243– 253, 2011. [12] Corrado Segre. Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici. Mathematische Annalen, 40(3):413–467, 1892. [13] Junesang Choi, Sanjib Kumar Datta, Tanmay Biswas, and Md Nazimul Islam. Some fixed point theorems in connection with two weakly compatible mappings in bicom- plex valued metric spaces. Honam Mathematical Journal, 39(1):115–126, 2017. [14] Ismat Beg, Sanjib Kumar Datta, and Dipankar Pal. Fixed point in bicomplex val- ued metric spaces. International Journal of Nonlinear Analysis and Applications, 12(2):717–727, 2021. [15] Ramaraj Hariharan and Ramalingam Udhayakumar. Existence of mild solution for fuzzy fractional differential equation utilizing the hilfer-katugampola fractional deriva- tive. An International Journal of Optimization and Control: Theories & Applications, 15(1):82–91, 2025. [16] S Sivasankar, K Nadhaprasadh, M Sathish Kumar, Shrideh Al-Omari, and R Ud- hayakumar. New study on cauchy problems of fractional stochastic evolution systems on an infinite interval. Mathematical Methods in the Applied Sciences, 48(1):890–904, 2025. [17] Zhaohui Gu, Gunaseelan Mani, Arul Joseph Gnanaprakasam, and Yongjin Li. Solv- ing a system of nonlinear integral equations via common fixed point theorems on bicomplex partial metric space. Mathematics, 9(14):1584, 2021. [18] Gunaseelan Mani, Arul Joseph Gnanaprakasam, Khalil Javed, Muhammad Arshad, and Fahd Jarad. Solving a fredholm integral equation via coupled fixed point on bicomplex partial metric space. 2022. [19] Sanjib Kumar Datta, Dipankar Pal, Rakesh Sarkar, and Arghyatanu Manna. On a common fixed point theorem in bicomplex valued b-metric space. Montes Taurus Journal of Pure and Applied Mathematics, 3(3):358–366, 2021. [20] Mohammad Esmael Samei. Convergence of an iterative scheme for multifunctions on fuzzy metric spaces. Sahand Communications in Mathematical Analysis, 15(1):91– 106, 2019. [21] Wasfi Shatanawi and Taqi AM Shatnawi. New fixed point results in controlled metric type spaces based on new contractive conditions. Aims Math, 8(4):9314–9330, 2023. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6314 28 of 28 [22] A Murali and K Muthunagai. Some theorems on fixed points in bi-complex valued metric spaces with an application to integral equations. Journal of the Nigerian Society of Physical Sciences, pages 1750–1750, 2024. [23] Sarra Guechi, Rajesh Dhayal, Amar Debbouche, and Muslim Malik. Analysis and optimal control of φ-hilfer fractional semilinear equations involving nonlocal impulsive conditions. Symmetry, 13(11):2084, 2021. [24] Gunaseelan Mani, Salma Haque, Arul Joseph Gnanaprakasam, Ozgur Ege, and Nabil Mlaiki. The study of bicomplex-valued controlled metric spaces with applications to fractional differential equations. Mathematics, 11(12):2742, 2023. [25] Amer Hassan Albargi, Amnah Essa Shammaky, and Jamshaid Ahmad. Common fixed point results in bicomplex valued metric spaces with application. Mathematics, 11(5):1207, 2023. [26] Stefan G Samko. Fractional integrals and derivatives. Theory and applications, 1993.