EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5982 ISSN 1307-5543 – ejpam.com Published by New York Business Global Common Fixed Point Theorems for Mappings Satisfying implicit Relation in Bipolar metric Space Penumarthy Parvateesam Murthy1, Chandra Prakash Dhuri1, Rajagopalan Ramaswamy2,∗, Khizar Hayat Khan2, Ola Ashour Abdelnaby2,3 1 Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur(CG), 495 009 India 2 Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt Abstract. In this article, we introduce the concept of compatible mappings of type (A) and weaken the same in the setting of Bipolar metric spaces and established fixed point results in the setting of Bipolar metric spaces, using implicit relation function. The derived results have been supplemented with suitable non-trivial examples. Our results have extended and generalized some results proven in the past and some open problems for future research has been given. 2020 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Fixed points, bipolar metric space, covariant map, contravariant map, compatible mapping of type (A), property (E.A.) Cauchy bisequence 1. Introduction It would be fair to say that the concept of metric fixed point theory started with the famous contraction mapping theorem of S. Banach [1]. This theory has seen rapid de- velopment in the past nineteenth and twentieth centuries. In the overlaps made in these centuries, while metric spaces and normed spaces developed, the domains were only taken as value regions with single variables and real positive numbers. In other words, new metric spaces are produced by taking the domains X,X2, and X3. However, bipolar met- ric space is defined as a new space by going beyond the conventional definition of metric spaces that have been defined for years. At the same time, this theory has been applied to real life and various fields of science, namely engineering, economics, medical sciences, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5982 Email addresses: ppmurthy@gmail.com (PP Murthy), cpdhuri@gmail.com (CP Dhuri), r.gopalan@psau.edu.sa (R Ramaswamy), drkhizar@gmail.com (KH Khan), o.abdelnaby@psau.edu.sa (Ola A.A) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 2 of 17 and computer, etc. In the sequel of generalisation of contractive conditions, Jungck proposed a very differ- ent type of generalization of the contraction condition introduced by Banach for a pair of compatible maps in metric spaces. (see [2]). The concept of compatible mappings of type (A) was described in [3] on complete metric spaces. The common fixed point theorems have been proved for the compatible mappings of type (A). This type (A) was shown to be equivalent to the context of compatible mappings defined by Jungck, with some restric- tions (see [3]). Valerie Popa demonstrated fixed point theorems for compatible mappings satisfying an implicit relation in [4]. In [5], the (E.A.) property in metric space was defined, which generalizes the concept of non-compatible mappings, and some common fixed point theorems were proved. Later, in 2016 Mutlu and Gürdal [6] introduced the concept of a bipolar metric space which is the generalization of a metric space. They have proved some generalizations of Banach Fixed Point Theorem [1] in bipolar metric spaces. Given the theorem proved herein, it is highly demanded to recall the most basic definitions and properties in bipolar metric spaces. Subsequently, in the recent past, various authors have reported fixed point results in the setting of bipolar metric spaces using various contractive conditions. For more details, [7–16]. Inspired, in this article we aim to prove common fixed point theorems with the con- cepts of maps of type (A) and property (E.A), and implicit relations in bipolar metric spaces, which generalizes some famous well-known results, namely Kannan [17], Reich [18] and Gaba [16]. The rest of the paper is organized as follows: In Section-2, we review some basic preliminaries and monograph. We present our main results in Section-3, establishing the fixed point results and generalising various fixed point results proven in the past. The derived results have been supported with suitable examples. Finally we conclude the manuscript presenting the scope for further research by presenting some open problems for future research. 2. Preliminaries The following are required in the sequel. Definition 1. [6] Let X and Y be two non nonempty sets and d : X × Y → [0,+∞) be a function. Then the triplet (X,Y, d) is called bipolar metric space and d is called bipolar P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 3 of 17 metric on (X,Y ), if the following conditions holds: (BP1) d(x, y) = 0 if and only if x = y where (x, y) ∈ X × Y , (BP2) If x, y ∈ X ∩ Y then d(x, y) = d(y, x), (BP3) d(x1, y2) ≤ d(x1, y1) + d(x2, y1) + d(x2, y2) for all x1, x2 ∈ X and y1, y2 ∈ Y . Definition 2. [6] Let (X,Y, d) be a bipolar metric space. A sequence {un} is said to be convergent to a point t if and only if {un} is a sequence in X, t is a point in Y and lim n→+∞ d(un, t) = 0; or {un} is a sequence in Y , t is a point in X and lim n→+∞ d(t, un) = 0. A sequence {(xn, yn)} in X × Y is called a bisequence on (X,Y ). This sequence is simply denoted by (xn, yn). If both the sequences {xn} and {yn} converge, then the bisequence (xn, yn) is said to be convergent. If both the sequences {xn} and {yn} converge to a same point u ∈ X ∩ Y then (xn, yn) is called biconvergent. If lim n,m→+∞ d(xn, ym) = 0 then the bisequence (xn, yn) is called a Cauchy bisequence. In a bipolar metric space, every convergent Cauchy bisequence is biconvergent. A bipolar metric space is called complete, if every Cauchy bisequence is convergent, hence biconver- gent. Remark 1. Limit of a convergent sequence in a bipolar metric space need not be unique, but if a limit is in X ∩ Y , then it is the unique limit of the sequence. Definition 3. [6] Let X1, Y1, X2 and Y2 be four sets. A function f : X1∪Y1 → X2∪Y2 is said to be a covariant map if f(X1) ⊆ X2 and f(Y1) ⊆ Y2 and is denoted as f : (X1, Y1) ⇒ (X2, Y2). In particular, if (X1, Y1, d1) and (X2, Y2, d2) are two bipolar metric space then we use the notaion f : (X1, Y1, d1) ⇒ (X2, Y2, d2) for covariant map f . A function g : X1 ∪ Y1 → X2 ∪ Y2 is said to be a contravariant map if g(X1) ⊆ Y2 and g(Y1) ⊆ X2 and is denoted as g : (X1, Y1) ⇄ (X2, Y2). Definition 4. Let (X1, Y1, d1) and (X2, Y2, d2) be two bipolar metric spaces. A covariant map f : (X1, Y1) ⇒ (X2, Y2) is continuous at v if and only if {un} converges to v on (X1, Y1, d1) implies {f(un)} converges to f(v) on (X2, Y2, d2). A contravariant map g : (X1, Y1, d1) ⇄ (X2, Y2, d2) is continuous if and only if it is continuous as a covariant map g : (X1, Y1, d1) ⇒ (Y2, X2, d̄2), where d̄2 is defined as d̄2(y, x) = d2(x, y), for all (y, x) ∈ Y2 ×X2. Definition 5. [7] If S and T are covariant or contravariant maps on X ∪ Y , then (i) u ∈ X ∪ Y is called fixed point of T if and only if Tu = u. (ii) u ∈ X ∪ Y is called common fixed point of S and T if and only if Tu = Su = u. (iii) u ∈ X ∪ Y is called coincidence point of S and T if and only if Tu = Su. . Popa [4] studied a new type of contraction condition by employing the implicit function to obtain fixed points. In the sequel, we are also going to prove a few fixed-point theorems by employing implicit relations in a bipolar metric space. P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 4 of 17 3. Main Results The concept of maps of type (A) was introduced initially by Jungck, Murthy, and Cho [3] in a metric space. Now we are ready to introduce the same concept in a bipolar metric space. The definition follows: Definition 6. Let (X,Y, d) be a bipolar metric space. Also, let T : (X,Y, d) ⇒ (X,Y, d) be a covariant map and S : (X,Y, d) ⇄ (X,Y, d) be a contravariant map. Then (i) S and T are said to be compatible mappings of type (A) with respect to X if and only if lim n→+∞ d(SSun, TSun) = 0 or lim n→+∞ d(TTun, STun) = 0, whenever {un} be a sequence in X such that lim n→+∞ Sun = lim n→+∞ Tun = t for some t ∈ X ∩ Y . (ii) S and T are said to be compatible mappings of type (A) with respect to Y if and only if lim n→+∞ d(TSun, SSun) = 0 or lim n→+∞ d(STun, TTun) = 0, whenever {un} is a sequence in Y such that lim n→+∞ Sun = lim n→+∞ Tun = t for some t ∈ X ∩ Y . (iii) S and T are said to be weak compatible mappings of type (A) if and only if Tu = Su for some u ∈ X ∩ Y , then TSu = SSu (or equivalently, STu = TTu.) Example 1. Let X = N∪ {0}, Y = [0, 1] and the metric d is defined by d(x, y) = |x− y|, where N is the set of positive integers. Then (X,Y, d) is a bipolar metric space. Let T (covariant) and S (contravariant map) are defined as S(n) = 1 n , for all n ∈ N, S(y) = 1, for all y ∈ Y, T (n) = 2n, for all n ∈ X − {1}, T1 = 0 Ty =  2y, if 0 ≤ y ≤ 1 2 1 2 , if 1 2 < y < 1 then the maps S and T are compatible of type (A) with respect to X vacuously as there is no sequence {xn} in X such that lim n→+∞ Sxn = lim n→+∞ Txn = 1 ∈ X ∩ Y or lim n→+∞ Sxn = lim n→+∞ Txn = 0 ∈ X ∩ Y, but it is not compatible of type (A) with respect to Y as the sequence {1 2 − 1 2n} in Y has the following property lim n→+∞ Syn = lim n→+∞ Tyn = 1 but lim n→+∞ d(TSyn, SSyn) = d(0, 1) ̸= 0 and lim n→+∞ d(STyn, TTyn) = d(1, 12) ̸= 0. P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 5 of 17 Now we extend the definition of property (E.A.) [5] to bipolar metric space. Definition 7. Let (X,Y, d) be a bipolar metric space and let T1, T2 : (X,Y, d) ⇒ (X,Y, d) be covariant maps and S1, S2 : (X,Y, d) ⇄ (X,Y, d) be contravariant maps. We say that (i) T1 and S1 satisfy the property (E.A.) if there exists a sequence {xn} in X and a sequence {yn} in Y such that limn→+∞ S1xn = lim n→+∞ T1xn = lim n→+∞ S1yn = lim n→+∞ T1yn = t for some t ∈ X ∩ Y (ii) T1 and S1 satisfy the weak form of property (E.A.) if there exists a sequence {un} in X or Y such that lim n→+∞ S1un = lim n→+∞ T1un = t for some t ∈ X ∩ Y (iii) the quadruple (S1, T1, S2, T2) satisfies the property (E.A.) if there exists a sequence {xn} in X and a sequence {yn} in Y such that lim n→+∞ S1xn = lim n→+∞ T1xn = lim n→+∞ S2yn = lim n→+∞ T2yn = t for some t ∈ X ∩ Y The following proposition gives the connection between compatible mappings of type (A) and a weak compatible mappings of type (A). Proposition 1. If S and T are mappings of type (A) with respect to X or Y , then they are weak compatible mappings of type (A). Proof. Let S and T are compatible mappings of type (A) with respect to X or Y with Su = Tu for some u ∈ X ∩ Y, then the proposition can be proved easily by taking un = u in the definitions of compatible mappings of type (A) with respect to X and Y. We now introduce the following class of implicit functions: Let Ψ be the collection of all real-valued function ψ : [0,+∞)4 → R satisfying the following conditions: (ψa): If ψ(a, b, a, b) ≤ 0 or ψ(a, b, b, a) ≤ 0 then there exists k ∈ [0, 1) such that a ≤ kb. (ψb): If ψ(a, a, 0, 0) > 0 for all a > 0. Remark 2. by definition of ψ, it is clear that the following implications hold: • ψ(a, a, 0, 0) ≤ 0 implies a = 0. • ψ(a, 0, 0, a) ≤ 0 implies a = 0. • ψ(a, a, a, a) ≤ 0 implies a = 0. Example 2. The following functions are members of Ψ. • ψ1(a, b, c, d) = a− k1b− k2c− k3d, k1, k2, k3 ≥ 0, k1 + k2 + k3 < 1 If ψ1(a, b, a, b) = a−k1b−k2a−k3b ≤ 0 then a ≤ ( k1 + k3 1− k2 ) b with 0 < k1 + k3 1− k2 < 1. If ψ1(a, b, b, a) = a−k1b−k2b−k3a ≤ 0 then a ≤ ( k1 + k2 1− k3 ) b with 0 < k1 + k2 1− k3 < 1. take k = max { k1 + k3 1− k2 , k1 + k2 1− k3 } P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 6 of 17 • ψ2(a, b, c, d) = a− k max{b, c, d}, k ∈ [0, 1) • ψ3(a, b, c, d) = a− k1b− k2 max{c, d} k1, k2 ≥ 0, k1 + k2 < 1 • ψ4(a, b, c, d) = a− k1 max{b, c} − k2d k1, k2 ≥ 0, k1 + k2 < 1 • ψ(a, b, c, d) = a − kF (max{b, c, d}) where F : [0,+∞) → [0,+∞) is a function satisfying the condition: F (t) ≤ t, for each t ∈ (0,+∞) and k ∈ [0, 1). Now we consider a super class of Ψ which will be denoted by Φ and it is the collection of all real valued functions ϕ : [0,+∞)4 → R satisfying the following condition: ϕ(a, a, 0, 0) > 0, for all a > 0. Now we prove a lemma that will be used in proving our theorems. Lemma 1. Let (X,Y, d) be a bipolar metric space and (xn, yn) is a bisequence in X × Y satisfying the following condition: There exists k ∈ [0, 1) such that d(xn+1, yn+1) ≤ kd(xn+1, yn) and d(xn+1, yn) ≤ kd(xn, yn) for all n ∈ N ∪ {0}. Then the bisequence (xn, yn) is Cauchy bisequence. Proof. First, we observe that the given condition implies the following condition d(xn+1, yn+1) ≤ k2d(xn, yn) for all n ∈ N ∪ {0} d(xn+1, yn+1) ≤ k2(n+1)d(x0, y0), taking limit as n→ +∞, we get lim n→+∞ d(xn, yn) = 0. (1) Let n, p ∈ N, then by (BP3) and given condition, we have d(xn, yn+p) ≤d(xn, yn) + d(xn+1, yn) + d(xn+1, yn+p) ≤d(xn, yn) + kd(xn, yn) + d(xn+1, yn+p) =(1 + k)d(xn, yn) + d(xn+1, yn+p) ≤(1 + k)k2nd(x0, y0) + d(xn+1, yn+p) ≤(1 + k)(k2n + k2(n+1) + k2(n+2) + · · ·+ k2(n+p−1))d(x0, y0) + d(xn+p, yn+p) ≤(1 + k)(k2n+k2(n+1)+k2(n+2) + · · · )d(x0, y0)+d(xn+p, yn+p) ≤(1 + k)k2n(1 + k2 + k4 + · · · )d(x0, y0) + k2(n+p)d(x0, y0) = (1 + k)k2n 1− k2 d(x0, y0) + k(2n+2p)d(x0, y0). P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 7 of 17 This implies lim n→+∞ d(xn, yn+p) = 0. (2) Now to prove that lim n→+∞ d(xn+p, yn) = 0, consider the inequality (by property (BP3)) d(xn+p, yn) ≤ d(xn+p, yn+p) + d(xn, yn+p) + d(xn, yn) and take the limit as n→ +∞ and use (1) and (2). Hence (xn, yn) is a Cauchy sequence. Our first main result is the following. Theorem 1. Let (X,Y, d) be a complete bipolar metric space and let T1, T2 : (X,Y, d) ⇒ (X,Y, d) be two covariant maps and S1, S2 : (X,Y, d) ⇄ (X,Y, d) be two contravariant maps satisfying the following conditions: (i) The mappings S2, T1 are compatible of type (A) with respect to Y. (ii) The mappings S1, T2 are compatible of type (A) with respect to X. (iii) S1(X ∪ Y ) ⊆ T1(X ∪ Y ) and S2(X ∪ Y ) ⊆ T2(X ∪ Y ). (iv) All the four mappings S1, S2, T1 and T2 are continuous. (v) There exists ψ ∈ Ψ such that ψ(d(S2y, S1x), d(T2x, T1y), d(T2x, S1x), d(S2y, T1y)) ≤ 0, (3) for all (x, y) ∈ X × Y . Then the functions S1, S2, T1 and T2 have a unique common fixed point. Proof. Let x0 ∈ X and choose x1 ∈ X and y1 ∈ Y such that S1x0 = T1y1 = v0 and S2y1 = T2x1 = u1. This can be done since S1(X ∪ Y ) ⊆ T1(X ∪ Y ) and S2(X ∪ Y ) ⊆ T2(X ∪Y ). In general we can choose (xn, yn) ∈ X ×Y such that S1xn = T1yn+1 = vn and S2yn+1 = T2xn+1 = un+1 for all n ∈ N ∪ {0}. Now putting x = xn+1 and y = yn+1 in (3), we get ψ(d(S2yn+1, S1xn+1), d(T2xn+1, T1yn+1), d(T2xn+1, S1xn+1), d(S2yn+1, T1yn+1)) ≤ 0 ψ(d(un+1, vn+1), d(un+1, vn), d(un+1, vn+1), d(un+1, vn)) ≤ 0. So by property of ψ, there exists k ∈ [0, 1) such that d(un+1, vn+1) ≤ kd(un+1, vn). (4) Again putting x = xn and y = yn+1 in (3), we get ψ((d(S2yn+1, S1xn), d(T2xn, T1yn+1), d(T2xn, S1xn), d(S2yn+1, T1yn+1)) ≤ 0 ψ(d(un+1, vn), d(un, vn), d(un, vn), d(un+1, vn)) ≤ 0. P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 8 of 17 So by property of ψ, we have d(un+1, vn) ≤ kd(un, vn). (5) By (4), (5) and Lemma 1, the sequence (un, vn) is Cauchy bisequence and since given bipolar metric space is complete, hence the sequence (un, vn) biconverges to a point t ∈ X ∩ Y . So lim n→+∞ S1xn = lim n→+∞ T1yn = lim n→+∞ S2yn = lim n→+∞ T2xn = t. (6) By using compatibility of type (A) with respect to Y of mappings S2 and T1 and (6), we get lim n→+∞ d(T1S2yn, S2S2yn) = 0 or lim n→+∞ d(S2T1yn, T1T1yn) = 0. By continuity of mappings S2 and T1, we have d(T1t, S2t) = 0 T1t = S2t. (7) Similarly by using Compatibility of type (A) with respect toX and Continuity of mappings S1 and T2 and (6), we get T2t = S1t. (8) Now Putting x = y = t in (3) and using (7) and (8), we get ψ(d(S2t, S1t), d(T2t, T1t), d(T2t, S1t), d(S2t, T1t)) ≤ 0 ψ(d(S2t, S1t), d(S1t, S2t), 0, 0) ≤ 0. So by property of ψ, this implies that S2t = S1t. Hence we get S2t = S1t = T2t = T1t = u (say) that is t is a coincidence point of S1, S2, T1 and T2. By Proposition 1 the pairs (S2, T1) and (S1, T2) are weak compatibility of type (A). So we get T1S2t = S2S2t and T2S1t = S1S1t. This implies T1u = S2u and T2u = S1u. (9) Now putting x = y = u in (3), we get ψ(d(S2u, S1u), d(T2u, T1u), d(T2u, S1u), d(S2u, T1u)) ≤ 0 ψ(d(S2u, S1u), d(S1u, S2u), 0, 0) ≤ 0 d(S2u, S1u) = 0 (by property of ψ) S2u = S1u. (10) P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 9 of 17 By (9) and (10), we get T1u = T2u = S1u = S2u. So u is also a coincidence point of S1, S2, T1 and T2. Now Putting y = u and x = t in (3), we get ψ(d(S2u, S1t), d(T2t, T1u), d(T2t, S1t), d(S2u, T1u)) ≤ 0 ψ(d(S2u, u), d(u, S2u), 0, 0) ≤ 0 ψ(d(S2u, u), d(S2u, u), 0, 0) ≤ 0 d(S2u, u) = 0 S2u = u. So u is a common fixed point of all the given four mappings. Now we prove that the fixed point is unique. For this let us assume that u1 is another common fixed point then putting y = u and x = u1 in (3), we get ψ(d(S2u, S1u1), d(T2u1, T1u), d(T2u1, S1u1), d(S2u, T1u)) ≤ 0 ψ(d(u, u1), d(u1, u), 0, 0) ≤ 0. By Remark 2, this implies d(u, u1) = 0 u = u1. So u is the unique common fixed point. Now we prove some corollaries derived from Theorem 1 Corollary 1. Let (X,Y, d) be a complete bipolar metric space and let T : (X,Y, d) ⇒ (X,Y, d) be a covariant map and S1, S2 : (X,Y, d) ⇄ (X,Y, d) be two contravariant maps satisfying the following conditions (i) S2 and T are compatible of type (A) with respect to Y. (ii) S1 and T are compatible of type (A) with respect to X. (iii) S1(X ∪ Y ) ⊆ T (X ∪ Y ) and S2(X ∪ Y ) ⊆ T (X ∪ Y ). (iv) All the three mappings S1, S2 and T are continuous. (v) There exists ψ ∈ Ψ such that ψ(d(S2y, S1x), d(Tx, Ty), d(Tx, S1x), d(S2y, Ty)) ≤ 0, (11) for all (x, y) ∈ X × Y . Then the functions S1, S2, and T have a unique common fixed point. P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 10 of 17 Proof. Take T1 = T2 = T in Theorem 1. Corollary 2. Let (X,Y, d) be a complete bipolar metric space and let T : (X,Y, d) ⇒ (X,Y, d) be a covariant map and S : (X,Y, d) ⇄ (X,Y, d) be a contravariant map satisfying the following conditions: (i) S and T are compatible of type (A) with respect to X or Y. (ii) S(X ∪ Y ) ⊆ T (X ∪ Y ). (iii) S and T are continuous. (iv) There exists ψ ∈ Ψ such that ψ(d(Sy, Sx), d(Tx, Ty), d(Tx, Sx), d(Sy, Ty)) ≤ 0, (12) for all (x, y) ∈ X × Y . Then the functions S and T have a unique common fixed point. Proof. Take T1 = T2 = T and S1 = S2 = S in Theorem 1. Corollary 3. Let (X,Y, d) be a complete bipolar metric space and let S : (X,Y, d) ⇄ (X,Y, d) be a contravariant map satisfying the following conditions: (i) S is continuous. (ii) There exists ψ ∈ Ψ such that ψ(d(Sy, Sx), d(x, y), d(x, Sx), d(Sy, y)) ≤ 0, (13) for all (x, y) ∈ X × Y . Then the function S has a unique fixed point. Proof. Take T1 = T2 = I and S1 = S2 = S in Theorem 1, where I is the identity mapping. In the following corollary, we take ψ as a continuous function and S need not be continuous. Corollary 4. Let (X,Y, d) be a complete bipolar metric space and let S : (X,Y, d) ⇄ (X,Y, d) be a contravariant map satisfying the following condition: ψ(d(Sy, Sx), d(x, y), d(x, Sx), d(Sy, y)) ≤ 0, for some continuous function ψ ∈ Ψ and for all (x, y) ∈ X × Y . Then the function S has a unique fixed point. P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 11 of 17 Proof. As in the proof of the previous corollary, we obtain a bisequence (Syn, Sxn) biconverging to a point t ∈ X ∩ Y , where Syn = xn and Sxn = yn+1. In the given condition, if we take y = yn and x = t, then we get, ψ(d(Syn, St), d(t, yn), d(t, St), d(Syn, yn)) ≤ 0, taking limit as n→ +∞, we get ψ(d(t, St), 0, d(t, St), 0) ≤ 0 So by property of ψ, we get d(t, St) = 0. This implies that t is a fixed point of S. Uniqueness can be proved as given in the previous corollary. The contraction used in the following corollary is Reich-type contraction (see [18], [16]). Corollary 5. Let (X,Y, d) be a complete bipolar metric space and let S : (X,Y, d) ⇄ (X,Y, d) be a continuous contravariant map satisfying the following condition: d(Sy, Sx) ≤ k1d(x, y) + k2d(x, Sx) + k3d(Sy, y)), for all (x, y) ∈ X × Y , where k1 + k2 + k3 < 1. Then the function S has a unique fixed point. Proof. In Corollary 4, take ψ(a, b, c, d) = a− (k1b+ k2c+ k3d). Corollary 6. Let (X,Y, d) be a complete bipolar metric space and let S : (X,Y, d) ⇄ (X,Y, d) be a contravariant map satisfying the following condition: d(Sy, Sx) ≤ k(d(x, y) + d(x, Sx) + d(Sy, y)), for all (x, y) ∈ X × Y , where k < 1 3 . Then the function S has a unique fixed point. Proof. In Corollary 5, take k1 = k2 = k3 = k. In the following corollary, Kannan-type contraction (see [16, 17]) is used. Corollary 7. Let (X,Y, d) be a complete bipolar metric space and let S : (X,Y, d) ⇄ (X,Y, d) be a contravariant map satisfying the following condition: d(Sy, Sx) ≤ k(d(x, Sx) + d(Sy, y)), for all (x, y) ∈ X × Y , where k < 1 2 . Then the function S has a unique fixed point. Proof. In Corollary 5, take k1 = 0, k2 = k3 = k. In our next theorem, we do not require the continuity of maps. Theorem 2. Let (X,Y, d) be a bipolar metric space and let T1, T2 : (X,Y, d) ⇒ (X,Y, d) be two covariant maps and S1, S2 : (X,Y, d) ⇄ (X,Y, d) be two contravariant maps satisfying the following conditions: P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 12 of 17 (i) The pairs (S2, T1) and (S1, T2) are weak compatible of type(A). (ii) S1(X) ⊆ T1(Y ) or S2(Y ) ⊆ T2(X). (iii) (T2(X), T1(Y ), d) or (S2(Y ), S1(X), d) is complete. (iv) There exists continuous function ψ ∈ Ψ such that ψ(d(S2y, S1x), d(T2x, T1y), d(T2x, S1x), d(S2y, T1y)) ≤ 0, (14) for all (x, y) ∈ X × Y . Then the functions S1, S2, T1 and T2 have a unique common fixed point. Proof. Let the bisequence (xn, yn) and (un, vn) be defined as in Theorem 1. By the same argument as given in the same theorem, (un, vn) is Cauchy bisequence in (T2(X), T1(Y ), d) and (S2(Y ), S1(X), d). The following two cases arise Case - I : If (T2(X), T1(Y ), d) is complete, then the sequence (un, vn) biconverges to some point in T2(X) ∩ T1(Y ). Case - II: If (S2(Y ), S1(X), d) is complete, then the sequence (un, vn) biconverges to a point in S2(Y ) ∩ S1(X). This implies that (un, vn) biconverges to a point in T2(X) ∩ T1(Y ) as S2(Y ) ∩ S1(X) ⊂ T2(X) ∩ T1(Y ). So in both the cases, it converges to a point t (say) in T2(X) ∩ T1(Y ). Hence, there exist p ∈ B and q ∈ A such that t = T1p = T2q. (15) Now putting y = yn and x = q in (14), we get ψ(d(S2yn, S1q), d(T2q, T1yn), d(T2q, S1q), d(S2yn, T1yn)) ≤ 0, taking limit as n→ +∞, we get ψ(d(t, S1q), d(t, t), d(t, S1q), 0) ≤ 0 ψ(d(t, S1q), 0, d(t, S1q), 0) ≤ 0 S1q = t. (16) Again putting y = p and x = xn in (14), we get ψ(d(S2p, S1xn), d(T2xn, T1p), d(T2xn, S1xn), d(S2p, T1p)) ≤ 0, taking limit as n→ +∞, we get ψ(d(S2p, t), d(t, t), d(t, t), d(S2p, t)) ≤ 0 ψ(d(S2p, t), 0, 0, d(S2p, t)) ≤ 0 P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 13 of 17 S2p = t. (17) From (15), (16) and (17), we get T1p = T2q = S1q = S2p = t. (18) Since the pairs (S2, T1) and (S1, T2) are weak compatible of type(A), equations (18) imply that T1S2p = S2S2p or S2T1p = T1T1p; and T2S1q = S1S1q or S1T2q = T2T2q. So T1t = S2t, T2t = S1t. Now putting x = y = t in (14), we get ψ(d(S2t, S1t), d(T2t, T1t), d(T2t, S1t), d(S2t, T1t)) ≤ 0 ψ(d(S2t, S1t), d(S1t, S2t), 0, 0) ≤ 0 d(S2t, S1t) = 0 S2t = S1t. So we get T1t = S2t = T2t = S1t. (19) That is, t is a coincidence point of T1, S2, T2 and S1. Now we show that t is a common fixed point of these four mappings. For this, substituting x = t and y = p in (14) and using (18) and (19), we get ψ(d(S2p, S1t), d(T2t, T1p), d(T2t, S1t), d(S2p, T1p)) ≤ 0 ψ(d(t, S1t), d(S1t, t), 0, 0) ≤ 0 S1t = t. So t is a common fixed point of given four mappings. The uniqueness of a common fixed point can be proved as in Theorem 1. Our next theorem is about the common fixed point of four mappings and is a gener- alization of the Theorem 1. Theorem 3. Let (X,Y, d) be a complete bipolar metric space and let T1, T2 : (X,Y, d) ⇒ (X,Y, d) be two covariant maps and S1, S2 : (X,Y, d) ⇄ (X,Y, d) be two contravariant maps satisfying the following conditions: (i) S2 and T1 are compatible of type (A) with respect to Y. (ii) S1 and T2 are compatible of type (A) with respect to X. (iii) The quadruple (S1, T2, S2, T1) satisfies the property (E.A.). (iv) All the four mappings S1, S2, T1 and T2 are continuous. (v) There exists ϕ ∈ Φ such that P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 14 of 17 ϕ(d(S2y, S1x), d(T2x, T1y), d(T2x, S1x), d(S2y, T1y)) ≤ 0, (20) for all (x, y) ∈ X × Y . Then the functions S1, S2, T1 and T2 have a unique common fixed point. Proof. Since the quadruple (S1, T2, S2, T1) satisfies the property (E.A.), so there exists a sequence {(xn, yn)} in X × Y such that lim n→+∞ S1xn = lim n→+∞ T2xn = lim n→+∞ S2yn = lim n→+∞ T1yn = t. (21) This is the equation (6) in Theorem 1. The remaining proof of the theorem is the same as the proof of the Theorem 1 with ψ replaced by ϕ. Like Theorem 1, many corollaries can be derived here also. One of the corollaries is given below: Corollary 8. Let (X,Y, d) be a complete bipolar metric space and let T : (X,Y, d) ⇒ (X,Y, d) be a covariant map and S : (X,Y, d) ⇄ (X,Y, d) be a contravariant map satisfying the following conditions: (i) S and T are compatible of type (A) with respect to X or Y. (ii) T and S satisfy the weak form of property (E.A.). (iii) S and T are continuous. (iv) There exists ϕ ∈ Φ such that ϕ(d(Sy, Sx), d(Tx, Ty), d(Tx, Sx), d(Sy, Ty)) ≤ 0 for all (x, y) ∈ X × Y . Then the functions S and T have a unique common fixed point. Example 3. Let a, d ∈ R, 0 ̸= b ∈ C with bb̄ − ad > 0 , where C is the set of com- plex numbers. Let us define two sets C(a, b, d) = {z ∈ C : azz̄ + bz̄ + b̄z + d = 0} and L(b, d) = {z ∈ C : bz̄+b̄z+d = 0}. It is clear that C(a.b, d) and L(b, d) represent a circle(if a ̸= 0) and a straight line respectively in a complex plane. Let X = {C(a, b, d) : a, d ∈ R} and Y = {L(b, d) : d ∈ R}. Hence Y ⊆ X. Let ρ : X × Y → [0,+∞) is defined as ρ(C(a, b, d), L(b, d1)) = |a| + |d− d1| for all C(a, b, d) ∈ X,L(b, d1) ∈ Y . Then (X,Y, ρ) is a complete bipolar metric space. Let T : (X,Y, ρ) ⇒ (X,Y, ρ) be a covariant map and S : (X,Y, ρ) ⇄ (X,Y, ρ) be a contravariant map defined as follows: S(C(a, b, d)) = L(b, d8), S(L(b, d)) = L(b, d 8 ), T (C(a, b, d)) = C ( a 2 , b, d 2 ) , T (L(b, d)) = L ( b, d 2 ) P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 15 of 17 Then S and T are continuous mappings, S(X ∪ Y ) = Y ⊆ X ∪ Y = T (X ∪ Y ). The map- pings S and T are compatible of type (A) with respect to Y, for let the sequence {L(b, dn)} in Y satisfies the condition: lim n→+∞ S(L(b, dn)) = lim n→+∞ T (L(b, dn)) = L(b, d0) ∈ X ∩ Y, then dn 8 → d0 and dn 2 → d0, so d0 = 0, this implies that ρ(TS(L(b, dn)), SS(L(b, dn))) → 0 and ρ(ST (L(b, dn)), TT (L(b, dn))) → 0. S and T satisfy the following condition ψ(d(Sy, Sx), d(Tx, Ty), d(Tx, Sx), d(Sy, Ty)) ≤ 0, for all (x, y) ∈ X × Y , where ψ(a, b, c, d) = a − 1 4 (b + c + d). So all the conditions of Corollary 2 are satisfied, so S and T have unique common fixed point. In fact , L(b, 0), (thatisbz̄ + b̄z = 0) is the unique common fixed point of S and T. Example 4. Let X = (−∞, 0] and Y = [0,+∞), then (X,Y, d) is a complete bipolar metric space where d is defined as d(x, y) = |x− y| . Let maps T1, T2, S1 and S2 be defined as S1(x) = −x 12 , S2(x) = −x 6 , T1(x) = x 2 and T2(x) = x 4 , for all x ∈ X ∪ Y. Then S1, S2 are two continuous contravariant maps and T1, T2 are continuous covariant maps. All these maps satisfy the condition ψ(d(S2y, S1x), d(T2x, T1y), d(T2x, S1x), d(S2y, T1y)) ≤ 0, for all (x, y) ∈ X×Y where ψ(a, b, c, d) = a−k1b−k2c−k3d with k1 = 2 3 , k2 = 1 12 , k3 = 1 12 . All the other conditions of Theorem 1 are also satisfied, so S1, S2, T1 and T2 have a unique common fixed point. Example 5. Let X = (−∞, 0] and Y = [0,+∞), then (X,Y, d) is a complete bipolar metric space where d is defined as d(x, y) = |x− y| . Let T (covariant) and S(contravariant map) are defined as S(x) = −x 3 and T (x) = x 2 , for all x ∈ X ∪ Y. Then S and T are continuous functions, compatible of type (A) with respect to X and Y both, S(X ∪ Y ) ⊆ T (X ∪ Y ), and satisfy the condition ψ(d(Sy, Sx), d(Tx, Ty), d(Tx, Sx), d(Sy, Ty)) ≤ 0, for all (x, y) ∈ X×Y where ψ(a, b, c, d) = a−k1b−k2c−k3d with k1 = 2 3 , k2 = 1 12 , k3 = 1 12 . So all the conditions of Corollary 2 are satisfied, so S and T have a unique common fixed point. P. P. Murthy et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5982 16 of 17 4. Conclusion In the article, fixed point results in the setting of Bipolar metric space generalising some famous results of Kannan [17], Reich [18] and Gaba [16] have been proven. Suitable non-trivial examples have been provided in support of the derived results. It will be an open problem to find some applications to examine the existence and uniqueness of solutions to Differential equations, integral equations and also extend the proven results using generalised contractive conditions. 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