EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6131 ISSN 1307-5543 – ejpam.com Published by New York Business Global Uniqueness of Stationary Distribution in Markov Processes: A Quintuple Fixed Point and Coincidence Point Approach Samina Batul1,∗, Sidra Fida1, Dur-e-Shehwar Sagheer1,, Hassen Aydi2,3, Saber Mansour4 1 Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan 2 Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia 3Department of Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa 4 Department of Mathematics, Umm Al-Qura University, Faculty of Sciences, P.O. Box 14035, Holly Makkah 21955, Saudi Arabia Abstract. This article introduces the concept of quintuple fixed points and coincidence points for matrix-related mappings in generalized metric spaces. Furthermore, the existence of quintuple coincidence points is established. This task is achieved by leveraging the structure of matrices. We derive several corollaries as special cases of our main results. These corollaries provide evidence for the authentication of the proven results. To validate the significance of our findings, we provide a selection of non-trivial examples. Eventually, we demonstrate the practical applicability of our established results by applying them to determine the stationary distribution of a Markov process. 2020 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Generalized metric space (GMs), Markov Process, Partially Ordered Metric Spaces (PoM), Tripled fixed point (TFp), Quintuple Fixed Point(QFP ) 1. Introduction Functional analysis has far-reaching applications in various fields, including linear and nonlinear analysis, calculus of variations, approximation theory, numerical analysis, and differential and integral equations. In nonlinear analysis, metric fixed point theory is a fundamental tool. Currently, finding solutions to differential and integral equations is a ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6131 Email addresses: samina.batul@cust.edu.pk (S. Batul), mmt221005@cust.pk (S. Fida), d.e.shehwar@cust.edu.pk (D. Sagheer), hassen.aydi@isima.rnu.tn (H. Aydi), samansour@uqu.edu.sa (S. Mansour) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 2 of 22 crucial research area. This task can be achieved by converting the equation into a fixed point problem for a suitable mapping and an appropriate domain. In 1922, Stephan Ba- nach [1] presented a vital result known as the Banach contraction principle (BCP), which erected a base for metric fixed-point theory. In 1964, Perov [2] extended the classical BCP on generalized metric spaces for contraction mappings. Filip and Petrusel [3] further gen- eralized these results on vector-valued metric spaces for self-mappings. A deeper concept of Perov-type contractions can be perceived from [4]. Later on, authors proved further extensions of BCP on generalized structures of metric spaces [5–7]. Working on a new track, Bhaskar and Lakshmikantham [8] established the coupled fixed point for mixed-monotone mappings under partially ordered metric spaces (PoM). Fur- thermore, a few important partial-order metric space results were presented in [9]. In 2011, Berinde and Borcut [10] extended the idea of coupled fixed point to tripled fixed point (TFp) for self-mappings and set up some significant results in PoMs. One can refer to [11] and [12] for a detailed review of these ideas. Subsequently, generalizing the concept of TFp in 2012, Karapınar [13] opened a gateway for researchers in a new direction by proposing the theory of quadruple fixed point of mappings and established exciting consequences in this regard, see also [14] and [15]. Working on a similar track, we extend the idea of a quadruple fixed point and intro- duced the notion of a quintuple fixed point(QFP ). Motivated by the work of Hammad [16] in 2022, we generalize crucial results for the existence of QFPs in generalized met- ric spaces and provide supportive examples and applications related to Markov process stationary distribution analysis. 2. Preliminaries Now, onward in this manuscript, Mn,n(R+), I, O are the symbol representations for the set of all n × n matrices over R+, identity and zero matrices respectively, and W = {0, 1, 2, 3, · · · } is the set of inetgers. Suppose that Γ̃ ∈ Mn,n(R+), then Γ̃ is said a convergent matrix, i.e, converges to zero matrix O if and only if lim n→∞ Γ̃n = O. A deeper concept can be built through [17]. Denote a set of all n×n matrices Γ̃ ∈ Mn,n(R+), with Γ̃n → O, whenever n → ∞ by ZM . Example 1. Let Γ̃ = ( ξ1 ξ2 ξ1 ξ2 ) be a matrix in M2,2(R+) with the condition that ξ1+ξ2 < 1, for some ξ1, ξ2 ∈ R+, then Γ̃ ∈ZM. Example 2. Suppose a matrix Υ̃ = ( ξ1 ξ2 ξ1 ξ2 ) ∈ M2,2(R+), such that ξ1 + ξ2 ≥ 1, for some ξ1, ξ2 ∈ R+, then Υ̃ does not belong to ZM. For a k dimensional vector space Rk, let 0, 1 be the zero vector and identity vector, respectively. Also, addition and multiplication in Rk are defined as under: ξ + ξ∗ = (ξ1 + ξ∗1 , ξ2 + ξ∗2 , ξ3 + ξ∗3 , · · · , ξk + ξ∗k) and ξ.ξ∗ = (ξ1.ξ ∗ 1 , ξ2.ξ ∗ 2 , · · · , ξk.ξ∗k), S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 3 of 22 for any ξ, ξ∗ ∈ Rk, where ξ = (ξ1, ξ2, ξ3, · · · , ξk) and ξ∗ = (ξ∗1 , ξ ∗ 2 , ξ ∗ 3 , · · · , ξ∗k). One can have a detailed study from [3]. The proof of the subsequent lemma is discussed in matrix analysis in [3]. Lemma 1. Suppose that Γ̃ is a square matrix with entries from R+, then the following statements are equivalent: (L1) Γ̃ → O; (L2) Γ̃n → O as n → ∞; (L3) for each z ∈ C, |z| < 1 with det(Γ̃− zI) = 0; (L4) for a non-singular matrix I − Γ̃ (I − Γ̃)−1 = I + Γ̃ + · · ·+ Γ̃n + · · · ; (L5) two matrices Γ̃nw and wΓ̃n tend to zero as n → ∞, for some w ∈ Rk. Definition 1. A mapping T : G2 → Rk, where G ̸= ∅, is named as a vector-valued metric over G, whenever the conditions below are fulfilled, that is, for each ξ1, ξ2, ξ3 ∈ G, (G1) T(ξ1, ξ2) ≥ 0, T(ξ1, ξ2) = 0 ⇔ ξ1 = ξ2, (G2) T(ξ1, ξ2) = T(ξ2, ξ1), (G3) T(ξ1, ξ2) ≤ T(ξ1, ξ3) + T(ξ3, ξ2). If ξ1, ξ2 ∈ Rk, where ξ1 = (ξ11 , ξ 2 1 , · · · , ξk1 ) and ξ2 = (ξ12 , ξ 2 2 , · · · , ξk2 ), then ξ1 ≤ ξ2 if and only if ξi1 ≤ ξi2, for 1 ≤ i ≤ k. Thus, (G,T) is a generalized metric space. [3] Bhaskar and Lakshmikantham [8] established the following concepts. Definition 2. An element (ξ1, ξ2) ∈ G2 is named as a coupled fixed point of the mapping P : G2 → G if P (ξ1, ξ2) = ξ1 and P (ξ2, ξ1) = ξ2. Definition 3. Two mappings P : G2 → G and p : G → G have a couple fixed point (ξ1, ξ2) ∈ G2 if P (ξ1, ξ2) = p(ξ1) and P (ξ2, ξ1) = p(ξ2). Definition 4. A mapping P : G2 → G on a partially ordered set (G,⪯) possesses the mixed-monotone property (MMP) if P (ξ1, ξ2) is non-decreasing in ξ1 and non-increasing in ξ2, i.e, for any ξ1, ξ2 ∈ G, ξ11 , ξ 2 1 ∈ G, ξ11 ⪯ ξ21 ⇒ P (ξ11 , ξ2) ⪯ P (ξ21 , ξ2) ξ12 , ξ 2 2 ∈ G, ξ12 ⪯ ξ22 ⇒ P (ξ1, ξ 1 2) ⪰ P (ξ1, ξ 2 2). Berinde and Borcut [10] constructed the idea of a tripled fixed point by generalizing the term of a coupled fixed point. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 4 of 22 Definition 5. An element (ξ1, ξ2, ξ3) ∈ G3 is termed as a triple fixed point of the mapping P : G3 → G if P (ξ1, ξ2, ξ3) = ξ1, P (ξ2, ξ3, ξ1) = ξ2,P (ξ3, ξ1, ξ2) = ξ3. The definition of a quadruple fixed point introduced by Karapınar [18] is stated as follows: Definition 6. [18] Let G ̸= ∅. An element (ξ1, ξ2, ξ3, ξ4) ∈ G4 is stated as the quadruple fixed point of the mapping P : G4 → G if P (ξ1, ξ2, ξ3, ξ4) = ξ1, P (ξ2, ξ3, ξ4, ξ1) = ξ2, P (ξ3, ξ4, ξ1, ξ2) = ξ3, P (ξ4, ξ1, ξ2, ξ3) = ξ4. Berinde extended the idea of mixed monotone property (MMP) [10] on G3, whereas Karapınar [18] introduced the concept of MMP on G4. Definition 7. A partially ordered set (G,⪯) is termed as regular if the conditions below are satisfied: (A) for m ≥ 0, ξm1 ⪯ ξ1 if a non-decreasing sequence ξm1 → ξ1, (B) for m ≥ 0, ξ2 ≤ ξm2 if a non-increasing sequence ξm2 → ξ2. [19] 3. Main Results Inspired by the results on couple, triple and quadruple fixed points, we initiate the term of quintuple fixed points to present some related fixed point results in the partially ordered complete generalized metric space (POCGMs) (G,T,⪯). Definition 8. Let P : G5 → G be a mapping. If P is monotonically non-increasing in ξ2, ξ4 and non-decreasing in ξ1, ξ3, ξ5, then P is said to have MMP. In other words, for any ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G, ξ11 , ξ 2 1 ∈ G, ξ11 ⪯ ξ21 ⇒ P (ξ11 , ξ2, ξ3, ξ4, ξ5) ⪯ P (ξ21 , ξ2, ξ3, ξ4, ξ5), ξ12 , ξ 2 2 ∈ G, ξ12 ⪯ ξ22 ⇒ P (ξ1, ξ 1 2 , ξ3, ξ4, ξ5) ⪰ P (ξ1, ξ 2 2 , ξ3, ξ4, ξ5), ξ13 , ξ 2 3 ∈ G, ξ13 ⪯ ξ23 ⇒ P (ξ1, ξ2, ξ 1 3 , ξ4, ξ5) ⪯ P (ξ1, ξ2, ξ 2 3 , ξ4, ξ5), ξ14 , ξ 2 4 ∈ G, ξ14 ⪯ ξ24 ⇒ P (ξ1, ξ2, ξ3, ξ 1 4 , ξ5) ⪰ P (ξ1, ξ2, ξ3, ξ 2 4 , ξ5), ξ15 , ξ 2 5 ∈ G, ξ15 ⪯ ξ25 ⇒ P (ξ1, ξ2, ξ3, ξ4, ξ 1 5) ⪯ P (ξ1, ξ2, ξ3, ξ4, ξ 2 5). The above definition can be generalized for two mappings as follows: Definition 9. Let P : G5 → G and p : G → G be two mappings. Then, P exhibits mixed p-monotone property (MpMP) if, for any ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G, ξ11 , ξ 2 1 ∈ G, p(ξ11) ⪯ p(ξ21) ⇒ P (ξ11 , ξ2, ξ3, ξ4, ξ5) ⪯ P (ξ21 , ξ2, ξ3, ξ4, ξ5), ξ12 , ξ 2 2 ∈ G, p(ξ12) ⪯ p(ξ22) ⇒ P (ξ1, ξ 1 2 , ξ3, ξ4, ξ5) ⪰ P (ξ1, ξ 2 2 , ξ3, ξ4, ξ5), ξ13 , ξ 2 3 ∈ G, p(ξ13) ⪯ p(ξ23) ⇒ P (ξ1, ξ2, ξ 1 3 , ξ4, ξ5) ⪯ P (ξ1, ξ2, ξ 2 3 , ξ4, ξ5), ξ14 , ξ 2 4 ∈ G, p(ξ14) ⪯ p(ξ24) ⇒ P (ξ1, ξ2, ξ3, ξ 1 4 , ξ5) ⪰ P (ξ1, ξ2, ξ3, ξ 2 4 , ξ5), ξ15 , ξ 2 5 ∈ G, p(ξ15) ⪯ p(ξ25) ⇒ P (ξ1, ξ2, ξ3, ξ4, ξ 1 5) ⪯ P (ξ1, ξ2, ξ3, ξ4, ξ 2 5). S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 5 of 22 Definition 10. An element (ξ1, ξ2, ξ3, ξ4, ξ5) ∈ G is called a quintuple fixed point (QFP ) of the mapping P : G5 → G if P (ξ1, ξ2, ξ3, ξ4, ξ5) = ξ1, P (ξ2, ξ3, ξ4, ξ5, ξ1) = ξ2, P (ξ3, ξ4, ξ5, ξ1, ξ2) = ξ3, P (ξ4, ξ5, ξ1, ξ2, ξ3) = ξ4, P (ξ5, ξ1, ξ2, ξ3, ξ4) = ξ5. For a generalized metric space (G,T), a function T̄ : G5 × G5 → Rk, given by T̄((ξ1, ξ2, ξ3, ξ4, ξ5), (q1, q2, q3, q4, q5)) = T(ξ1, q1)+T(ξ2, q2)+T(ξ3, q3)+T(ξ4, q4)+T(ξ5, q5), is a GMs on G5, that is, (G5, T̄) is a GMs induced by T. Definition 11. An element (ξ1, ξ2, ξ3, ξ4, ξ5) ∈ G5 is called a quintuple coincidence point (QCP ) or a common quintuple fixed point of the mappings P : G5 → G and p : G → G if P (ξ1, ξ2, ξ3, ξ4, ξ5) = p(ξ1), P (ξ2, ξ3, ξ4, ξ5, ξ1) = p(ξ2), P (ξ3, ξ4, ξ5, ξ1, ξ2) = p(ξ3), P (ξ4, ξ5, ξ1, ξ2, ξ3) = p(ξ4), P (ξ5, ξ1, ξ2, ξ3, ξ4) = p(ξ5). Moreover, for p : G → G being the identity map, Definition 9 and Definition 11 reduce into Definition 8 and Definition 10, respectively. Definition 12. Let P : G5 → G and p : G → G be two mappings. Then, P and p are said to be commutable if p(P (ξ1, ξ2, ξ3, ξ4, ξ5)) = P (p(ξ1), p(ξ2), p(ξ3), p(ξ4), p(ξ5)), for all ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G. Definition 13. The mappings P : G5 → G and p : G → G over a metric space (G,T) are compatible if the following conditions hold: lim n→+∞ T(p(U1), P (V 1)) = 0, where U1 = P (ξn1 , ξ n 2 , ξ n 3 , ξ n 4 , ξ n 5 ) and V 1 = (p(ξn1 ), p(ξ n 2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 )), lim n→+∞ T(p(U2), P (V 2)) = 0, where U2 = P (ξn2 , ξ n 3 , ξ n 4 , ξ n 5 , ξ n 1 ) and V 2 = (p(ξn2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 ), p(ξ n 1 )), lim n→+∞ T(p(U3), P (V 3)) = 0, where U3 = P (ξn3 , ξ n 4 , ξ n 5 , ξ n 1 , ξ n 2 ) and V 3 = (p(ξn3 ), p(ξ n 4 ), p(ξ n 5 ), p(ξ n 1 ), p(ξ n 2 )), lim n→+∞ T(p(U4), P (V 4)) = 0, where U4 = P (ξn4 , ξ n 5 , ξ n 1 , ξ n 2 , ξ n 3 ) and V 4 = (p(ξn4 ), p(ξ n 5 ), p(ξ n 1 ), p(ξ n 2 ), p(ξ n 3 )), lim n→+∞ T(p(U5), P (V 5)) = 0, where U5 = P (ξn5 , ξ n 1 , ξ n 2 , ξ n 3 , ξ n 4 ) and V 5 = (p(ξn5 ), p(ξ n 1 ), p(ξ n 2 ), p(ξ n 3 ), p(ξ n 4 )), S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 6 of 22 whenever {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } are sequences in G in such a way that lim n→+∞ U1 = lim n→+∞ p(ξn1 ) = ξ1, lim n→+∞ U2 = lim n→+∞ p(ξn2 ) = ξ2, lim n→+∞ U3 = lim n→+∞ p(ξn3 ) = ξ3, lim n→+∞ U4 = lim n→+∞ p(ξn4 ) = ξ4, lim n→+∞ U5 = lim n→+∞ p(ξn5 ) = ξ5, for some ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G. Definition 14. The mappings P : G5 → G and p : G → G are termed as reciprocally continuous if for some ξi ∈ G, where 1 ≤ i ≤ 5, we have lim n→+∞ p(U1) = p(ξ1) and lim n→+∞ P (V 1) = P (ξ1, ξ2, ξ3, ξ4, ξ5), lim n→+∞ p(U2) = p(ξ2) and lim n→+∞ P (V 2) = P (ξ2, ξ3, ξ4, ξ5, ξ1), lim n→+∞ p(U3) = p(ξ3) and lim n→+∞ P (V 3) = P (ξ3, ξ4, ξ5, ξ1, ξ2), lim n→+∞ p(U4) = p(ξ4) and lim n→+∞ P (V 4) = P (ξ4, ξ5, ξ1, ξ2, ξ3), lim n→+∞ p(U5) = p(ξ5) and lim n→+∞ P (V 5) = P (ξ5, ξ1, ξ2, ξ3, ξ4), whenever {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } are sequences in G such that lim n→+∞ U1 = lim n→+∞ p(ξn1 ) = ξ1, lim n→+∞ U2 = lim n→+∞ p(ξn2 ) = ξ2, , lim n→+∞ U3 = lim n→+∞ p(ξn3 ) = ξ3, lim n→+∞ U4 = lim n→+∞ p(ξn4 ) = ξ4, lim n→+∞ U5 = lim n→+∞ p(ξn5 ) = ξ5, for some ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G. Definition 15. Two mappings P : G5 → G and p : G → G are known as weakly reciprocally continuous if for some ξi ∈ G, where 1 ≤ i ≤ 5, we have lim n→+∞ p(U1) = p(ξ1) or lim n→+∞ P (V 1) = P (ξ1, ξ2, ξ3, ξ4, ξ5), lim n→+∞ p(U2) = p(ξ2) or lim n→+∞ P (V 2) = P (ξ2, ξ3, ξ4, ξ5, ξ1), lim n→+∞ p(U3) = p(ξ3) or lim n→+∞ P (V 3) = P (ξ3, ξ4, ξ5, ξ1, ξ2), lim n→+∞ p(U4) = p(ξ4) or lim n→+∞ P (V 4) = P (ξ4, ξ5, ξ1, ξ2, ξ3), lim n→+∞ p(U5) = p(ξ5) or lim n→+∞ P (V 5) = P (ξ5, ξ1, ξ2, ξ3, ξ4), whenever {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } are few sequences within the set G in such a way that lim n→+∞ U1 = lim n→+∞ p(ξn1 ) = ξ1, lim n→+∞ U2 = lim n→+∞ p(ξn2 ) = ξ2, , lim n→+∞ U3 = lim n→+∞ p(ξn3 ) = ξ3, lim n→+∞ U4 = lim n→+∞ p(ξn4 ) = ξ4, lim n→+∞ U5 = lim n→+∞ p(ξn5 ) = ξ5, for some ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 7 of 22 Example 3. Let G = [0, 1]. Let T(ξ1, ξ2) = |ξ1 − ξ2| and “⪯” be the partial order on G defined for all ξ1, ξ2 ∈ G, ξ1 ⪯ ξ2 ⇔ ξ1 ≤ ξ2. Let P : G5 → G and p : G → G be two mappings defined as P (ξ1, ξ2, ξ3, ξ4, ξ5) = ξ1ξ2 − ξ3ξ4 + ξ5 5 and p(ξ1) = ξ1, ∀ ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G. Consider the sequences {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } defined by ξn1 = 1 n , ξn2 = 1 n3 + 1 , ξn3 = 1√ n3 + 1 , ξn4 = 1 n3 , and ξn5 = 1 n3 + 2 ∀ n ∈ N. Clearly, (G,T) is a partially ordered metric space and G is complete. In addition, lim n→+∞ U1 = lim n→+∞ p(ξn1 ) = 0, lim n→+∞ U2 = lim n→+∞ p(ξn2 ) = 0, lim n→+∞ U3 = lim n→+∞ p(ξn3 ) = 0, lim n→+∞ U4 = lim n→+∞ p(ξn4 ) = 0, lim n→+∞ U5 = lim n→+∞ p(ξn5 ) = 0. Moreover, defined sequences, functions and metrics satisfy compatibility conditions, recip- rocal continuity and weakly reciprocal continuity for P and p. In light of this, both P and p exhibit compatibility, reciprocal continuity and weak reciprocal continuity. Definition 16. For two mappings ζn : G5 → G and p : G → G defined on the metric space (G,T), the sequence {ζn}n∈W and p are known as compatible if lim n→+∞ T(p(Ū1), ζn(V̄ 1)) = 0, where Ū1 = ζn(ξn1 , ξ n 2 , ξ n 3 , ξ n 4 , ξ n 5 ) and V̄ 1 = (p(ξn1 ), p(ξ n 2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 )), lim n→+∞ T(p(Ū2), ζn(V̄ 2)) = 0, where Ū2 = ζn(ξn2 , ξ n 3 , ξ n 4 , ξ n 5 , ξ n 1 ) and V̄ 2 = (p(ξn2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 ), p(ξ n 1 )), lim n→+∞ T(p(Ū3), ζn(V̄ 3)) = 0, where Ū3 = ζn(ξn3 , ξ n 4 , ξ n 5 , ξ n 1 , ξ n 2 ) and V̄ 3 = (p(ξn3 ), p(ξ n 4 ), p(ξ n 5 ), p(ξ n 1 ), p(ξ n 2 )), lim n→∞ T(p(Ū4), ζn(V̄ 4)) = 0, where Ū4 = ζn(ξn4 , ξ n 5 , ξ n 1 , ξ n 2 , ξ n 3 ) and V̄ 4 = (p(ξn4 ), p(ξ n 5 ), p(ξ n 1 ), p(ξ n 2 ), p(ξ n 3 )), lim n→∞ T(p(Ū5), ζn(V̄ 5)) = 0, where Ū5 = ζn(ξn5 , ξ n 1 , ξ n 2 , ξ n 3 , ξ n 4 ) and V̄ 5 = (p(ξn5 ), p(ξ n 1 ), p(ξ n 2 ), p(ξ n 3 ), p(ξ n 4 )), whenever {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } are sequences in G such that lim n→+∞ Ū1 = lim n→∞ p(ξn+1 1 ) = ξ1, lim n→+∞ Ū2 = lim n→∞ p(ξn+1 2 ) = ξ2, lim n→+∞ Ū3 = lim n→∞ p(ξn+1 3 ) = ξ3 lim n→+∞ Ū4 = lim n→∞ p(ξn+1 4 ) = ξ4, lim n→+∞ Ū5 = lim n→∞ p(ξn+1 5 ) = ξ5, for some ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 8 of 22 Definition 17. Let ζn : G5 → G and p : G → G be two mappings on a metric space (G,T), then the following conditions describe weak reciprocal continuity of {ζn}n∈N and p. lim n→+∞ p(Ū1) = p(ξ1), lim n→+∞ p(Ū2) = p(ξ2), lim n→+∞ p(Ū3) = p(ξ3) lim n→+∞ p(Ū4) = p(ξ4), lim n→+∞ p(Ū5) = p(ξ5), whenever {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } are sequences in G such that lim n→+∞ Ū1 = lim n→∞ p(ξn+1 1 ) = ξ1, lim n→+∞ Ū2 = lim n→∞ p(ξn+1 2 ) = ξ2, lim n→+∞ Ū3 = lim n→∞ p(ξn+1 3 ) = ξ3 lim n→+∞ Ū4 = lim n→∞ p(ξn+1 4 ) = ξ4, lim n→+∞ Ū5 = lim n→∞ p(ξn+1 5 ) = ξ5, for some ξi in G, where 1 ≤ i ≤ 5. Example 4. Let G = [0, 1] be endowed with the metric T(ξ1, ξ2) = |ξ1−ξ2|. Let ζn : G5 → G and p : G → G be two mappings defined as ζn(ξ1, ξ2, ξ3, ξ4, ξ5) = 1 5n − ξ1ξ2ξ3ξ4ξ5 5 and p(ξ1) = ξ1, ∀ ξ1, ξ2, ξ3, ξ4, ξ5 ∈ G. Consider the five sequences {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } ∈ G defined as ξn1 = 1 n2 + 1 , ξn2 = 1√ n2 + 1 , ξn3 = 1 n+ 1 , ξn4 = 1√ n+ 1 and ξn5 = 1 n3 + 1 , ∀ n ∈ N. Then, lim n→+∞ Ū1 = lim n→∞ p(ξn+1 1 ) = 0, lim n→+∞ Ū2 = lim n→∞ p(ξn+1 2 ) = 0, lim n→+∞ Ū3 = lim n→∞ p(ξn+1 3 ) = 0, lim n→+∞ Ū4 = lim n→∞ p(ξn+1 4 ) = 0, lim n→+∞ Ū5 = lim n→∞ p(ξn+1 5 ) = 0. Also, lim n→+∞ T(p(Ū1), ζn(V̄ 1)) = 0, lim n→+∞ T(p(Ū2), ζn(V̄ 2)) = 0, lim n→+∞ T(p(Ū3), ζn(V̄ 3)) = 0, lim n→+∞ T(p(Ū4), ζn(V̄ 4)) = 0, lim n→+∞ T(p(Ū5), ζn(V̄ 5)) = 0, and lim n→+∞ p(Ū1) = p(ξ1) = 0, lim n→+∞ p(Ū2) = p(ξ2) = 0, lim n→+∞ p(Ū3) = p(ξ3) = 0 lim n→+∞ p(Ū4) = p(ξ4) = 0, lim n→+∞ p(Ū5) = p(ξ5) = 0, which proves the compatibility and weak reciprocal continuity of {ζn}n∈N and p. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 9 of 22 Definition 18. Let (G,⪯) be a partially ordered set (PoS) and ζn : G5 → G and p : G → G be two mappings on G, then {ζn}n∈W is said to have MpMP if for any ξ1, ξ2, ξ3, ξ4, ξ5, q1, q2, q3, q4, q5 ∈ G, p(ξ1) ⪯ p(q1) ⇒ ζn(ξ1, ξ2, ξ3, ξ4, ξ5) ⪯ ζn+1(q1, q2, q3, q4, q5), p(ξ2) ⪰ p(q2) ⇒ ζn(ξ2, ξ3, ξ4, ξ5, ξ1) ⪰ ζn+1(q2, q3, q4, q5, q1), p(ξ3) ⪯ p(q3) ⇒ ζn(ξ3, ξ4, ξ5, ξ1, ξ2) ⪯ ζn+1(q3, q4, q5, q1, q2), p(ξ4) ⪰ p(q4) ⇒ ζn(ξ4, ξ5, ξ1, ξ2, ξ3) ⪰ ζn+1(q4, q5, q1, q2, q3), p(ξ5) ⪯ p(q5) ⇒ ζn(ξ5, ξ1, ξ2, ξ3, ξ4) ⪯ ζn+1(q5, q1, q2, q3, q4). Definition 19. Let ζi : G5 → G and p : G → G be two mappings. Then, {ζi}i∈W and p are said to satisfy the (C) condition if T(ζi(ξ1, ξ2, ξ3, ξ4, ξ5), ζ j(q1, q2, q3, q4, q5)) ≤ Γ̃[T(p(ξ1), ζ i(ξ1, ξ2, ξ3, ξ4, ξ5)) + T(p(q1), ζ j(q1, q2, q3, q4, q5))] + Υ̃[T(p(ξ1), p(q1))], (1) for some ξi, qi ∈ G, where 1 ≤ i ≤ 5, provided that p(ξi) ⪯ p(qi), for 1 ≤ i ≤ 5, or p(ξi) ⪰ p(qi), for 1 ≤ i ≤ 5. In addition, I ≠ Γ̃ = (Γ̃ij) and I ≠ Υ̃ = (Υ̃ij) ∈ ZM satisfy the condition that (Γ̃ + Υ̃)(I − Γ̃)−1 ∈ ZM. Example 5. Let G = [0, 1] be equipped with metric T(ξ1, ξ2) = |ξ1 − ξ2|, for all ξ1, ξ2 ∈ G. (1) Let Γ̃ = ( 1 5 0 0 1 5 ) and Υ̃ = ( 0 1 5 1 5 0 ) ∈ ZM . Then, (Γ̃ + Υ̃)(I − Γ̃)−1 ∈;ZM . (2) Let Γ̃ = ξI, and Υ̃ = ((1−ξ)3−ξ)I ∈ ZM such that ξ = 1 4 , 1 5 , 1 7 , 1 8 , then (Γ̃+Υ̃)(I− Γ̃)−1 ∈ ZM . Definition 20. Let ζo : G5 → G and p : G → G be two mappings, then ζo and p are said to have mixed quintuple transcendence point (MQTP) if there exists some ξo1, ξ o 2, ξ o 3, ξ o 4, ξ o 5 ∈ G such that ζo(ξo1, ξ o 2, ξ o 3, ξ o 4, ξ o 5) ⪰ p(ξo1), ζo(ξo2, ξ o 3, ξ o 4, ξ o 5, ξ o 1) ⪯ p(ξo2), ζo(ξo3, ξ o 4, ξ o 5, ξ o 1, ξ o 2) ⪰ p(ξo3), ζo(ξo4, ξ o 5, ξ o 1, ξ o 2, ξ o 3) ⪯ p(ξo4), ζo(ξo5, ξ o 1, ξ o 2, ξ o 3, ξ o 4) ⪰ p(ξo5), (2) given that ζo and p have non-decreasing transcendent point in ξo1, ξ o 3, ξ o 5 and a non-increasing transcendence point in ξo2, ξ o 4. Lemma 2. Let ζi : G5 → G and p : G → G be two mappings in the setting of a partially ordered complete generalized metric space (POCGMs) (G,T,⪯). Suppose that {ζi}i∈W have MpMP such that ζi(G5) ⊆ p(G). If ζo and p have MQTp, then S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 10 of 22 (a): ∃ sequences {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } ∈ G such that p(ξn1 ) = ζn−1(ξn−1 1 , ξn−1 2 , ξn−1 3 , ξn−1 4 , ξn−1 5 ), p(ξn2 ) = ζn−1(ξn−1 2 , ξn−1 3 , ξn−1 4 , ξn−1 5 , ξn−1 1 ), p(ξn3 ) = ζn−1(ξn−1 3 , ξn−1 4 , ξn−1 5 , ξn−1 1 , ξn−1 2 ), p(ξn4 ) = ζn−1(ξn−1 4 , ξn−1 5 , ξn−1 1 , ξn−1 2 , ξn−1 3 ), p(ξn5 ) = ζn−1(ξn−1 5 , ξn−1 1 , ξn−1 2 , ξn−1 3 , ξn−1 4 ). (b): {p(ξn1 )}, {p(ξn3 )}, {p(ξn5 )} are non-decreasing sequences and {p(ξn2 )}, {p(ξn4 )} are non- increasing sequences. Proof. (a): Suppose that condition in (2) is fulfilled for some ξo1, ξ o 2, ξ o 3, ξ o 4, ξ o 5 ∈ G. Since ζo(G5) ⊆ p(G), then some elements ξ11 , ξ 1 2 , ξ 1 3 , ξ 1 4 , ξ 1 5 ∈ G p(ξ11) =ζo(ξo1, ξ o 2, ξ o 3, ξ o 4, ξ o 5), p(ξ12) = ζo(ξo2, ξ o 3, ξ o 4, ξ o 5, ξ o 1), p(ξ13) =ζo(ξo3, ξ o 4, ξ o 5, ξ o 1, ξ o 2), p(ξ14) = ζo(ξo4, ξ o 5, ξ o 1, ξ o 2, ξ o 3), p(ξ15) =ζo(ξo5, ξ o 1, ξ o 2, ξ o 3, ξ o 4). (3) Since ζo(G5) ⊆ p(G), then some elements can be set as ξ21 , ξ 2 2 , ξ 2 3 , ξ 2 4 , ξ 2 5 ∈ G such that p(ξ21) =ζ1(ξ11 , ξ 1 2 , ξ 1 3 , ξ 1 4 , ξ 1 5), p(ξ22) = ζ1(ξ12 , ξ 1 3 , ξ 1 4 , ξ 1 5 , ξ 1 1), p(ξ23) =ζ1(ξ13 , ξ 1 4 , ξ 1 5 , ξ 1 1 , ξ 1 2), p(ξ24) = ζ1(ξ14 , ξ 1 5 , ξ 1 1 , ξ 1 2 , ξ 1 3), p(ξ25) =ζ1(ξ15 , ξ 1 1 , ξ 1 2 , ξ 1 3 , ξ 1 4). Proceeding similarly, we obtain p(ξn1 ) = ζn−1(ξn−1 1 , ξn−1 2 , ξn−1 3 , ξn−1 4 , ξn−1 5 ), p(ξn2 ) = ζn−1(ξn−1 2 , ξn−1 3 , ξn−1 4 , ξn−1 5 , ξn−1 1 ), p(ξn3 ) = ζn−1(ξn−1 3 , ξn−1 4 , ξn−1 5 , ξn−1 1 , ξn−1 2 ), p(ξn4 ) = ζn−1(ξn−1 4 , ξn−1 5 , ξn−1 1 , ξn−1 2 , ξn−1 3 ), p(ξn5 ) = ζn−1(ξn−1 5 , ξn−1 1 , ξn−1 2 , ξn−1 3 , ξn−1 4 ). (4) (b): Now, from (2) and (3), p(ξo1) ⪯ p(ξ11), p(ξo3) ⪯ p(ξ13), p(ξo5) ⪯ p(ξ15), p(ξo2) ⪰ s(ξ12) and s(ξo4) ⪰ s(ξ14). Then, ∀ n ≥ 0, by mathematical induction, it is obtained that p(ξn1 ) ⪯ p(ξn+1 1 ), p(ξn3 ) ⪯ p(ξn+1 3 ), p(ξn5 ) ⪯ p(ξn+1 5 ), p(ξn2 ) ⪰ p(ξn+1 2 ), and p(ξn4 ) ⪰ p(ξn+1 4 ). (5) Hence, (4) and (5) complete the required result. Our next theorem is the core part of this section. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 11 of 22 Theorem 1. Suppose that all suppositions of Lemma 2 hold, let {ζi}i∈W and p be two monotonically decreasing mappings such that they satisfy (C) condition. In addition, suppose that both mappings are compatible and weakly reciprocally continuous and p is continuous. If p(G) ⊆ G is complete and regular, then there exists a quintuple coincidence point (QCP ) of {ζi}i∈W and p provided that Γ̃, Υ̃ ̸= O belongs to ZM. Proof. Let {ξn1 }, {ξn2 }, {ξn3 }, {ξn4 } and {ξn5 } be the sequences in G constructed by Lemma 2, then from (2) it follows that T(p(ξn1 ), (ξ n+1 1 )) = T(ζn−1(ξn−1 1 , ξn−1 2 , ξn−1 3 , ξn−1 4 , ξn−1 5 ), ζn(ξn1 , ξ n 2 , ξ n 3 , ξ n 4 , ξ n 5 )) ≤ Γ̃[T(p(ξn−1 1 ), ζn−1(ξn−1 1 , ξn−1 2 , ξn−1 3 , ξn−1 4 , ξn−1 5 )) + T(p(ξn1 ), ζ n(ξn1 , ξ n 2 , ξ n 3 , ξ n 4 , ξ n 5 ))] + Υ̃(T(p(ξn−1 1 ), p(ξn1 ))) = (Γ̃ + Υ̃)T(p(ξn−1 1 ), p(ξn1 )) + Γ̃T(p(ξn1 ), p(ξ n+1 1 )). It results in T(p(ξn1 ), p(ξ n+1 1 )) ≤ (Γ̃ + Υ̃)(I − Γ̃)−1T(p(ξn−1 1 ), p(ξn1 )). (6) Similar operations generate T(p(ξn2 ), p(ξ n+1 2 )) ≤ (Γ̃ + Υ̃)(I − Γ̃)−1T(p(ξn−1 2 ), p(ξn2 )), T(p(ξn3 ), p(ξ n+1 3 )) ≤ (Γ̃ + Υ̃)(I − Γ̃)−1T(p(ξn−1 3 ), p(ξn3 )), T(p(ξn4 ), p(ξ n+1 4 )) ≤ (Γ̃ + Υ̃)(I − Γ̃)−1T(p(ξn−1 4 ), p(ξn4 )), (7) and T(p(ξn5 ), p(ξ n+1 5 )) ≤ (Γ̃ + Υ̃)(I − Γ̃)−1T(p(ξn−1 5 ), p(ξn5 )). (8) Adding (3.6) - (3.8), one writes µn = T(p(ξn1 ), p(ξ n+1 1 )) + T(p(ξn2 ), p(ξ n+1 2 )) + T(p(ξn3 ), p(ξ n+1 3 )) + T(p(ξn4 ), p(ξ n+1 4 )) + T(p(ξn5 ), p(ξ n+1 5 )) ≤ (Γ̃ + Υ̃)(I − Γ̃)−1[T(p(ξn−1 1 ), p(ξn1 )) + T(p(ξn−1 2 ), p(ξn2 )) + T(p(ξn−1 3 ), p(ξn3 )) + T(p(ξn−1 4 ), P (ξn4 )) + T(p(ξn−1 5 ), P (ξn5 ))] = (Γ̃ + Υ̃)(I − Γ̃)−1µn−1. Take (Γ̃ + Υ̃)(I − Γ̃)−1 = Q, hence for n ∈ N, O ≤ µn ≤ Qµn−1 ≤ Q2µn−2 ≤ · · · ≤ Qnµo. In light of triangular inequality, for m > 0, T(p(ξn1 ), p(ξ n+m 1 )) + T(p(ξn2 ), p(ξ n+m 2 )) + T(p(ξn3 ), p(ξ n+m 3 )) + T(p(ξn4 ), p(ξ n+m 4 )) + T(p(ξn5 ), p(ξ n+m 5 )) S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 12 of 22 ≤ T(p(ξn1 ), p(ξ n+1 1 )) + T(p(ξn2 ), (ξ n+1 2 )) + T(p(ξn3 ), p(ξ n+1 3 )) + T(p(ξn4 ), p(ξ n+1 4 )) + T(p(ξn5 ), p(ξ n+1 5 )) + T(p(ξn+1 1 ), p(ξn+2 1 )) + T(p(ξn+1 2 ), p(ξn+2 2 )) + T(p(ξn+1 3 ), p(ξn+2 3 )) + T(p(ξn5 ), p(ξ n+2 5 )) + T(p(ξn+1 4 ), p(aξ+2 4 )) + · · ·+ T(p(ξn+m−1 1 ), p(ξn+m 1 )) + T(p(ξn+m−1 2 ), p(ξn+m 2 )) + T(p(ξn+m−1 3 ), p(ξn+m 3 )) + T(p(ξn+m−1 4 ), p(ξn+m 4 )) + T(p(ξn+m−1 5 ), p(ξn+m 5 )) = µn + µn+1 + · · ·+ µn+m−1 ≤ (Qn +Qn+1 + · · ·+Qn+m−1)µo = Qn(I +Q+ · · ·+Qm−1 + · · · )µo = Qn(I −Q)−1µo. This implies that lim n→+∞ [T(p(ξn1 ), p(ξ n+m 1 )) + T(p(ξn2 ), p(ξ n+m 2 )) + T(p(ξn3 ), p(ξ n+m 3 )) + T(p(ξn4 ), p(ξ n+m 4 )) + T(p(ξn5 ), p(ξ n+m 5 ))] ≤ [(Γ̃ + Υ̃)(I − Γ̃)−1]n[I − (Γ̃ + Υ̃)(I − Γ̃)−1]−1µo = [(Γ̃ + Υ̃)(I − Γ̃)−1]n[I − (Γ̃ + Υ̃)(I − Γ̃)−1]−1µo. Applying lim n→+∞ on both sides yields that lim n→+∞ [T(p(ξn1 ), p(ξ n+m 1 )) + T(p(ξn2 ), p(ξ n+m 2 )) + T(p(ξn3 ), p(ξ n+m 3 )) + T(p(ξn4 ), p(ξ n+m 4 )) + T(p(ξn5 ), p(ξ n+m 5 ))] = 0, or lim n→+∞ T(p(ξn1 ), p(ξ n+m 1 )) = lim n→+∞ T(p(ξn2 ), p(ξ n+m 2 )) = lim n→+∞ T(p(ξn3 ), p(ξ n+m 3 )) = = lim n→+∞ T(p(ξn4 ), p(ξ n+m 4 ))] = lim n→+∞ T(p(ξn5 ), p(ξ n+m 5 ))] = 0. Hence, {p(ξn1 )}, {p(ξn2 )}, {p(ξn3 )}, {p(ξn4 )} and {p(ξn5 )} are Cauchy sequences within the set G. Moreover, by completeness of p(G), there must exist (ξ∗1 , ξ ∗ 2 , ξ ∗ 3 , ξ ∗ 4 , ξ ∗ 5) ∈ G5 such that lim n→+∞ p(ξn1 ) = p(ξ∗1) = ξ1, lim n→+∞ p(ξn2 ) = p(ξ∗2) = ξ2, lim n→+∞ p(ξn3 ) = p(ξ∗3) = ξ3, lim n→+∞ p(ξn4 ) = p(ξ∗4) = ξ4, lim n→+∞ p(ξn5 ) = p(ξ∗5) = ξ5, which results in lim n→+∞ p(ξn+1 1 ) = lim n→+∞ ζn(ξn1 , ξ n 2 , ξ n 3 , ξ n 4 , ξ n 5 ), lim n→+∞ p(ξn+1 2 ) = lim n→+∞ ζn(ξn2 , ξ n 3 , ξ n 4 , ξ n 5 , ξ n 1 ), lim n→+∞ p(ξn+1 3 ) = lim n→+∞ ζn(ξn3 , ξ n 4 , ξ n 5 , ξ n 1 , ξ n 2 ), lim n→+∞ p(ξn+1 4 ) = lim n→+∞ ζn(ξn4 , ξ n 5 , ξ n 1 , ξ n 2 , ξ n 3 ), lim n→+∞ p(ξn+1 5 ) = lim n→+∞ ζn(ξn5 , ξ n 1 , ξ n 2 , ξ n 3 , ξ n 4 ). S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 13 of 22 Also, from the weak reciprocal continuity and compatibility of {ζi}i∈W and p, it is derived that lim n→+∞ ζn(p(ξn1 ), p(ξ n 2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 )) = p(ξ1), lim n→+∞ ζn(p(ξn2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 ), p(ξ n 1 )) = p(ξ2), lim n→+∞ ζn(p(ξn3 ), p(ξ n 4 ), p(ξ n 5 ), p(ξ n 1 ), p(ξ n 2 )) = p(ξ3), lim n→+∞ ζn(p(ξn4 ), p(ξ n 5 ), p(ξ n 1 ), p(ξ n 2 ), p(ξ n 3 )) = p(ξ4), lim n→+∞ ζn(p(ξn5 ), p(ξ n 1 ), p(ξ n 2 ), s(ξ n 3 ), p(ξ n 4 )) = p(ξ5). Since {p(ξn1 )}, {p(ξn3 )}, {p(ξn5 )} are non-decreasing sequences and {p(ξn2 )}, {p(ξn4 )} are non-increasing sequences, from the regularity of G, for all n ≥ 0, it is obtained that p(ξn1 ) ⪯ ξ1, ξ2 ⪯ p(ξn2 ), p(ξ n 3 ) ⪯ ξ3, ξ4 ⪯ p(ξn4 ), p(ξ n 5 ) ⪯ ξ5. Then, from (1), it is computed as T(ζi(ξ1, ξ2, ξ3, ξ4, ξ5), ζ n(p(ξn1 ), p(ξ n 2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 )) ≤ Γ̃[T(p(ξ1), ζ i(ξ1, ξ2, ξ3, ξ4, ξ5)) + T(p(p(ξn1 )), ζ n(p(ξn1 ), p(ξ n 2 ), p(ξ n 3 ), p(ξ n 4 ), p(ξ n 5 ))] + Υ̃T(p(ξ1), p(p(ξ n 1 ))). Hence, taking n → +∞, one gets T(ζi(ξ1, ξ2, ξ3, ξ4, ξ5), p(ξ1)) ≤ Γ̃T(p(ξ1), ζ i(ξ1, ξ2, ξ3, ξ4, ξ5)), which only holds if T(ζi(ξ1, ξ2, ξ3, ξ4, ξ5), p(ξ1)) = 0 ⇒ ζi(ξ1, ξ2, ξ3, ξ4, ξ5) = p(ξ1). Similar operation generates ζi(ξ2, ξ3, ξ4, ξ5, ξ1) = p(ξ2), ζi(ξ3, ξ4, ξ5, ξ1, ξ2) = p(ξ3), ζi(ξ4, ξ5, ξ1, ξ2, ξ3) = p(ξ4) and ζi(ξ5, ξ1, ξ2, ξ3, ξ4) = p(ξ5). Hence, (ξ1, ξ2, ξ3, ξ4, ξ5) is a QCP of {ζi}i∈W and p. The next result is an extension of Theorem 1 by introducing s = Id as an identity map. Corollary 1. Let {ζi}i∈W : G5 → G be a mixed-monotone sequence over a POCGMs (G,T,⪯) such that {ζi}i∈W and Id : G → G satisfy (C) condition and Id(ℵ) is regular. If Id and ζo have MQTP, then ∃ (ξ1, ξ2, ξ3, ξ4, ξ5) ∈ G5 such that ζi(ξ1, ξ2, ξ3, ξ4, ξ5) = ξ1, ζi(ξ2, ξ3, ξ4, ξ5, ξ1) = ξ2, ζi(ξ3, ξ4, ξ5, ξ1, ξ2) = ξ3, ζi(ξ4, ξ5, ξ1, ξ2, ξ3) = ξ4, and ζi(ξ5, ξ1, ξ2, ξ3, ξ4) = ξ5. for i ∈ W. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 14 of 22 By excluding some of the conditions from Corollary 1, taking Γ̃ as a zero matrix and expanding the distance T(ξ1, ξ2), we conclude an important outcome. Corollary 2. Let F : G5 → G be a mixed-monotone mapping in the setting of a POCGMs (G,T,⪯) such that F has a MQTP and F satisfy the condition T(F(ξ1, ξ2, ξ3, ξ4, ξ5),F(q1, q2, q3, q4, q5)) ≤ Υ̃(T((ξ1, ξ2, ξ3, ξ4, ξ5), (q1, q2, q3, q4, q5))). Then, their exists a QFP of F in G. Definition 21. Two points (ξ1, ξ2, ξ3, ξ4, ξ5) and (q1, q2, q3, q4, q5) ∈ G are called quintuple comparable (QC) if and only if ξ1 ⪯ q1, ξ2 ⪰ q2, ξ3 ⪯ q3, ξ4 ⪰ q4, ξ5 ⪯ q5 or ξ1 ⪰ q1, ξ2 ⪯ q2, ξ3 ⪰ q3, ξ4 ⪯ q4, ξ5 ⪰ q5 or ξ1 ⪯ q2, ξ2 ⪰ q3, ξ3 ⪯ q4, ξ4 ⪰ q5, ξ5 ⪯ q1 or ξ1 ⪰ q2, ξ2 ⪯ q3, ξ3 ⪰ q4, ξ4 ⪯ q5, ξ5 ⪰ q1 or ξ1 ⪯ q3, ξ2 ⪰ q4, ξ3 ⪯ q5, ξ4 ⪰ q1, ξ5 ⪯ q2 or ξ1 ⪰ q3, ξ2 ⪯ q4, ξ3 ⪰ q5, ξ4 ⪯ q1, ξ5 ⪰ q2 or ξ1 ⪯ q4, ξ2 ⪰ q5, ξ3 ⪯ q1, ξ4 ⪰ q2, ξ5 ⪯ q3 or ξ1 ⪰ q4, ξ2 ⪯ q5, ξ3 ⪰ q1, ξ5 ⪯ q2, ξ5 ⪰ q3 or ξ1 ⪯ q5, ξ2 ⪰ q1, ξ3 ⪯ q2, ξ4 ⪰ q3, ξ5 ⪯ q4 or ξ1 ⪰ q5, ξ2 ⪯ q1, ξ3 ⪰ q2, ξ4 ⪯ q3, ξ5 ⪰ q4. If we replace (ξ1, ξ2, ξ3, ξ4, ξ5) and (q1, q2, q3, q4, q5) with (p(ξ1), p(ξ2), p(ξ3), p(ξ4), p(ξ5)) and (p(q1), p(q2), p(q3), p(q4), p(q5)) in above Definition, then we say that (ξ1, ξ2, ξ3, ξ4, ξ5) a QC with (q, q2, q3, q4, q5) with respect to (w.r.t) p. Theorem 2. Let {ζi}i∈W : G5 → G and p : G → G be two mappings over a POCGMs (G,T,⪯) such that {ζi}i∈W and p have quintuple coincidence point with quintuple compa- rable (w.r.t) p and satisfy (C) condition. Then, there is a unique quintuple coincidence point {ζi}i∈W and p. Proof. Theorem 1 shows that the existence of a QCP s of mappings is ensured. Let (ξ1, ξ2, ξ3, ξ4, ξ5) and ((q1, q2, q3, q4, q5) be QCP s, that is, if p(ξ1) = ζi(ξ1, ξ2, ξ3, ξ4, ξ5), p(ξ2) = ζi(ξ2, ξ3, ξ4, ξ5, ξ1), p(ξ3) = ζi(ξ3, ξ4, ξ5, ξ1, ξ2), p(ξ4) = ζi(ξ4, ξ5, ξ1, ξ2, ξ3), p(ξ5) = ζi(ξ5, ξ1, ξ2, ξ3, ξ4), p(q1) = ζi((q1, q2, q3, q4, q5), p(q2) = ζi(q2, q3, q4, q5, q1), p(q3) = ζi(q3, q4, q5, q1, q2), p(q4) = ζi(q4, q5, q1, q2, q3), p(q5) = ζi(q5, q1, q2, q3, q4), then, p(ξ1) = p(q1), p(ξ2) = p(q2), p(ξ3) = p(q3), p(ξ4) = p(q4) and p(ξ5) = p(q5). Since, QCP s are also QC, then from (1), it is obtained that T(p(ξ1), p(q1)) = T(ζi(ξ1, ξ2, ξ3, ξ4, ξ5), ζ j(q1, q2, q3, q4, q5)) ≤ Γ̃[T(p(ξ1), ζ i(ξ1, ξ2, ξ3, ξ4, ξ5)) + T(p(q1), ζ j(q1, q2, q3, q4, q5))] + Υ̃[T(p(ξ1), p(q1))], ⇒ T(p(ξ1), p(q1)) ≤ Υ̃[T(p(ξ1), p(q1))]. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 15 of 22 Since I ̸= Υ̃ ∈ ZM, then T(p(ξ1), p(q1)) = 0, or p(ξ1) = p(q1). Similarly, it is obtained that p(ξ2) = p(q2), p(ξ3) = p(q3), p(ξ4) = p(q4) and p(ξ5) = p(q5). Hence, p(ξ1) = p(ξ2) = p(ξ3) = p(ξ4) = p(ξ5) = p(q1) = p(q2) = p(q3) = p(q4) = p(q5). Which shows that (p(ξ1), p(ξ2), p(ξ3), p(ξ4), p(ξ5)) is a unique quintuple coincidence point of {ζi}i∈W and p. Moreover, {ζi}i∈W and p being compatible are also commutable, which proves the uniqueness of the QFP (ξ1, ξ2, ξ3, ξ4, ξ5) of {ζi}i∈W and p. Example 6. Let G = [0, 1] be the non-empty set equipped with the metric T(ξ1, ξ2) =( |ξ1 − ξ2| |ξ1 − ξ2| ) and Γ̃ = ( 1 5 0 0 1 5 ) , Υ̃ = ( 0 1 25 1 25 0 ) ∈ ZM. Hence, (G,T,≤) is clearly a POCGMs. Let ζi : G5 → G and s : G → G be two mappings defined as ζi(ξ1, ξ2, ξ3, ξ4, ξ5) = ξ1 5i and p(ξ1) = 5ξ1 respectively. Then, T(ζi(ξ1, ξ2, ξ3, ξ4, ξ5), ζ j(q1, q2, q3, q4, q5)) = ( | ξ1 5i − q1 5j | | ξ1 5i − q1 5j | ) = 1 5 ( |(6ξ1 5i − ξ1 5i ) + ( q1 4j − 6q1 5j )| |(6ξ1 5i − ξ1 5i ) + ( q1 4j − 6q1 5j )| ) ≤ 1 5 ( |(5ξ1 − ξ1 5i ) + ( q1 5j − 5q1) + (ξ1 − q1)| |(5ξ1 − ξ1 5i ) + ( q1 4j − 4q1) + (ξ1 − q1)| ) ≤ 1 5 (( |5ξ1 − ξ1 5i | |5ξ1 − ξ1 5i | ) + ( |5q1 − q1 5j | |q1 − q1 5j | )) + 1 25 ( |5ξ1 − 5q1| |5ξ1 − 5q1| ) = ( 1 5 0 0 1 5 )( T(p(ξ1), ζ i(ξ1, ξ2, ξ3, ξ4, ξ5)) + T(p(q1), ζ j(q1, q2, q3, q4, q5)) ) + ( 0 1 25 1 25 0 ) T(p(ξ1), p(q1)). Consequently, T(ζi(ξ1, ξ2, ξ3, ξ4, ξ5), ζ j(q1, q2, q3, q4, q5)) ≤ Γ̃[T(p(ξ1), ζ i(ξ1, ξ2, ξ3, ξ4, ξ5)) + T(p(q1), ζ j(q1, q2, q3, q4, q5))] + Υ̃T(p(ξ1), p(q1)). Hence, the (C) condition is fulfilled. All the conditions of Theorem 1 are accomplished. Moreover, (0, 0, 0, 0, 0) being a QCP of {ζi}i∈W and p is also a unique quintuple coinci- dence point of both mappings according to Theorem 2. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 16 of 22 4. Application Suppose that Rn + = {ξ1 = (ξ11 , ξ 2 1 , ξ 3 1 , · · · , ξn1 ) : ξi ≥, i ≥ 1} and Θ5 n−1 being 5(n − 1) dimensional unit simplex defined as Θ5 n−1 ={θ = (ξ1, ξ2, ξ3, ξ4, ξ5) ∈ Rn + × Rn + × Rn + × Rn + × Rn + : n∑ i=1 θi = n∑ i=1 (ξi1 + ξi2 + ξi3 + ξi4 + ξi5) = 1}. Let θ ∈ Θ5 n−1 be the probability over 5n prospective states. As a stochastic process, the Markov process asserts that 5n states are achieved in each period π = 1, 2, 3, · · · with the probability events over the currently attained states. For each π = 1, 2, 3, · · · , eij shows the probability event achieved by state i in the next period starting from state j. Then, the preceding probability vector θπ and the succeeding probability vector θπ+1 in the period π and π + 1 respectively, written as θπ+1 i = ∑ j eijθ π j , for each j ≥ 1. Let θπ be a column vector, then to obtain matrix form, consider the mapping θπ+1 = Fθπ. In addition with that for all eij ≥ 0, ∑n i=1 eij = 1, required for conditional probability. Finding the stationary distribution for the Markov process is the same as finding the fixed point of F , i.e., there exists some θ ∈ Θ5 n−1 such that Fθπ = θπ, whenever θπ+1 = θπ. The period θπ is called the stationary distribution of the Markov process. Suppose that ϕi = minj eij , for each i and ϕ = ∑n i=1 ϕi. Now, the major part of this section is stated here. Theorem 3. By supposition eij ≥ 0, the Markov process has a unique stationary distri- bution. Proof. Let T : Θ5 n−1 ×Θ5 n−1 → R2 be a mapping defined as T(M,N) =T((ξ1, ξ2, ξ3, ξ4, ξ5), (q1, q2, q3, q4, q5)) = ( n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|), n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|) ) , where M = (ξ1, ξ2, ξ3, ξ4, ξ5) and N = (q1, q2, q3, q4, q5) belongs to Θ5 n−1. Since, T(M,N) ≥ (0, 0) for all M and N in Θ5 n−1. Also, if T(M,N) = (0, 0), then this implies that( n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|), n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|) ) = (0, 0), S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 17 of 22 or |ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5| = 0, ⇒ |ξi1 − qi1| = |ξi2 − qi2| = |ξi3 − qi3| = |ξi4 − qi4|+ |ξi5 − qi5| = 0, ⇒ ξi1 = qi1, ξi2 = qi2, ξi3 = qi3, ξi4 = qi4, ξi5 = qi5. Hence, M = N . Conversely, let M = N , then ξi1 = qi1, ξi2 = qi2, ξi3 = qi3, ξi4 = qi4, ξi5 = qi5, ⇒ |ξi1 − qi1| = |ξi2 − qi2| = |ξi3 − qi3| = |ξi4 − qi4| = |ξi5 − qi5| = 0. Hence, ( n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξ5i − qi5|), n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|) ) = (0, 0) ⇒ T(M,N) = (0, 0). Moreover, T(M,N) = ( n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|), n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|) ) = ( n∑ i=1 (|qi1 − ξi1|+ |qi2 − ξi2|+ |qi3 − ξi3|+ |qi4 − ξi4|+ |qi5 − ξi5|), n∑ i=1 (|qi1 − ξi1|+ |qi2 − ξi2|+ |qi3 − ξi3|+ |qi4 − ξi4|+ |qi5 − ξi5|) ) =T(N,M). Now, T(M,N) = ( n∑ i−1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|), n∑ i=1 (|ξi1 − qi1|+ |ξi2 − qi2|+ |ξi3 − qi3|+ |ξi4 − qi4|+ |ξi5 − qi5|) ) = ( n∑ i=1 ( |(ξi1 − κi1) + (κi1 − qi1)|+ |(ξi2 − κi2) + (κi2 − qi2)|+ |(ξi3 − κi3) +(κi3 − qi3)|+ |(ξi4 − κi4) + (κi4 − qi4)|+ |(ξi5 − κi5) + (κi5 − qi5)| ) , S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 18 of 22 n∑ i=1 ( |(ξi1 − κi1) + (κi1 − qi1)|+ |(ξi2 − κi2) + (κi2 − qi2)|+ |(ξi3 − κi3) +(κi3 − qi3)|+ |(ξi4 − κi4) + (κi4 − qi4)|+ |(ξi5 − κi5) + (κi5 − qi5)| )) ≤ ( n∑ i−1 ( |ξi1 − κi1|+ |κi1 − qi1|+ |ξi2 − κi2|+ |κi2 − qi2|+ |ξi3 − κi3| +|κi3 − qi3|+ |ξi4 − κi4|+ |κi4 − qi4|+ |ξi5 − κi5|+ |κi5 − qi5| ) , n∑ i=1 ( |ξi1 − κi1|+ |κi1 − qi1|+ |ξi2 − κi2|+ |κi2 − qi2|+ |ξi3 − κi3| +|κi3 − qi3|+ |ξi4 − κi4|+ |κi4 − qi4|+ |ξi5 − κi5|+ |κi5 − qi5| )) = { ( n∑ i=1 (|ξi1 − κi1|+|ξi2 − κi2|+ |ξi3 − κi3|+ |ξi4 − κi4|+ |ξi5 − κi5|), n∑ i=1 (|ξi1 − κi1|+|ξi2 − κi2|+ |ξi3 − κi3|+ |ξi4 − κi4|+ |ξi5 − κi5|) ) + ( n∑ i=1 (|κi1 − qi1|+ |κi2 − qi2|+ |κi3 − qi3|+ |κi4 − qi4|+ |κi5 − qi5|), n∑ i=1 (|κi1 − qi1|+ |κi2 − qi2|+ |κi3 − qi3|+ |κi4 − qi4|+ |κi5 − qi5|) ) } = T(M,L) + T(L,N), where, L = (κ1, κ2, κ3, κ4, κ5) ∈ Θ5 n−1. Hence, (Θ 5 n−1, T) is a generalized metric space. Completeness of Θ5 n−1 can be easily proved. Moreover, define the partial order on Θ5 n−1 as for all (ξ1, ξ2, ξ3, ξ4, ξ5), (q1, q2, q3, q4, q5) ∈ Θ5 n−1, (ξ1, ξ2, ξ3, ξ4, ξ5) ⪯ (q1, q2, q3, q4, q5) ⇐⇒ ξ1 ⪯ q1, ξ2 ⪰ q2, ξ3 ⪯ q3, ξ4 ⪰ q4, ξ5 ⪯ q5. Hence, (Θ5 n−1, T, ⪯) is a POCGMs. Let F : Θ5 n−1 → Θ5 n−1 be a mapping defined as for all θ ∈ Θ5 n−1, Fθ = αj such that for each j, αj = ∑n i=1 eijθj . As, n∑ j=1 αj = n∑ j=1 n∑ i=1 eijθj = n∑ i=1 eij n∑ j=1 (ξj1 + ξj2 + ξj3 + ξj4, ξ j 5) = n∑ j=1 (ξj1 + ξj2 + ξj3 + ξj4, ξ j 5) = 1, therefore, αj ∈ Θ5 n−1, so mapping is defined. Now, we have to show that F satis- fies the contraction condition. For this, let αi be the ith row of α. Then, for all (ξ1, ξ2, ξ3, ξ4, ξ5), (q1, q2, q3, q4, q5) in Θ5 n−1, we have T(F(ξ1, ξ2, ξ3, ξ4, ξ5),F(q1, q2, q3, q4, q5)) =  n∑ i=1 ∣∣∣∣∣∣ n∑ j=1 (eij(ξ j 1 + ξj2 + ξj3 + ξj4 + ξj5)− eij(q j 1 + qj2 + qj3 + qj4 + qj5) ∣∣∣∣∣∣  , S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 19 of 22 n∑ i=1 ∣∣∣∣∣∣ n∑ j=1 ( eij(ξ j 1 + ξj2 + ξj3 + ξj4 + ξj5)− eij(q j 1 + qj2 + qj3 + qj4 + qj5) )∣∣∣∣∣∣ ) = ( n∑ i=1 ( | n∑ j=1 (eij − ϕi){(ξj1 + ξj2 + ξj3 + ξj4 + ξj5)− (qj1 + qj2 + qj3 + qj4 + qj5)} + ϕi{(ξj1 + ξj2 + ξj3 + ξj4, ξ j 5)− (qj1 + qj2 + qj3 + qj4 + qj5)}| ) , n∑ i=1 ( | n∑ j=1 (eij − ϕi){(ξj1 + ξj2 + ξj3 + ξj4 + ξj5)− (qj1 + qj2 + qj3 + qj4 + qj5)} + ϕi{(ξj1 + ξj2 + ξj3 + ξj4 + ξj5)− (qj1 + qj2 + qj3 + qj4 + qj5)}| )) ≤ (( n∑ i=1 n∑ j=1 |(eij − ϕi){(ξj1 + ξj2 + ξj3 + ξj4 + ξj5)− (qj1 + qj2 + qj3 + qj4 + qj5)}| + n∑ i=1 ∣∣∣∣∣∣ϕi n∑ j=1 {(ξj1 + ξj2 + ξj3 + ξj4 + ξj5)− (qj1 + qj2 + qj3 + qj4 + qj5)} ∣∣∣∣∣∣ ) , ( n∑ i=1 n∑ j=1 ∣∣∣(eij − ϕi){(ξj1 + ξj2 + ξj3 + ξj4 + ξj5)− (qj1 + qj2 + qj3 + qj4 + qj5)} ∣∣∣ + n∑ i=1 ∣∣∣∣∣∣ϕi n∑ j=1 {(ξj1 + ξj2 + ξj3 + ξj4 + ξj5)− (qj1 + qj2 + qj3 + qj4 + qj5)} ∣∣∣∣∣∣ )) ≤ (∑n i=1 ∑n j=1(|ξ j 1 − qj1|+ |ξj2 − qj2|+ |ξj3 − qj3|+ |ξj4 − qj4|+ |ξj5 − qj5|)× |eij − ϕi|),∑n i=1 ∑n j=1(|ξ j 1 − qj1|+ |ξj2 − qj2|+ |ξj3 − qj3|+ |ξj4 − qj4|+ |ξj5 − qj5| × |eij − ϕi|) ) = ( I − ϕ )(∑n j=1(|ξ j 1 − qj1|+ |ξj2 − qj2|+ |ξj3 − qj3|+ |ξj4 − qj4|+ |ξj5 − qj5|),∑n j=1(|ξ j 1 − qj1|+ |ξj2 − qj2|+ |ξj3 − qj3|2 + |ξj4 − qj4|+ |ξj5 − qj5|) ) =Υ̃T((ξ1, ξ2, ξ3, ξ4, ξ5), (q1, q2, q3, q4, q5)), where (I − ϕ) = Υ̃ ∈ ZM, therefore all conditions of Corollary 2 have been met. Then, there is a unique quintuple fixed point for F or, in other words, a unique stationary distribution of the Markov process. Moreover, the sequence {Fnθk} converges to a unique stationary distribution for any θk ∈ Θ5 n−1. 5. Conclusion In this work, we found some Quadruple fixed point theorems for solving integral equa- tions involved with matrices and the Markov process in generalized metric spaces. In this study, the notions introduced in [16] are structured with a mapping defined on quintuples. S. Batul et a. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6131 20 of 22 The research study investigated the existence of quintuple fixed points (QFPs) for map- pings in generalized metric spaces, utilizing matrix-based methods. Several definitions of (QFPs) are formulated, and new fixed point theorems is established. To illustrate the findings, an example is provided. Additionally, an application was developed to verify the results by determining the stationary distribution of a Markov process. Future research on quintuple fixed points could explore alternative approaches, such as: (a): By working on different structures, e.g., by generalizing the obtained results in the setting of “ b-metric spaces”. (b): Different contraction conditions can be adopted by involving new parameters and introducing more properties of contraction mappings. Acknowledgements The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4331214GSSR04. Declarations Availability of Data and Materials Data sharing not applicable to this article as no data sets were generated or analyzed during the current study. Funding This Research work was funded by Umm Al-Qura University, Saudi Arabia under grant number : 25UQU4331214GSSR04. 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