EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 1350-1366 ISSN 1307-5543 – ejpam.com Published by New York Business Global Coincidence and Common Fixed Point Theorems in Gb-Cone Metric Spaces Saadia Benchabane1,∗, Smäıl Djebali1,2 1 Laboratoire ”Théorie du Point Fixe et Applications”, ENS, BP 92 Kouba. Algiers, 16006. Algeria 2 Department of Mathematics & Statistics, College of Sciences, Imam Mohammad Ibn Saud Islamic University (IMSIU), PB 90950. Riyadh 11623, Saudi Arabia Abstract. In this article, some coincidence and common fixed point results were obtained for four mappings satisfying some special contractive conditions and defined on a Gb-cone metric space (with or without the assumption of normality). Our results generalize recent results in the literature. Two illustrative examples are included and some consequences are provided. 2020 Mathematics Subject Classifications: 47H10, 54E50, 54H25 Key Words and Phrases: Coincidence and common fixed point, Gb-Cone Metric Spaces. 1. Introduction and preliminaries The notion of cone metric space is an important generalization of the classical con- cept of metric. It was introduced in 2007 by Huang and Zhang [18] who extended the Banach contraction principle in the setting of cone metric spaces. Their result has been generalized in several directions by many authors (see, e.g., [14], [8], [5], [1], [4], [12], [17]) and there has been a number of generalizations of the notion of a cone metric space. One such generalization is that of the G-cone metric space initiated by Ismat Beg et al. [10]. Recently, Ughade and Daheriya [3] introduced the concept of Gb-cone metric space as a generalization of G-cone metric space and obtained some fixed point (and common fixed point) results. In this paper, our aim is to establish some coincidence and common fixed point results for four mappings, defined on a Gb-cone metric space without the assumption of normality and also with the assumption of normality. The mappings satisfy some special contractive conditions. We first collect some basic notions and primary results needed to develop our existence results. N will refer to the set of nonnegative integers. Our main existence results are Theorem 1 (without the assumption of normality) and Theorem 2 (with the assumption of normality). In each case, an example of application and a corollary are supplied. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4126 Email addresses: benchabane.saadia@ymail.com (S. Benchabane), djebali@hotmail.com (S. Djebali) http://www.ejpam.com 1350 © 2021 EJPAM All rights reserved. S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1351 Definition 1. Let E be a real Banach space with norm ∥ · ∥ and P a subset of E. Then P is called a cone if and only if (i) P is closed, nonempty, and P ̸= {θ}, where θ is the zero vector in E, (ii) if a, b ≥ 0, and x, y ∈ P , then ax+ by ∈ P , (iii) if x ∈ P and −x ∈ P , then x = θ. Given a cone P in a Banach space E, a partial ordering ⪯ with respect to P is given by x ⪯ y if and only if y − x ∈ P. We write x ≺ y whenever x ⪯ y and x ̸= y, while x ≪ y will stand for y − x ∈ int(P ), where int(P ) designates the interior of P . If int(P ) ̸= ∅, then P is called a solid cone. The cone P is called normal if there is a number k > 0, such that for all x, y ∈ E, we have θ ⪯ x ⪯ y implies ∥x∥ ≤ k∥y∥. The least positive number satisfying this inequality is called the normal constant of P . In [6], it is proved that there are no normal cones with a normal constant k < 1. Lemma 1. [13] For cones which are not normal, the following properties hold (PT1) If u ⪯ v and v ≪ w, then u ≪ w. (PT2) If u ≪ v and v ⪯ w, then u ≪ w. (PT3) If u ≪ v and v ≪ w, then u ≪ w. (PT4) If θ ⪯ u ≪ c for each c ∈ intP , then u = θ. (PT5) If a ⪯ b+ c for each c ∈ intP , then a ⪯ b. (PT6) If E be a real Banach space with a cone P , and if a ⪯ λa, where a ∈ P and 0 ≤ λ < 1, then a = θ. (PT7) If c ∈ intP , an ∈ E and an → θ, then there exists an n0 ∈ N such that, for all n > n0, an ≪ c. Definition 2. [3] Let X be a nonempty set and E a real Banach space equipped with the partial ordering ⪯ with respect to the cone P . A vector-valued function G : X×X×X → X is said to be a generalized cone b-metric function on X with the constant s ≥ 1 if the fol- lowing conditions are satisfied (GBC1) G(x, y, z) = θ if x = y = z, (GBC2) θ ≺ G(x, y, z), for y ̸= z, for all x, y, z ∈ X, (GBC3) G(x, x, y) ⪯ G(x, y, z), whenever y ̸= z, for all x, y, z ∈ X, (GBC4) G(x, y, z) = G(y, x, z) = · · · (symmetric in all three variables), (GBC5) G(x, y, z) ⪯ s(G(x, a, a) +G(a, y, z)), for all x, y, z, a ∈ X (the rectangle inequal- ity). Then the pair (X,G) is called a generalized b-cone metric space or, more specifically, a Gb-cone metric space. The concept of a Gb-cone metric space is more general than that of a Gb-metric space, a G-cone metric space, and a cone metric space. For the definition of Gb-metric, G-cone metric, cone metric spaces, and related concepts we refer the reader to [15], [10], [18]. S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1352 Definition 3. [3] Let (X,G) be a Gb-cone metric space. A sequence (xn) in X is said to be (1) a Gb-cone Cauchy sequence if, for every c ∈ E with θ ≪ c, there exists n0 ∈ N such that for all n,m, l > n0, G(xn, xm, xl) ≪ c. (2) a Gb-cone convergent sequence if, for every c ∈ E with θ ≪ c, there exists n0 ∈ N such that for all m,n > n0, G(xn, xm, x) ≪ θ for some fixed x in X. Here x is called the Gb-limit of (xn) and is denoted by Gb- lim n→+∞ xn = x or xn → x as n → +∞. Definition 4. [3] A Gb-cone metric space X is a Gb-complete cone metric space, if every Gb-cone Cauchy sequence in X is Gb-cone convergent in X. Next, we state some Gb-cone convergence results. Proposition 1. [3] Let (X,G) be a Gb-cone metric space. Then the following conditions are equivalent (1) (xn) is Gb-cone Cauchy in X. (2) For every c ∈ E with θ ≪ c, there is n0 ∈ N such that for all n,m > n0, G(xn, xm, xm) ≪ θ. Lemma 2. [3] Let (X,G) be a Gb-cone metric space and P a normal cone with normal constant k. A sequence (xn) ⊂ X is Gb-cone convergent to x if and only if G(xn, xm, x) → θ, as n,m → +∞. Proposition 2. [3] Let (X,G) be a Gb-cone metric space and P be a normal cone with normal constant k. The following conditions are equivalent (1) (xn) is Gb-cone convergent to x, (2) G(xn, xn, x) → θ,as n → +∞, (3) G(xn, x, x) → θ,as n → +∞, (4) G(xm, xn, x) → θ,as n,m → +∞. Lemma 3. [3] Let (X,G) be a complete Gb-cone metric space and P be a normal cone with normal constant k. If (xn) ⊂ X Gb-cone converges to x and Gb-cone converges to y, then x = y (the limit of (xn) is unique). Proposition 3. [3] Let (X,G) be a Gb-cone metric space and P be a normal cone with normal constant k. Then sequence (xn) is Gb-cone Cauchy if and only if G(xn, xm, xl) → θ,as n,m, l → +∞. Definition 5. Let f and F be self mappings of a set X. If y = fx = Fx for some x in X, then x is called a coincidence point of f and F and y is called a point of coincidence of f and F . Definition 6. The self-mappings f and F of a set X are said to be weakly compatible if they commute at their coincidence points, that is, if fu = Fu for some u in X, then fFu = Ffu. Proposition 4. [7] Let f and F be weakly compatible self mappings of a set X. If f and F have a unique point of coincidence, that is, y = fx = Fx, then y is the unique common fixed point of f and F . S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1353 2. Main results Our existence first results for coincidence common fixed points no condition of nor- mality assumed. Theorem 1. Let (X,G) be a cone Gb-metric space with the coefficient s ≥ 1 relative to a solid cone P . Suppose that the mappings F, T,R, f : X → X satisfy the condition that for some constant λ ∈ [0, 12) and for all x, y, z ∈ X, there exists M(x, y, z) ∈ {G(fx, fy, fz), G(Fx, fy, fz), G(fx, Ty, fz), G(fx, fy,Rz), G(fx, Ty,Rz), G(Fx, fy,Rz), G(Fx, Ty, fz), G(Fx, Fx, fx), G(Ty, Ty, fy), G(Rz,Rz, fz)} such that s2G(Fx, Ty,Rz) ⪯ λM(x, y, z). If F (X) ∪ T (X) ∪ R(X) ⊂ f(X) and f(X) is a Gb-complete subspace of X, then F, T,R and f have a unique point of coincidence in X. If the pairs (f, F ), (f, T ) and (f,R) are further weakly compatible, then F, T,R and f have a unique common fixed point. Proof. Let x0 in X be an arbitrary point. Since F (X) ∪ T (X) ∪ R(X) ⊂ f(X), there exist two sequences (xn) and (yn) in X such that y3n = fx3n+1 = Fx3n, y3n+1 = fx3n+2 = Tx3n+1, y3n+2 = fx3n+3 = Rx3n+2, for all n ∈ N. By the contractive condition, for all n ∈ N, there exists M(x3n, x3n+1, x3n+2) ∈ {G(fx3n, fx3n+1, fx3n+2), G(Fx3n, fx3n+1, fx3n+2), G(fx3n, Tx3n+1, fx3n+2), G(fx3n, fx3n+1, Rx3n+2), G(fx3n, Tx3n+1, Rx3n+2), G(Fx3n, fx3n+1, Rx3n+2), G(Fx3n, Tx3n+1, fx3n+2), G(Fx3n, Fx3n, fx3n), G(Tx3n+1, Tx3n+1, fx3n+1), G(Rx3n+2, Rx3n+2, fx3n+2)} = {G(y3n−1, y3n, y3n+1), G(y3n, y3n, y3n+1), G(y3n−1, y3n+1, y3n+1), G(y3n−1, y3n, y3n+2), G(y3n−1, y3n+1, y3n+2), G(y3n, y3n, y3n+2), G(y3n, y3n+1, y3n+1), G(y3n, y3n, y3n−1), G(y3n+1, y3n+1, y3n), G(y3n+2, y3n+2, y3n+1)} such that G(y3n, y3n+1, y3n+2) = G(Fx3n, Tx3n+1, Rx3n+2) ⪯ λ s2 M(x3n, x3n+1, x3n+2). We discuss three cases. Case 1. M(x3n, x3n+1, x3n+2) ∈ {G(y3n, y3n, y3n+2), G(y3n+2, y3n+2, y3n+1)}. By (GBC3) and (GBC4), G(y3n, y3n+1, y3n+2) ⪯ λG(y3n, y3n+1, y3n+2). S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1354 By Lemma 1 (PT6) and (GBC1), y3n = y3n+1 = y3n+2. By applying the contractive condition, there exists M(x3n+3, x3n+1, x3n+2) ∈ {G(fx3n+3, fx3n+1, fx3n+2), G(Fx3n+3, fx3n+1, fx3n+2), G(fx3n+3, Tx3n+1, fx3n+2), G(fx3n+3, fx3n+1, Rx3n+2), G(fx3n+3, Tx3n+1, Rx3n+2), G(Fx3n+3, fx3n+1, Rx3n+2), G(Fx3n+3, Tx3n+1, fx3n+2), G(Fx3n+3, Fx3n+3, fx3n+3), G(Tx3n+1, Tx3n+1, fx3n+1), G(Rx3n+2, Rx3n+2, fx3n+2)} = {G(y3n+2, y3n, y3n+1), G(y3n+3, y3n, y3n+1), G(y3n+2, y3n+1, y3n+1), G(y3n+2, y3n, y3n+2), G(y3n+2, y3n+1, y3n+2), G(y3n+3, y3n, y3n+2), G(y3n+3, y3n+1, y3n+1), G(y3n+3, y3n+3, y3n+2), G(y3n+1, y3n+1, y3n), G(y3n+2, y3n+2, y3n+1)} = {θ,G(y3n+3, y3n+2, y3n+2), θ, θ, θ,G(y3n+3, y3n+2, y3n+2), G(y3n+3, y3n+2, y3n+2), G(y3n+3, y3n+3, y3n+2), θ, θ} such that G(y3n+3, y3n+1, y3n+2) = G(Fx3n+3, Tx3n+1, Rx3n+2) ⪯ λ s2 M(x3n+3, x3n+1, x3n+2). Hence G(y3n+3, y3n+1, y3n+2) ⪯ λ s2 G(y3n+3, y3n+1, y3n+2). By a similar argument, we obtain y3n+2 = y3n+3. Thus (yn) is a constant sequence, which implies that (yn) is a Gb-Cauchy sequence. Case 2. M(x3n, x3n+1, x3n+2) ∈ {G(y3n−1, y3n, y3n+1), G(y3n, y3n, y3n+1), G(y3n−1, y3n+1, y3n+1), G(y3n, y3n+1, y3n+1), G(y3n, y3n, y3n−1), G(y3n+1, y3n+1, y3n)}. By (GBC3) and (GBC4), G(y3n, y3n+1, y3n+2) ⪯ λ s2 G(y3n−1, y3n, y3n+1). Case 3. M(x3n, x3n+1, x3n+2) ∈ {G(y3n−1, y3n, y3n+2), G(y3n−1, y3n+1, y3n+2)}. Assump- tions (GBC5), (GBC3) and (GBC4) imply G(y3n, y3n+1, y3n+2) ⪯ λ s− λ G(y3n−1, y3n, y3n+1). Hence for all n ∈ N, G(y3n, y3n+1, y3n+2) ⪯ αG(y3n−1, y3n, y3n+1), (1) where α = max{ λ s−λ , λ s2 } = λ s−λ ∈ [0, 1). Similarly we can prove that G(y3n+1, y3n+2, y3n+3) ⪯ αG(y3n, y3n+1, y3n+2) (2) and G(y3n+2, y3n+3, y3n+4) ⪯ αG(y3n+1, y3n+2, y3n+3). (3) S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1355 From (1), (2), and (3) that, for all n ∈ N, we get G(yn, yn+1, yn+2) ⪯ αG(yn−1, yn, yn+1). (4) From the inequality in (4), we infer that for all n ∈ N, G(yn, yn+1, yn+2) ⪯ αG(yn−1, yn, yn+1) ⪯ · · · ⪯ αnG(y0, y1, y2). (5) For every n,m ∈ N with m > n, using (GBC5), (GBC3), (GBC4), and (5), we deduce that G(yn, ym, ym) ⪯ sG(yn, yn+1, yn+1) + s2G(yn+1, yn+2, yn+2) + · · ·+ sm−n G(ym−1, ym, ym) ⪯ sG(yn, yn+1, yn+2) + s2G(yn+1, yn+2, yn+3) + · · ·+ sm−n G(ym−1, ym, ym+1) ⪯ sαnG(y0, y1, y2) + s2αn+1G(y0, y1, y2) + · · ·+ sm−nαm−1 G(y0, y1, y2) = sαnG(y0, y1, y2)(1 + sα+ · · ·+ (sα)m−n−1) ⪯ sαn 1−sαG(y0, y1, y2) → θ, as n → +∞. By Lemma 1 (PT7), for every c ∈ E with c ≫ θ, there exists n0 ∈ N such that for all n > n0, sαn 1− sα G(y0, y1, y2) ≪ c. Moreover, for any m > n > n0, by Lemma 1 (PT1), G(yn, ym, ym) ≪ c. Using Proposition 1, we conclude that (yn) is a Gb-Cauchy sequence in f(X). Since f(X) is Gb-complete subspace of X, there exists v ∈ f(X) such that f(xn) → v, as n → +∞. Consequently, there is an u ∈ X such that fu = v. Since the sequences (fx3n+1) = (Fx3n), (fx3n+2) = (Tx3n+1), and (fx3n+3) = (Rx3n+2) are subsequences of (yn), they converge to u. Next we prove that fu = Fu. By (GBC5) and (GBC3), we have for all n ∈ N G(Fu, fu, fu) ⪯ sG(Fu, y3n+1, y3n+1) + sG(y3n+1, fu, fu) ⪯ sG(Fu, y3n+1, y3n+2) + sG(y3n+1, fu, fu). The contractive condition yields that for all n ∈ N, there exists M(u, x3n+1, x3n+2) ∈ {G(fu, fx3n+1, fx3n+2), G(Fu, fx3n+1, fx3n+2), G(fu, Tx3n+1, fx3n+2), G(fu, fx3n+1, Rx3n+2), G(fu, Tx3n+1, Rx3n+2), G(Fu, fx3n+1, Rx3n+2), G(Fu, Tx3n+1, fx3n+2), G(Fu, Fu, fu), G(Tx3n+1, Tx3n+1, fx3n+1), G(Rx3n+2, Rx3n+2, fx3n+2)} = {G(fu, y3n, y3n+1), G(Fu, y3n, y3n+1), G(fu, y3n+1, y3n+1), G(fu, y3n, y3n+2), G(fu, y3n+1, y3n+2), G(Fu, y3n, y3n+2), G(Fu, y3n+1, y3n+1)G(Fu, Fu, fu), G(y3n+1, y3n+1, y3n), G(y3n+2, y3n+2, y3n+1)} S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1356 such that G(Fu, fu, fu) ⪯ sG(Fu, Tx3n+1, Rx3n+2) + sG(y3n+1, fu, fu) ⪯ λ sM(u, x3n+1, x3n+2) + sG(y3n+1, fu, fu). We distinguish between three cases. Case 1. M(u, x3n+1, x3n+2) ∈ {G(fu, y3n, y3n+1), G(fu, y3n+1, y3n+1), G(fu, y3n, y3n+2), G(fu, y3n+1, y3n+2), G(y3n+1, y3n+1, y3n), G(y3n+2, y3n+2, y3n+1)} If, e.g., M(u, x3n+1, x3n+2) = G(fu, y3n, y3n+1), then for all n ∈ N, G(Fu, fu, fu) ⪯ λ s G(fu, y3n, y3n+1) + sG(y3n+1, fu, fu). Since yn → fu, as n → +∞, then for c ≫ θ, there exists n0 ∈ N such that for all n > n0, G(fu, y3n, y3n+1) ≪ sc 2λ , G(y3n+1, fu, fu) ≪ c 2s . Hence θ ⪯ G(Fu, fu, fu) ⪯ λ s G(fu, y3n, y3n+1) + sG(y3n+1, fu, fu) ≪ c. By Lemma 1 (PT4), G(Fu, fu, fu) = θ, that is Fu = fu. The remaining cases are readily dealt with in a similar way. Case 2. M(u, x3n+1, x3n+2) ∈ {G(Fu, y3n, y3n+1), G(Fu, y3n, y3n+2), G(Fu, y3n+1, y3n+1)}. If M(u, x3n+1, x3n+2) = G(Fu, y3n, y3n+1), then by (GBC5), for all n ∈ N, we get G(Fu, fu, fu) ⪯ λ sG(Fu, y3n, y3n+1) + sG(y3n+1, fu, fu) ⪯ λG(Fu, fu, fu) + λG(fu, y3n, y3n+1) + sG(y3n+1, fu, fu), which implies that (1− λ)G(Fu, fu, fu) ⪯ λG(fu, y3n, y3n+1) + sG(y3n+1, fu, fu). Since yn → fu as n → +∞, then for c ≫ θ, there exists n0 ∈ N such that for all n > n0, G(fu, y3n, y3n+1) ≪ c 2λ , G(y3n+1, fu, fu) ≪ c 2s . Hence θ ⪯ (1− λ)G(Fu, fu, fu) ⪯ λG(fu, y3n, y3n+1) + sG(y3n+1, fu, fu) ≪ c. By Lemma 1 (PT4), (1− λ)G(Fu, fu, fu) = θ, that is Fu = fu. The remaining cases are analogous. Case 3. M(u, x3n+1, x3n+2) = G(Fu, Fu, fu). By (GBC5) and (GBC4), for all n ∈ N, we have (1− 2λ)G(Fu, fu, fu) ⪯ sG(y3n+1, fu, fu). S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1357 Since yn → fu as n → +∞, then for c ≫ θ, there exists n0 ∈ N such that for all n > n0, G(y3n+1, fu, fu) ≪ c s . Hence θ ⪯ (1− 2λ)G(Fu, fu, fu) ⪯ sG(y3n+1, fu, fu) ≪ c. By Lemma 1 (PT4), (1− 2λ)G(Fu, fu, fu) = θ, that is Fu = fu. The other cases are identical. To sum up, Fu = fu. We can check that Tu = fu and Ru = fu, too. So, fu = Fu = Tu = Ru = v, that is v is a point of coincidence of f , F , T and R. To show that f , F , T , and R have a unique point of coincidence in X, assume that there exists another coincidence point v⋆ ∈ X such that fu⋆ = Fu⋆ = Tu⋆ = Ru⋆ = v⋆, for some u⋆ ∈ X. By the contractive condition, there exists M(u⋆u, u) ∈ {G(fu⋆, fu, fu), G(Fu⋆, fu, fu), G(fu⋆, Tu, fu), G(fu⋆, fu,Ru), G(fu⋆, Tu,Ru), G(Fu⋆, fu,Ru), G(Fu⋆, Tu, fu), G(Fu⋆, Fu⋆, fu⋆), G(Tu, Tu, fu), G(Ru,Ru, fu)} = {G(v⋆, v, v)} such that G(v⋆, v, v) = G(Fu⋆, Tu,Ru) ⪯ λ s2 M(u⋆u, u). Therefore, G(v⋆, v, v) ⪯ λ s2 G(v⋆, v, v). By Lemma 1 (PT6), G(v⋆, v, v) = θ, that is v⋆ = v. Finally, since the pairs (f, F ), (f, T ) and (f,R) are weakly compatible, we have Ffu = fFu, Tfu = fTu, Rfu = fRu. This implies that Fv = Tv = Rv = fv = t, i.e., t is a point of coincidence of f , F , T and R.So t = v by uniqueness. Finally Proposition 4 tells us that v is the unique common fixed point of f , F , T , and R. Theorem 1 is illustrated by the following example inspired from [1, Example 2.12]. Example 1. Let X = [0, 1], E = C[0, 1] be endowed with the strongly locally convex topology τ(E,E∗) and let P = {x ∈ E : x(t) ≥ 0, t ∈ [0, 1]}. Then the cone is τ(E,E∗)- solid but not normal with respect to the topology τ(E,E∗). Define the mapping G : X × X ×X → E by G(x, y, z)(t) = max{|x− y|2, |y − z|2, |x− z|2}et. S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1358 Then G is a Gb-cone metric on X with the coefficient s = 2. On X, define the self-maps f, F, T , and R by F (x) = { 1 4 , x ∈ [0, 23), 1 7 , x ∈ [23 , 1], T (x) = { 1 4 , x ∈ [0, 23), 1 5 , x ∈ [23 , 1], R(x) = { 1 4 , x ∈ [0, 23), 1 6 , x ∈ [23 , 1], f(x) = { x, x ∈ [0, 23), 2 3 , x ∈ [23 , 1]. Note that for all x, y, z ∈ X, there exists M(x, y, z) ∈ {G(fx, fy, fz), G(Fx, fy, fz), G(fx, Ty, fz), G(fx, fy,Rz), G(fx, Ty,Rz), G(Fx, fy,Rz), G(Fx, Ty, fz), G(Fx, Fx, fx), G(Ty, Ty, fy), G(Rz,Rz, fz)} such that 4G(Fx, Ty,Rz)et ≤ 2 5 M(x, y, z)et. Moreover f(X) = [0, 23 ] is a Gb-complete subspace of X, F (X) ∪ T (X) ∪ R(X) ⊂ f(X), and the pairs (f, F ), (f, T ), and (f,R) are weakly compatible. Therefore all the conditions of Theorem 1 are fulfilled. Finally, 1 4 is the unique point of coincidence and the unique common fixed point for all of the mappings F, T,R, and f . The following result is immediately derived from Theorem 1. Corollary 1. Let (X,G) be a cone G-metric space relative to a solid cone P . Suppose that the mappings F, T,R, f : X → X satisfy for some constant λ ∈ [0, 12) and for all x, y, z ∈ X, there exists M(x, y, z) ∈ {G(fx, fy, fz), G(Fx, fy, fz), G(fx, Ty, fz), G(fx, fy,Rz), G(fx, Ty,Rz), G(Fx, fy,Rz), G(Fx, Ty, fz), G(Fx, Fx, fx), G(Ty, Ty, fy), G(Rz,Rz, fz)} such that G(Fx, Ty,Rz) ⪯ λM(x, y, z). If F (X) ∪ T (X) ∪ R(X) ⊂ f(X) and f(X) is a G-complete subspace of X, then F, T,R and f have a unique point of coincidence in X. If further the pairs (f, F ), (f, T ), and (f,R) are weakly compatible, then F, T,R and f have a unique common fixed point. S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1359 For x, y, z ∈ X, the distance Gk(x, y, z) between x, y and z is defined by Gk(x, y, z) = ∥G(x, y, z)∥. When the assumption of normality is assumed, the existence results for coincidence common fixed points are collected in the following Theorem 2. Let (X,G) be a cone Gb-metric space with the coefficient s ≥ 1 relative to a normal constant k ≥ 1. Suppose that the mappings F, T,R, f : X → X satisfy sGk(Fx, Ty,Rz) ≤ λMk(x, y, z) + LNk(x, y, z), (6) where Mk(x, y, z) = max{Gk(fx, fy, fz), Gk(Fx, fy, fz), Gk(fx, Ty, fz), Gk(fx, fy,Rz), Gk(fx, Ty,Rz), Gk(Fx, fy,Rz), Gk(Fx, Ty, fz), Gk(Fx, Fx, fx), Gk(Ty, Ty, fy), Gk(Rz,Rz, fz), Gk(Fx, fx, fx), Gk(Ty, fy, fy), Gk(Rz, fz, fz)}, Nk(x, y, z) = min{Gk(Fx, fy, fy), Gk(Fx, fz, fz), Gk(Ty, fz, fz), Gk(Ty, fx, fx), Gk(Rz, fx, fx), Gk(Rz, fy, fy)}, for all x, y, z ∈ X, λ ∈ [0, 1 2k ), and L ≥ 0. If F (X)∪T (X)∪R(X) ⊂ f(X) and f(X) is a Gb-complete subspace of X, then F, T,R, and f have a unique point of coincidence in X. Moreover if the pairs (f, F ), (f, T ), and (f,R) are weakly compatible, then F, T,R and f have a unique common fixed point. Proof. Let x0 in X be an arbitrary point since F (X) ∪ T (X) ∪ R(X) ⊂ f(X). There exist sequences (xn) and (yn) in X such that y3n = fx3n+1 = Fx3n, y3n+1 = fx3n+2 = Tx3n+1, y3n+2 = fx3n+3 = Rx3n+2, for all n = 0, 1, 2, . . .. Then from (6), we have for all n ∈ N Gk(y3n, y3n+1, y3n+2) = Gk(Fx3n, Tx3n+1, Rx3n+2) ≤ λ sMk(x3n, x3n+1, x3n+2) + L s2 Nk(x3n, x3n+1, x3n+2), (7) S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1360 where Mk(x3n, x3n+1, x3n+2) = max{Gk(fx3n, fx3n+1, fx3n+2), Gk(Fx3n, fx3n+1, fx3n+2), Gk(fx3n, Tx3n+1, fx3n+2), Gk(fx3n, fx3n+1, Rx3n+2), Gk(fx3n, Tx3n+1, Rx3n+2), Gk(Fx3n, fx3n+1, Rx3n+2), Gk(Fx3n, Tx3n+1, fx3n+2), Gk(Fx3n, Fx3n, fx3n), Gk(Tx3n+1, Tx3n+1, fx3n+1), Gk(Rx3n+2, Rx3n+2, fx3n+2) Gk(Fx3n, fx3n, fx3n), Gk(Tx3n+1, fx3n+1, fx3n+1), Gk(Rx3n+2, fx3n+2, fx3n+2)} = max{Gk(y3n−1, y3n, y3n+1), Gk(y3n, y3n, y3n+1), Gk(y3n−1, y3n+1, y3n+1), Gk(y3n−1, y3n, y3n+2), Gk(y3n−1, y3n+1, y3n+2), Gk(y3n, y3n, y3n+2), Gk(y3n, y3n+1, y3n+1), Gk(y3n, y3n, y3n−1), Gk(y3n+1, y3n+1, y3n), Gk(y3n+2, y3n+2, y3n+1) Gk(y3n, y3n−1, y3n−1), Gk(y3n+1, y3n, y3n), Gk(y3n+2, y3n+1, y3n+1)} and Nk(x3n, x3n+1, x3n+2) = min{Gk(Fx3n, fx3n+1, fx3n+1), Gk(Fx3n, fx3n+2, fx3n+2), Gk(Tx3n+1, fx3n+2, fx3n+2), Gk(Tx3n+1, fx3n, fx3n), Gk(Rx3n+2, fx3n, fx3n), Gk(Rx3n+2, fx3n+1, fx3n+1)} = min{Gk(y3n, y3n, y3n), Gk(y3n, y3n+1, y3n+1), Gk(y3n+1, y3n+1, y3n+1), Gk(y3n+1, y3n−1, y3n−1), Gk(y3n+2, y3n−1, y3n−1), Gk(y3n+2, y3n, y3n)} = 0. By (GBC5), (GBC3), and (GBC4), we have G(y3n, y3n, y3n+1) ⪯ G(y3n−1, y3n, y3n+1), G(y3n−1, y3n+1, y3n+1) ⪯ G(y3n−1, y3n, y3n+1), G(y3n−1, y3n, y3n+2) ⪯ sG(y3n−1, y3n, y3n+1) + sG(y3n, y3n+1, y3n+2), G(y3n−1, y3n+1, y3n+2) ⪯ sG(y3n−1, y3n, y3n+1) + sG(y3n, y3n+1, y3n+2), G(y3n, y3n, y3n+2) ⪯ G(y3n, y3n+1, y3n+2), G(y3n, y3n+1, y3n+1) ⪯ G(y3n−1, y3n, y3n+1), G(y3n, y3n, y3n−1) ⪯ G(y3n−1, y3n, y3n+1), G(y3n+1, y3n+1, y3n) ⪯ G(y3n, y3n+1, y3n+2), G(y3n+2, y3n+2, y3n+1) ⪯ G(y3n, y3n+1, y3n+2), G(y3n, y3n−1, y3n−1) ⪯ G(y3n−1, y3n, y3n+1), G(y3n+1, y3n, y3n) ⪯ G(y3n−1, y3n, y3n+1), G(y3n+2, y3n+1, y3n+1) ⪯ G(y3n, y3n+1, y3n+2). Using the normality of the cone and the fact that ∥.∥ satisfies the triangle inequality, we S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1361 obtain Gk(y3n, y3n, y3n+1) ≤ kGk(y3n−1, y3n, y3n+1), Gk(y3n−1, y3n+1, y3n+1) ≤ kGk(y3n−1, y3n, y3n+1), Gk(y3n−1, y3n, y3n+2) ≤ skGk(y3n−1, y3n, y3n+1) + skGk(y3n, y3n+1, y3n+2), Gk(y3n−1, y3n+1, y3n+2) ≤ skGk(y3n−1, y3n, y3n+1) + skGk(y3n, y3n+1, y3n+2), Gk(y3n, y3n, y3n+2) ≤ kGk(y3n, y3n+1, y3n+2), Gk(y3n, y3n+1, y3n+1) ≤ kGk(y3n−1, y3n, y3n+1), Gk(y3n, y3n, y3n−1) ≤ kGk(y3n−1, y3n, y3n+1), Gk(y3n+1, y3n+1, y3n) ≤ kGk(y3n, y3n+1, y3n+2), Gk(y3n+2, y3n+2, y3n+1) ≤ kGk(y3n, y3n+1, y3n+2), Gk(y3n, y3n−1, y3n−1) ≤ kGk(y3n−1, y3n, y3n+1), Gk(y3n+1, y3n, y3n) ≤ kGk(y3n−1, y3n, y3n+1), Gk(y3n+2, y3n+1, y3n+1) ≤ kGk(y3n, y3n+1, y3n+2). (8) By (7) and (8), for all n ∈ N Gk(y3n, y3n+1, y3n+2) ≤ αGk(y3n−1, y3n, y3n+1), (9) where α = kλ 1−kλ ∈ [0, 1). We can also prove that Gk(y3n+1, y3n+2, y3n+3) ≤ αGk(y3n, y3n+1, y3n+2) (10) and Gk(y3n+2, y3n+3, y3n+4) ≤ αGk(y3n+1, y3n+2, y3n+3). (11) (9), (10), and (11) imply that, for all n ∈ N, Gk(yn, yn+1, yn+2) ≤ αGk(yn−1, yn, yn+1). (12) From the inequality in (12), we infer that for all n ∈ N, Gk(yn, yn+1, yn+2) ≤ αGk(yn−1, yn, yn+1) ≤ · · · ≤ αnGk(y0, y1, y2). (13) Hence for each n,m, l ∈ N with l > m > n, by (GBC5), (GBC3), and (GBC4), we have G(yn, ym, yl) ⪯ sG(yn, yn+1, yn+1) + s2G(yn+1, yn+2, yn+2) + · · ·+ sl−n G(yl−1, yl, yl) ⪯ sG(yn, yn+1, yn+2) + s2G(yn+1, yn+2, yn+3) + · · ·+ sl−n G(yl−1, yl, yl+1). By the normality of the cone, Equation (13), and since ∥.∥ satisfies the triangle inequality, we find Gk(yn, ym, yl) ≤ k(sGk(yn, yn+1, yn+2) + s2Gk(yn+1, yn+2, yn+3) + · · ·+ sl−n Gk(yl−1, yl, yl+1) ≤ k(sαnGk(y0, y1, y2) + s2αn+1Gk(y0, y1, y2) + · · ·+ sl−nαl−1 Gk(y0, y1, y2) = ksαnGk(y0, y1, y2)(1 + sα+ · · ·+ (sα)l−n−1) ≤ ksαn 1−sαGk(y0, y1, y2) → 0, as n → +∞, S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1362 which implies that G(yn, ym, yl) → θ, as n,m, l → +∞. Proposition 3 implies that (yn) is a Gb-Cauchy sequence in f(X). Since f(X) is Gb-complete subspace of X, there exists a point v in f(X) such that f(xn) → v as n → +∞. Consequently, there is some u ∈ X such that fu = v. Since the sequences (fx3n+1) = (Fx3n), (fx3n+2) = (Tx3n+1), and (fx3n+3) = (Rx3n+2) are subsequences of (yn), they converge to the same limit v. We show that fu = Fu. By normality of the cone, Equation (6), and since ∥.∥ satisfies the triangle inequality, for all n ∈ N, we have Gk(Fu, y3n+1, y3n+2) = Gk(Fu, Tx3n+1, Rx3n+2) ≤ λ sMk(u, x3n+1, x3n+2) + L sNk(u, x3n+1, x3n+2), (14) where Mk(u, x3n+1, x3n+2) = max{Gk(fu, fx3n+1, fx3n+2), Gk(Fu, fx3n+1, fx3n+2), Gk(fu, Tx3n+1, fx3n+2), Gk(fu, fx3n+1, Rx3n+2), Gk(fu, Tx3n+1, Rx3n+2), Gk(Fu, fx3n+1, Rx3n+2), Gk(Fu, Tx3n+1, fx3n+2), Gk(Fu, Fu, fu), Gk(Tx3n+1, Tx3n+1, fx3n+1), Gk(Rx3n+2, Rx3n+2, fx3n+2), Gk(Fu, fu, fu), Gk(Tx3n+1, fx3n+1, fx3n+1), Gk(Rx3n+2, fx3n+2, fx3n+2)} = max{Gk(fu, y3n, y3n+1), Gk(Fu, y3n, y3n+1), Gk(fu, y3n+1, y3n+1), Gk(fu, y3n, y3n+2), Gk(fu, y3n+1, y3n+2), Gk(Fu, y3n, y3n+2), Gk(Fu, y3n+1, y3n+1), Gk(Fu, Fu, fu), Gk(y3n+1, y3n+1, y3n), Gk(y3n+2, y3n+2, y3n+1), Gk(Fu, fu, fu), Gk(y3n+1, y3n, y3n), Gk(y3n+2, y3n+1, y3n+1)} and Nk(u, x3n+1, x3n+2) = min{Gk(Fu, fx3n+1, fx3n+1), Gk(Fu, fx3n+2, fx3n+2), Gk(Tx3n+1, fx3n+2, fx3n+2), Gk(Tx3n+1, fu, fu), Gk(Rx3n+2, fu, fu), Gk(Rx3n+2, fx3n+1, fx3n+1)} = min{Gk(Fu, y3n, y3n), Gk(Fu, y3n+1, y3n+1), Gk(y3n+1, y3n+1, y3n+1), Gk(y3n+1, fu, fu), Gk(y3n+2, fu, fu), Gk(y3n+2, y3n, y3n)} = 0. As a consequence, lim n→+∞ Mk(u, x3n+1, x3n+2) = max{0, sGk(Fu, fu, fu), 0, 0, 0, sGk(Fu, fu, fu), sGk(Fu, fu, fu), Gk(Fu, Fu, fu), 0, 0, Gk(Fu, fu, fu), 0, 0} = max{sGk(Fu, fu, fu), Gk(Fu, Fu, fu)} ≤ max{sGk(Fu, fu, fu), 2skGk(Fu, fu, fu)} = 2skGk(Fu, fu, fu). As n → +∞ in (14), we find sGk(Fu, fu, fu) ≤ 2kλGk(Fu, fu, fu). S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1363 Then Gk(Fu, fu, fu) = 0 which implies that G(Fu, fu, fu) = θ, that is Fu = fu. In the same way, we can prove that Tu = fu and Ru = fu. Then, fu = Fu = Tu = Ru = v, and so v is a point of coincidence of f , F , T and R. To show that f , F , T and R have a unique point of coincidence in X, assume that there exists a second coincidence point v⋆ ∈ X such that fu⋆ = Fu⋆ = Tu⋆ = Ru⋆ = v⋆, for some u⋆ ∈ X. Condition (6) yields Gk(v ⋆, v, v) = Gk(Fu⋆, Tu,Ru) ≤ λ sMk(u ⋆u, u) + L sNk(u ⋆u, u), where Mk(u ⋆, u, u) = max{Gk(fu ⋆, fu, fu), Gk(Fu⋆, fu, fu), Gk(fu ⋆, Tu, fu), Gk(fu ⋆, fu,Ru), Gk(fu ⋆, Tu,Ru), Gk(Fu⋆, fu,Ru), Gk(Fu⋆, Tu, fu), Gk(Fu⋆, Fu⋆, fu⋆), Gk(Tu, Tu, fu), Gk(Ru,Ru, fu), Gk(Fu⋆, fu⋆, fu⋆), Gk(Tu, fu, fu), Gk(Ru, fu, fu)} = Gk(v ⋆, v, v) and Nk(u ⋆, u, u) = min{Gk(Fu⋆, fu, fu), Gk(Fu⋆, fu, fu), Gk(Tu, fu, fu), Gk(Tu, fu ⋆, fu⋆), Gk(Ru, fu⋆, fu⋆), Gk(Ru, fu, fu)} = 0. Therefore, Gk(v ⋆, v, v) ≤ λ s Gk(v ⋆, v, v). So Gk(v ⋆, v, v) = 0, which implies that G(v⋆, v, v) = θ, that is v⋆ = v. Finally, since the pairs (f, F ), (f, T ), and (f,R) are weakly compatible, we obtain Ffu = fFu, Tfu = fTu, Rfu = fRu. This implies that Fv = Tv = Rv = fv = t, i.e., t is a point of coincidence of f , F , T , and R and t = v by uniqueness. Making use of Proposition 4, we conclude that v is the unique common fixed point of f , F , T , and R. Example 2. Let X = [−1, 1], E = R2 and P = {(x, y) ∈ R2 : x, y ≥ 0}. Define G : X ×X ×X → E by G(x, y, z) = ( 1 3 max{|x− y|2, |y − z|2, |x− z|2}, 2 3 max{|x− y|2, |y − z|2, |x− z|2} ) . S. Benchabane, S. Djebali / Eur. J. Pure Appl. Math, 14 (4) (2021), 1350-1366 1364 Then G is a Gb-cone metric on X with the coefficient s = 2. Let Gk(x, y, z) = ∥G(x, y, z)∥1 = max{|x− y|2, |y − z|2, |x− z|2}. Define the self-maps f, F, T , and R on X by F (x) = { −1 4 , x ∈ [−1, 23), −1 6 , x ∈ [23 , 1], T (x) = { −1 4 , x ∈ [−1, 23), −1 7 , x ∈ [23 , 1], R(x) = { −1 4 , x ∈ [−1, 23), −1 5 , x ∈ [23 , 1], f(x) = { x, x ∈ [−1, 23), 2 3 , x ∈ [23 , 1]. Note that F, T,R, and f satisfy 2Gk(Fx, Ty,Rz) ≤ 1 5 Mk(x, y, z) + LNk(x, y, z), for all x, y, z ∈ X and L ≥ 0. In addition f(X) = [−1, 23 ] is a Gb-complete subspace of X, F (X) ∪ T (X) ∪ R(X) ⊂ f(X), and the pairs (f, F ), (f, T ), and (f,R) are weakly compatible. All the conditions of Theorem 2 are thus verified. As a consequence −1 4 is the unique point of coincidence and the unique common fixed point for all of the mappings F, T,R, and f . The following result derives from Theorem 2. Corollary 2. Let (X,G) be a cone G-metric space relative to a normal constant k ≥ 1. Suppose that the mappings F, T,R, f : X → X satisfy Gk(Fx, Ty,Rz) ≤ λMk(x, y, z) + LNk(x, y, z), for all x, y, z ∈ X, λ ∈ [0, 1 2k ) and L ≥ 0. If F (X) ∪ T (X) ∪ R(X) ⊂ f(X) and f(X) is a G-complete subspace of X, then F, T,R, and f have a unique point of coincidence in X. If further the pairs (f, F ), (f, T ) and (f,R) are weakly compatible, then F, T,R, and f have a unique common fixed point. Remark 1. (1) Let F = R = S and f = I in Theorem 1. Then Theorem 1 improves [3, Theorem 3.2]. (2) Corollary 1 with F = R = S extends [9, Theorem 2.1]. REFERENCES 1365 (3) If in Corollary 1, we set F = R = S and f = I, then we obtain an extension of [11, Theorem 2.7]. 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