Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 159 https://internationalpubls.com Contractive Fixed Point Theorems in Cone G- Metric Spaces M. Uma1, P.Thirunavukarasu2 1 Research Scholar, PG & Research Department of Mathematics, Thanthai Periyar Government Arts and Science College, Thiruchirapalli, Tamil Nadu, India (Affiliated to Bharathidasan University) e-mail: umaleelu@gmail.com 2Assistant Professor, PG & Research Department of Mathematics, Thanthai Periyar Government Arts and Science College, Thiruchirapalli, Tamil Nadu, India (Affiliated to Bharathidasan University) e-mail : ptavinash1967@gmail.com Article History: Received: 28-07-2024 Revised: 09-09-2024 Accepted: 17-09-2024 Abstract: In partially ordered cone G- metre spaces, we provide certain fixed point and coincidence theorems for mappings that meet contractive criteria under θ -maps. Keywords: Cone G- metric space, Complete Cone G- metric space, Partial order Comparable elements 1. Introduction A novel concept known as G-metric space was introduced in 2006 by Z. Mustafa and B. Sims as a generalised metric space [1]. Initiated in [2], fixed point theory in such spaces was investigated in [3]. Specifically, these studies developed the principle of Banach contraction mapping. Cone metric spaces are not a particularly new idea. Kurepa proposed the concept of metric spaces in 1934 [5], where the metric takes values in an ordered space. One can find examples of Huang- Zhang's definition [6] of a cone metric space in Chung's works [7]. These spaces were dubbed "cone-valued metric spaces" by Chung In such spaces, additional fixed point solutions were achieved by a number of writers [9, 10]. Cone G-metric spaces, a generalisation of G-metric spaces and cone metric spaces, were recently introduced by Beg et al. [11]. They demonstrated a few fixed point theorems in terms of specific contractive requirements. Fixed points for ϕ-maps in G-metric spaces were studied by Shatanawi [4], and these fixed points are extended to cone G-metric spaces for two maps by Ozturk and Basarir [12]. Additionally, partially ordered G-metric spaces [15] and partially ordered cone metric spaces [14] have been studied with fixed point issues. In this research, we investigate common fixed point theorems in partially ordered cone G-metric spaces for mappings that meet contractive criteria associated with a nondecreasing θ-map [8,9]. Our findings are an ordered cone G-version extension of research by Ozturk and Basarir [12] and Shatanawi [4]. Preliminaries Let P be a real Banach space and A be a subset of P. By we denote the zero element of P and by in A the interior of A . The subset A is called an order cone if: 1. A is closed, nonempty and A { }; 2. x,y ∈ Q, x,y ≥ 0, a,b ∈ A ⇒ xa + yb ∈ A ; 3. a ∈ a and −a ∈ A ⇒ a = . mailto:umaleelu@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 160 https://internationalpubls.com Definition 1 Let S be a nonempty set, P be a real Banach space and A ⊂ P be an order cone. Suppose a mapping T : S × S × S → P satisfies (T1) T(a,b,c) = if a = b = c; (T2) < T(a,a,b) for all a,b ∈ S with a b; (T3) T(a,a,b) ≤ T(a,b,c) for all a,b,c ∈ S with c b; (T4) T(a,b,c) = T(a,c,b) = T(b.c.a) = · · · (symmetry in all three variables); (T5) T(a,b,c) ≤ T(a,x,x) + T(x,b,c) for all a,b,c,x ∈ S (rectangle inequality). Then the function T is called a generalized cone metric on S and S is called a generalized cone metric space or, shortly, a cone G- metric space. It is obvious that the concept of a cone G- metric space is more general than that of a G-metric space or a cone metric space. If P = Q and A = [0, +∞) then a cone G- metric space becomes a G- metric space. Definition 2 Let (S , T) be a Cone G- metric space. (1) A sequence {am} in S is said to converge to a ∈ S if for every z ∈ P with ≪ z there is N ∈ N such that for all n, m ≥ N , T(an, am, a) ≪ z. (2) A sequence {an} in S is called a Cauchy sequence if for every z ∈ P with ≪ z there is a positive integer N such that T(an, am, aℓ) ≪ z, for all n, m, ℓ ≥ N . (3) (S , T) is said to be complete if every Cauchy sequence in S is convergent in S . Lemma 1 [11]Let S be a cone G- metric space over a normal cone, a ∈ S and let {an} be a sequence in S. Then the following are equivalent: (1) {an} is convergent to a; (2) T(an, an, a) → as n → ∞; (3) T(an, a, a) → as n → ∞; (4) T(am, an, a) → as m, n → ∞. Definition 3 Let S be a nonempty set. Then (S , T, ) is called an ordered cone G-metric space if: (i) (S , T) is a cone G- metric space, (ii) (S , ) is a partially ordered set. Let (S , ) be a partially ordered set. Then a,b ∈ S are called comparable if a b or b a holds. In [13], Nashine and Samet introduced the following concept. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 161 https://internationalpubls.com Let S be a non-empty set and let Q : S → S be a given mapping. For every a ∈ S , we denote by Q−1(a) the subset of S defined by Q−1(a) = { p ∈ S : Qp = a }. Definition 4 Let (S , ) be a partially ordered set and let G , T, Q : S → S be given mappings such that GS ⊆ QS and XS ⊆ QS . We say that X and G are weakly increasing with respect to Q if for all a ∈ S , we have: Ga Xb, ∀ b ∈ Q−1(Ga) and Xa Gb, ∀ b ∈ Q−1(Xa). If G = X, we say that G is weakly increasing with respect to Q. Definition 5 ([9,10]). Let A be an order cone. A non decreasing function : P → P is called a - map if: (i) ( ) = θ and < (ω) < ω for ω ∈ A \ { }, (ii) ω ∈ in A implies ω − (ω) ∈ in A , (iii) if ω ∈ A \ { } and z ∈ in A , then there exists n0 ∈ N such that n(ω) ≪ z for each n n0. Theorem 1 Let (S , ) be a partially ordered set, A be an order cone and let T be a cone G- metric on S. Let G , Q : S → S be two mappings such that T(Ga, Gb, Gc) (T(Qa, Qb, Qc)) (1) for all a,b,c ∈ S with Qa Qb Qc, where θ is a θ -map. We suppose the following: (i) G is weakly increasing with respect to Q; (ii) QS is a complete subspace of S; (iii) S is regular. Then G and Q have a coincidence point. Proof. Let a0 be an arbitrary point in S . Since GS ⊆ QS (by Definition 4), we can construct a sequence {an} in S defined by Qan+1 = Gan, ∀ n ∈ N0. Now, since a1 ∈ Q−1(Ga0) and a2 ∈ Q−1(Ga1), using that G is weakly increasing with respect to Q, we obtain that Qa1 = Ga0 Ga1 = Qa2 Ga2 = Qa3. Continuing this process, we get that Qa1 Qa2 Qa3 · · · Qan Qan+1 · · · . We will prove that {Qan} is a Cauchy sequence in (Q(S ), T). We distinguish two cases. First case. There exists n ∈ N such that Qan = Qan+1. Using the considered contractive condition, we get Gan = Gan+1, thats is, Qan+1 = Qan+2. So, for every m ≥ n, we have Qam = Qan. This implies that {Qan} is a Cauchy sequence. Second case. The successive terms of {Qan} are different. From (1), we have T(Qan, Qan+1, Qan+1) = T(Gan−1, Gan, Qan) ≤ θ (T(Qan−1, Qan, Qan)) ≤ θ 2(T(Qan−2, Qan−1, Qan−1))… Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 162 https://internationalpubls.com ≤ θ n(T(Qa0, Qa1, Qa1)). Fix z, ≪ z. According to property (iii) of function θ, there is n0 ∈ N such that θ n(T(Qa0, Qa1,GQa1)) ≪ z for n ≥ n0. We get that T(Qan, Qan+1, Qan+1) ≪ z for n ≥ n0. In a similar way, there is N1 ∈ N such that T(Qam, Qam+1, Qam+1) < z − θ (c) for all m ≥ N1. (2) We claim that T(Qan, Qam, Qam) ≪ z ∀ m > n ≥ N1 (3) and prove it by induction on m. The inequality (3) holds for m = n + 1 by using (2) and the fact that z − θ (z) < z. Assume that (3) holds for m = d. For m = d + 1, T(Qan, Qad+1, Qad+1) ≤ Q(Qan, Qan+1, Qan+1) + T(Qan+1, Qad+1, Qad+1) ≪ z − θ (z) + θ (T(Qan, Qad, Qad)) ≪ z − θ (z) + θ (z) = z. By induction on m, we conclude that (3) holds for all m > n ≥ N1. Now axiom (T5) of G-metric implies that T(am, an, aℓ) ≤ T(am, an, an) + T(an, an, aℓ) ≪ 2z holds for m, n, ℓ ≥ N1. Hence {Qan} is a G-Cauchy sequence in (QS , T) which is complete by assumption. Then, there exist p = Qq, c ∈ S such that lim Qan = p = Qc. (4) n→∞ Since {Qan} is a non-decreasing sequence and S is regular, it follows from (4) that Qan ≤ Qc for all n ∈ N. Assume Qan Qc. Fix z, ≪ z, and choose a natural number n such that T(Qan, Qan, Qc) ≪ and T(Qan+1, Qc, Qc) ≪ . Hence, we can apply the considered contractive condition to obtain T(Gc, Qc, Qc) ≤ T(Gc, Gan, Gan) + T(Gan, Qc, Qc) ≤ θ (T(Qan, Qan, Qc)) + T(Qan+1, Qc, Qc) (by (1)) < T(Qan, Qan, Qc) + T(Qan+1, Qc, Qc) ≪ + = z . Since z ∈ in A is arbitrary, it follows that T(Gc, Qc, Qc) = which by axiom (T2) implies that Gc = Qc. Then c is a coincidence point for the mappings G and Q. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 163 https://internationalpubls.com Corollary 1 Let (S , ≤) be a partially ordered set, A be an order cone and suppose there is a metric T on S such that (S , T) is a complete cone G- metric space. Let G : S → S be a mapping such that T(Ga, Gb, Gc) ≤ θ (T(a,b,c)) holds for all a,b,c ∈ S with a ≥ b ≥ c where θ is a θ -map. We suppose the following: (i) Ga ≤ G (Ga) for all a ∈ S; (ii) S is regular. Then G has a fixed point. Theorem 2 Let (S , ≤) be a partially ordered set, A be an order cone and suppose there is a cone G- metric T on S such that (S , T) is a complete cone G- metric space. Let G , Q : S → S be nondecreasing mappings such that for all a,b,c ∈ S with Qa ≥ Qq ≥ Qc there exists Θ(a,b,c) ∈ {T(Qa, Qb, Qc), T(Qa, Ga, Ga), T(Qb, Gb, Gb), T(Ga, Gb, Gc)} such that T(Ga, Gb, Gc) ≤ θ (Θ(a,b,c)), where θ is a θ -map. We suppose the following: (i) G is weakly increasing with respect to Q, (ii) S is regular. Then G and Q have a coincidence point. Proof. Let a0 be an arbitrary point in S . Since GS ⊆ QS (by Definition 4), we can construct a sequence {an} in S defined by: San+1 = Gan, ∀ n ∈ N. Now, since a1 ∈ Q−1(Ga0) and a2 ∈ Q−1(Ga1), using that G is weakly increasing with respect to Q, we obtain that Qa1 = Ga0 ≤ Ga1 = Qa2 ≤ Ga2 = Qa3. Continuing this process, we get that Qa1 ≤ Qa2 ≤ Qa3 ≤ · · · ≤ Qan ≤ Qan+1 ≤ · · · . If there exists n0 ∈ {1, 2, . . .} such that Θ(an0 , an0 −1, an0 −1) = θ then it is clear that Qan0 −1 = Qan0 = Gan0 −1 and so we are finished. Now we can suppose Θ(an, an−1, an−1) > for all n ≥ 1. Assume Qan Qan−1, for each n ∈ N. Thus for n ∈ N, we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 164 https://internationalpubls.com T(Qan, Qan+1, Qan+1) = T(Gan−1, Gan, Gan) ≤ θ (Θ(an−1, an, an)) where Θ(an−1, an, an) ∈ {T(Qan−1, Qan, Qan), T(Qan−1, Gan−1, Gan−1), T(Qan, Gan, Gan), T(Gan−1, Qan, Qan)} = {T(Qan−1, Qan, Qan), T(Qan−1, Qan, Qan), T(Qan, Qan+1, Qan+1), T(Qan, Qan, Qan)} = {T(Qan−1, Qan, Qan), T(Qan, Qan+1, Qan+1), }. • If Θ(an−1, an, an) = T(Qan, Qan+1, Qan+1), then T(Qan, Qan+1, Qan+1) ≤ θ (T(Qan, Qan+1, Qan+1)) and by the property of θ we have T(Qan, Qan+1, Qan+1) < T(Qan, Qan+1, Qan+1) which is impossible. • If Θ(an−1, an, an) = , then T(Qan, Qan+1, Qan+1) ≤ (θ) < which is a contradiction. Therefore, Θ(an−1, an, an) = T(Qan−1, Qan, Qan), and then T(Qan, Qan+1, Qan+1) ≤ θ (T(Qan−1, Qan, Qan)). Thus for n ∈ N , we have T(Qan, Qan+1, Qan+1) = T(Gan−1, Gan, Gan) ≤ θ (T(Qan−1, Qan, Qan)) ≤ θ 2(T(Qan−2, Qan−1, Qan−1)) ≤ θ n(T(Qa0, Qa1, Qa1)). By an argument similar to that in the proof of theorem, one can show that {Qan} is a Cauchy sequence. Since S is G-complete,Qan is convergent to p ∈ S . Now we show that Qp = Gp. Since {Qan} is a nondecreasing sequence and Qan → p, by regularity of S we have Qan ≤ p for all n. If Qan = p for some n, then, by construction, Qan+1 = p and p is a fixed point. So we assume that Qan . Then, for n ∈ N, we have T(Qp, Qp, Gp) ≤ T(Qp, Qp, Qan) + T(Qan, Qan, Gp) = T(Qp, Qp, Qan) + T(Gan−1, Gan−1, Gp) ≤ T(Qp, Qp, Qan) + θ (Θ(an−1, an−1, p)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 165 https://internationalpubls.com where Θ(an−1, an−1, p) ∈ {T(Qan−1, Qan−1, Qp), T(Qan−1, Gan−1, Gan−1), T(Qan−1, Gan−1, Gan−1), T(Gan−1, Qan−1, Qp)} = {T(Qan−1, Qan−1, Qp), T(Qan−1, Qan, Qan), T(Qan, Qan−1, Qp)}. Fix z, ≪ z. Choose a natural number N1 such that T(Qp, Qp, Qan) ≪ and T(Qan−1, Qan−1, Qp) ≪ , for all n ≥ N1. We investigate these situations as follows: Case 1. If Θ(an−1, an−1, p) = T(Qan−1, Qan−1, Qp), then we have T(Qp, Qp, Gp) ≤ T (Qp, Qp, Qan) + θ (T(Qan−1, Qan−1, Qp)) < T(Qp, Qp, Qan) + T(Qan−1, Qan−1, Qp) ≪ + = z . Case 2. If Θ(an−1, an−1, p) = T(Qan−1, Qan, Qan), then we have T(Qp, Qp, Qp) ≤ T(Qp, Qp, Qan) + θ (T(Qan−1, Qan, Qan)) < T(Qp, Qp, Qan) + T(Qan−1, Qan, Qan) ≪ z. Case 3. If Θ(an−1, an−1, p) = T(Qan, Qan−1, Qp), then we have T(Qp, Qp, Gp) ≤ T(Qp, Qp, Qan) + θ (T(Qan, Qan−1, Qp)) < T(Qp, Qp, Qan) + T(Qan, Qan−1, Qp) ≤ T(Qp, Qp, Qan) + T (Qan, Qan−1, Qan−1) + T(Qan−1, Qan−1, Qp) ≪ z whenever n ∈ N. Thus in all cases T(Qp, Qp, Gp) ≪ z for arbitrary Z ∈ in A . It follows that T(Qp, Qp, Gp)= θ which implies that Gp = Qp. Then p is a coincidence point for the mappings G and Q. Conclusion We examine common fixed point theorems in partially ordered cone G-metric spaces for mappings that meet contractive criteria associated with a nondecreasing θ -map [8,9]. The work by Shatanawi [4] and Ozturk and Basarir [12] is extended to our results in an ordered cone G-version. References: [1] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006) 289–297. [2] Z. Mustafa, H. Obiedat, F. Awawdeh, Some of fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl. 2008 (2008) 12. Article ID 189870. [3] Z. Mustafa, B. Sims, Fixed point theorems for contractive mappings in complete G-metric space, Fixed Point Theory Appl. 2009 (2009) 10. Article ID 917175. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 166 https://internationalpubls.com [4] W. Shatanawi, Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces, Fixed Point Theory Appl. 2010 (2010) 9. Article ID 181650. [5] Ð.R. Kurepa, Tableaux ramifiés d’ensembles. Espace pseudo-distanciés, C.R. Acad. Sci. Paris 198 (1934) 1563–1565. [6] L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476. [7] K.J. Chung, Remarks on nonlinear contractions, Pacific J. Math. 101 (1982) 41–48. [8] I. Aranđelović, Z. Kadelburg, S. Radenović, Boyd-Wong-type common fixed point results in cone metric spaces, Appl. Math. Comput. 217 (2011) 7167–7171. [9] C. Di Bari, P. Vetro, ϕ-pairs and common fixed points in cone metric spaces, Rend. Circolo Mat. Palermo 57 (2008) 279–285. [10] Z. Kadelburg, S. Radenović, V. Rakočević, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett. 24 (2011) 370–374. [11] I. Beg, M. Abbas, T. Nazir, Generalized cone metric spaces, J. Nonlinear Sci. Appl. 3 (2010) 21–31. [12] M. Ozturk, M. Basarir, On some common fixed point theorems with ϕ-maps on G-cone metric spaces, Bull. Math. Anal. Appl. 3 (2011) 121–133. [13] H.K. Nashine, B. Samet, Fixed point results for mappings satisfying (ψ, ϕ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 2201–2209. [14] Z. Kadelburg, M. Pavlović, S. Radenović, Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl. 59 (2010) 3148–3159. [15] Saadati, S.M. Vaezpour, P. Vetro, B.E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Math. Comput. Modelling 52 (2010) 797–801.