EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6659 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Fixed Points in Complex Valued Controlled S-Metric Spaces and an Application Haitham Qawaqneh1, Divyanshu Chamoli2, Shivam Rawat3, Hassen Aydi4,5,∗, Monika Bisht6 1 Al-Zaytoonah University of Jordan, Amman 11733, Jordan 2 Department of Mathematics, H.N.B. Garhwal University, Srinagar (Garhwal), Uttarakhand 246174, India. 3 Department of Mathematics, Graphic Era Deemed to be University, Dehradun, Uttarakhand, 248002, India. 4 Institut Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, H. Sousse 4000, Tunisia 5 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa 6 Department of Mathematics, Graphic Era Hill University, Dehradun Campus, Uttarakhand, 248001, India Abstract. In this paper, we present complex valued controlled S-metric spaces, a new general- isation of controlled S-metric spaces. This generalization is also a new extension of the notion of a complex valued Sb-metric space, which differs from the complex valued extended Sb-metric space. Moreover, in this newly generalized notion, we derive certain fixed point results along with an example. Many results from the existing literature are also derived as corollaries of our main results. As an application of our result, we demonstrate the existence of a solution of a Volterra integral equation. 2020 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Fixed point, Complex valued controlled S-metric spaces, Volterra integral equation 1. Introduction The framework of fixed point theory serves as a fundamental tool in understanding the behavior of mappings and has found extensive applications in diverse branches of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6659 Email addresses: h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), chamoli.divyanshu25@gmail.com (D. Chamoli), rawat.shivam09@gmail.com (S. Rawat), hassen.aydi@isima.rnu.tn (H. Aydi), monikabisht391@gmail.com (M. Bisht), monikabisht391@gmail.com (M. Bisht) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 2 of 16 mathematics, including numerical analysis, optimization, and mathematical modeling, see [1–4, 4–7]. Fréchet [8] initially put forth the metric space idea in 1906. Subsequently, numerous researchers have extended the concept of metric space by altering the metric function and weakening different conditions (see, for example, [9–14]). In 1993, Czerwik [12] introduced the notion of a b-metric space as an extension of the concept of a metric space. To broaden the definition of b-metric space, Kamran et al. [15] presented the notion of an extended b-metric space in the year 2017. Mlaiki et al. [16] further introduced controlled metric space, which is a novel type of extended b-metric space. Sedghi et al. [17] presented S-metric spaces in 2012 as a generalization of G-metric spaces [18] and D∗-metric spaces [19]. Several fixed point theorems for S-metric spaces were also found by them. In 2019, Rezaee et al. [20] introduced a new class of generalized metric spaces, called partial S-metric spaces. In 2017, Rohen et al. [21] modified the definition of Sb-metric introduced by Souayan and Mlaiki [22], and proved some coupled common fixed point theorems in Sb-metric spaces. The concept of controlled-S metric spaces was first developed by Gangwar et al. [23] in 2023. Recently, Azmi [24] extended the idea of a controlled S-metric type space to present the concept of a triple controlled S-metric type space, characterized by three control functions: β, µ, and γ. They also proved that fixed points of multivalued mappings exist within the context of controlled S-metric spaces. The concept of complex valued metric spaces was introduced by Azam et al. [25] in 2011. They established certain fixed point theorems for a pair of mappings involving a contraction condition expressed through a rational function. Additionally, Kang et al. [26] introduced the idea of complex valued G-metric spaces and defined contraction mappings in this domain. Further, Mlaiki [27] presented the notion of a new metric space, named, a complex valued S-metric space and established the existence and the uniqueness of a common fixed point for two self mappings. Recently, Ozgur [28] presented the notion of complex valued Gb-metric spaces and presented Kannan fixed point theorem and Banach contraction principle in this setting. In 2017, Priyobarta et al. [29] defined complex valued Sb-metric spaces and proved very interesting fixed point theorems. Motivated by the existing research in the fields of fixed point theory, complex-valued metric spaces and controlled S-metric spaces, the notion of complex valued controlled S-metric spaces (CVCS-metric spaces) is introduced as a means of incorporating complex- valued functions and providing a more flexible framework for addressing the challenges posed by complex-valued systems. This research article aims to present and analyze a series of novel fixed point results within the context of this specialized setting. This exten- sion has proved to be instrumental in capturing the intricacies of complex-valued dynamics and has facilitated the study of various properties, such as convergence, continuity, and completeness, within this complex-valued setting. By establishing crucial fixed point the- orems and exploring their implications in the broader context of CVCS-metric spaces, we aim to provide a deeper understanding of the underlying dynamics and behaviors of mappings within this complex domain. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 3 of 16 2. Preliminaries If C denotes the set of complex numbers and ϖ1, ϖ2 ∈ C, then the partial order ≾ is defined on C in the following manner: ϖ1 ≺ ϖ2 if and only if Re(ϖ1) < Re(ϖ2), Im(ϖ1) < Im(ϖ2) and ϖ1 ≾ ϖ2 if and only if Re(ϖ1) ≤ Re(ϖ2), Im(ϖ1) ≤ Im(ϖ2). We also express ϖ1 ≾ ϖ2 if any one of the following conditions is met: (i) Im(ϖ1) < Im(ϖ2) and Re(ϖ1) = Re(ϖ2), (ii) Im(ϖ1) = Im(ϖ2) and Re(ϖ1) < Re(ϖ2), (iii) Im(ϖ1) = Im(ϖ2) and Re(ϖ1) = Re(ϖ2). Note that 0 ≾ ϖ1 ⪵ ϖ2 implies |ϖ1| < |ϖ2|; and ϖ1 ≾ ϖ2, ϖ2 ≺ ϖ3 implies ϖ1 ≺ ϖ3. Now, we recall some definitions as well as established lemmas which are outlined in the references. Definition 1. [27] Consider a nonempty set Γ. Then a function S : Γ× Γ× Γ → C is a complex valued S-metric on Γ if it satisfies the following conditions for all ϖ, ρ, q, t ∈ Γ: (i) S(ϖ, ρ, q) ⪰ 0, (ii) S(ϖ, ρ, q) = 0 if and only if ϖ = ρ = q, (iii) S(ϖ, ρ, q) ≾ S(ϖ,ϖ, t) + S(ρ, ρ, t) + S(q, q, t). A complex valued S-metric space is denoted by (Γ, S). Definition 2. [27] Consider a complex valued S-metric space (Γ, S). (i) A sequence {τω} ∈ Γ converges to τ if and only if for all ϵ such that 0 ≺ ϵ ∈ C there exists a natural number ω0 such that for all ω ≥ ω0, we have S(τω, τω, τ) ≾ ϵ and it is denoted by lim lim ω→+∞ τω = τ . (ii) A sequence τω ∈ Γ is called a Cauchy sequence if for all ϵ such that 0 ≺ ϵ ∈ C there exists a natural number ω0 such that for all ω, ϱ ≥ ω0, we have S(τω, τω, τϱ) ≺ ϵ. (iii) A complex valued S-metric space (Γ, S) is called complete if every Cauchy sequence in Γ is convergent. Lemma 1. [27] Consider a complex valued S-metric space (Γ, S) and a sequence {τω} in Γ. Then {τω} is said to converge to τ if and only if |S(τω, τω, τ)| → 0 as ω → +∞. Lemma 2. [27] Consider a complex valued S-metric space (Γ, S) and a sequence {τω} in Γ. Then {τω} is a Cauchy sequence if and only if |S(τω, τω, τω+ϱ)| → 0 as ω, ϱ → +∞. Lemma 3. [27] Consider a complex valued S-metric space (Γ, S), then S(ς, ς, τ) = S(τ, τ, ς) for all τ, ς ∈ Γ. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 4 of 16 3. Fixed Point Results The need to generalize S-metric spaces to handle complex-valued distances under ad- ditional control conditions motivates the introduction of the concept of a CVCS-metric space as follows. Definition 3. Consider a set Γ ̸= ∅ and let S : Γ×Γ×Γ → C and α : Γ×Γ×Γ → [1,+∞) be two functions adhering to the following conditions for all τ, ς, ρ,ϖ ∈ Γ: (CV CS1) 0 ≾ S(τ, ς, ρ); (CV CS2) S(τ, ς, ρ) = 0 if and only if τ = ς = ρ; (CV CS3) S(τ, ς, ρ) ≾ α(τ, τ,ϖ)S(τ, τ,ϖ) + α(ς, ς,ϖ)S(ς, ς,ϖ) + α(ρ, ρ,ϖ)S(ρ, ρ,ϖ). Then, the tuple (Γ, S, α) is called a CVCS-metric space. Definition 2.2, Lemma 2.3, 2.4 and 2.5 can be stated and proved in a similar way for a CVCS-metric space. Definition 4. Consider a CVCS-metric space (Γ, S, α). (i) A sequence {τω} ∈ Γ is CVCS-convergent to τ if and only if for all ϵ such that 0 ≺ ϵ ∈ C there exists a natural number ω0 such that for all ω ≥ ω0, we have S(τω, τω, τ) ≾ ϵ and it is denoted by lim lim ω→+∞ τω = τ . (ii) A sequence τω ∈ Γ is called a CVCS-Cauchy sequence if for all ϵ such that 0 ≺ ϵ ∈ C there exists a natural number ω0 such that for all ω, ϱ ≥ ω0, we have S(τω, τω, τϱ) ≺ ϵ. (iii) A CVCS-metric space (Γ, S, α) is called complete if every CVCS-Cauchy sequence in Γ is convergent. Lemma 4. Consider a CVCS-metric space (Γ, S, α) and a sequence {τω} in Γ. Then {τω} is said to be CVCS-convergent to τ if and only if |S(τω, τω, τ)| → 0 as ω → +∞. Lemma 5. Consider a CVCS-metric space (Γ, S, α) and a sequence {τω} in Γ. Then {τω} is a CVCS-Cauchy sequence if and only if |S(τω, τω, τω+ϱ)| → 0 as ω, ϱ → +∞. Lemma 6. Consider a CVCS-metric space (Γ, S, α), then S(ς, ς, τ) = S(τ, τ, ς) for all τ, ς ∈ Γ. Theorem 1. Consider a CVCS-metric space (Γ, S, α) and a sequence {τω} in Γ. Then, {τω} is a CVCS-Cauchy sequence if and only if |S(τω, τϱ, τl)| → 0 as ω, ϱ, l → +∞. Proof. The proof is on the similar lines as in Theorem 3.3 of [29]. Theorem 2. Consider a complete CVCS-metric space (Γ, S, α). Define a function f : Γ → Γ such that S(fτ, fτ, fς) ≾ θS(τ, τ, ς), (1) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 5 of 16 where θ ∈ (0, 1). For τ0 ∈ Γ, choose a sequence τω = f(τω−1), ω ∈ N. Suppose that sup ϱ≥1 lim i→+∞ α(τi+1, τi+1, τi+2).α(τϱ, τϱ, τi+1) α(τi, τi, τi+1) < 1 2θ , (2) and lim ω→+∞ α(τω, τω, τω+1) exists. Then there is a unique fixed point of f . Proof. Consider an arbitrary element τ0 of Γ. Define a sequence {τω} in Γ by τω = f(τω−1) = fω(τ0). Let τω ̸= τω+1 for all ω. From (1), we obtain S(τω, τω, τω+1) ≾ θS(τω−1, τω−1, τω) ≾ θ2S(τω−2, τω−2, τω−1) ... ≾ θωS(τ0, τ0, τ1). Considering ϱ > ω and using triangle inequality, we obtain S(τϱ, τϱ, τω) ≾ α(τϱ, τϱ, τω+1)S(τϱ, τϱ, τω+1) + α(τϱ, τϱ, τω+1)S(τϱ, τϱ, τω+1) + α(τω, τω, τω+1)S(τω, τω, τω+1) ≾ 2α(τϱ, τϱ, τω+1)S(τϱ, τϱ, τω+1) + α(τω, τω, τω+1)S(τω, τω, τω+1). Once more, applying the triangle inequality to S(uϱ, uϱ, uω+1), we obtain S(τϱ, τϱ, τω) ≾ 2α(τϱ, τϱ, τω+1)(2α(τϱ, τϱ, τω+2)S(τϱ, τϱ, τω+2) + α(τω+1, τω+1, τω+2)S(τω+1, τω+1, τω+2)) + α(τω, τω, τω+1)S(τω, τω, τω+1) ≾ α(τω, τω, τω+1)S(τω, τω, τω+1) + 2α(τϱ, τϱ, τω+1)α(τω+1, τω+1, τω+2)S(τω+1, τω+1, τω+2) + 22α(τϱ, τϱ, τω+1) α(τϱ, τϱ, τω+2)S(τϱ, τϱ, τω+2). Proceeding with the continuous application of triangle inequality in the same manner, we obtain S(τϱ, τϱ, τω) ≾ α(τω, τω, τω+1)S(τω, τω, τω+1) + ϱ−2∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1)S(τi, τi, τi+1) + 2ϱ−ω−1 ϱ−1∏ k=ω+1 α(τϱ, τϱ, τk)S(τϱ, τϱ, τϱ−1) ≾ α(τω, τω, τω+1)(θ ωS(τ0, τ0, τ1)) + ϱ−1∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1)(θ iS(τ0, τ0, τ1)). H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 6 of 16 Now, consider the series Sϱ = ∑ϱ−1 i=ω 2i−ω ∏i j=ω α(τϱ, τϱ, τj)α(τi, τi, τi+1)(θ i). This series converges by ratio test under the condition: supϱ≥i lim i→+∞ α(τi+1,τi+1,τi+2).α(τϱ,τϱ,τi+1) α(τi,τi,τi+1) < 1 2θ . Letting ϱ, ω → +∞, we obtain lim ϱ,ω→+∞ S(τϱ, τϱ, τω) = 0. By CV CS3, we have S(τω, τϱ, τl) ≾ α(τϱ, τϱ, τω)S(τϱ, τϱ, τω) + α(τl, τl, τω)S(τl, τl, τω) for all ω, ϱ, l ∈ N. Thus, |S(τω, τϱ, τl)| ≤ α(τϱ, τϱ, τω)|S(τϱ, τϱ, τω)|+ α(τl, τl, τω)|S(τl, τl, τω)|. Taking limit ω, ϱ, l → +∞, we obtain |S(τω, τϱ, τl)| → 0. So {τω} is a CVCS-Cauchy sequence. Completeness of (Γ, S, α) gives us that there is an element ς∗ ∈ Γ such that {τω} is CVCS-convergent to ς∗. Now, we’ll prove that f(ς∗) = ς∗. Consider, S(τω+1, τω+1, f(ς ∗)) ≾ θS(τω, τω, ς ∗), that is, |S(τω+1, τω+1, f(ς ∗))| ≤ θ|S(τω, τω, ς∗)|. Letting ω → +∞, we obtain f(ς∗) = ς∗. Now towards the end, the uniqueness will be proved. Let τ∗ be some other fixed point of f . Consider S(ς∗, ς∗, τ∗) = S(f(ς∗), f(ς∗), f(τ∗)) ≾ θS(ς∗, ς∗, τ∗), that is, |S(f(ς∗), f(ς∗), f(τ∗))| ≤ θ|S(ς∗, ς∗, τ∗)|. This implies that |S(ς∗, ς∗, τ∗)| ≤ 0. Thus, ς∗ = τ∗, that is, τ∗ is the unique fixed point of f . Example 1. Consider the complex plane Γ = C equipped with a controlled function α : Γ3 → [1,+∞) defined by α(τ, ς,ϖ) = 1 + |τ |+ |ς|+ |ϖ| 1 + |τ |+ |ς|+ |ϖ| , and a complex-valued S-metric S : Γ3 → C given by S(τ, ς,ϖ) = max { |τ −ϖ|, |ς −ϖ|, |τ − ς| 2 } . H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 7 of 16 Then (Γ, S, α) is a complete CVCS-metric space. Define the mapping f : Γ → Γ as f(τ) = τ 5 . Now, for any τ, ς ∈ Γ, S(fτ, fτ, fς) = max {∣∣∣τ 3 − ς 3 ∣∣∣ , |τ − ς| 2 } = |τ − ς| 3 ≾ θS(τ, τ, ς), for any θ ∈ (15 , 1). Also, let τ0 = 1 and τω = fω(τ0) = 1 5ω . Then, α(τω, τω, τω+1) = 1 + 2 5ω + 1 5ω+1 1 + 2 5ω + 1 5ω+1 → 1 as ω → +∞, and sup ϱ≥1 lim ω→+∞ α(τω+1, τω+1, τω+2)α(τϱ, τϱ, τω+1) α(τω, τω, τω+1) = 1 < 1 2 · 1 3 = 3 2 . Thus, (2) holds and the sequence {τω} converges to 0, and f(0) = 0 is the unique fixed point. Corollary 1. Let (Γ, S) be a complete complex valued S-metric space and f : Γ → Γ be a mapping, such that for τ, ς ∈ Γ SH(f(τ), f(τ), f(ς)) ≾ θS(τ, τ, ς), where θ ∈ (0, 1). Then, f has a unique fixed point. Proof. The required result is obtained by taking α(τ, ς,ϖ) = 1 and following the same procedures as in the preceding theorem. Theorem 3. Let (Γ, S, α) be a complete CVCS-metric space and f : Γ → Γ satisfy the following for every τ, ς ∈ Γ: S(f(τ), f(τ), f(ς)) ≾ aS(τ, τ, f(τ)) + bS(ς, ς, f(ς)), (3) where a, b ∈ (0, 1) with a + b < 1. For τ0 ∈ Γ, choose a sequence τω = f(τω−1), ω ∈ N. Suppose that sup ϱ≥1 lim i→+∞ α(τi+1, τi+1, τi+2).α(τϱ, τϱ, τi+1) α(τi, τi, τi+1) < 1− b 2a , (4) and lim ω→+∞ α(τω, τω, τω+1) exists. Then, f admits a unique fixed point. Proof. Consider an arbitrary τ0 ∈ Γ. Define a sequence {τω} in Γ by τω = f(τω−1) = fω(τ0). Suppose that τω ̸= τω+1 for all ω. From (3), we obtain S(τω, τω, τω+1) ≾ aS(τω−1, τω−1, τω) + bS(τω, τω, τω+1) that is S(τω, τω, τω+1) ≾ a 1− b S(τω−1, τω−1, τω) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 8 of 16 ≾ a2 (1− b)2 S(τω−2, τω−2, fτω−1) ... ≾ aω (1− b)ω S(τ0, τ0, τ1). Let ϱ > ω, where ϱ, ω ∈ N. Using the triangle inequality, we obtain S(τϱ, τϱ, τω) ≾ α(τϱ, τϱ, τω+1)S(τϱ, τϱ, τω+1) + α(τϱ, τϱ, τω+1)S(τϱ, τϱ, τω+1) + α(τω, τω, τω+1)S(τω, τω, τω+1) = 2α(τϱ, τϱ, τω+1)S(τϱ, τϱ, τω+1) + α(τω, τω, τω+1)S(τω, τω, τω+1). Again, using the triangle inequality on S(τϱ, τϱ, τω+1), one writes S(τϱ, τϱ, τω) ≾ 2α(τϱ, τϱ, τω+1)(2α(τϱ, τϱ, τω+2)S(τϱ, τϱ, τω+2) + α(τω+1, τω+1, τω+2)S(τω+1, τω+1, τω+2)) + α(τω, τω, τω+1)S(τω, τω, τω+1) = α(τω, τω, τω+1)S(τω, τω, τω+1) + 2α(τϱ, τϱ, τω+1)α(τω+1, τω+1, τω+2)S(τω+1, τω+1, τω+2) + 22α(τϱ, τϱ, τω+1) α(τϱ, τϱ, τω+2)S(τϱ, τϱ, τω+2). Using triangle inequality again in a same way, we obtain S(τϱ, τϱ, τω) ≾ α(τω, τω, τω+1)S(τω, τω, τω+1) + ϱ−2∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1)S(τi, τi, τi+1) + 2ϱ−ω−1 ϱ−1∏ k=ω+1 α(τϱ, τϱ, τk)S(τϱ−1, τϱ−1, τϱ) ≾ α(τω, τω, τω+1) ( a 1− b )ω S(τ0, τ0, τ1) + ϱ−1∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1) ( a 1− b )i S(τ0, τ0, τ1). As ω, ϱ → +∞, one has ϱ−1∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1) ( a 1− b )i → 0, H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 9 of 16 if supϱ≥1 lim i→+∞ α(τi+1,τi+1,τi+2).α(τϱ,τϱ,τi+1) α(τi,τi,τi+1) < 1−b 2a . For ϱ > n, we obtain lim ϱ,ω→+∞ S(τϱ, τϱ, τω) = 0. By CV CS3, we have S(τω, τϱ, τl) ≾ α(τϱ, τϱ, τω)S(τϱ, τϱ, τω) + α(τl, τl, τω)S(τl, τl, τω) for all ω, ϱ, l ∈ N. Thus, |S(τω, τϱ, τl)| ≤ α(τϱ, τϱ, τω)|S(τϱ, τϱ, τω)|+ α(τl, τl, τω)|S(τl, τl, τω)|. Considering the limit as ω, ϱ, l → +∞, we have |S(τω, τϱ, τl)| → 0. So {τω} is a CVCS- Cauchy sequence. Completeness of (Γ, S, α) gives us that there is an element ς∗ ∈ Γ such that {τω} is CVCS-convergent to ς∗. Now, we’ll prove that f(ς∗) = ς∗. Consider, S(τω+1, τω+1, f(ς ∗)) ≾ aS(τω, τω, τω+1) + bS(ς∗, ς∗, f(ς∗)) that is, |S(τω+1, τω+1, f(ς ∗))| ≤ a|S(τω, τω, τω+1)|+ b|S(ς∗, ς∗, f(ς∗))|. Letting ω → +∞, we obtain |S(ς∗, ς∗, f(ς∗))| ≤ b|S(ς∗, ς∗, f(ς∗))|, which implies f(ς∗) = ς∗. Now towards the end, the uniqueness will be proved. Let τ∗ be some other fixed point of f . Consider S(ς∗, ς∗, τ∗) = S(f(ς∗), f(ς∗), f(τ∗)) ≾ aS(ς∗, ς∗, f(ς∗)) + bS(τ∗, τ∗, f(τ∗)), that is, |S(f(ς∗), f(ς∗), f(τ∗))| ≤ 0. This implies that |S(ς∗, ς∗, τ∗)| ≤ 0. Thus, ς∗ = τ∗, i.e., τ∗ is the unique fixed point of f . Corollary 2. Let (Γ, S, α) be a complete complex valued S-metric space. Let f : Γ → Γ be a mapping such that, for τ, ς ∈ Γ, S(f(τ), f(τ), f(ς)) ≾ aS(τ, τ, f(τ)) + bS(ς, ς, f(ς)), (5) where a ∈ [0, 12), with 2a+ b < 1. Then, there exists a fixed point for f in Γ. Proof. If we take α(τ, ς,ϖ) = 1 and proceed with the same steps as outlined in the proof of Theorem 3, the corollary is proved. Theorem 4. Let (Γ, S, α) be a complete complex valued S-metric space. Let f and g be two self mappings on Γ that meet the contraction condition given below: S(fτ, fτ, gς) ≾ γS(τ, τ, ς)+β S(τ, τ, fτ)S(ς, ς, gς) 2α(gς, gς, τ)S(τ, τ, gς) + α(fτ, fτ, ς)S(ς, ς, fτ) + α(ς, ς, τ)S(τ, τ, ς) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 10 of 16 for all τ, ς ∈ Γ such that τ ̸= ς, S(τ, τ, gς) + S(ς, ς, fτ) + S(τ, τ, ς) ̸= 0 for any two nonnegative real numbers γ, β satisfying the condition γ + β < 1 or S(fτ, fτ, gς) = 0 if S(τ, τ, gς) + S(ς, ς, fτ) + S(τ, τ, ς) = 0. Suppose that sup ϱ≥1 lim i→+∞ α(τi+1, τi+1, τi+2).α(τϱ, τϱ, τi+1) α(τi, τi, τi+1) < 1 2(γ + β) , (6) and lim ω→+∞ α(τω, τω, τω+1) exists. Then there exists a unique common fixed point for f and g. Proof. Let τ0 ∈ Γ and τ2κ+1 = fτ2κ, τ2κ+2 = gτ2κ+1, κ ∈ {0, 1, 2, }. Thus, S(τ2κ+1, τ2κ+1, τ2κ+2) = S(fτ2κ, fτ2κ, gτ2κ+1) ≾ γS(τ2κ, τ2κ, τ2κ+1) + βS(τ2κ, τ2κ, fτ2κ)S(τ2κ+1, τ2κ+1, gτ2κ+1) (2α(gτ2κ+1, gτ2κ+1, τ2κ)S(τ2κ, τ2κ, gτ2κ+1) + α(fτ2κ, fτ2κ, τ2κ+1)S(τ2κ+1, τ2κ+1, fτ2κ) + α(τ2κ+1, τ2κ+1, τ2κ)S(τ2κ, τ2κ, τ2κ+1)) = γS(τ2κ, τ2κ, τ2κ+1) + βS(τ2κ, τ2κ, τ2κ+1)S(τ2κ+1, τ2κ+1, τ2κ+2) (2α(τ2κ+2, τ2κ+2, τ2κ)S(τ2κ, τ2κ, τ2κ+2) + α(τ2κ+1, τ2κ+1, τ2κ+1)S(τ2κ+1, τ2κ+1, τ2κ+1) + α(τ2κ+1, τ2κ+1, τ2κ)S(τ2κ, τ2κ, τ2κ+1)) . Hence, |S(τ2κ+1, τ2κ+1, τ2κ+2)| ≤ γ|S(τ2κ, τ2κ, τ2κ+1)| + β|S(τ2κ, τ2κ, τ2κ+1)||S(τ2κ+1, τ2κ+1, τ2κ+2)| (|2α(τ2κ+2, τ2κ+2, τ2κ)S(τ2κ, τ2κ, τ2κ+2) + α(τ2κ+1, τ2κ+1, τ2κ)S(τ2κ, τ2κ, τ2κ+1)|) . By CV CS3 and Lemma 3, we see that |S(τ2κ+1, τ2κ+1, τ2κ+2)| = |S(τ2κ+2, τ2κ+2, τ2κ+1)| ≤ |2α(τ2κ+2, τ2κ+2, τ2κ)S(τ2κ+2, τ2κ+2, τ2κ) + α(τ2κ+1, τ2κ+1, τ2κ)S(τ2κ+1, τ2κ+1, τ2κ)| = |2α(τ2κ+2, τ2κ+2, τ2κ)S(τ2κ, τ2κ, τ2κ+2) + α(τ2κ+1, τ2κ+1, τ2κ)S(τ2κ, τ2κ, τ2κ+1)|. Thus, |S(τ2κ+1, τ2κ+1, τ2κ+2)| ≤ γ|S(τ2κ, τ2κ, τ2κ+1)|+ β|S(τ2κ, τ2κ, τ2κ+1)| = (γ + β)|S(τ2κ, τ2κ, τ2κ+1)|. Likewise, we obtain |S(τ2κ+2, τ2κ+2, τ2κ+3)| = (γ + β)|S(τ2κ+1, τ2κ+1, τ2κ+2)|. Therefore, |S(τω, τω, τω+1)| ≤ (γ + β)|S(τω−1, τω−1, τω| ≤ ... ≤ (γ + β)ω|S(τ0, τ0, τ1). H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 11 of 16 Hence, for any ϱ > ω, we have S(τϱ, τϱ, τω) ≾ 2α(τϱ, τϱ, τω+1)(2α(τϱ, τϱ, τω+2)S(τϱ, τϱ, τω+2) + α(τω+1, τω+1, τω+2)S(τω+1, τω+1, τω+2)) + α(τω, τω, τω+1)S(τω, τω, τω+1) = α(τω, τω, τω+1)S(τω, τω, τω+1) + 2α(τϱ, τϱ, τω+1)α(τω+1, τω+1, τω+2)S(τω+1, τω+1, τω+2) + 22α(τϱ, τϱ, τω+1) α(τϱ, τϱ, τω+2)S(τϱ, τϱ, τω+2). Using triangle inequality again in a same way, we obtain S(τϱ, τϱ, τω) ≾ α(τω, τω, τω+1)S(τω, τω, τω+1) + ϱ−2∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1)S(τi, τi, τi+1) + 2ϱ−ω−1 ϱ−1∏ k=ω+1 α(τϱ, τϱ, τk)S(τϱ−1, τϱ−1, τϱ) ≾ α(τω, τω, τω+1)(γ + β)ωS(τ0, τ0, τ1) + ϱ−1∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1)(γ + β)iS(τ0, τ0, τ1). As ω, ϱ → +∞, one has ϱ−1∑ i=ω+1 2i−ω i∏ j=ω+1 α(τϱ, τϱ, τj)α(τi, τi, τi+1)(γ + β)i → 0, if supϱ≥1 lim i→+∞ α(τi+1,τi+1,τi+2).α(τϱ,τϱ,τi+1) α(τi,τi,τi+1) < 1 2(γ+β) . For ϱ > ω, we obtain lim ϱ,ω→+∞ S(τω, τω, τϱ) = 0. By CV CS3, we have S(τω, τϱ, τl) ≾ α(τϱ, τϱ, τω)S(τϱ, τϱ, τω) + α(τl, τl, τω)S(τl, τl, τω) for all ω, ϱ, l ∈ N. Thus, |S(τω, τϱ, τl)| ≤ α(τϱ, τϱ, τω)|S(τϱ, τϱ, τω)|+ α(τl, τl, τω)|S(τl, τl, τω)|. Considering the limit as ω, ϱ, l → +∞, we have |S(τω, τϱ, τl)| → 0. So {τω} is a CVCS- Cauchy sequence. Completeness of (Γ, S, α) gives us that there is an element ς∗ ∈ Γ such H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 12 of 16 that {τω} is CVCS-convergent to ς∗. Now, we’ll prove that f(ς∗) = g(ς∗) = ς∗. Assume that fς∗ ̸= ς∗. Thus, 0 ≺ ϖ = S(ς∗, ς∗, fς∗). Therefore, ϖ ≾ 2α(ς∗, ς∗, τ2κ+2)S(ς ∗, ς∗, τ2κ+2) + α(τ2κ+2, τ2κ+2, fς ∗)S(τ2κ+2, τ2κ+2, fς ∗) ≾ 2α(ς∗, ς∗, τ2κ+2)S(ς ∗, ς∗, τ2κ+2) + α(τ2κ+2, τ2κ+2, fς ∗)S(gτ2κ+1, gτ2κ+1, fς ∗) ≾ 2α(ς∗, ς∗, τ2κ+2)S(ς ∗, ς∗, τ2κ+2) + γα(τ2κ+2, τ2κ+2, fς ∗)S(τ2κ+1, τ2κ+1, ς ∗) + βα(τ2κ+2, τ2κ+2, fς ∗)S(τ2κ+1, τ2κ+1, τ2κ+2)S(ς ∗, ς∗, fς∗) 2α(fς∗, fς∗, τ2κ+1)S(τ2κ+1, τ2κ+1, fς ∗) + α(τ2κ+2, τ2κ+2, ς ∗)S(ς∗, ς∗, τ2κ+2) + α(ς∗, ς∗, τ2κ+1)S(τ2κ+1, τ2κ+1, ς ∗). Hence, |ϖ| ≤ 2α(ς∗, ς∗, τ2κ+2)|S(ς∗, ς∗, τ2κ+2)|+ γα(τ2κ+2, τ2κ+2, fς ∗)|S(τ2κ+1, τ2κ+1, ς ∗)| + βα(τ2κ+2, τ2κ+2, fς ∗)|S(τ2κ+1, τ2κ+1, τ2κ+2)||z| 2α(fς∗, fς∗, τ2κ+1)|S(τ2κ+1, τ2κ+1, fς ∗)|+ α(τ2κ+2, τ2κ+2, ς ∗)|S(ς∗, ς∗, τ2κ+2)|+ α(ς∗, ς∗, τ2κ+1)|S(τ2κ+1, τ2κ+1, ς ∗)|. Our assumption on ϖ is contradicted by the fact that the right-hand side of the preceding inequality tends to 0 as κ → +∞. Consequently, it is possible to demonstrate that fς∗ = ς∗ and that gς∗ = ς∗. As a result, f and g have a common fixed point. In order to demonstrate uniqueness assume that there is another common fixed point of f and g, say ρ∗. Therefore, S(ς∗, ς∗, ρ∗) = S(fς∗, fς∗, gρ∗) ≾ γS(ς∗, ς∗, ρ∗) + β S(ς∗, ς∗, fς∗)S(ρ∗, ρ∗, gρ∗) 2α(gρ∗, gρ∗, ς∗)S(ς∗, ς∗, gρ∗) + α(fς∗, fς∗, ρ∗)S(ρ∗, ρ∗, fς∗) + α(ρ∗, ρ∗, ς∗)S(ς∗, ς∗, ρ∗) = γS(ς∗, ς∗, ρ∗). This implies that |S(ς∗, ς∗, ρ∗)| ≤ γ|S(ς∗, ς∗, ρ∗)|. However, since γ < 1, we can thus conclude that S(ς∗, ς∗, ρ∗) = 0, and as a result, ς∗ = ρ∗ as intended. To complete our proof, suppose that for every natural number κ, if we obtain S(τ2κ, τ2κ, gτ2κ+1) + S(τ2κ+1, τ2κ+1, fτ2κ) + S(τ2κ, τ2κ, τ2κ+1) = 0, then S(fτ2κ, fτ2κ, gτ2κ+1) = 0, which implies τ2κ = fτ2κ = τ2κ+1 = gτ2κ+1 = τ2κ+2. Therefore, τ2κ+1 = fτ2κ = τ2κ, hence there exist ω1, ϱ1 such that ω1 = fϱ1 = ϱ1. Similarly, there exist ω2, ϱ2 such that ω2 = gϱ2 = ϱ2. We know that S(ϱ1, ϱ1, gϱ2) + S(ϱ2, ϱ2, fϱ1) + S(ϱ1, ϱ1, ϱ2) = 0. We deduce that S(fϱ1, fϱ1, gϱ2) = 0, which implies that ω1 = fϱ1 = gϱ2 = ω2. Therefore, ω1 = fϱ1 = fω1. Similarly, we get ω2 = gϱ2 = gω2. Since ω1 = ω2, we deduce that fω1 = gω1 = ω1. As a result, f and g have a common fixed point, namely ω1. To prove uniqueness, assume that f and g have τ and ς as common fixed points. We know that S(τ, τ, gς) + S(ς, ς, fu) + S(τ, τ, ς) = 0. Thus, S(τ, τ, ς) = S(fτ, fτ, gς) = 0 which implies that τ = ς as required. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 13 of 16 Corollary 3. Let (Γ, S, α) be a complete complex valued S-metric space. Let f and g be two self mappings on Γ that meet the contraction condition given below: S(fτ, fτ, gς) ≾ γS(τ, τ, ς) + βS(τ, τ, fτ)S(ς, ς, gς) b(2S(τ, τ, gς) + S(ς, ς, fτ) + S(τ, τ, ς)) for all τ, ς ∈ Γ such that τ ̸= ς, S(τ, τ, gς) + S(ς, ς, fτ) + S(τ, τ, ς) ̸= 0, where γ, β are two real numbers which are non- negative and satisfy the condition γ+β < 1 b or S(fτ, fτ, gς) = 0 if S(τ, τ, gς)+S(ς, ς, fτ)+S(τ, τ, ς) = 0. Then there exists a unique common fixed point for f and g. Proof. If we take α(τ, ς,ϖ) = b (b is some constant) and proceed with the same steps as outlined in the proof of Theorem 4, the corollary can be proved. Corollary 4. Let (Γ, S, α) be a complete complex valued S-metric space. Let f and g be two self mappings on Γ that meet the contraction condition given below: S(fτ, fτ, gς) ≾ γS(τ, τ, ς) + βS(τ, τ, fτ)S(ς, ς, gς) 2S(τ, τ, gς) + S(ς, ς, fτ) + S(τ, τ, ς) for all τ, ς ∈ Γ such that τ ̸= ς, S(τ, τ, gς) + S(ς, ς, fτ) + S(τ, τ, ς) ̸= 0, where γ, β are two real numbers which are non- negative and satisfy the condition γ+β < 1 or S(fτ, fτ, gς) = 0 if S(τ, τ, gς)+S(ς, ς, fτ)+S(τ, τ, ς) = 0. Then there exists a unique common fixed point for f, g. Proof. If we take α(τ, ς,ϖ) = 1 and proceed with the same steps as outlined in the proof of Theorem 4, the corollary can be proved. 4. An application We examine a nonlinear Volterra-type integral equation defined for complex-valued functions u(τ) = τ2 + i 4 ∫ τ 0 u(ς)3 1 + |u(ς)| dς, τ ∈ [0, 1], (7) where u(τ) ∈ C is the unknown function and the kernel K(τ, ς, u) = iu3 4(1+|u|) is Lipschitz continuous in u. Let Γ be the space of continuous functions that map from [0, 1] to the complex numbers. We define the operator T : Γ → Γ by T (u(τ)) := τ2 + i 4 ∫ τ 0 u(ς)3 1 + |u(ς)| dς. Consider a closed ball BR(0) = {u ∈ Γ : ∥u∥∞ ≤ R} for some R > 0. Also, define the metric as S(u, v, w) = max τ∈[0,1] { |u(τ)− w(τ)|, |v(τ)− w(τ)|, |u(τ)− v(τ)| 2 } H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 14 of 16 and the control function as α(u, v, w) = 1 + ∥u∥2∞ + ∥v∥∞ + ∥w∥3∞. Theorem 5. The integral equation (7) admits a unique solution u∗ ∈ Γ under the stated assumptions. Proof. For u, v ∈ BR(0) with R = 0.35, we have |T (u(τ))− T (v(τ))| ≤ 1 4 ∫ τ 0 ∣∣∣∣ u3 1 + |u| − v3 1 + |v| ∣∣∣∣ dς ≤ 1 4 ∫ τ 0 ( 3R2 +R3 ) |u− v| dς (using |K(τ, ς, u)−K(τ, ς, v)| ≤ ( 3R2 +R3 ) |u− v|) ≤ 3(0.35)2 + (0.35)3 4 max τ∈[0,1] |u(τ)− v(τ)| < θS(u, u, v), where θ = 0.11 and hence, the contraction condition is satisfied. For the Picard iterates τω = T ωτ0, we obtain ∥τω∥∞ ≤ 1 + (0.35)3 4(1 + 0.35) ≈ 0.0079. That is, α(τω, τω, τω+1) ≤ 1 + (0.0079)2 + 0.0079 + (0.0079)3 ≈ 1.008. Thus, sup ϱ≥1 lim ω→+∞ α(τω+1, τω+1, τω+2)α(τϱ, τϱ, τω+1) α(τω, τω, τω+1) ≤ 1.008 < 1 2× 0.11 ≈ 4.54 By Theorem 2, T admits a unique fixed point in Γ, i.e., the integral equation has a unique solution. 5. Conclusion In this paper, we introduced the concept of CVCS-metric spaces, which generalizes both complex valued S-metrics and controlled S-metric spaces. Within this new framework, we established several fixed point theorems under various contractive conditions. Our results not only extend existing theorems from the literature but also unify them under a broader and more flexible setting. Furthermore, we demonstrated the applicability of our theoretical results by proving the existence and uniqueness of a solution to a nonlinear Volterra integral equation involving complex-valued functions. These findings open new avenues for future research, particularly in analyzing nonlinear problems and integral equations within complex metric frameworks. Potential directions include extending these results to multivalued mappings and exploring their implications in applied mathematics and computational analysis. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 15 of 16 Authors’ Contributions All authors contribute equally in this paper. Conflict of interest The authors declare that they have no conflict of interest. Acknowledgements The authors acknowledge the financial support from Al-Zaytoonah University of Jor- dan, Amman 11733, Jordan. References [1] D. Judeh and M. Abu Hammad. Applications of conformable fractional pareto prob- ability distribution. International Journal of Advances in Soft Computing and Its Applications, 14:116–124, 2022. [2] T. Kanan, M. Elbes, K. Abu Maria, and M. Alia. Exploring the potential of iot-based learning environments in education. [3] H. Qawaqneh, M. S. Noorani, H. Aydi, A. Zraiqat, and A. H. Ansari. On fixed pointresults in partial b-metric spaces. Journal of Function Spaces, 8769190:9 pages, 2021. [4] H. Qawaqneh, M. S. Noorani, and H. Aydi. Some new characterizations and results for fuzzy contractions in fuzzy b-metric spaces and applications. AIMS Mathematics, 8:6682–6696, 2023. [5] H. Qawaqneh, H. A. Hammad, and H. Aydi. Exploring new geometric contraction mappings and their applications in fractional metric spaces. Advances in Fixed Point Theory, 9:521–541, 2024. [6] M. Nazam, H. Aydi, M.S. Noorani, and H. Qawaqneh. Existence of fixed points of four maps for a new generalized f−contraction and an application. Journal of Function Spaces, 5980312:8 pages, 2019. [7] H. Qawaqneh, M. S. Noorani, H. Aydi, and W. Shatanawi. , on common fixed point results for new contractions with applications to graph and integral equations. Mathematics, 7:1082, 2019. [8] M. Fréchet. Sur quelques points du calcul fonctionnel. Rendiconti del Circolo Matem- atico di Palermo, 22(1):1–72, 1906. [9] T. Van An, N. Van Dung, Z. Kadelburg, and S. Radenović. Various generalizations of metric spaces and fixed point theorems. Revista de la Real Academia de Ciencias Exactas, F́ısicas y Naturales. Serie A. Matemáticas, 109:175–198, 2015. [10] I. A. Bakhtin. The contraction mapping principle in almost metric spaces. Functional Analysis and Its Applications, 30:26–37, 1989. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6659 16 of 16 [11] A. Branciari. A fixed point theorem of banach–caccioppoli type on a class of gener- alized metric spaces. Publications Mathématiques, 57(1–2):31–37, 2000. [12] S. Czerwik. Contraction mappings in b-metric spaces. Acta Mathematica Universitatis Ostraviensis, 1(1):5–11, 1993. [13] S. G. Matthews. Partial metric topology. Annals of the New York Academy of Sciences, 728:183–197, 1994. [14] S. Shukla. Partial rectangular metric spaces and fixed point theorems. The Scientific World Journal, 2014. Article ID 756298. [15] T. Kamran, M. Samreen, and Q. U. Ain. A generalization of b-metric space and some fixed point theorems. Mathematics, 5:19, 2017. [16] N. Mlaiki, H. Aydi, N. Souayah, and T. Abdeljawad. Controlled metric type spaces and the related contraction principle. Mathematics, 6:194, 2018. [17] S. Sedghi, N. Shobe, and A. Aliouche. A generalization of fixed point theorem in S-metric spaces. Matematicki Vesnik, 64:258–266, 2012. [18] B. C. Dhage. Generalized metric spaces mappings with fixed point. Bulletin of the Calcutta Mathematical Society, 84:329–336, 1992. [19] S. Sedghi, N. Shobe, and H. Zhou. A common fixed point theorem in d∗-metric space. Fixed Point Theory and Applications, 2007. Article ID 1–13. [20] M. M. Rezaee, S. Sedghi, A. Muckheimer, K. Abodayeh, and Z. D. Mitrović. Some fixed point results in partial s-metric spaces. Australian Journal of Mathematical Analysis and Applications, 16(2), 2019. Article No. 16, 19 pages. [21] Y. Rohen, T. Došenović, and S. Radenović. A note on the paper “a fixed point theorems in sb-metric spaces”. Filomat, 31(11):3335–3346, 2017. [22] N. Souayah and N. Mlaiki. A fixed point theorem in sb-metric spaces. Journal of Mathematical and Computer Sciences, 16:131–139, 2016. [23] A. Gangwar, S. Rawat, and R. C. Dimri. Solution of differential inclusion problem in controlled s-metric spaces via new multivalued fixed point theorem. Journal of Analysis, 31:2459–2472, 2023. [24] F. M. Azmi. Wardowski contraction on controlled s-metric type spaces with fixed point results. International Journal of Analysis and Applications, 22:151–151, 2024. [25] A. Azam, B. Fisher, and M. Khan. Common fixed point theorems in complex valued metric spaces. Numerical Functional Analysis and Optimization, 32(3):243–253, 2011. [26] S. M. Kang, B. Singh, V. Gupta, and S. Kumar. Contraction principle in complex valued g-metric spaces. International Journal of Mathematical Analysis, 7(52):2549– 2556, 2013. [27] N. Mlaiki. Common fixed points in complex S-metric space. Advances in Fixed Point Theory, 4:509–524, 2014. [28] E. Ozgur. Complex valued gb-metric space. Journal of Computational Analysis and Applications, 21(2):363–368, 2016. [29] N. Priyobarta, Y. Rohen, and N. Mlaiki. Complex valued sb-metric spaces. Journal of Mathematical Analysis, 8(3):13–24, 2017.