EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6029 ISSN 1307-5543 – ejpam.com Published by New York Business Global Common Fixed Point Results on Double-Composed Cone-Metric-Like Spaces for Generalized Rational Contractions with Applications Anas A. Hijab1,2, Laith K. Shaakir1, Sarah Aljohani3,∗, Nabil Mlaiki3 1 Department of Mathematics, College of Computer Sciences and Mathematics, Tikrit University, Tikrit, Iraq 2 Department of Mathematics, Education for Pure Sciences College, Tikrit University, Tikrit, Iraq 3 Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia Abstract. This paper presents a novel concept, known as a double-composed cone-metric-like space, which extends the idea of a double-composed cone-metric space. In this new space, the self- distance may not necessarily be zero; however, if the distance metric is zero, it must be for identical points. Additionally, this text introduces several results pertaining to this innovative concept, including theorems that demonstrate the existence of common fixed points for two mappings that satisfy generalized non-linear rational contractions within our new space. Various examples are provided to illustrate the main results and their relationship with other cone metric spaces. Finally, we demonstrate applications to nonlinear integral equations and boundary value problems (BVPs) to validate our findings. 2020 Mathematics Subject Classifications: 47H10, 54E50, 54H25 Key Words and Phrases: Fixed point, Cone metric space, Double-controlled metric-like spaces, Double-composed cone metric spaces, Double-composed cone-metric-like spaces. 1. Introduction Fixed-point theory is a fundamental branch of functional and mathematical analysis that addresses the existence and uniqueness of solutions to integral-differential equations. Building upon the renowned Banach contraction principle [1], numerous scholars have made significant contributions to this field. Various results have emerged concerning map- pings that satisfy different contractive conditions across diverse types of metric spaces. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6029 Email addresses: anas abass@tu.edu.iq (A. A. Hijab), Dr.LaithKhaleel@tu.edu.iq (L. K. Shaakir), sjohani@psu.edu.sa (S. Aljohani), nmlaiki@psu.edu.sa (N. Mlaiki) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 2 of 23 One such extension is b-metric spaces, which were introduced independently by Czerwik [2] and Bakhtin [3]. In recent years, there have been several generalizations of b-metric spaces, such as bv(s)-metric spaces proposed by Mitrović et al. [4]. Kamran et al. [5] extended b-metric spaces, while in 2018, Mlaiki [6], and Abdeljawad et al. [7] introduced the concept of controlled metric-type spaces and double-controlled metric spaces, respec- tively. Amini-Harandi [8] (and independence Hitzler et al. [9]) have expanded the concept of partial metric spaces by defining metric-like spaces, also known as (dislocated metric spaces). The most comprehensive generalization, the b-metric-like space, was introduced by Alghamdi et al. [10]. Several fixed-point results have been explored in b-metric and their predecessors (see [11, 12]). In addition, in 2020, Mlaiki [13] and Ayoob et al. [14] introduced double-controlled metric-like spaces as a further extension of double-controlled metric-type spaces. In 2023, Ayoob et al. [15] proposed an extension of metric spaces known as double-composed metric spaces, which involve two composed functions in the triangular inequality. Huang et al. [16] introduced the concept of cone metric spaces as an extension of traditional metric spaces. Following this, Hussain et al. [17] introduced cone b-metric spaces and Shateri [18] presented fixed-point theorems on double-controlled cone metric spaces. Subsequently, Anas et al. [19] introduced type I and II composed cone metric spaces and [20] extended double-composed metric spaces to double-composed metric-like spaces (see [21–27]). In 2020, Lateef [28] proved Fisher-type fixed point results in controlled metric spaces, with subsequent discussion by authors including Dass and Gupta [29] and Jaggi [30] utilizing a contraction condition of the rational-types. Additionally, Ahmad et al. [31] provided a generalization of rational contractions in double-controlled metric spaces for common fixed point theorems. For further details, see [25, 32–34]. The objective of the current study is to establish common fixed point results for new generalized rational contractions, serving as a generalization of various types of metric spaces mentioned previously. This study introduces a new class known as double-composed cone metric-like spaces (for short, DCCML-space). The goal is to present common fixed point results involving various types of generalized rational contractions, accompanied by examples. Finally, the manuscript introduces applications of nonlinear integral equations and boundary value problems (BVPs) that support our fixed-point theorems within these new spaces. 2. Preliminaries This section revisits some notations basic concepts, definitions, and lemmas from prior research that will be utilized throughout the remainder of this manuscript. Definition 1. [16] Let E be a real Banach space and P ⊂ E. P is called a cone if it satisfies the following conditions: (P1) {0E} ≠ P is nonempty and closed, (P2) α1a+ α2b ∈ P for all a, b ∈ P , where α1, α2 ≥ 0, A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 3 of 23 (P3) P ∩ (−P ) = {0E}, where 0E is the zero element of E. Consider a cone P , we can define a partial ordering ⪯ on E with respect to P by a ⪯ b if and only if b − a ∈ P . Here, a ≺ b indicates that a ⪯ b and a ̸= b, but a ≪ b stands for b− a ∈ intP , such that intP denotes the interior of P . Let E be a Banach space, P be a cone in E such as intP ̸= ϕ and ⪯ is the partial ordering of P . The cone P is called normal if there exists a constant number M > 0 such that for all a, b ∈ E and 0E ⪯ a ⪯ b⇒ ∥a∥ ≤M∥b∥ it holds or equivalently, if inf{∥a+ b∥ : a, b ∈ P, ∥a∥ = ∥b∥ = 1} > 0. For a non-normal cone (see [18]). Moreover, P is called a solid if intP ̸= ϕ. Now, we present some basic notations of cone b-metric spaces and their properties. Definition 2. [17] Let Γ be a non-empty set and s ≥ 1. Assume that a mapping db : Γ× Γ → E satisfies the following conditions: For all a, b, c ∈ Γ, (Cb1) db(a, b) = 0E if and only if a = b, (Cb2) db(a, b) = db(b, a), (Cb3) db(a, b) ⪯ s (db(a, c) + db(c, a)). The pair (Γ, db) is called a cone b-metric space. If changing condition (Cb1) in Definition 2 to db(a, b) = 0E, implies a = b, then (Γ, db) is called a cone b-metric-like space. Obviously, cone b-metric-like spaces are generalized to cone b-metric spaces, cone metric-like spaces, cone metric spaces and metric spaces, respectively, but the same is not true vice versa (see [17, 19, 21–25, 27]). Abdeljawad et al. [7] introduced double-controlled type-metric spaces. Mlaiki et al. [13] generalized double-controlled metric-type spaces (DCMTS) to double-controlled metric-like spaces (DCMLS). Moreover, we expand on the expanded on cone metric space as follows: Definition 3. [18] Consider a set Γ ̸= ϕ and non-comparable functions ω1, ω2 : Γ× Γ → [1,∞). Assume that a mapping σ : Γ × Γ → E satisfies the conditions below: For all a, b, c ∈ Γ, (C1) σ(a, b) = 0E implies a = b, (C2) σ(a, b) = σ(b, a), (C3) σ(a, b) ⪯ ω1(a, c)σ(a, c) + ω2(c, b)σ(c, b). The pair (Γ, σ) is referred to as a double controlled cone-metric-like space (DCCMLS) (see [18, 26, 28, 35]). A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 4 of 23 Ayoob et al. [15] introduced generalizations of DCMTS and named it a double- composed metric space (abbreviated as DCMS). In the same vein, Anas et al. [19] extended DCMS to a type II composed cone-metric space (C2CMS) and [20] further generalized DCMS to a double composed metric-like space known as DCML-space. In this context, we present the double composed cone-metric-like space, DCCML-space, as outlined below: Definition 4. [19] Let Γ be a non-empty set and f, g : P → P be nonconstant functions. Consider the mapping Dc : Γ × Γ → E that adheres to the following conditions: For all a, b, c ∈ Γ, (D1) Dc(a, b) = 0E if and only if a = b, (D2) Dc(a, b) = Dc(b, a), (D3) Dc(a, b) ⪯ f ( Dc(a, c) ) + g ( Dc(c, b) ) . Then the pair (Γ,Dc) is referred to as a C2CMS. Example 1. Let E = R2, P = {u = (r, s) ∈ E : r, s ≥ 0}, and Γ = R. Let Dc : Γ× Γ → E be defined by Dc(a, b) = (e1(a− b)3, e2(a− b)2), where e1, e2 ≥ 0. Define f, g : P → P by f(u) = (e4r − 1, e2s − 1) and g(u) = (4r, 2s), u ∈ P . It is not difficult to see that (a − b)3 ≤ 4a3 + 4b3 ≤ (e4a 3 − 1) + 4b3. Also, (a − b)2 ≤ 2a2 + 2b2 ≤ (e4a 2 − 1) + 2b2. Therefore, (Γ, Dc) is a C2CMS. Now we introduce our generalization of the DCML-spaces. Definition 5. Let Γ be a non-empty set and f, g : P → P be nonconstant functions. Consider the mapping Lc : Γ × Γ → E that adheres to the following conditions: For all a, b, c ∈ Γ, (L1) Lc(a, b) = 0E ⇒ a = b, (L2) Lc(a, b) = Lc(b, a), (L3) Lc(a, b) ⪯ f ( Lc(a, c) ) + g ( Lc(c, b) ) . The pair (Γ,Lc) is known as a double-composed cone metric-like space (DCCML-space). The following examples demonstrate that every C2CMS is a DCCML-space; however, the converse is not always true. Example 2. Let E = R2, P = {u = (r, s) ∈ E : r, s ≥ 0}, and Γ = R. Then Lc : Γ×Γ → E is defined as Lc(a, b) = (e1(a+ b)p, e2(a+ b)q), where e1, e2 ≥ 0 and p, q > 1 and p ̸= q. Define f, g : P → P by f(u) = ( (1+2p−1rp)n−1 n , (1+2q−1rq)n−1 n ) and g(u) = ( (1+2p−1rp)m−1 m , (1+2q−1rq)m−1 m ) , u ∈ P, n ≥ A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 5 of 23 m ≥ 1. Obviously, conditions (L1) and (L2) of Definition 5 are satisfied. Note that if a, b are two nonnegative real numbers, then (a + b)q ≤ 2q−1aq + 2q−1bq, q > 1, and 1 + na ≤ (1 + a)n is Bernoulli’s inequality. Hence, a ≤ (1+a)n−1 n for any n ≥ 1, by the same way for p > 1,m > 1. We can easily deduce that the condition (L3) is satisfied. Therefore, (Γ,Lc) is a DCCML-space. Obviously, (Γ,Lc) is not a C2CMS because it does not satisfy (D1). Example 3. Consider E = R2, P = {u = (r, s) ∈ E : r, s ≥ 0} and Γ = R. Define Lc : Γ× Γ → E by Lc(a, b) = ( sinh(eσ(a, b)), e(a+b) − 1 ) , where e ≥ 0, and σ(a, b) is a DCMLS with two controlled functions ω1, ω2 : Γ× Γ → [1,∞). Take f, g : P → P defined by f(u) = ( sinh(2ω1(a, c)r), s2+2u 2 ) and g(u) = ( sinh(2ω2(c, b)r), s2+2s 2 ) , where u ∈ P . Evidently, (L1) and (L2) are satisfied. Since sinh(r) is an increasing function, for all a, b ≥ 0, sinh(a+ b) ≤ sinh(2max{a, b}) ≤ sinh(2a) + sinh(2b). Therefore, for each a, b, c ∈ Γ, sinh (eσ(a, b)) ≤ sinh (eω1(a, c)σ(a, c) + eω2(c, b)σ(c, b)) ≤ sinh (ω1(a, c) sinh(eσ(a, c)) + ω2(c, b) sinh(eσ(c, b))) ≤ sinh (2ω1(a, c)Lc(a, c)) + sinh (2ω2(c, b)Lc(c, b)) , (1) and e(a+b) − 1 ≤ ea+2c+b − 1 = ea+ceb+c − 1 ≤ e2(a+c) + e2(c+b) 2 − 1 = e2(a+c) − 1 2 + e2(c+b) − 1 2 . (2) Thus, from (1), (2), we get Lc(a, b) ⪯ f(Lc(a, c)) + g(Lc(c, b)). Then, (Γ,Lc) is a DCCML-space. Clearly, it is not C2CMS or DCML-space. Afterwards, we define the topology of the DCCML-space on Γ. Definition 6. Let (Γ,Lc) be a DCCML-space, where P is a normal cone with normal constant M , and {an} be a sequence in Γ. (i) The sequence {an} is called convergent to a0 ∈ Γ if lim n→∞ Lc(an, a0) = Lc(a0, a0), i.e., lim n→∞ an = a0. (ii) {an} in Γ is called Lc-Cauchy if lim n,m→∞ Lc(an, am) is a converges in Γ, i.e., for every c ∈ E with 0E ≪ c, there is a positive integer n0 ∈ N such that Lc(an, am) ≪ c for all n,m > n0. A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 6 of 23 (iii) The space (Γ,Lc) is said to be Lc-complete if every Lc-Cauchy sequence in Γ con- verges to a point in Γ, i.e., lim n→∞ Lc(an, a0) = Lc(a0, a0) = lim n,m→∞ Lc(an, am). Definition 7. Let (Γ,Lc) be a DCCML-space via f and g, where P is a normal cone with normal constant M . Suppose a0 ∈ Γ and 0E ≺ c. Then Lc-ball with center a0 and radius c is B(a0, c) = {b ∈ Γ : |Lc(a0, b) − Lc(a0, a0)| ≺ c}, and put B = {B(a0, c) : a0 ∈ Γ and 0E ≪ c}. Lemma 1. The collection B = {B(a0, c) : a0 ∈ Γ and 0E ≪ c} of all open balls forms a basis for a topology τLc on Γ. Proof. Let a0 ∈ Γ. So, a0 ∈ B(a0, c) for 0E ≺ c, which implies that a0 ∈ B(a0, c) ⊆⋃ a0∈Γ 0E≺c B(a0, c). Now, assume that b ∈ B(a0, c1) ∩ B(a0, c2). Then there exists 0E ≺ c such that B(a0, c) ⊆ B(a0, c1) and B(a0, c) ⊆ B(a0, c2). Let d ∈ B(b, c), then LC(b, d) − LC(b, b) ≪ c. Thus, B(b, c) ⊆ B(a0, c1) ∩B(a0, c2). Definition 8. [36] Presume P is a solid cone in a Banach space E. A sequence {an} ⊂ P is said to converge if for each 0E ≪ c there exists N such that an ≪ c for all n > N . Lemma 2. [32] If E is a real Banach space with a solid cone P and {an} ⊂ P is a sequence with ∥an∥ → 0, as n→ +∞, then {an} is a convergent sequence. Lemma 3. [32] Let E be a real Banach space with a solid cone P . (i) If r, s, t ∈ E and r ⪯ s ≪ t, then r ≪ t. (ii) If r ∈ P and r ≪ t for each t ≫ 0E, then r = 0E. In general, the limit of a convergent sequence in DCCML-space may not be unique. Proposition 1. Let E be a real Banach space with a solid cone P and {an} ⊂ P . Let (Γ,Lc) be a DCCML-space via f and g. Assume lim n→∞ Lc(an, a0) = 0E. Then, every convergent sequence has a unique limit, i.e., ∥Lc(an, a0)∥ → 0 implies that lim n→∞ an = a0 is unique. Proof. The proof is omitted. Let (Γ,Lc) be a DCCML-space. Define L̂c : Γ 2 → E by L̂c(a, b) = |2Lc(a, b)− Lc(a, a)− Lc(b, b)|, ∀a, b ∈ Γ. Obviously, L̂c(a, a) = 0E , ∀a ∈ Γ. Let Ψ be the family of all onto mappings ψ : [0,∞) → [0,∞) under the following necessities: r ≤ ψ(r) for each r ∈ [0,∞), and ψ ′ (the derivative of ψ) increases [37]. Next, we present the following lemma, utilizing results from the literature. Lemma 4. [20] Let ψ ∈ Ψ, then for all x ∈ [0, 1] and 0 < q ≤ 1 ≤ p, we have (i) ( ψ(x p) ) 1 p ≤ ψ(x ) ≤ ( ψ(x q) ) 1 q ; (ii) ( ψ−1(x q) ) 1 q ≤ ψ−1(x ) ≤ ( ψ−1(x p) ) 1 p . A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 7 of 23 3. Main results This section presents some fixed-point results within the framework of DCCML-space. In this work, the first theorem for common fixed points is analogous to the non-linear generalization rational contraction, with the self-mapping of DCCML-space, see [31, 38]. Motivated by Ahmad et al. [31], we denote by ∆ the family of all mappings λ : Γ2 → [0, 1) with any mapping (say) T : Γ → Γ satisfying the following conditions: (i) λ(Ta, b) ≤ λ(a, b) for each a, b ∈ Γ. (ii) λ(a, Tb) ≤ λ(a, b) for each a, b ∈ Γ. Clearly, since λ ∈ ∆ the iterative λj(a, b) → 0 as j → +∞. Now, we state and prove the common fixed point results in DCCML-space. Theorem 1. Assume (Γ,Lc) is an Lc-complete DCCML-space with two non-constant functions f, g : P → P , where P is a normal cone via normal constant M . Let T1, T2 : Γ → Γ be a mappings and there exists λ ∈ ∆ such that Lc(T1a, T2b) ⪯ λ(a, b)M̃(a, b), for all a, b ∈ Γ, (3) where M̃(a, b) = max { Lc(a, b),Lc(a, T1a),Lc(b, T2b), Lc(a, T1a)Lc(b, T2b) 1 + Lc(a, b) , Lc(b, T2b) [ 1 + Lc(a, T1a) ] 1 + Lc(a, b) , [ Lc(a, T1a) + Lc(b, T2b) ] Lc(T1a, T2b) 1 + Lc(a, b) + Lc(T1a, T2b) } . For a0 ∈ Γ, we set a sequence {an} defined as a2n+1 = T1a2n and a2n+2 = T2a2n+1 for every n ≥ 0. Suppose (i) f and g are bounded and non-decreasing, g is sub-additive and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiLc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Lc(a0, a1) ) ∥ = 0, where ξ = λ(a0, a1) < 1. If for every fixed point a, we conclude that Lc(a, a) = 0E, then T1 and T2 have a unique common fixed point. Proof. Let a0 ∈ Γ. Then, {an} is constructed in Γ by a2n+1 = T1a2n and a2n+2 = T2a2n+1, for all n ∈ N. If there exists n0 ∈ N for which an0+1 = an0 , then T1an0 = an0 . Therefore, there is nothing to prove. Thus, we assume that an+1 ̸= an for all n ∈ N. From inequality (3), we obtain Lc(a2n+1, a2n+2) = Lc(T1a2n, T2a2n+1) ⪯ λ(a2n, a2n+1))M̃(a2n, a2n+1), A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 8 of 23 where M̃(a2n, a2n+1) = max {Lc(a2n, a2n+1),Lc(a2n, T1a2n),Lc(a2n+1, T2a2n+1), Lc(a2n, T1a2n)Lc(a2n+1, T2a2n+1) 1 + Lc(a2n, a2n+1) , Lc(a2n+1, T2a2n+1)[1 + Lc(a2n, T1a2n)] 1 + Lc(a2n, a2n+1) ,[ Lc(a2n, T1a2n) + Lc(a2n+1, T2a2n+1) ] Lc(T1a2n, T2a2n+1) 1 + Lc(a2n, a2n+1) + Lc(T1a2n, T2a2n+1) } = max {Lc(a2n, a2n+1),Lc(a2n, a2n+1),Lc(a2n+1, a2n+2), Lc(a2n, a2n+1)Lc(a2n+1, a2n+2) 1 + Lc(a2n, a2n+1) , Lc(a2n+1, a2n+2)[1 + Lc(a2n, a2n+1)] 1 + Lc(a2n, a2n+1) ,[ Lc(a2n, a2n+1) + Lc(a2n+1, a2n+2) ] Lc(a2n+1, a2n+2) 1 + Lc(a2n, a2n+1) + Lc(a2n+1, a2n+2) } ⪯ max {Lc(a2n, a2n+1),Lc(a2n+1, a2n+2)} . Hence, Lc(a2n+1, a2n+2) ⪯ λ(a2n, a2n+1)M̃(a2n, a2n+1), where M̃(a2n, a2n+1) = max{Lc(a2n, a2n+1),Lc(a2n+1, a2n+2)}. By the properties of the function λ we deduce that Lc(a2n+1, a2n+2) ⪯ λ(a2n, a2n+1)M̃(a2n, a2n+1) = λ(T2T1a2n−2, a2n+1)M̃(a2n, a2n+1) ⪯ λ(a2n−2, a2n+1)M̃(a2n, a2n+1) ⪯ · · · ⪯ λ(a0, a2n+1)M̃(a2n, a2n+1) = λ(a0, T1T2a2n−1)M̃(a2n, a2n+1) ⪯ λ(a0, a2n−1)(M̃(a2n, a2n+1) ⪯ · · · ⪯ λ(a0, a1)M̃(a2n, a2n+1). Thus, Lc(a2n+1, a2n+2) ⪯ λ(a0, a1)M̃(a2n, a2n+1). (4) If M̃(a2n, a2n+1) = Lc(a2n+1, a2n+2), then by (4) we get Lc(a2n+1, a2n+2) ⪯ λ(a0, a1)Lc(a2n+1, a2n+2) ≺ Lc(a2n+1, a2n+2), which is a contradiction. On the other hand, if M̃(a2n, a2n+1) = Lc(a2n, a2n+1), then by (4) we have Lc(a2n+1, a2n+2) ⪯ λ(a0, a1)Lc(a2n, a2n+1) ⪯ ( λ(a0, a1) )2Lc(a2n−1, a2n) ⪯ · · · ⪯ ( λ(a0, a1) )nLc(a0, a1). Applying it recursively, we have Lc(an, an+1) ⪯ ξnLc(a0, a1), where ξ = λ(a0, a1). (5) A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 9 of 23 For m < n, and n,m ∈ N and condition (i), we deduce that Lc(am, an) ⪯ f ( Lc(am, am+1) ) + g ( Lc(am+1, an) ) ⪯ f ( Lc(am, am+1) ) + gf ( Lc(am+1, am+2) ) + g2 ( Lc(am+2, an) ) ... ⪯ n−2∑ i=m gi−mf ( Lc(ai, ai+1) ) + gn−m−1 ( Lc(an−1, an) ) . (6) Substituting (5) in (6) implies that ∥Lc(am, an)∥ ⪯M [ ∥ n−2∑ i=m gi−mf ( ξiLc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Lc(a0, a1) ) ∥ ] . Thus, as n,m→ ∞, and condition (ii), we have ∥Lc(am, an)∥ = 0. Since the sequence {an} is Lc-Cauchy in Γ, which is Lc-complete DCCML-space, there exists an element a ∈ Γ such that {an} → a. Hence Lc(an, a) = Lc(a, a) = Lc(an, am) = 0E . (7) Now, we prove that T1a = T2a = a. Since {an} → a, as n → +∞, from condition (i) and (L3), we deduce that Lc(a, T1a) ⪯ f ( Lc(a, a2n+2) ) + g ( Lc(a2n+2, T1a) ) = f ( Lc(a, a2n+2) ) + g ( Lc(T1a, T2a2n+1) ) , implying that Lc(T1a, T2a2n+1) ⪯ λ(a, a2n+1)M̃(a, a2n+1), and M̃(a, a2n+1) =max {Lc(a, a2n+1),Lc(a, T1a),Lc(a2n+1, T2a2n+1), Lc(a, T1a)Lc(a2n+1, T2a2n+1) 1 + Lc(a, a2n+1) , Lc(a2n+1, T2a2n+1)[1 + Lc(a, T1a)] 1 + Lc(a, a2n+1) ,[ Lc(a, T1a) + Lc(a2n+1, T2a2n+1) ] Lc(T1a, T2a2n+1) 1 + Lc(a, a2n+1) + Lc(T1a, T2a2n+1) } =max {Lc(a, a2n+1),Lc(a, T1a),Lc(a2n+1, a2n+2), Lc(a, T1a)Lc(a2n+1, a2n+2) 1 + Lc(a, a2n+1) , Lc(a2n+1, a2n+2)[1 + Lc(a, T1a)] 1 + Lc(a, a2n+1) ,[ Lc(a, T1a) + Lc(a2n+1, a2n+2) ] Lc(T1a, a2n+2) 1 + Lc(a, a2n+1) + Lc(T1a, a2n+2) } . A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 10 of 23 By assuming f and g are bounded and taking the limit in the above relationship and (7), we obtain that 0E ≺ Lc(a, T1a) ⪯ f(0) + g ( λ(a, a2n+1)Lc(a, T1a) ) . We use the fact that λ(a, a2n+1) ∈ ∆ and condition (i), we can take g(λ(a, a2n+1)Lc(a, T1a)) ≺ Lc(a, T1a). Therefore, ∥Lc(a, T1a)∥ = 0, i.e., T1a = a. Using the same way as for the proof of T1, we get T2a = a. Hence, T1 and T2 have a common fixed point a. Let a and a∗ be two fixed points of T1 and T2, where T1a = T2a = a and T1a ∗ = T2a ∗ = a. Since a ̸= a∗, it implies Tia ̸= Tja ∗, i, j = 1, 2. By (3) we obtain Lc(a ∗, a) = Lc(T1a ∗, T2a) ⪯ λ(a∗, a)M̃(a∗, a), where M̃(a∗, a) = max { Lc(a ∗, a),Lc(a ∗, T1a ∗),Lc(a, T2a), Lc(a ∗, T1a ∗)Lc(a, T2a) 1 + Lc(a∗, a) , Lc(a, T2a)[1 + Lc(a ∗, T1a ∗)] 1 + Lc(a∗, a) , [ Lc(a ∗, T1a ∗) + Lc(a, T2a) ] Lc(T1a ∗, T2a) 1 + Lc(a∗, a) + Lc(T1a∗, T2a) } = max { Lc(a ∗, a),Lc(a ∗, a∗),Lc(a, a), Lc(a ∗, a∗)Lc(a, a) 1 + Lc(a∗, a) , Lc(a, a)[1 + Lc(a ∗, a∗)] 1 + Lc(a∗, a) , [ Lc(a ∗, a∗) + Lc(a, a) ] Lc(a ∗, a) 1 + Lc(a∗, a) + Lc(a∗, a) } . By assuming any fixed point a and a∗, Lc(a ∗, a∗) = Lc(a, a) = 0E and the fact that λ(a∗, a) ∈ [0, 1), the result is Lc(a ∗, a) ⪯ λ(a∗, a)M̃(a∗, a) ≺ Lc(a ∗, a). Thus, ∥Lc(a ∗, a)∥ = 0, which is a contradiction. Therefore, a∗ = a. Corollary 1. Assume (Γ,Dc) is a complete C2CMS with two non-constant functions f, g : P → P , where P is a normal cone via normal constant M . Let T1, T2 : Γ → Γ be a mappings and there exists λ ∈ ∆ such that Dc(T1a, T2b) ⪯ λ(a, b)M̃(a, b), for all a, b ∈ Γ, (8) where M̃(a, b) = max { Dc(a, b),Dc(a, T1a),Dc(b, T2b), Dc(a, T1a)Dc(b, T2b) 1 +Dc(a, b) , Dc(b, T2b) [ 1 +Dc(a, T1a) ] 1 +Dc(a, b) , [ Dc(a, T1a) +Dc(b, T2b) ] Dc(T1a, T2b) 1 +Dc(a, b) +Dc(T1a, T2b) } . For a0 ∈ Γ, we set a sequence {an} defined as a2n+1 = T1a2n and a2n+2 = T2a2n+1 for every n ≥ 0. Suppose A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 11 of 23 (i) f and g are bounded and non-decreasing, g is sub-additive and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiDc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Dc(a0, a1) ) ∥ = 0, where ξ = λ(a0, a1) < 1. Then T1 and T2 have a unique common fixed point. Proof. The proof follows from Theorem 1 by taking (Γ,Lc) in C2CMS. Then, some special cases of Theorem 1 are presented, and since every C2CMS is a DCCML-space, the last cases of Corollary 1 are investigated, while the early cases of Corollary 1 are omitted. Corollary 2. Let (Γ,Lc) be a Lc-complete DCCML-space with two non-constant functions f, g : P → P , where P is a normal cone via the normal constant M . Suppose that T1, T2 : Γ → Γ is a mappings and λj ∈ ∆, j = 1, · · · , 6, such that Lc(T1a, T2b) ⪯λ1(a, b)Lc(a, b) + λ2(a, b)Lc(a, T1a) + λ3(a, b)Lc(b, T2b) + λ4(a, b) Lc(a, T1a)Lc(b, T2b) 1 + Lc(a, b) + λ5(a, b) Lc(b, T2b) [ 1 + Lc(a, T1a) ] 1 + Lc(a, b) + λ6(a, b) [ Lc(a, T1a) + Lc(b, T2b) ] Lc(T1a, T2b) 1 + Lc(a, b) + Lc(T1a, T2b) , (9) for all a, b ∈ Γ, where ∑6 j=1 λj(a, b) < 1. For a0 ∈ Γ, take the sequence {an} as a2n+1 = T1a2n and a2n+2 = T2a2n+1 for every n ≥ 0. Let ξ = λ1(a0,a1)+λ2(a0,a1) 1− ∑6 j=3 λj(a0,a1) < 1. Suppose (i) f and g are bounded and non-decreasing, g is sub-additive, and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiLc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Lc(a0, a1) ) ∥ = 0. If for every fixed point a, we conclude that Lc(a, a) = 0E, then T1 and T2 have a unique common fixed point. Proof. It is observed that for each a, b ∈ Γ, there exist λj ∈ ∆, j = 1, · · · , 6, such that λ(a, b) = ∑6 j=1 λj(a, b) < 1, resulting in Lc(T1a, T2b) ⪯λ1(a, b)Lc(a, b) + λ2(a, b)Lc(a, T1a) + λ3(a, b)Lc(b, T2b) + λ4(a, b) Lc(a, T1a)Lc(b, T2b) 1 + Lc(a, b) + λ5(a, b) Lc(b, T2b) [ 1 + Lc(a, T1a) ] 1 + Lc(a, b) + λ6(a, b) [ Lc(a, T1a) + Lc(b, T2b) ] Lc(T1a, T2b) 1 + Lc(a, b) + Lc(T1a, T2b) ⪯ [ 6∑ j=1 λj(a, b) ] max { Lc(a, b),Lc(a, T1a),Lc(b, T2b), Lc(a, T1a)Lc(b, T2b) 1 + Lc(a, b) , Lc(b, T2b) [ 1 + Lc(a, T1a) ] 1 + Lc(a, b) , [ Lc(a, T1a) + Lc(b, T2b) ] Lc(T1a, T2b) 1 + Lc(a, b) + Lc(T1a, T2b) } =λ(a, b)M̃(a, b). A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 12 of 23 Therefore, through Theorem 1 we obtain the desired result. Moreover, let a0 ∈ Γ. A sequence {an} in Γ defined as a2n+1 = T1a2n and a2n+2 = T2a2n+1,∀n ∈ N. Then, Lc(a2n+1, a2n+2) =Lc(T1a2n, T2a2n+1) ⪯ [ λ1(a0, a1) + λ2(a0, a1) 1− ∑6 j=3 λj(a0, a1) ] Lc(a2n, a2n+1). Remark 1. In Theorem 1 and Corollaries 1 and 2, if P = R+ is taken, the condition (i) is interchanged to f and g are continuous and non-decreasing, g is sub-additive, and g(λa) < a, λ ∈ (0, 1); and condition (ii) to n−2∑ i=m gi−mf ( ξiLc(a0, a1) ) + gn−m−1 ( ξn−1Lc(a0, a1) ) → 0, as n,m → ∞. Then, the study reveals the same special results in the spaces DCMS and DCML-spaces, respectively. By providing T1 = T2 = T in Theorem 1, and Corollaries 1 and 2, respectively, we derive the following corollaries: Corollary 3. Assume (Γ,Lc) is an Lc-complete DCCML-space, where P is a normal cone via normal constant M . Let T : Γ → Γ be a mapping and there exists λ ∈ ∆ such that Lc(Ta, Tb) ⪯ λ(a, b)M̃(a, b), for all a, b ∈ Γ, (10) where M̃(a, b) = max { Lc(a, b),Lc(a, Ta),Lc(b, T b), Lc(a, Ta)Lc(b, T b) 1 + Lc(a, b) , Lc(b, T b) [ 1 + Lc(a, Ta) ] 1 + Lc(a, b) , [ Lc(a, Ta) + Lc(b, T b) ] Lc(Ta, Tb) 1 + Lc(a, b) + Lc(Ta, Tb) } . For a0 ∈ Γ, we take the sequence {an} defined as an+1 = Tna0 for every n ≥ 0. Suppose (i) f and g are bounded and non-decreasing, g is sub-additive, and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiLc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Lc(a0, a1) ) ∥ = 0, where ξ = λ(a0, a1) < 1. If for every fixed point a, we conclude that Lc(a, a) = 0E, then T has a unique fixed point. Proof. The proof follows from Theorem 1 by taking the self-map T : Γ → Γ as T1 = T2 = T . A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 13 of 23 Corollary 4. Suppose that (Γ,Lc) is a complete C2CMS, where P is a normal cone via normal constant M . Let T : Γ → Γ be a mapping and there exists λ ∈ ∆ such that Dc(Ta, Tb) ⪯ λ(a, b)M̃(a, b), for all a, b ∈ Γ, (11) where M̃(a, b) = max { Dc(a, b),Dc(a, Ta),Dc(b, T b), Dc(a, Ta)Dc(b, T b) 1 +Dc(a, b) , Dc(b, T b) [ 1 +Dc(a, Ta) ] 1 +Dc(a, b) , [ Dc(a, Ta) +Dc(b, T b) ] Dc(Ta, Tb) 1 +Dc(a, b) +Dc(Ta, Tb) } . For a0 ∈ Γ, we take the sequence {an} defined as an+1 = Tna0 for every n ≥ 0. Assume that (i) f and g are bounded and non-decreasing, g is sub-additive, and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiDc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Dc(a0, a1) ) ∥ = 0, where ξ = λ(a0, a1) < 1. Then T ensures a unique fixed point. Proof. The proof follows from Corollary 1 by taking the self-map T : Γ → Γ as T1 = T2 = T . Corollary 5. Suppose that (Γ,Lc) is an Lc-complete DCCML-space, where P is a normal cone via normal constant M . Let T : Γ → Γ be a mapping and there exists λj ∈ ∆, j = 1, · · · , 5, such that Lc(Ta, Tb) ⪯λ1(a, b)Lc(a, b) + λ2(a, b)Lc(a, Ta) + λ3(a, b)Lc(b, T b) + λ4(a, b) Lc(a, Ta)Lc(b, T b) 1 + Lc(a, b) + λ5(a, b) Lc(b, T b) [ 1 + Lc(a, Ta) ] 1 + Lc(a, b) + λ6(a, b) [ Lc(a, Ta) + Lc(b, T b) ] Lc(Ta, Tb) 1 + Lc(a, b) + Lc(Ta, Tb) , (12) for all a, b ∈ Γ with ∑6 j=1 λj(a, b) < 1. For a0 ∈ Γ, we take the sequence {an} as an+1 = Tna0 for every n ≥ 0. Let ξ = λ1(a0,a1)+λ2(a0,a1) 1− ∑6 j=3 λj(a0,a1) < 1. Suppose that (i) f and g are bounded and non-decreasing, g is sub-additive, and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiLc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Lc(a0, a1) ) ∥ = 0. If for every fixed point a, we conclude that Lc(a, a) = 0E, then T ensures a unique fixed point. A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 14 of 23 Proof. The proof follows from Corollary 2 by setting the self-map T : Γ → Γ as T1 = T2 = T . Remark 2. It is worth noting that the fourth member Lc(a,Ta)Lc(b,T b) Lc(a,b) in the sources [35, 39] raises some doubts. Indeed, it follows from the proof of the previous references and others results, as well as some examples in these works, from which we obtain the form 0 0 , a division by zero. This is incorrect since 0 is the unique fixed point of map T or the distance metric approaches zero. The main motivation for our new results complements and provides entirely new observations. Remark 3. In the following we show that: (i) Our results represent an improvement and generalization of the findings of Lateef [28]. On one hand, if λ2 = λ3 = λ5 = λ6 = 0, the new generalized contraction becomes a Fisher contraction or an improvement of Jaggi work [29, 30] regarding common fixed points or merely fixed points. Moreover, the conclusion still holds, i.e., T has a fixed point. On the other hand, we extend the result in DCCML-spaces and C2CMS, instead of DCCMLS and DCCMTS. In other words, we broaden the result to DCCML-spaces. (ii) Special Cases for Corollaries 2 and 5: Case 1. if λ2 = λ3 = λ4 = λ6 = 0, then we obtain the result of the Dass and Gupta contraction, where λ1(a, b) = k1 and λ5(a, b) = k2, k1, k2 ∈ (0, 1), (see, [29]). Case 2. if λ2 = λ3 = λ4 = λ5 = λ6 = 0, then we obtain the result of extending Banach contraction principle. Case 3. if λ1 = λ4 = λ5 = λ6 = 0, then we obtain the result of extending Kannan’s contraction. Case 4. if λ4 = λ5 = λ6 = 0, then we obtain the result of extending the Riech-type contraction (see [19]). (iii) Through Remark 1, we know that every C2CMS is a DCCML-space, and the self- distance in the latter does not need to be zero. Thus, the new results are still valid in C2CMS. (iv) Towards the six-member in Corollary 2 we can that obtained as the form Lc(a,T1a)Lc(b,T1a)+Lc(b,T2b)Lc(a,T2b) 1+Lc(b,T1a)+Lc(a,T2b) , which implies that it is less than and equal to Lc(a, T1a) +Lc(b, T2b). So, we go again to the second and third members. Thus, we discuss the result obtained. We present some examples below to verify our theorems. Example 4. Consider E = R2, P = {u = (r, s) ∈ E : r, s ≥ 0}, and Γ = [0, 1]. Now, Lc : Γ× Γ → E is defined by Lc(a, b) = ( sinh ( (a+b)2 2 ) , 0 ) . Then Lc is a DCCML-space with two functions, f(u) =( sinh((a+b+2)r), 0 ) and g(u) = ( sinh((a2+b2+1)r), 0 ) , where u ∈ P , by the same process A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 15 of 23 as for Example 3, where the function (a+b)2 2 is a DCMLS. Moreover, (a1, b1)(a2, b2) = (a1a2, b1b2) is defined in [40]. Define T1, T2 : Γ → Γ by T1a = a 2 and T2a = a 3 , for a ∈ R. Choose λj : Γ× Γ → [0, 1), for j = 1, · · · , 6 by λ1(a, b) = 5+a+b 36 , λ2(a, b) = 3+a+b 36 , λ3(a, b) = 2+a+b 36 , λ4(a, b) = 4+a+b 36 , λ5(a, b) = 6+a+b 36 and λ6(a, b) = 0. Then, evidently, ∑6 j=1 λj(a, b) = 20+5a+5b 36 < 1. Also, λj(a, b) ∈ ∆ with two maps T1 and T2 for all j = 1, · · · , 6. Now, consider a0 = 0 and a, b ∈ Γ. Then, Lc(T1a, T2b) = ( sinh ((3a+ 2b)2 72 ) , 0 ) ⪯ 5 + a+ b 36 ( sinh ((a+ b)2 2 ) , 0 ) + 3 + a+ b 36 ( sinh (9a2 8 ) , 0 ) + 2 + a+ b 36 ( sinh (16b2 18 ) , 0 ) + 4 + a+ b 36 ( sinh ( 9a2 8 ) sinh ( 16b2 18 ) , 0 ) 1 + ( sinh ( (a+b)2 2 ) , 0 ) + 6 + a+ b 36 ( sinh ( 16b2 18 ) , 0 ) + ( sinh ( 9a2 8 ) sinh ( 16b2 18 ) , 0 ) 1 + ( sinh ( (a+b)2 2 ) , 0 ) = λ1(a, b)Lc(a, b) + λ2(a, b)Lc(a, Ta) + λ3(a, b)Lc(b, T b) + λ4(a, b) Lc(a, Ta)Lc(b, T b) 1 + Lc(a, b) + λ5(a, b) Lc(b, T b) [ 1 + Lc(a, Ta) ] 1 + Lc(a, b) . Hence, Corollary 2 is fulfilled and a = 0 ∈ Γ is a common fixed point such that T1a = T2a = a. Example 5. Consider E = R, and P = R+. Let Γ = {1, 2, 3}. Define a symmetric map Lc : Γ× Γ → E by Lc(1, 1) = Lc(2, 2) = 0,Lc(3, 3) = 2, and Lc(1, 2) = 11,Lc(1, 3) = 6,Lc(3, 2) = 3. Take f, g : P → P ; it is defined by f(u) = sinh ( 12 11u ) ; and g(u) = ( 3 11u ) , where u ∈ P . Evidently, valid that (Γ,Lc) is a Lc-complete DCCML-space regarding f, g, and Lc(3, 3) ̸= 0. Therefore, (Γ,Lc) is not a C2CMS, see [20]. Further, let us define a map T : Γ → Γ as T (a) = { 2 if a ∈ {2, 3} 3 if a = 1. Then, T ensures a unique fixed point. Proof. Let us take λ(a, b) = 3 2+a+b ∈ ∆, for all a, b ∈ Γ. Now, consider the following cases to show that Corollary 3 satisfies: Case 1. a = 1, b = 2, Lc(T1, T2) = Lc(3, 2) = 3 ⪯ 33 5 = 3 5 × 11 = λ(1, 2)max{11, 6, 0, 6×0 1+11 , 0×[1+6] 1+11 , [6+0]×3 1+11+3}; A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 16 of 23 Case 2. a = 1, b = 3, Lc(T1, T3) = Lc(3, 2) = 3 ⪯ 3 = 3 6 × 6 = λ(1, 3)max{6, 6, 3, 6×3 1+6 , 3×[1+6] 1+6 , [6+3]×3 1+6+3 }; Case 3. a = 2, b = 3, Lc(T2, T3) = Lc(2, 2) = 0 ⪯ 9 7 = 3 7 × 3 = λ(2, 3)max{3, 0, 3, 0×3 1+3 , 3×[1+0] 1+3 , [0+3]×0 1+3+0 }. Since Lc(3, 3) ̸= 0, we further take Case 4. a = 1, b = 1, Lc(T1, T1) = Lc(3, 3) = 2 ⪯ 126 4 = 3 4 × 42 = λ(1, 2)max{0, 6, 6, 6×6 1+0 , 6×[1+6] 1+0 , [6+6]×2 1+0+2 }. Consider a0 = 2 ∈ Γ. Thus, an = Tna0 = 2 for each n ≥ 1. For condition (i) in Corollary 3, we reach that u ≺ f(u) = sinh ( 12 11u ) , ( 3 11u ) = g(u) ≺ u, 0E ≺ u, and g(λ(a, b)u) ≺ u, u ∈ P . Moreover, we see that lim n→∞ a,b∈Γ ( λ(a, b) )n = 0. Therefore, all the conditions of Corollary 3 are satisfied and a fixed point is given as a = 2. Afterwards, we present the iterative fixed point T k, k > 1 as follows. Theorem 2. Suppose (Γ,Lc) is an Lc-complete DCCML-space with two non-constant functions f, g : P → P , where P is a normal cone via normal constant M . Let T : Γ → Γ be a mapping and there exists λ ∈ ∆ such that Lc(T ka, T kb) ⪯ λ(a, b)M̃(a, b), for all a, b ∈ Γ, (13) where M̃(a, b) = max { Lc(a, b),Lc(a, T ka),Lc(b, T kb), Lc(a, T ka)Lc(b, T kb) 1 + Lc(a, b) , Lc(b, T kb) [ 1 + Lc(a, T ka) ] 1 + Lc(a, b) , [ Lc(a, T ka) + Lc(b, T kb) ] Lc(T ka, T kb) 1 + Lc(a, b) + Lc(T ka, T kb) } . For a0 ∈ Γ, we set a sequence {an} defined as an+1 = Tna0 for every n ≥ 0. Suppose that (i) f and g are bounded and non-decreasing, g is sub-additive, and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiLc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Lc(a0, a1) ) ∥ = 0, where ξ = λ(a0, a1) < 1. If for every fixed point a, we conclude that Lc(a, a) = 0E, then T k has a unique fixed point. Proof. The proof follows from Corollary 3 by taking T ka = a. Subsequently, we observe T k(Ta) = T (T ka) = Ta. Hence, T k has a fixed point Ta and Ta = a, which means that T k has a unique fixed point, where T has a fixed point a. A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 17 of 23 Corollary 6. Suppose (Γ,Lc) is a complete C2CMS with two non-constant functions f, g : P → P , where P is a normal cone via normal constant M . Let T : Γ → Γ be a mapping and there exists λ ∈ ∆ such that Dc(T ka, T kb) ⪯ λ(a, b)M̃(a, b), for all a, b ∈ Γ, (14) where M̃(a, b) = max { Dc(a, b),Dc(a, T ka),Dc(b, T kb), Dc(a, T ka)Dc(b, T kb) 1 +Dc(a, b) , Dc(b, T kb) [ 1 +Dc(a, T ka) ] 1 +Dc(a, b) , [ Dc(a, T ka) +Dc(b, T kb) ] Dc(T ka, T kb) 1 +Dc(a, b) +Dc(T ka, T kb) } . For a0 ∈ Γ, take the sequence {an} defined as an+1 = Tna0 for every n ≥ 0. Suppose (i) f and g are bounded and non-decreasing, g is sub-additive, and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiDc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Dc(a0, a1) ) ∥ = 0, where ξ = λ(a0, a1) < 1. Then T k ensures a unique fixed point. Corollary 7. Assume (Γ,Lc) is an Lc-complete DCCML-space with two non-constant functions f, g : P → P , where P is a normal cone via normal constant M . Let T : Γ → Γ be a mapping and there exists λj ∈ ∆, j = 1, · · · , 6, such that Lc(T ka, T kb) ⪯λ1(a, b)Lc(a, b) + λ2(a, b)Lc(a, T ka) + λ3(a, b)Lc(b, T kb) + λ4(a, b) Lc(a, T ka)Lc(b, T kb) 1 + Lc(a, b) + λ5(a, b) Lc(b, T kb) [ 1 + Lc(a, T ka) ] 1 + Lc(a, b) + λ6(a, b) [ Lc(a, T ka) + Lc(b, T kb) ] Lc(T ka, T kb) 1 + Lc(a, b) + Lc(T ka, T kb) , (15) for all a, b ∈ Γ, where ∑6 j=1 λj(a, b) < 1. For a0 ∈ Γ, take the sequence {an} as an+1 = Tna0 for every n ≥ 0. Let ξ = λ1(a0,a1)+λ2(a0,a1) 1− ∑6 j=3 λj(a0,a1) < 1. Suppose that (i) f and g are bounded and non-decreasing, g is sub-additive, and g(λa) ≺ a, λ ∈ (0, 1); (ii) lim n,m→∞ ∑n−2 i=m ∥gi−mf ( ξiLc(a0, a1) ) ∥+ ∥gn−m−1 ( ξn−1Lc(a0, a1) ) ∥ = 0. If for every fixed point a, we conclude that Lc(a, a) = 0E, then T k possesses a unique fixed point. 4. Applications The fixed-point results play a vital role in the existence of theory of various classes of equations, particularly, for solving differential equations, integral equations, and fractional differential equations. This has led to significant improvements in the applications of fixed- point techniques. A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 18 of 23 4.1. Non-linear Integral Equations Consider Γ = C[0, 1], the class of all continuous functions on [0, 1]. Let E = C[0, 1] so that P = {h(t) ∈ E : h(t) ≥ 0, t ∈ [0, 1]} is equipped via the norm ∥η∥ = ∥η∥∞ + ∥η′∥∞. We endow Γ with DCCML-space as follows, Lc(η1, η2)(t) = ( sup t∈[0,1] sinh (|η1(t)|+ |η2(t)|)p ) 1 p et, for each η1, η2 ∈ Γ, and p ≥ 1. (16) Evidently, (Γ,Lc) is an Lc-completeDCCML-space, where f(u) = [sinh((1+η1+η2)(2u) p)] 1 p , and g(u) = [sinh((2 + η21 + η22)(2u) p)] 1 p , u ∈ P . Theorem 3. Assume that for each η1, η1 ∈ Γ = C[0, 1], (i) There exist a function λ ∈ ∆, and 0 < β < 1, such that, |Ξ(t, ν, η1(ν))|+ |Ξ(t, ν, η2(ν))| < β 2 λ(η1(ν), η2(ν))(|η1(ν)|+ |η2(ν)|); (17) (ii) Ξ ( t, ν, ∫ 1 0 Ξ(t, ν, η(ν))dν ) < Ξ(t, ν, η(ν)) for some t, ν ∈ [0, 1]. Then, this integral equation η(ν) = ∫ 1 0 Ξ(t, ν, η(ν))dν, admits a unique solution in C[0, 1]. Proof. Let T : Γ → Γ be continuous defined by Tη(ν) = ∫ 1 0 Ξ(t, ν, η(ν))dν. Then Lc(Tη1, Tη2)(t) = ( sup t∈[0,1] sinh ( |Tη1(t)|+ |Tη2(t)| )p) 1 p et, from Lemma 4, we have ( sinh ( |Tη1(t)|+ |Tη2(t)| )p) 1 p ⪯ sinh |Tη1(t)|+ |Tη2(t)| ⪯ 2|Tη1(t)|+ |Tη2(t)| = 2 ∣∣∣∣∫ 1 0 Ξ(t, ν, η1(ν))dν ∣∣∣∣+ ∣∣∣∣∫ 1 0 Ξ(t, ν, η2(ν))dν ∣∣∣∣ ⪯ 2 ∫ 1 0 |Ξ(t, ν, η1(ν))| dν + ∫ 1 0 |Ξ(t, ν, η2(ν))| dν = 2 ∫ 1 0 |Ξ(t, ν, η1(ν))|+ |Ξ(t, ν, η2(ν))| dν ⪯ 2 ∫ 1 0 β 2 λ(η1(ν), η2(ν))(|η1(ν)|+ |η2(ν)|)dν A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 19 of 23 = β ∫ 1 0 λ(η1(ν), η2(ν)) [(|η1(ν)|+ |η2(ν)|)p] 1 p dν ⪯ β ∫ 1 0 λ(η1(ν), η2(ν)) [sinh(|η1(ν)|+ |η2(ν)|)p] 1 p dν ⪯ βLc(η1, η2)(t) ∫ 1 0 λ(η1(ν), η2(ν))dν ⪯ βλ(η1(ν), η2(ν))Lc(η1, η2)(t). We observe that Lc(Tη1, Tη2)(t) ⪯ βλ(η1(ν), η2(ν))Lc(η1, η2)(t), where 0 < β < 1 and λ ∈ ∆. Hence, all of the requirements for Corollary 5 have been met. We obtain the desired results. 4.2. Boundary Value Problems The current study results will be applied to solve the first-order periodic BVPs: ϖ ′ (t) =h(t, ϖ(t)), t ∈ [0, 1] (18) ϖ(0) =ϖ(1), where h : [0, 1] × R → R is a continuous function on [0, 1]. The problem above can be formulated as: ϖ ′ (t) + δϖ(t) =h(t, ϖ(t)) + δϖ(t), t ∈ [0, 1] (19) ϖ(0) =ϖ(1). The problem (19) is equivalent to the following integral equation: ϖ(t) = ∫ 1 0 G(t, u) ( h(u, ϖ(u)) + δϖ(u) ) du, (20) where G is a Green function defined by G(t, u) = { eδ(u−t+1) eδ−1 0 ≤ u ≤ t eδ(u−t) eδ−1 t ≤ u ≤ 1. Thus, it is noticed that ∫ 1 0 G(t, u)du = 1 δ . Let Γ = C[0, 1]. Define Lc : Γ× Γ → E, where E = C[0, 1], P = {φ(t) ∈ E : φ(t) ≥ 0, t ∈ [0, 1]} is a DCCML-space, by Lc(ϖ1, ϖ2)(t) = ( e|ϖ1(t)|+|ϖ2(t)| − 1 ) φ(t), (21) where φ(t) = et > 0, and (Γ,Lc) is an Lc-complete DCCML-space via f(u) = g(u) =( u2+2u 2 ) et. Moreover, let T : Γ → Γ be a mapping defined by Tϖ(t) = ∫ 1 0 G(t, u) ( h(u, ϖ(u)) + δϖ(u) ) du, (22) A. A. Hijab et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6029 20 of 23 Corollary 3 is utilized to show that T has a unique fixed point, which is the solution for the BVP (18). Theorem 4. Assume that there exists δ > 0 such that, for each ϖ1, ϖ2 ∈ Γ, |h(t, ϖ1(t)) + δϖ1(t)|+ |h(t, ϖ2(t)) + δϖ2(t)| ≤ δ 3 (|ϖ1(t)|+ |ϖ2(t)|). Then, BVP (18) possesses a unique solution in Γ. Proof. Let Lc be a mapping given in (21), T be the operator function in (22). Then Lc(Tϖ1, Tϖ2)(t) = ( e|Tϖ1(t)|+|Tϖ2(t)| − 1 ) et = ( e| ∫ 1 0 G(t,u)(h(u,ϖ1(u))+δϖ1(u))du|+| ∫ 1 0 G(t,u)(h(u,ϖ2(u))+δϖ2(u))du| − 1 ) et ⪯ ( e ∫ 1 0 G(t,u)|h(u,ϖ1(u)+δϖ1(u))|+|h(u,ϖ2(u))+δϖ2(u)|du − 1 ) et ⪯ ( e ∫ 1 0 G(t,u) δ 3 (|ϖ1(u)|+|ϖ2(u)|)du − 1 ) et ⪯ ( e δ 3 (|ϖ1(t)|+|ϖ2(t)|) ∫ 1 0 G(t,u)du − 1 ) et ⪯ ( eδ(|ϖ1(t)|+|ϖ2(t)|) 1δ − 1 ) et 3 , (since ert − 1 ≤ r(et − 1), r = 1 3 ∈ (0, 1)). ⪯ Lc(ϖ1, ϖ2) sup t∈[0,1] et 3 . (Since 1 ≤ et, t ∈ [0, 1]). We deduce that Lc(Tϖ1, Tϖ2) ⪯ λ(t)Lc(ϖ1, ϖ2) ⪯ λ(t)M̃(ϖ1, ϖ2), where M̃(ϖ1, ϖ2) in (10), and λ(t) = sup t∈[0,1] et 3 ∈ (0, 1). Therefore, T is a generalized rational contraction, and all the conditions in Corollary 3 hold. Thus, we obtain the desired result. 5. Conclusions This research introduces a novel concept in the realm of generalized metric spaces, called double-composed cone-metric-like spaces, which is illustrated through a series of examples. We derived generalization rational-type contraction theorems for a variety of mappings, termed common fixed points in double-composed cone-metric-like spaces and provided a number of related results to support our theorems. Furthermore, we presented numerous examples to substantiate the main results of our study. The study demonstrates applications of nonlinear integral equations and BVPs, proving the existence of solutions. This particular new generalization provides valuable tools for studying fixed point theorems. The following points outline potential open problems and avenues for future research: • Explore new generalizations of double-composed cone-metric-like spaces, such as fuzzy double-composed metric-like spaces, fuzzy double-composed cone-metric-like spaces, and neutrosophic double-composed cone-metric-like spaces. A. A. Hijab et al. / Eur. J. Pure Appl. 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