EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6705 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Fixed Point Results for Monotone Multivalued and Integral Type Contractive Mappings Samina Batul1, Haitham Qawaqneh2,∗, Arbab Sikandar1, Usman Shehzad1, Hassen Aydi3,4 1 Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan 2 Al-Zaytoonah University of Jordan, Amman 11733, Jordan 3 Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, Sousse 4000, Tunisia 4 Department of Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa Abstract. This study focuses on establishing fixed point results for monotone multivalued map- pings within the framework of partially ordered complete Gb-metric spaces. The partial order on the set (X ,≼) is defined through a functional pair (κ,Θ). The research further explores conditions under which coupled fixed points exist and are unique, particularly for mappings that meet certain contractive requirements. These investigations are carried out using the notion of integral-type contractions tailored to the structure of partially ordered Gb-metric spaces. In addition to the core results, several corollaries are derived as specific instances. To enhance the reliability and relevance of the findings, the paper includes a number of illustrative examples. 2020 Mathematics Subject Classifications: 47H10, 54H25, 54C60 Key Words and Phrases: Fixed point, Gb-metric space, monotone multivalued functions 1. Introduction The concept of a metric space was first introduced by Fréchet [1] in 1906, and later extended by his student Kurepa [2] in 1934 to more abstract spaces where the metric takes values in an ordered vector space. Consider a complete metric space (X , d). A mapping T : X → X is said to be a contraction if d(T (u),T (v)) ≤ α d(u, v) for all u, v ∈ X , ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6705 Email addresses: samina.batul@cust.edu.pk (S. Batul), h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), arbabsikandar873@gmail.com (A. Sikandar), dmt211001@cust.pk (U. Shehzad), hassen.aydi@isima.rnu.tn (H. Aydi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 2 of 23 where α ∈ (0, 1). According to the Banach fixed point theorem, such a mapping T posseses a unique fixed point in X . The Banach fixed point principle has undergone significant expansions due to its efficacy in resolving existence and uniqueness problems in integral and differential equations. Researchers have built upon this foundation, in- troducing novel generalizations. Notably, Edelstein [3] work on subsequences of iterates led to a relaxation of the contraction condition, Subsequently, Boyd and Wong [4] intro- duced a continuous function ð : [0,∞) → [0,∞), replacing the linear contraction condition qd(u, v) ∀ q ∈ (0, 1) with ð(d(u, v)), thereby presenting a more general version of the Ba- nach fixed point theorem. Fixed point theorems in a partially ordered metric space play a vital role in determining the existence and uniqueness of solutions to specific equations. Moreover, multivalued mappings have gained significant attention due to its wide ranging applications in fields such as convex optimization, optimal control theory, and differential inclusions. For more related works, see [5–16]. As a generalization of metric space the con- cept b-metric space was first introduced by Bakhtin [17]. He also established several fixed point results for mappings satisfying specific contractive conditions within this framework. Later on, Selma Gulyaz Ozyurt [18] defined α-admissible contraction mappings on Bran- ciari b-metric spaces. Conditions for the existence and uniqueness of fixed points for these mappings were discussed, and related theorems were proved. Aydi et al. [19] established a fixed point theorem for set-valued quasi-contraction mappings in b-metric spaces. Further generalizations in such spaces can be found in [20, 21]. In 1976, Caristi [22] formulated a new class of fixed point results based on the concept of weakly inward mappings. A variant of the Banach contraction principle tailored to partially ordered sets was later established by Ran and Reurings [23], which is now widely referred to as the Ran-Reurings fixed point theorem. However, an unsuccessful attempt to generalize the Banach principle was made by Dhage et al. [24], who gave the concept of D-metric space topology. More precisely, Sedghi et al. [25] proposed a revised framework in 2007, introducing the notion of G-metric spaces as a modification of the original D-metric structure. Since then, numerous fixed point results have been established within this improved framework by various authors [26, 27]. Researchers have investigated coupled fixed point results for mixed monotone mappings in ordered metric spaces [28, 29]. For comprehensive insights into coupled fixed points and n-tupled fixed points theorems, readers can refer to [30]. Recently, Rajagopalan Ra- maswamy and Gunaseelan Mani [31] introduced graphical Branciari ℵ-metric spaces and proved a fixed point theorem for Ω-Q contractions on complete graphical Branciari ℵ- metric spaces. Additionally, fixed point problems have been extensively studied in the setting of partially ordered complete metric spaces. Notably, Al-Jumaili [32] utilized the concept of these spaces to establish coincidence fixed point theorems for functions sat- isfying certain contractive properties involving monotone increasing η-mappings, thereby advancing the field. Ghasab et al. [33] used the notion of integral-type contractions to establish coupled fixed point results in ordered G-metric spaces. Majid et al. [34] developed and investigated new fixed point theorems for multivalued functions in partially ordered complete D-metric spaces, where the order is defined by a pair of functions (κ,Θ) while Aghajani et al. [35] S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 3 of 23 introduced a new type of metric, called the Gb-metric. Ramaswamy et al. [36] introduced a new notion of (β, ϕ)-admissible hybrid contractions in metric spaces and established fixed point results in this setting. Recently, Samuel et al. [37] introduced integral-type contractions on orthogonal S-metric spaces and established common fixed point results. In this article, we aim to develop and explore several novel fixed point results for mono- tone multivalued mappings within the framework of partially ordered complete Gb-metric spaces. The partial order on the set (X ,≼) is defined through a functional pair (κ,Θ). Additionally, we establish existence and uniqueness results for coupled fixed points of mappings that satisfy specific contractive conditions, utilizing the notion of integral type contractions. To support our findings, appropriate examples are provided as practical applications, see related application [38, 39]. 2. Preliminaries The following are some definitions and results which are useful for the proof of our main theorems. Definition 1. [35] A function Gb : X × X × X → [0,∞) is a Gb-metric on X if for all u, v, w, x ∈ X , the following conditions are satisfied: (Gb1) Gb(u, v, w) = 0 ⇔ u = v = w. (Gb2) 0 < Gb(u, u, v) for all u, v ∈ X with u ̸= v. (Gb3) Gb(u, u, v) ≤ Gb(u, v, w) for all u, v, w ∈ X with v ̸= w. (Gb4) Gb(u, v, w) is invariant under permutations of its arguments, i.e., Gb(u, v, w) = Gb(p{u, v, w}) for any permutation p. (Gb5) Gb(u, v, w) ≤ l[Gb(u, v, x) +Gb(x,w,w)] for some constant l ≥ 1. The pair (X ,Gb) is called a Gb-metric space. Example 1. Let X = [0,∞) and Gb : X ×X ×X → [0,∞) be a mapping defined by Gb(u, v, w) = |u− v|q + |u− w|q + |v − w|q, where q ≥ 1. We have: (i) Gb(u, v, w) ≥ 0. (ii) Gb(u, v, w) = 0 ⇔ |u− v|q + |u− w|q + |v − w|q = 0 ⇔ |u− v|q = 0, |u− w|q = 0, |v − w|q = 0 ⇔ |u− v| = 0. It yields that u = v. Also, |u− w| = 0 ⇒ u = w. Moreover, |v − w| = 0 ⇒ v = w. We have u = v = w. (iii) Trivial. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 4 of 23 (iv) Recall that (a+ b)q ≤ 2q−1(aq + bq), q ≥ 1 and 2n(a+ b) + c ≤ 2n(a+ b+ c), n ≥ 1. Now, Gb(u, v, w) = |u− v|q + |u− w|q + |v − w|q = |u− v|q + |v − x+ x− w|q + |w − x+ x− u|q ≤ |u− v|q + 2q−1(|v − x|q + |x− w|q) + 2q−1(|w − x|q + |x− u|q) ≤ 2q−1(|u− v|q + |v − x|q + |x− w|q + |w − x|q + |x− u|q) = 2q−1(Gb(u, v, x) +Gb(x,w,w)) = l(Gb(u, v, x) +Gb(x,w,w)). Since all conditions are satisfied, therefore (X ,Gb) is a Gb-metric space. Definition 2. [35] Let (X ,Gb) be a Gb-metric space. A sequence (us) in X is said to converge to u ∈ X if and only if Gb(us, us, u) = Gb(u, u, us) → 0 as s→ ∞. Definition 3. [35] A sequence (us) ∈ X is called a Cauchy sequence if for any ϵ > 0 there exists a positive integer s0 such that for all s, r ≥ s0,Gb(us, us, ur) < ϵ. Definition 4. [35] A Gb-metric space (X ,Gb) is a complete Gb-metric if every Cauchy sequence in (X ,Gb) converges in (X ,Gb). Definition 5. Let (X ,Gb,≼) be a Gb-metric space, and Θ : X → [0,∞) be a functional. We define the relation ≼ as follows: u ≼ v ⇔ κ(Gb(u, u, v)) ≤ Θ(u)−Θ(v) ∀ u, v ∈ X , where κ : [0,∞) → [0,∞) is so that (i) κ is increasing and continuous. (ii) κ−1({0}) = {0}. (iii) κ(l(k+ j)) ≤ κ(k) + κ(j) ∀ k, j ∈ [0,∞). The triplet (X ,Gb,≼) with this partial order is called an ordered Gb-metric space induced via (κ,Θ). Proposition 1. Suppose that (X ,Gb) is a Gb-metric space, then ≼ is a partial order on X and (X ,≼) is a partially ordered set. Proof. Let start by showing that the relation ≼ is reflexive, meaning that every element is ≼ itself. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 5 of 23 Since κ(Gb(u, u, u)) = Θ(u)−Θ(u) for all u ∈ X , this implies that ≼ is reflexive. Next to show that the relation ≼ is antisymmetric. If u, v ∈ X with u ≼ v and v ≼ u, then κ(Gb(u, u, v)) ≤ Θ(u)−Θ(v) and κ(Gb(v, v, u)) ≤ Θ(v)−Θ(u). It implies κ(Gb(u, u, v)) + κ(Gb(v, v, u)) = 0. Thus, κ(Gb(u, u, v)) = κ(Gb(v, v, u)) = 0. That is, κ(Gb(u, u, v)) = 0, and so u = v, which shows that ≼ is antisymmetric. Lastly, we prove that ≼ is transitive. If u, v, w ∈ X such that u ≼ v and v ≼ w, then κ(Gb(u, u, v)) ≤ Θ(u)−Θ(v). (1) Also, κ(Gb(v, v, u)) ≤ Θ(v)−Θ(u). (2) Hence, combining (1) and (2), one writes κ(Gb(u, u, v)) + κ(Gb(v, v, u)) ≤ Θ(u)−Θ(w). By the definition of Gb-metric space, one has κ(Gb(u, u, w)) = κ[l (Gb(u, u, v) +Gb(v, v, w))] ≤ κ (Gb(u, u, v)) + κ (Gb(v, v, w)) = Θ(u)−Θ(v) + Θ(v)−Θ(w) ≤ Θ(u)−Θ(w). Thus, u ≼ w. The triplet (X ,Gb,≼) is called partially ordered Gb-metric space induced via (κ,Θ). Definition 6. Let (X ,Gb,≼) be an ordered Gb-metric space induced by (κ,Θ). The fol- lowing defines the ordered intervals in X : (i) [u, v] = {w ∈ X : u ≼ w ≼ v}. (ii) [u,∞) = {w ∈ X : u ≼ w}. (iii) (−∞, u] = {w ∈ X : w ≼ u}. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 6 of 23 Definition 7. [40] Let u ∈ X . u is said to be a fixed point of a multivalued mapping T : X → 2X if u ∈ T (u). Definition 8. [40] Let T : X → 2X be a multivalued mapping, then T is termed as upper semi-continuous if whenever (us) ∈ X and (vs) ∈ T (us) with us → m ∈ X and vs → e ∈ X , then e ∈ T (m) . Definition 9. [41] An element (a, b) ∈ X × X is said to be a coupled fixed point of a mapping T : X ×X → X if T (a, b) = a and T (b, a) = b. Definition 10. [40] A function U : X → R is called a lower semi-continuous if for any {un} ⊂ X and u ∈ X un → u⇒ U(u) ≤ lim n→∞ inf U(un). Definition 11. [41] Let (X ,≼) be a partial order set, then T : X × X → X is said to have mixed monotone property if T (u, v) is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument; i.e, for all u1, u2 ∈ X , u1 ≼ u2 ⇒ T (u1, v) ≼ T (u2, v) ∀ v ∈ X and for all v1, v2 ∈ X , v1 ≼ v2 ⇒ T (u, v1) ≽ T (u, v2) ∀ u ∈ T. Definition 12. [42] Denote by σ the collection of all functions κ : [0,∞) → [0,∞) such that: (i) κ is continuous. (ii) κ is non-decreasing with κ(τ) = 0 ⇔ τ = 0. Definition 13. [42] Denote by ℵ the collection of all functions Θ : [0,∞) → [0,∞) such that: (i) Θ is lower semi-continuous. (ii) Θ(τ) > 0 for all τ > 0 and Θ(0) = 0. 3. Multivalued Functions and Gb-Metric Spaces This section introduces and explores new fixed point theorems for monotone multival- ued functions, with a specific focus on their applications within partially ordered complete Gb-metric spaces. Theorem 1. Let (X ,Gb,≼) be a partially ordered complete Gb-metric space generated by (κ,Θ), where Θ : X → [0,∞) is a mapping which is bounded below. Let T : X → 2X be a multivalued mapping and M = {u ∈ X : T (u) ∩ [u,∞) ̸= ∅}. Assume that: (i) T is upper semi-continuous. (ii) If u ∈ M, then v ∈ M for all v ∈ T (u) ∩ [u,∞). S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 7 of 23 (iii) T (m) ∩ [m,∞) ̸= ∅ for some m ∈ X . Then there is a sequence (us) such that us−1 ≼ us ∈ T (us−1) for all s ∈ N, and T has a fixed point u0 such that us → u0. In addition, if Θ is lower semi-continuous, then us ≼ u0 for all s. Proof. By using (iii), there is m ∈ X that belongs to M. Then choose n ∈ T (m) ∩ [m,∞), and we have m ≼ n. By condition (ii), n ∈ M. Choose τ ∈ T (n) ∩ [n,∞) such that n ≼ τ . By repeating the process, we get a sequence (us) ∈ X such that us−1 ≼ us ∈ T (us−1) ∀ s ∈ N. Since (X ,Gb,≼) is a partially ordered Gb-metric space induced via (κ,Θ) κ(Gb(us−1, us−1, us)) ≤ Θ(us−1)−Θ(us). (3) The mapping κ is non-negative, so for all s ∈ N, Θ(us−1)−Θ(us) ≥ 0. That is, for all s ∈ N, Θ(us−1) ≥ Θ(us). Since Θ is bounded below, the sequence Θ(us) is both decreasing and bounded below. Therefore, by the completeness property of R, lim s→∞ Θ(us) = inf{us : s ∈ N}. Thus, by equation (3), lim s,r→∞ κ(Gb(us, us, ur)) ≤ lim s→∞ Θ(us)− lim r→∞ Θ(ur). Therefore, lim s,r→∞ κ(Gb(us, us, ur)) = 0. By exploiting the continuity of κ and the fact that κ−1({0}) = {0}, it follows that lim s,r→∞ Gb(us, us, ur) = 0. Therefore, (us) is a Cauchy sequence in X . Since X is a complete Gb-metric space, there is ∃ u0 ∈ X such that (us) is Gb-convergent to u0. Since us−1 ∈ X , us ∈ T (us−1), us−1 → u0, and us → u0, via the definition of upper semi-continuity of T , we have u0 ∈ T (u0). Now, assuming Θ is lower semi-continuous, then for each s ∈ N, κ(Gb(us, us, u0)) = lim r→∞ κ(Gb(us, us, ur)) ≤ lim r→∞ inf{Θ(us)−Θ(ur)} = Θ(us)− lim r→∞ inf Θ(ur) ≤ Θ(us)−Θ(u0). Thus, us ≤ u0 for all s ∈ N. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 8 of 23 Corollary 1. Suppose that (X ,Gb,≼) is a partially ordered complete Gb-induced via (κ,Θ), where Θ : X → [0,∞) is a bounded below mapping, and let T : X → 2X be a multivalued mapping be so that: (i) T is upper semi-continuous. (ii) T satisfies the condition of monotonic sequence: for all u, v ∈ X and u ≼ v and every α ≼ T (u), there exists β ≼ T (v) such that α ≼ β. (iii) There is ∃ m ∈ X such that T (m) ∩ [0,∞) ̸= ∅. Then there exists a sequence (us) ∈ X with us−1 ≼ us ∈ T (us−1) for all s ∈ N, and T has a fixed point u0 such that us → u0. Furthermore, if Θ is lower semi-continuous, then us ≼ u0 for all s. Proof. By property (ii), m ∈ M. Now, consider v ∈ T (m) ∩ [0,∞), then by the condition of T , there exists w ∈ T (v) such that v ≼ w. Equivalently, w ∈ T (v)∩ [0,∞) ̸= ∅. This implies that v ∈ M and then by Theorem 1, the proof is completed. Corollary 2. Let (X ,Gb,≼) be a partially ordered complete Gb-metric space induced by (κ,Θ) such that Θ : X → [0,∞) is a bounded below mapping, and let S : X → X satisfy the following: (i) S is a continuous function. (ii) For any α ∈ S (u), there is β ∈ S (v) such that α ≼ β. (iii) There is m ∈ X such that m ≼ S (m). Then there is a sequence (us) ∈ X with us−1 ≼ us ∈ S (us−1) for all s ∈ N, and S has a fixed point u0 such that us → u0. Also, if Θ is lower semi-continuous, then us ≼ u0 for all s. Proof. Define the multivalued mapping T : X → 2X via T (u) = {S (u)}, then T and X satisfy all the conditions of Theorem 1. Therefore, the proof follows from Theorem 1. By replacing the conditions bounded below with the conditions of bounded above, we obtain the following results. Theorem 2. Let (X ,Gb,≼) be a partially ordered complete Gb-metric space induced via (κ,Θ), where Θ : X → (−∞, 0] is a bounded above mapping. Presume that T : X → 2X is a multivalued mapping and M = {u ∈ X : T (u) ∩ (−∞, u] ̸= ∅}. Assume that (i) T is upper semi-continuous. (ii) For all u ∈ M,T (u) ∩M∩ (−∞, u] ̸= ∅. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 9 of 23 Then there is a sequence (us) such that us−1 ≽ us ∈ T (us−1) for all s ∈ N, and T has a fixed point u0 such that us → u0. Also, if Θ is lower semi-continuous, then us ≽ u0 for all s. Proof. By using condition (ii), there exists m ∈ X such that m ∈ M. By choosing n ∈ T (m) ∩ (−∞,m], and we get m ≽ n. By condition (ii), n ∈ M. Choose τ ∈ T (n) ∩ (−∞, n], ⇒ n ≽ τ . By proceeding in this way, there is a sequence (us) ∈ X s.t. us−1 ≽ us ∈ T (us−1) for all s ∈ N. Since (X ,Gb,≼) is a partially ordered Gb metric space induced via (κ,Θ), ⇒ κ(Gb(us−1, us−1, us)) ≤ Θ(us−1)−Θ(us). Given that κ is non-negative mapping, Θ(us−1)−Θ(us) ≥ 0 ∀ s ∈ N. ⇒ Θ(us−1) ≥ Θ(us) ∀ s ∈ N. As Θ is bounded above, we get Θ(us) is an increasing sequence which is bounded above. By the completeness of R, lim s→−∞ Θ(us) = inf{us : s ∈ N}, thus lim s,r→−∞ κ(Gb(us, us, ur)) ≤ lim s→−∞ Θ(us)− lim r→−∞ Θ(ur). Therefore, lim s,r→−∞ κ(Gb(us, us, ur)) = 0. Now, since κ is continuous κ−1({0}) = {0}, we get lim s,r→−∞ Gb(us, us, ur) = 0. Therefore, (us) is a Cauchy sequence in X . Since X is complete, there exists u0 ∈ X such that (us) is Gb convergent. to u0. Since us−1 ∈ X , us ∈ T (us−1), us−1 → u0, and us → u0, via the definition of upper semi-continuity of T , we have u0 ∈ T (u0). Now, if Θ is lower semi-continuous, then for all s ∈ N, κ(Gb(us, us, u0)) = lim r→∞ κ(Gb(us, us, ur)) ≤ lim r→∞ {inf Θ(us)−Θ(ur)} = Θ(us)− lim r→∞ Θ(ur) ≤ Θ(us)−Θ(u0). Thus, us ≽ u0 for all s ∈ N. Corollary 3. Suppose that (X ,Gb,≼) is a partially ordered complete Gb-metric space induced via (κ,Θ), where Θ : X → (−∞, 0] is bounded above, and let T : X → 2X be a multivalued mapping so that: (i) T is upper semi-continuous. (ii) For all u, v ∈ X and u ≽ v and every α ∈ T (u), there exists β ∈ T (v) such that α ≽ β. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 10 of 23 (iii) There is m ∈ X such that T (m) ∩ [0,∞) ̸= ∅. Then there exists a sequence (us) ∈ X with us−1 ≽ us ∈ T (us−1) ∀ s ∈ N, and T has a fixed point u0 such that us → u0. Furthermore, if Θ is lower semi-continuous, then us ≽ u0 for all s. Corollary 4. Assume that (X ,Gb,≼) is a partially ordered complete Gb-metric space induced via (κ,Θ) such that Θ : X → (−∞, 0] is bounded above, and let S : X → X satisfy the following: (i) S is continuous. (ii) For any α ∈ S (u), there exists β ∈ S (v) such that α ≽ β. (iii) There is m ∈ X such that m ≽ S (m). Then there is a sequence (us) ∈ X with us−1 ≽ us ∈ S (us−1) for all s ∈ N, and T has a fixed point u0 such that us → u0. Also, if Θ is lower semi-continuous, then us ≽ u0 for all s. 4. Coupled Fixed Point Theorems in Gb-Metric Spaces Theorem 3. Assume that (X ,Gb,≼) is a partially ordered complete Gb-metric space, and let T : X × X → X be a continuous mapping with the mixed monotone property on X such that ∫ Gb(T (u,v),T (m,n),T (f,w)) 0 g(t)dt ≤ σ (∫ Gb(u,m,f)+Gb(v,n,w) 0 g(t)dt ) , (4) where u, v, w,m, n, f ∈ X and g : [0,∞) → [0,∞) is a Lebesgue integrable mapping with f ≼ m ≼ u and v ≼ n ≼ w, where either m ̸= f or n ̸= w. If there exist u0, v0 ∈ X such that u0 ≼ T (u0, v0) and T (v0, u0) ≼ v0 , then T has a coupled fixed point in X . Proof. By hypothesis, there are u0, v0 ∈ X such that u0 ≼ T (u0, v0) and T (v0, u0) ≼ v0. Define u1, v1 ∈ X as u0 ≼ T (u0, v0) = u1 and v1 = T (v0, u0) ≼ v0. Suppose that u2 = T (u1, v1) and v2 = T (v1, u1), therefore u2 = T (u1, v1) = T (T (u0, v0),T (v0, u0)) = T 2(u0, v0). v2 = T (v1, u1) = T (T (v0, u0),T (u0, v0)) = T 2(v0, u0). Utilizing the mixed monotonicity for the mapping T , one writes u2 = T 2(u0, v0) = T (u1, v1) ≽ T (u0, v0) = u1 ≽ u0, v2 = T 2(v0, u0) = T (v1, u1) ≼ T (v0, u0) = v1 ≼ v0. Repeatedly applying the above process for all s ≥ 0 leads to S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 11 of 23 u0 ≤ u1 ≼ u2 ≼ ... ≼ us+1 ≼ ..., v0 ≽ v1 ≽ v2 ≽ ... ≽ vs+1 ≽ ... such that us+1 = T s+1(u0, v0) = T (T s(u0, v0),T s(v0, u0)), vs+1 = T s+1(v0, u0) = T (T s(v0, u0),T s(u0, v0)). If (us+1, vs+1) = (u0, v0), then a coupled fixed point exists for the mapping T . Now, we assume that (us+1, vs+1) ̸= (us, vs) for all s ≥ 0, that is, let either us+1 = T (us, vs) ̸= us or vs+1 = T (v0, u0) ̸= vs. By equation (4), it follows that∫ Gb(us,us,us+1) 0 g(t)dt = ∫ Gb(T (us−1,vs−1),T (us−1,vs−1),T (us,vs)) 0 g(t)dt ≤ σ (∫ Gb(us−1,us−1,us),Gb(vs−1,vs−1,vs) 0 g(t)dt ) . (5) In the same way, it can be proved that∫ Gb(vs,vs,vs+1) 0 g(t)dt = ∫ Gb(T (vs−1,us−1),T (vs−1,us−1),T (vs,us)) 0 g(t)dt ≤ σ (∫ Gb(us−1,us−1,us),Gb(vs−1,vs−1,vs) 0 g(t)dt ) . (6) Since g is non-increasing mapping, then for each k, j ≥ 0,∫ k+j 0 g(t)dt ≤ ∫ k 0 g(t)dt+ ∫ j 0 g(t)dt. (7) Additionally, since σ is a linear and monotonically increasing mapping, it follows from (4), (5) and (7) that for all s ≥ 0∫ Gb(us,us,us+1) 0 g(t)dt = ∫ Gb(T (us−1,vs−1),T (us−1,vs−1),T (us,vs)) 0 g(t)dt ≤ σ (∫ Gb(us−1,us−1,us)+Gb(vs−1,vs−1,vs) 0 g(t)dt ) ≤ σ (∫ Gb(us−1,us−1,us) 0 g(t)dt ) + σ (∫ Gb(vs−1,vs−1,vs) 0 g(t)dt ) = σ (∫ Gb(T (us−2,vs−2),T (us−2,vs−2),T (us−1,vs−1)) 0 g(t)dt ) + σ (∫ Gb(T (vs−2,us−2),T (vs−2,us−2),T (vs−1,us−1)) 0 g(t)dt ) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 12 of 23 ≤ σ ( σ (∫ Gb(us−2,us−2,us−1)+Gb(vs−2,vs−2,vs−1) 0 g(t)dt )) + σ ( σ (∫ Gb(vs−2,vs−2,vs−1)+Gb(us−2,us−2,us−1) 0 g(t)dt )) ≤ σ2 (∫ Gb(us−2,us−2,us−1) 0 g(t)dt ) + σ2 (∫ G(vs−2,vs−2,vs−1) 0 g(t)dt ) + σ2 (∫ Gb(vs−2,vs−2,vs−1) 0 g(t)dt ) + σ2 (∫ Gb(us−2,us−2,us−1) 0 g(t)dt ) = 2σ2 (∫ Gb(us−2,us−2,us−1) 0 g(t)dt ) + 2σ2 (∫ Gb(vs−2,vs−2,vs−1) 0 g(t)dt ) = 2σ2 (∫ Gb(us−2,us−2,us−1) 0 g(t)dt+ ∫ Gb(vs−2,vs−2,vs−1) 0 g(t)dt ) ≤ 2σ2 (∫ Gb(us−2,us−2,us−1)+Gb(vs−2,vs−2,vs−1) 0 g(t)dt ) ... ≤ sσs (∫ Gb(u0,u0,u1)+Gb(v0,v0,v1) 0 g(t)dt ) . Following the same steps, it can be proved that∫ Gb(vs,vs,vs+1) 0 g(t)dt = ∫ Gb(T (vs−1,us−1),T (vs−1,us−1),T (vs,us)) 0 g(t)dt ≤ σ (∫ Gb(vs−1,vs−1,vs)+Gb(us−1,us−1,us) 0 g(t)dt ) ≤ σ (∫ Gb(vs−1,vs−1,vs) 0 g(t)dt ) + σ (∫ Gb(us−1,us−1,us) 0 g(t)dt ) = σ (∫ Gb(T (vs−2,us−2),T (vs−2,us−2),T (vs−1,us−1)) 0 g(t)dt ) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 13 of 23 + σ (∫ Gb(T (us−2,vs−2),T (us−2,vs−2),T (us−1,vs−1)) 0 g(t)dt ) ≤ σ ( σ (∫ Gb(vs−2,vs−2,vs−1)+Gb(us−2,us−2,us−1) 0 g(t)dt )) + σ ( σ (∫ Gb(us−2,us−2,us−1)+Gb(vs−2,vs−2,vs−1) 0 g(t)dt )) ≤ σ2 (∫ Gb(vs−2,vs−2,vs−1) 0 g(t)dt ) + σ2 (∫ Gb(us−2,us−2,us−1) 0 g(t)dt ) + σ2 (∫ Gb(us−2,us−2,us−1) 0 g(t)dt ) + σ2 (∫ Gb(vs−2,vs−2,vs−1) 0 g(t)dt ) = 2σ2 (∫ Gb(vs−2,vs−2,vs−1) 0 g(t)dt ) + 2σ2 (∫ Gb(us−2,us−2,us−1) 0 g(t)dt ) = 2σ2 (∫ Gb(vs−2,vs−2,vs−1) 0 g(t)dt+ ∫ Gb(us−2,us−2,us−1) 0 g(t)dt ) ≤ 2σ2 (∫ Gb(vs−2,vs−2,vs−1)+Gb(us−2,us−2,us−1) 0 g(t)dt ) ... ≤ sσs (∫ Gb(v0,v0,v1)+Gb(u0,u0,u1) 0 g(t)dt ) . Let r, s ∈ N such that r > s, then from the definition of Gb-metric space,∫ Gb(us,us,ur) 0 g(t)dt ≤ ∫ l[Gb(us,us,us+1)+Gb(us+1,us+1,ur)] 0 g(t)dt = ∫ lGb(us,us,us+1)+lGb(us+1,us+1,ur) 0 g(t)dt ≤ ∫ lGb(us,us,us+1) 0 g(t)dt+ ∫ lGb(us+1,us+1,ur) 0 g(t)dt S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 14 of 23 ≤ ∫ lGb(us,us,us+1) 0 g(t)dt + ∫ l[l(Gb(us+1,us+1,us+2)+Gb(us+2,us+2,ur))] 0 g(t)dt ≤ ∫ lGb(us,us,us+1) 0 g(t)dt+ ∫ l2Gb(us+1,us+1,us+2) 0 g(t)dt + ∫ l2Gb(us+2,us+2,ur) 0 g(t)dt ≤ ∫ lGb(us,us,us+1) 0 g(t)dt+ ∫ l2Gb(us+1,us+1,us+2) 0 g(t)dt + ∫ l3Gb(us+2,us+2,us+3) 0 g(t)dt+ . . .+ ∫ lr−sGb(ur−2,ur−2,ur−1) 0 g(t)dt + ∫ lr−sGb(ur−1,ur−1,ur) 0 g(t)dt ≤ sσs (∫ l[Gb(u0,u0,u1)+Gb(v0,v0,v1)] 0 g(t)dt ) + (s+ 1)σs+1 (∫ l2[Gb(u0,u0,u1)+Gb(v0,v0,v1)] 0 g(t)dt ) + (s+ 2)σs+2 (∫ l3[Gb(u0,u0,u1)+Gb(v0,v0,v1)] o g(t)dt ) + . . .+ (r − 2)σr−2 (∫ lr−s[Gb(u0,u0,u1)+Gb(v0,v0,v1)] 0 g(t)dt ) + (r − 1)σr−1 (∫ lr−s[Gb(u0,u0,u1)+Gb(v0,v0,v1)] 0 g(t)dt ) = i=r−1∑ i=s iσi (∫ li−s+1[Gb(u0,u0,u1)+Gb(v0,v0,v1)] 0 g(t)dt ) ≤ ∞∑ i=s iσi (∫ li−s+1[Gb(u0,u0,u1)+Gb(v0,v0,v1)] 0 g(t)dt ) . Since ∞∑ i=s iσi(t) < ∞ for all t > 0, this implies that lim s,r→∞ Gb(us, us, ur) = 0 and (us) is a Cauchy sequence in X . In a similar manner, the following result can be obtained∫ Gb(vs,vs,vr) 0 g(t)dt ≤ ∫ l[Gb(vs,vs,vs+1)+Gb(vs+1,vs+1,vr)] 0 g(t)dt S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 15 of 23 = ∫ lGb(vs,vs,vs+1)+lGb(vs+1,vs+1,vr) 0 g(t)dt ≤ ∫ lGb(vs,vs,vs+1) 0 g(t)dt+ ∫ lGb(vs+1,vs+1,vr) 0 g(t)dt ≤ ∫ lGb(vs,vs,vs+1) 0 g(t)dt + ∫ l[l(Gb(vs+1,vs+1,vs+2)+Gb(vs+2,vs+2,vr))] 0 g(t)dt ≤ ∫ lGb(vs,vs,vs+1) 0 g(t)dt+ ∫ l2Gb(vs+1,vs+1,vs+2) 0 g(t)dt + ∫ l2Gb(vs+2,vs+2,vr) 0 g(t)dt ≤ ∫ lGb(vs,vs,vs+1) 0 g(t)dt+ ∫ l2Gb(vs+1,vs+1,vs+2) 0 g(t)dt + ∫ l3Gb(vs+2,vs+2,vs+3) 0 g(t)dt+ . . .+ ∫ lr−sGb(vr−2,vr−2,vr−1) 0 g(t)dt + ∫ lr−sGb(vr−1,vr−1,vr) 0 g(t)dt ≤ sσs (∫ l[Gb(v0,v0,v1)+Gb(u0,u0,u1)] 0 g(t)dt ) + (s+ 1)σs+1 (∫ l2[Gb(v0,v0,v1)+Gb(u0,u0,u1)] 0 g(t)dt ) + (s+ 2)σs+2 (∫ l3[Gb(v0,v0,v1)+Gb(u0,u0,u1)] o g(t)dt ) + . . .+ (r − 2)σr−2 (∫ lr−s[Gb(v0,v0,v1)+Gb(u0,u0,u1)] 0 g(t)dt ) + (r − 1)σr−1 (∫ lr−s[Gb(v0,v0,v1)+Gb(u0,u0,u1)] 0 g(t)dt ) = i=r−1∑ i=s iσi (∫ li−s+1[Gb(v0,v0,v1)+Gb(u0,u0,u1)] 0 g(t)dt ) ≤ ∞∑ i=s iσi (∫ li−s+1[Gb(v0,v0,v1)+Gb(u0,u0,u1)] 0 g(t)dt ) . Since ∞∑ i=s iσi(t) < ∞ for all t ∈ [0,+∞), then lim s,r→∞ Gb(vs, vs, vr) = 0 and (vs) is a S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 16 of 23 Cauchy sequence in X , which is a complete Gb-metric space, there exist u, v ∈ X such that lim s→∞ us = u and lim s→∞ vs = v. Since T is continuous, it follows that T (u, v) = u and T (v, u) = v, that is, (u, v) is a coupled fixed point of T . Theorem 4. Let (X ,Gb,≼) be a partially ordered complete Gb-metric space satisfying the following conditions: (i) If (us) is non-decreasing sequence which converges to u ∈ X , then us ≼ u ∀ s. (ii) If (vs) is non-increasing sequence which converges to v ∈ X , then vs ≽ v ∀ s. Also suppose that T : X × X → X is a continuous function having the mixed monotone property on X such that ∫ Gb(T (u,v),T (m,n),T (f,w)) 0 g(t)dt ≤ σ (∫ Gb(u,m,f)+Gb(v,n,w) 0 g(t)dt ) , (8) where u, v, w,m, n, f ∈ X and g : [0,∞) → [0,∞) is a Lebesgue integrable mapping with f ≼ m ≼ u and v ≼ n ≼ w, where either m ̸= f or n ̸= w. If there exist u0, v0 ∈ X such that u0 ≼ H(u0, v0) and (v0, u0) ≼ v0 , then T has a coupled fixed point in X . Proof. Using the similar approach to that in the proof of Theorem 3, gives two Cauchy sequences (us) and (vs) ∈ X . Conditions (i) and (ii) implies that there exist u, v ∈ X such that us ≼ u and vs ≽ v for all s ≥ 0. If us = u and vs = v for some s, then us+1 = u and vs+1 = v; that is, (u, v) is a coupled fixed point . Now, without loss of generality, let either us ̸= u or vs ̸= v. By using (8),∫ Gb(T (u,v),T (u,v),u) 0 g(t)dt ≤ ∫ Gb(T (u,v),T (u,v),T (us,vs))+Gb(T (us,vs),T (us,vs),u) 0 g(t)dt ≤ ∫ Gb(T (u,v),T (u,v),T (us,vs)) 0 g(t)dt+ ∫ Gb(T (us,vs),T (us,vs),u) 0 g(t)dt ≤ σ (∫ Gb(u,u,us)+Gb(v,v,vs) 0 g(t)dt ) + ∫ Gb(us+1,us+1,u) 0 g(t)dt. Hence, from above with s → ∞, we get G(T (u, v),T (u, v), u) = 0, which gives that T (u, v) = u. Likewise, the same approach can be applied to write S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 17 of 23 ∫ Gb(T (v,u),T (v,u),v) 0 g(t)dt ≤ ∫ Gb(T (v,u),T (v,u),T (vs,us))+Gb(T (vs,us),T (vs,us),v) 0 g(t)dt ≤ ∫ Gb(H(v,u),H(v,u),H(vs,us)) 0 g(t)dt + ∫ Gb(T (vs,us),T (vs,us),v) 0 g(t)dt ≤ σ (∫ Gb(v,v,vs)+Gb(u,u,us) 0 g(t)dt ) + ∫ Gb(vs+1,vs+1,v) 0 g(t)dt. (9) Hence, via (9) with s → ∞, we get Gb(T (v, u), H(v, u), v) = 0 and thus H(v, u) = v. Therefore, (u, v) is a coupled fixed point of the mapping T. The following theorem shows that the coupled fixed point of T can be unique. Theorem 5. Suppose that (X ,Gb,≼) is a partially ordered complete Gb-metric space satisfying the following: (i) If (us) is a non-decreasing sequence that converges to some point u ∈ X , then us ≼ u for all s ∈ N. (ii) If (vs) is a non-increasing sequence that converges to a point v ∈ X , then vs ≽ v for all s ∈ N. (iii) For any two pairs (u, v), (u1, v1) ∈ X ×X , there exists a pair (w1, w2) ∈ X ×X that is comparable with both (u, v) and (u1, v1). Assume that T : X ×X → X is a continuous mapping which satisfies the mixed monotone property on X such that∫ Gb(T (u,v),T (m,n),T (f,w)) 0 g(t)dt ≤ σ (∫ Gb(u,m,f)+Gb(v,n,w) 0 g(t)dt ) , (10) where u, v, w,m, n, f ∈ X and g : [0,∞) → [0,∞) is a Lebesgue integrable mapping with f ≼ m ≼ u and v ≼ n ≼ w, where either m ̸= f or n ̸= w. If ∃ u0, v0 ∈ X such that u0 ≼ T (u0, v0) and T (v0, u0) ≼ v0 , then T has a unique coupled fixed point in (X ,G). S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 18 of 23 Proof. Suppose that (u1, v1) is another fixed point of T . The following cases are now considered. Case 1: Let (u, v) and (u1, v1) be elements in X × X that are comparable that (u, v) ≼ (u1, v1) i.e. u ≼ u1 and v ≼ v1. Now, by using (10),∫ Gb(T s(u,v),Ts(u1,v1),Ts(u1,v1)) 0 g(t)dt ≤ ∞∑ s=0 sσs (∫ Gb(u,u1,u1)+Gb(v,v1,v1) 0 g(t)dt ) . (11) Taking lim s→∞ and by the use of (11), u = u1. Following a similar pattern, it is clear that ∫ Gb(T s(v,u),Ts(v1,u1),Ts(v1,u1)) 0 g(t)dt ≤ ∞∑ s=0 sσs (∫ Gb(v,v1,v1)+Gb(u,u1,u1) 0 g(t)dt ) . (12) Taking lim s→∞ and by the use of (12), we obtain u = u1. Case 2: Let (u, v) be not comparable with (u1, v1). So by condition (iii) there exist (w1, w2) ∈ X × X , which is comparable to (u, v) and (u1, v1). We can assume that w1 ≼ u,w2 ≼ v, w1 ≼ u1 and w2 ≼ v1. Again, by using (10),∫ Gb(T s(u,v),Ts(w1,w2),Ts(w1,w2)) 0 g(t)dt ≤ ∞∑ s=0 sσs (∫ Gb(u,w1,w1)+Gb(v,w2,w2) 0 g(t)dt ) . (13) Taking s→ ∞ and by (13), Gb(T s(u, v),T s(w1, w2),T s(w1, w2)) = 0. That is, lim s→∞ T s(u, v) = lim s→∞ T s(w1, w2) = u.∫ Gb(T s(u1,v1),Ts(w1,w2),Ts(w1,w2)) 0 g(t)dt ≤ ∞∑ s=0 sσs (∫ Gb(u1,w1,w1)+Gb(v1,w2,w2) 0 g(t)dt ) . (14) From (14) lim s→∞ T s(u1, v1) = lim s→∞ T s(w1, w2) = u1, and so u = u1. Preceding in the same way, one has∫ Gb(T s(v,u),Ts(w2,w1),Ts(w2,w1)) 0 g(t)dt ≤ ∞∑ s=0 sσs (∫ Gb(v,w2,w2)+Gb(u,w1,w1) 0 g(t)dt ) . (15) Assume that s→ ∞, then (15) gives Gb(T s(v, u),T s(w2, w1),T s(w2, w1)) = 0. We have lim s→∞ T s(v, u) = lim s→∞ T s(w2, w1) = v. Similarly, it can be proved that∫ Gb(T s(v1,u1),Ts(w2,w1),Ts(w2,w1)) 0 g(t)dt ≤ ∞∑ s=0 sσs (∫ Gb(v1,w2,w2)+Gb(u1,w1,w1) 0 g(t)dt ) . (16) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 19 of 23 Also lim s→∞ T s(v1, u1) = lim s→∞ T s(w2, w1) = v1 by using (16). That is, v = v1. Hence, in all cases, (u, v) = (u1, v1), which means that the coupled fixed point of the mapping T is unique. Theorem 6. Let (X ,Gb,≼) be a partially ordered complete Gb-metric space satisfying the following conditions: (i) If (us) is a non-decreasing sequence which converges to u ∈ X , then us ≼ u ∀ s. (ii) If (vs) is a non-increasing sequence which converges to v ∈ X , then vs ≽ v ∀ s. (iii) Every pair of the element X has an upper bound and a lower bound in X . Also, let T : X ×X → X be a continuous having the mixed monotone property on X such that ∫ Gb(T (u,v),T (m,n),T (f,w)) 0 g(t)dt ≤ σ (∫ Gb(u,m,f)+Gb(v,n,w) 0 g(t)dt ) , (17) where u, v, w,m, n, f ∈ X and g : [0,∞) → [0,∞) is a Lebesgue integrable mapping with f ≼ m ≼ u and v ≼ n ≼ w, where either m ̸= f or n ̸= w. If there exist u0, v0 ∈ X such that u0 ≼ T (u0, v0) and T (v0, u0) ≼ v0, then u = v. Proof. Assume that u and v are comparable under the partial ordering ≼ in X , allowing us to assume that u ≼ v and v ≼ v. Using the same argument as in Theorem 3, we arrive at u = v. Next assume that u and v are incomparable. Then there is a common upper bound w ∈ X that is comparable with both u and v. So suppose that u ≼ w and v ≼ w. By applying Theorem 3, (u, v) = (w,w). Thus, w = v. The following examples validate our result. Example 2. Consider the set X = [0, 1] and define a mapping Gb : X ×X ×X → R+ by Gb(u, v, w) = |u− v|2+ |u−w|2+ |v−w|2 ∀ u, v, w ∈ X . Therefore (X ,Gb) is a complete Gb-metric space. Now, assume that σ(t) = t 2 for all t ∈ [0,∞), and let T : X × X → X be a mapping defined by T (g, h) = 3(g+h) 16 . Thus the conditions of Theorem 3 are satisfied. That is,∫ Gb(T (g,h),T (m,n),T (c,k)) 0 g(t)dt = ∫ |T (g,h)−T (m,n)|2+|T (g,h)−T (c,k)|2+|T (m,n)−T (c,k)|2 0 g(t)dt = ∫ | 3(g+h) 16 − 3(m+n) 16 |2+| 3(g+h) 16 − 3(c+l) 16 |2+| 3(m+n) 16 − 3(c+k) 16 |2 0 g(t)dt ≤ ∫ 9(2) 256 (|g−m|2+|h−n|2+|g−c|2+|h−k|2+|m−c|2+|n−k|2) 0 g(t)dt ≤ 1 256 ∫ l[|g−m|2+|h−n|2+|g−c|2+|h−k|2+|m−c|2+|n−k|2] 0 g(t)dt ≤ σ (∫ l[Gb(g,m,c)+Gb(h,n,k)] 0 g(t)dt ) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6705 20 of 23 where g, h, c,m, n, k ∈ X . Thus T has a coupled fixed point. Example 3. Let X = [0,∞) and Gb : X ×X ×X → R+ be a mapping defined by: Gb(u, v, w) = |u− v|q + |v − w|q + |w − u|q. Then Gb is Gb-metric space (by Example 1). Now, suppose that σ(t) = 1 2 t for all t ∈ [0,∞], and let T : X ×X → X be a mapping defined by T (g, h) = g+h 16 . We have∫ Gb(T (g,h),T (m,n),T (c,k)) 0 g(t)dt = ∫ |T (g,h)−T (m,n)|q+|T (g,h)−T (c,k)|q+|T (m,n)−T (c,k)|q 0 g(t)dt = ∫ | g+h 16 −m+n 16 |q+| g+h 16 − c+k 16 |q+|m+n 16 − c+k 16 |q 0 g(t)dt ≤ ∫ ( 1 16 )q(2)q−1(|g−m|q+|h−n|q+|g−c|q+|h−k|q+|m−c|q+|n−k|q) 0 g(t)dt ≤ ( 1 16 )q ∫ l(|g−m|q+|h−n|q+|g−c|q+|h−k|q+|m−c|q+|n−k|q) 0 g(t)dt ≤ σ (∫ l[Gb(g,m,c)+Gb(h,n,k)] 0 g(t)dt ) where g, h, c,m, n, k ∈ X . Clearly, T fulfills all the axioms of Theorem 3, so T possesses a coupled fixed point. 5. 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