EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 1, 2018, 90-109 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Some common fixed points of six mappings on Gb- metric spaces using (E.A) property Z. Mustafa1,∗, M.M.M. Jaradat2, H. Aydi3,4, A. Alrhayyel5 1 Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar Department of Mathematics, The Hashemite University, Zarqa- Jordan 2 Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar 3 Imam Abdulrahman Bin Faisal University, Department of Mathematics, College of Education of Jubail, P.O: 12020, Industrial Jubail 31961, Saudi Arabia 4 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan 5 Department of Mathematics. Faculty of Science, Yarmouk University, Irbid, Jordan Abstract. The aim of this manuscript is to present a unique common fixed point theorem for six mappings satisfying (φ, ψ)-contractions using (E.A) property in the framework of Gb- metric spaces. An illustrative example is also given to justify the established result. 2010 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Complete Gb-metric space, Cauchy sequence, (φ, ψ)-contraction, (E.A) property, common fixed point, weakly compatible. 1. Introduction In recent years, many authors studied common fixed points of mappings having dif- ferent contractive conditions. This area has variety of important applications in applied mathematics and sciences. In 1976, Jungck [17] proved a common fixed point theorem for commuting maps under the assumption that one of maps must be continuous. In 1982, the concept of weak commutativity for a pair of self maps was introduced by Sessa [47]. He also proved that weakly commuting pairs of maps in a metric space are commuting, but the converse need not be true. Later, Jungck [18] introduced the notion of compatible mappings in order to generalize the concepts of weak commutativity and showed that weak commuting maps are compatible, but the reverse implication may not hold. ∗Corresponding author. Email addresses: zead@qu.edu.qa, zmagablh@hu.edu.jo (Z. Mustafa), mmjst4@qu.edu.qa (M.M.M. Jaradat), hmaydi@iau.edu.sa, hassen.aydi@isima.rnu.tn (H. Aydi), Al-Rhayyel@yu.edu.jo (A. Alrhayyel) http://www.ejpam.com 90 c© 2018 EJPAM All rights reserved. Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 91 In 1996, Jungck [20] defined a pair of self mappings to be weakly compatible if they commute at their coincidence points. Therefore, we have one way implication namely, Commuting maps⇒Weakly Commut- ing maps ⇒ Compatible maps ⇒ Weakly Compatible maps. Recently, various authors have introduced a coincidence points results for various classes of mappings on metric spaces. For more details on coincidence point theory and related results, see [19, 21, 43]. However, the study of common fixed points of non-compatible mappings has recently been initiated by Pant [44]. In 2002, Amari and El Moutawakil [1] defined a new property called (E.A) property which generalizes the concept of non-compatible mappings and they proved some common fixed point theorem. Yan et al. [48] gave the idea of (φ, ψ)-contractions and proved a fixed point theorem of a contraction mapping in a complete metric space endowed with a partial order by using altering distance functions [22]. Different authors used (φ, ψ)-contractions to obtain common fixed point results in different spaces. Some of the works on (φ, ψ)-contractions are given in [4, 5, 8, 10, 26, 27, 42, 23, 41]. Mustafa and Sims [28] introduced a new generalizations of a metric space by assigning to every (x, y, z) ∈ X ×X ×X a real number and is named as a G-metric space. In 2008, Mustafa et al. [29] obtained some fixed point results in G-metric spaces for mappings satis- fying different contractive conditions. After that several fixed point results were obtained. Among these works, we mention ([6],[7],[11],[14],[15], [16],[24]-[40]). In 2014, Aghajani et al. [2] introduced a new generalization of a metric space. They combined the definition of a G-metric and a b-metric and generated a new definition called a Gb-metric space. They also pointed out that the class of Gb-metric spaces is effectively larger than that of G-metric spaces. Note that a G-metric space becomes a particular case of a Gb-metric space when s = 1. Further, they showed that every Gb-metric space is equivalent to a b-metric space topologically. In the current work, we will obtain a unique common fixed point result in Gb- metric spaces involving (φ, ψ)-contractions and using the (E.A) property. Also, an example to illustrate the main result is given. 2. Preliminaries First, we present some definitions from the literature. Definition 1. ([13]) Let X be a nonempty set and s ≥ 1 be a given real number. A function d : X × X → [0,∞) is called a b-metric provided that, for all a, b, c ∈ X, the following conditions are satisfied: (B1) d(a, b) = 0 if and only if a = b; (B2) d(a, b) = d(b, a); (B3) d(a, c) ≤ s[d(a, b) + d(b, c)]. The pair (X, d) is called a b-metric space with parameter s. The following definition was given by Mustafa and Sims [28] Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 92 Definition 2. ([28]) Let X be a nonempty set and G : X ×X ×X → [0,∞) satisfies: (G1) G(a, b, c) = 0 if a = b = c; (G2) G(a, a, b) > 0 for all a, b ∈ X with a 6= b; (G3) G(a, b, b) ≤ G(a, b, c) for all a, b, c ∈ X with a 6= c; (G4) G(a, b, c) = G(b, c, a) = G(c, a, b) = · · · (symmetry in a, b, c); (G5) G(a, b, c) ≤ G(a, d, d) +G(d, b, c) for all a, b, c, d ∈ X. Then function G is called a G-metric on X, and the pair (X,G) is called a G-metric space. As a combination of the two above definitions, Aghajani et al. [2] (see also [3]) introduced the following. Definition 3. ([2]) Let X be a nonempty set and s ≥ 1 be a given real number. Suppose that a mapping Gb : X ×X ×X → [0,∞) satisfies: (Gb1) Gb(x, y, z) = 0 if x = y = z; (Gb2) Gb(x, x, y) > 0 for all x, y ∈ X with x 6= y; (Gb3) Gb(x, y, y) ≤ Gb(x, y, z) for all x, y, z ∈ X with x 6= z; (Gb4) Gb(x, y, z) = Gb(p{x, y, z}) where p is a permutation of x, y, z (symmetry); (Gb5) Gb(x, y, z) ≤ s(Gb(x, a, a) +Gb(a, y, z)) for all x, y, z, a ∈ X. Then Gb is called a generalized b-metric ( named as a Gb-metric) on X, and the pair (X,Gb) is called a Gb-metric space. Note that every G-metric space is a Gb-metric space, but the converse need not to be true as its clear from the following example. Example 1. ([46]) Let X={1, 2, 3, 4}. Define Gb : X ×X ×X → [0,∞) by Gb(1, 1, 1) = Gb(2, 2, 2) = Gb(3, 3, 3) = Gb(4, 4, 4) = 0, Gb(1, 1, 2) = Gb(1, 2, 2) = Gb(1, 1, 3) = Gb(1, 3, 3) = Gb(1, 1, 4) = Gb(1, 4, 4) = 1, Gb(2, 2, 3) = Gb(2, 3, 3) = Gb(2, 4, 4) = Gb(2, 2, 4) = 2, Gb(3, 4, 4) = Gb(3, 3, 4) = 3, Gb(1, 2, 3) = 4, Gb(1, 3, 4) = 5, Gb(1, 2, 4) = 6, Gb(2, 3, 4) = 7. Evidently, the above is a Gb-metric on X with s = 7 5 , but not a G-metric. In fact, the rectangle inequality is violated, for instant 7 = Gb(2, 3, 4) � Gb(2, 1, 1)+Gb(1, 3, 4) = 1+5. The following example can be founded in [45]. Example 2. Let (X,G) be a G-metric space. Take Gb(x, y, z) = Gp(x, y, z), where p > 1 is a real number. Note that Gb is a Gb-metric with s = 2p−1. In general (X,Gb) is not necessary a G-metric space. For instant, let X = R and the G-metric be defined by G(x, y, z) = 1 3(|x−y|+ |y−z|+ |x−z|) for all x, y, z ∈ R. Then Gb(x, y, z) = G2(x, y, z) = 1 9(|x − y| + |y − z| + |x − z|)2 is a Gb-metric on R with s = 22−1 = 2, but it is not a G-metric on R. Example 3. ([2]) Let X = R. Take the Gb-metric defined by Gb(x, y, z) = max { |x− y|2, |y − z|2, |z − x|2 } , ∀ x, y, z ∈ X. Then (X,Gb) is a complete Gb-metric space with s = 2, but not a G-metric. Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 93 Proposition 1. ([2]) Let X be a Gb-metric space. Then for each x, y, z, a ∈ X, it follows that (1) If Gb(x, y, z) = 0, then x = y = z, (2) Gb(x, y, z) ≤ s(Gb(x, x, y) +Gb(x, x, z)), (3) Gb(x, y, y) ≤ 2sGb(y, x, x), (4) Gb(x, y, z) ≤ s(Gb(x, a, z) +Gb(a, y, z)). Definition 4. ([2]) Let X be a Gb-metric space. A sequence {xn} in X is said to be: (1) Gb-Cauchy sequence if for each ε > 0, there exists a positive integer n0 such that for all m,n, l ≥ n0, Gb(xn, xm, xl) < ε; (2) Gb-convergent to a point x ∈ X if for each ε > 0, there exists a positive integer n0 such that, for all m,n ≥ n0, Gb(xn, xm, x) < ε. Proposition 2. ([2, 9]) Let X be a Gb-metric space. The following are equivalent: (1) {xn} is Gb-convergent to x; (2) Gb(xn, xn, x)→ 0 as n→∞; (3) Gb(xn, x, x)→ 0 as n→∞. Definition 5. ([2]) A Gb-metric space X is called Gb-complete if every Gb-Cauchy se- quence is Gb-convergent in X. The following definition was given by Jungck [19]. Definition 6. ([19]) Two maps f and g are said to be weakly compatible if they commute at their coincidence points, that is if f(x) = g(x) for some x ∈ X, then f(g(x)) = g(f(x)). The following definition was introduced by Amari and El Moutawakil [1] in 2002. Definition 7. ([1]) Two self mappings S and T of a metric space (X, d) are said to satisfy an (E.A) property if there exists a sequence {xn} in X such that lim n→∞ Sxn = lim n→∞ Txn = r for some r ∈ X. This concept was extended to G-metric spaces in [24]. The following lemma is useful in the proof of our main result. Lemma 1. ([45]) Let (X,Gb) be a Gb-metric space with s > 1. Suppose that {xn}, {yn} and {zn} are Gb-convergent sequences to x, y and z, respectively. Then we have (i) 1 s3 Gb(x, y, z) ≤ lim inf n→∞ Gb(xn, yn, zn) ≤ lim sup n→∞ Gb(xn, yn, zn) ≤ s3Gb(x, y, z). (ii) If {zn} = c is constant, then 1 s2 Gb(x, y, c) ≤ lim inf n→∞ Gb(xn, yn, c) ≤ lim sup n→∞ Gb(xn, ync) ≤ s2Gb(x, y, c). Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 94 (iii) If {zn} = c and {yn} = b are constant, then 1 s Gb(x, b, c) ≤ lim inf n→∞ Gb(xn, b, c) ≤ lim sup n→∞ Gb(xn, b, c) ≤ sGb(x, b, c). In particular, if x = y = z, then we have limn→∞Gb(xn, yn, zn) = 0. 3. Main results We start this section with the following definition and lemma which will play a major role in our main result. Lemma 2. Let (X,Gb) be a Gb-metric space with s > 1. Suppose that {xn} is a Gb- convergent sequence to x. Then for y ∈ X we have 1 s Gb(y, x, x) ≤ lim inf n→∞ Gb(y, xn, xn) ≤ lim sup n→∞ Gb(y, xn, xn) ≤ sGb(y, x, x). Proof. Using the rectangle inequality for the Gb-metric, we obtain that Gb(y, x, x) ≤ s[Gb(y, xn, xn) +Gb(xn, x, x)] (1) and Gb(y, xn, xn) ≤ s[Gb(y, x, x) +Gb(x, xn, xn)]. (2) Taking the limit inferior as n → ∞ in (1) and the limit superior as n → ∞ in (2), the proof is completed. Definition 8. A mapping ψ : [0,∞)→ [0,∞) is called a super-altering distance function if the following properties are satisfied: 1. ψ is continuous and increasing. 2. ψ(t) = 0 if and only if t = 0. We denoted by Ψ to be the set of all super-altering distance functions. Note that the class of altering distance functions was defined in [22], where ψ is considered non- decreasing (not necessarily increasing). Any super-altering distance function is of course a function in the sense of [22]. In the following example, the given mapping is just an altering distance function, but not in Ψ. Example 4. Let ψ : [0,∞)→ [0,∞) be such that{ ψ(t) = t if t ∈ [0, 1] ψ(t) = 1 if t ≥ 1. Theorem 1. Let (X,Gb) be a complete Gb-metric space and let f, g, h,R, S, T : X → X be self mappings such that Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 95 (i) (f, S) and (g,R) satisfy the (E.A) property; (ii) f(X) ⊆ T (X), g(X) ⊆ S(X) and h(X) ⊆ R(X); (iii) R(X) is a closed subspace of X; (iv) (f, S), (g,R) and (h, T ) are weakly compatible pairs of mappings; (v) ψ ( s2Gb(fx, gy, hz) ) ≤ ψ ( M(x, y, z) ) − φ ( M(x, y, z) ) ,∀x, y, z ∈ X (3) where ψ, φ ∈ Ψ and M(x, y, z) = max { Gb(fx, Sx, Tz), Gb(gy,Ry,Ry), Gb(fx, fx, hz), Gb(Tz, Tz, hz) +Gb(fx, Sx, Sx) 2s } . Then f, g, h,R, S and T have a unique common fixed point in X. Proof. Since the pair (f, S) satisfies the (E.A) property, there exists a sequence {xn} such that lim n→∞ fxn = lim n→∞ Sxn = q1, for some q1 ∈ X. As f(X) ⊆ T (X), there exists a sequence {zn} ∈ X such that fxn = Tzn and lim n→∞ fxn = lim n→∞ Tzn = lim n→∞ Sxn = q1. (4) Again the pair (g,R) satisfies the (E.A) property, so there exists a sequence {yn} such that lim n→∞ gyn = lim n→∞ Ryn = q2, for some q2 ∈ X. (5) But g(X) ⊆ S(X), so there exists a sequence {αn} ∈ X such that gyn = Sαn, and lim n→∞ gyn = lim n→∞ Sαn = lim n→∞ Ryn = q2. (6) Now, we shall show that lim n→∞ hzn = q1. From (3), (Gb3) and the fact that ψ is an increasing mapping, we have ψ ( sGb(fxn, fxn, hzn) ) ≤ ψ ( s2Gb(fxn, gyn, hzn) ) ≤ ψ ( M(xn, yn, zn) ) − φ ( M(xn, yn, zn) ) (7) where, M(xn, yn, zn) = max { Gb(fxn, Sxn, T zn), Gb(gyn, Ryn, Ryn), Gb(fxn, fxn, hzn), Gb(Tzn, T zn, hzn) +Gb(fxn, Sxn, Sxn) 2s } , Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 96 = max { Gb(fxn, Sxn, fxn), Gb(gyn, Ryn, Ryn), Gb(fxn, fxn, hzn), Gb(fxn, fxn, hzn) +Gb(fxn, Sxn, Sxn) 2s } . Taking lim supn→∞ and using (4) together with (6), we obtain lim sup n→∞ M(xn, yn, zn) = lim sup n→∞ Gb(fxn, fxn, hzn). (8) Taking again lim supn→∞ in (7) and substituting (8), we get ψ ( lim sup n→∞ sGb(fxn, fxn, hzn) ) ≤ ψ ( lim sup n→∞ s2Gb(fxn, gyn, hzn) ) ≤ ψ ( lim sup n→∞ Gb(fxn, fxn, hzn) ) − φ ( lim inf n→∞ M(xn, yn, zn) ) . ≤ ψ ( lim sup n→∞ Gb(fxn, fxn, hzn) ) . (9) Since s > 1 and being ψ is an increasing mapping, we deduce from (9) that lim sup n→∞ Gb(fxn, fxn, hzn) = 0, which implies that lim n→∞ Gb(fxn, fxn, hzn) = 0, (10) and so by (8), we conclude that lim n→∞ M(xn, yn, zn) = 0. (11) Now, by (Gb4), (10) and (4), we have Gb(hzn, q1, q1) ≤ s [ Gb(hzn, fxn, fxn) +Gb(fxn, q1, q1) ] → 0 as n→∞. (12) Thus, lim n→∞ Gb(hzn, q1, q1) = 0 which gives that limhzn = q1 as n → ∞. Now, we shall prove that q1 = q2. By applying (3) and using (Gb3), we find that ψ ( sGb(fxn, gyn, gyn) ) ≤ ψ ( s2Gb(fxn, gyn, hzn) ) ≤ ψ ( M(xn, yn, zn) ) − φ ( M(xn, yn, zn) ) . (13) Taking the limit as n→∞ in (13) and recalling (11), we obtain lim n→∞ Gb(fxn, gyn, gyn) = 0. (14) Thus, by using (Gb4), (4) and (14), Gb(q1, Sαn, Sαn) = Gb(q1, gyn, gyn) Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 97 ≤ s [ Gb(q1, fxn, fxn) +Gb(fxn, gyn, gyn) ] → 0 as n→∞. This implies that limn→∞ Sαn = q1. On the other hand, from (6) we have lim n→∞ Sαn = q2, hence by uniqueness of limits, we obtain that q1 = q2. Therefore lim n→∞ fxn = lim n→∞ hzn = lim n→∞ Tzn = lim n→∞ Sxn = lim n→∞ gyn = lim n→∞ Sαn = lim n→∞ Ryn = q (15) for some q ∈ X. Since R(X) is a closed subspace of X, there exists u ∈ X such that Ru = q. Now we shall prove that gu = q. Observe that M(xn, u, zn) = max { Gb(fxn, Sxn, T zn), Gb(gu,Ru,Ru), Gb(fxn, fxn, hzn), Gb(Tzn, T zn, hzn) +Gb(fxn, Sxn, Sxn) 2s } , = max { Gb(fxn, Sxn, T zn), Gb(gu,Ru,Ru), Gb(fxn, fxn, hzn), Gb(fxn, fxn, hzn) +Gb(fxn, Sxn, Sxn) 2s } , = max { Gb(fxn, Sxn, T zn), Gb(gu, q, q), Gb(fxn, fxn, hzn), Gb(fxn, fxn, hzn) +Gb(fxn, Sxn, Sxn) 2s } . (16) By taking limit superior as n → ∞ and taking into account (4), (6) and (15), then (16) becomes lim sup n→∞ M(xn, u, zn) = Gb(gu, q, q). (17) By the help of Lemma 2, we obtain that 1 s Gb(q, gu, q) ≤ lim inf n→∞ Gb(gu, fxn, fxn) ≤ lim sup n→∞ Gb(gu, fxn, fxn) ≤ sGb(gu, q, q). (18) Also from (Gb3), we have Gb(gu, fxn, fxn) ≤ Gb(fxn, gu, hzn). (19) Thus, from (3), together with (17), (18), (19) and properties of ψ, we get that ψ ( sGb(q, gu, q) ) ≤ ψ ( lim sup n→∞ s2Gb(gu, fxn, fxn) ) ≤ ψ ( lim sup n→∞ s2Gb(fxn, gu, hzn) ) = lim sup n→∞ ψ ( s2Gb(fxn, gu, hzn) ) ≤ lim sup n→∞ ψ ( M(xn, u, zn) ) − lim inf n→∞ φ ( M(xn, u, zn) ) , Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 98 = ψ ( lim sup n→∞ M(xn, u, zn) ) − φ ( lim inf n→∞ M(xn, u, zn) ) , ≤ ψ ( Gb(q, gu, q) ) − φ ( lim inf n→∞ M(xn, u, zn) ) , ≤ ψ(Gb(q, gu, q)). (20) Since s > 1 and ψ is increasing, the above inequality gives that Gb(q, gu, q) = 0, which implies that gu = q. But g(X) ⊆ S(X), so there exists a point p ∈ X such that gu = Sp = q. We shall show that fp = q. Now M(p, u, zn) = max { Gb(fp, Sp, Tzn), Gb(gu,Ru,Ru), Gb(fp, fp, hzn), Gb(Tzn, T zn, hzn) +Gb(fp, Sp, Sp) 2s } , = max { Gb(fp, q, Tzn), Gb(q, q, q), Gb(fp, fp, hzn), Gb(Tzn, T zn, hzn) +Gb(fp, q, q) 2s } = max { Gb(fp, q, Tzn), Gb(fp, fp, hzn), Gb(Tzn, T zn, hzn) +Gb(fp, q, q) 2s } (21) ≤ max { Gb(fp, q, Tzn), Gb(fp, Tzn, hzn), Gb(fp, Tzn, hzn) +Gb(fp, q, Tzn) 2s } , ≤ max { Gb(fp, q, Tzn), Gb(fp, Tzn, hzn) } . (22) Now, taking the limit superior in (22) as n → ∞ and using Lemma 1, parts (2) and (3), we obtain lim sup n→∞ M(p, u, zn) = lim sup n→∞ max { Gb(fp, q, Tzn), Gb(fp, Tzn, hzn) } = max { lim sup n→∞ Gb(fp, q, Tzn), lim sup n→∞ Gb(fp, Tzn, hzn) } ≤ max{sGb(fp, q, q), s2Gb(fp, q, q)} = s2Gb(fp, q, q). (23) Now, taking the limit infimum in (21) as n → ∞ and using Lemma 1, parts (2) and (3), we get lim inf n→∞ M(p, u, zn) = lim inf n→∞ max { Gb(fp, q, Tzn), Gb(fp, fp, hzn), Gb(Tzn, T zn, hzn) +Gb(fp, q, q) 2s } = max { lim inf n→∞ Gb(fp, q, Tzn), lim inf n→∞ Gb(fp, fp, hzn), lim infn→∞Gb(Tzn, T zn, hzn) + lim infn→∞Gb(fp, q, q) 2s } Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 99 ≥ max{1 s Gb(fp, q, q), 1 s Gb(fp, fp, q), Gb(fp, q, q) 2s } = max{1 s Gb(fp, q, q), 1 s Gb(fp, fp, q)}. (24) Thus, from (3), (Gb3) and the fact that ψ and φ are increasing, we have ψ ( s2Gb(fp, q, q) ) ≤ ψ ( s2Gb(fp, q, hzn) ) = ψ ( s2Gb(fp, gu, hzn) ) ≤ ψ ( M(p, u, zn) ) − φ ( M(p, u, zn) ) . (25) Therefore, by taking the limit superior in (25) as n→∞ and using (23) and (24), ψ ( s2Gb(fp, q, q) ) ≤ ψ ( lim sup n→∞ M(p, u, zn) ) − φ ( lim inf n→∞ M(p, u, zn) ) , ≤ ψ ( s2Gb(fp, q, q) ) − φ ( max{1 s Gb(fp, q, q), 1 s Gb(fp, fp, q)} ) , . (26) So, φ ( max{1 s Gb(fp, q, q), 1 s Gb(fp, fp, q)} ) = 0, or equivalently, max{1 s Gb(fp, q, q), 1 s Gb(fp, fp, q)} = 0, which implies that Gb(fp, q, q) = Gb(fp, fp, q) = 0. Hence fp = Sp = q. We conclude that p is a coincidence point of f and S. Also fp = Sp = gu = Ru = q. (27) Again, since h(X) ⊆ R(X), there exists w ∈ X such that hw = Ru = q. Now, we shall show that Tw = hw. From the definition of M(x, y, z) and by the help of (27), we get M(p, u, w) = max { Gb(fp, Sp, Tw), Gb(gu,Ru,Ru), Gb(fp, fp, hw), Gb(Tw, Tw, hw) +Gb(fp, Sp, Sp) 2s } , = max { Gb(q, q, Tw), Gb(q, q, q), Gb(q, q, q), Gb(Tw, Tw, q) +Gb(q, q, q) 2s } , = max { Gb(q, q, Tw), Gb(Tw, Tw, q) 2s } . But, by part 3 of Proposition 1, we have Gb(Tw,Tw,q) 2s ≤ Gb(q, q, Tw) and so the above inequality becomes M(p, u, w) = Gb(q, q, Tw). (28) Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 100 Thus, applying (3) for x = q, y = q and z = Tw and using (Gb3), (28) and properties of ψ, we obtain ψ ( Gb(q, q, Tw) ) ≤ ψ ( s2Gb(q, q, Tw) ) = ψ ( s2Gb(fp, gu, Tw) ) ≤ ψ ( M(p, u, w) ) − φ ( M(p, u, w) ) , = ψ ( Gb(q, q, Tw) ) − φ ( Gb(q, q, Tw) ) . (29) So, φ ( Gb(q, q, Tw) ) = 0, which implies that Gb(q, q, Tw) = 0. Hence Tw = q = hw and so w is a coincidence point of h and T . Therefore fp = Sp = gu = Ru = Tw = hw = q. (30) Now, we shall show that q is a common fixed point of f, g, h,R, S and T . Since the pairs (f, S), (g,R) and (h, T ) are weakly compatible, the functions of each pair commute at their coincidence point, that is f(q) = f(Sp) = S(fp) = S(q), R(q) = R(gu) = g(Ru) = g(q), T (q) = T (hw) = h(Tw) = h(q).  (31) Using (30) and (31), we obtain M(q, u, w) = max { Gb(fq, Sq, Tw), Gb(gu,Ru,Ru), Gb(fq, fq, hw), Gb(Tw, Tw, hw) +Gb(fq, Sq, Sq) 2s } , = max { Gb(fq, Sq, q), 0, Gb(fq, fq, q), 0 } , = Gb(fq, fq, q). Also, from (3) and (Gb3), we get ψ ( s2Gb(fq, fq, q) ) ≤ ψ ( s2Gb(fq, gu, q) ) = ψ ( s2Gb(fq, gu, hw) ) ≤ ψ ( M(q, u, w) ) − φ ( M(q, u, w) ) , = ψ ( Gb(fq, fq, q)) ) − φ ( Gb(fq, fq, q) ) , ≤ ψ ( Gb(fq, fq, q) ) . (32) Since s2 > s > 1 and ψ is increasing, the inequality above yields that Gb(fq, fq, q) = 0 and so fq = q = Sq. We shall prove that gq = Rq = q. As in the above, using (30) and (31), we find that M(p, q, w) = max { Gb(fp, Sp, Tw), Gb(gq,Rq,Rq), Gb(fp, fp, hw), Gb(Tw, Tw, hw) +Gb(fp, Sp, Sp) 2s } , Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 101 = max { Gb(q, q, q), Gb(Rq,Rq,Rq), Gb(q, q, q), Gb(q, q, q) +Gb(q, q, q) 2s } , = 0. Applying (3), ψ ( s2Gb(fp, gq, hw) ) ≤ ψ ( M(p, q, w) ) − φ ( M(p, q, w) ) , = ψ ( 0 ) − φ ( 0 ) = 0. (33) Consequently, Gb(fp, gq, hw) = Gb(q, gq, q) = 0 and so gq = q. Hence gq = Rq = q. Now we shall prove that hq = Tq = q. Similarly, using (30) and (31), we obtain that M(p, u, q) = max { Gb(fp, Sp, Tq), Gb(gu,Ru,Ru), Gb(fp, fp, hq), Gb(Tq, Tq, hq) +Gb(fp, Sp, Sp) 2s } , = max { Gb(q, q, T q), Gb(q, q, q), Gb(q, q, hq), Gb(Tq, Tq, T q) +Gb(q, q, q) 2s } , = max{Gb(q, q, T q), 0, Gb(q, q, T q), 0} = Gb(q, q, T q). By specifying x = z = q and y = u in (3) and using (27), ψ ( s2Gb(q, q, T q) ) = ψ ( s2Gb(fq, gu, hq) ) ≤ ψ ( M(p, u, q) ) − φ ( M(p, u, q) ) , = ψ ( Gb(q, q, T q)) ) − φ ( Gb(q, q, T q)) ) , ≤ ψ ( Gb(q, q, T q)) ) . (34) Again, s2 > s > 1 and ψ is increasing, so Gb(q, q, T q) = 0, that is, Tq = q = Rq. Thus fq = Sq = gq = Rq = hq = Tq = q. Then q is a common fixed point of f, g, h,R, S and T . Now, we shall prove that the obtained fixed point is unique. Suppose that v is another common fixed point of f, g, h,R, S and T , that is fv = gv = hv = Rv = Sv = Tv = v. Then M(q, q, v) = max { Gb(fq, Sq, Tv), Gb(gq,Rq,Rq), Gb(fq, fq, hv), Gb(Tv, Tv, hv) +Gb(fq, Sq, Sq) 2s } , = max { Gb(q, q, v), 0, Gb(q, q, v), 0 } , = Gb(q, q, v). From (3) we have that ψ ( s2Gb(q, q, v) ) = ψ ( s2Gb(fq, gq, hv) ) Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 102 ≤ ψ ( M(q, q, v) ) − φ ( M(q, q, v) ) , = ψ ( Gb(q, q, v)) ) − φ ( Gb(q, q, v)) ) , ≤ ψ ( Gb(q, q, v)) ) . (35) Again, since s2 > s > 1 and being ψ is increasing, the above inequality implies that Gb(q, q, v) = 0 and so q = v. That is, q is the unique common fixed point for f, g, h, S,R and T . The following result is an immediate consequence of Theorem 1 by taking φ(t) = t. Corollary 1. Let (X,Gb) be a complete Gb-metric space and let f, g, h,R, S, T : X → X be self mappings such that (1) (f, S) and (g,R) satisfy the (E.A) property; (2) f(X) ⊆ T (X), g(X) ⊆ S(X) and h(X) ⊆ R(X); (3) R(X) is a closed subspace of X; (4) (f, S), (g,R) and (h, T ) are weakly compatible pairs of mappings; (5) ψ ( s2Gb(fx, gy, hz) ) ≤ ψ ( M(x, y, z) ) −M(x, y, z) for all x, y, z ∈ X where ψ ∈ Ψ and M(x, y, z) = max { Gb(fx, Sy, Tz), Gb(gy,Ry,Ry), Gb(fx, fx, hz), Gb(Tz, Tz, hz) +Gb(fx, Sx, Sx) 2s } . Then f, g, h,R, S and T have a unique common fixed point in X. As in the above corollary, the following result follows from Theorem 1 by taking ψ(t) = t. Corollary 2. Let (X,Gb) be a complete Gb-metric space and let f, g, h,R, S, T : X → X be self mappings such that (1) (f, S) and (g,R) satisfy the (E.A) property; (2) f(X) ⊆ T (X), g(X) ⊆ S(X) and h(X) ⊆ R(X); (3) R(X) is a closed subspace of X; (4) (f, S), (g,R) and (h, T ) are weakly compatible pairs of mappings; (5) s2Gb(fx, gy, hz) ≤M(x, y, z)−φ ( M(x, y, z) ) for each x, y, z ∈ X where φ ∈ Ψ and M(x, y, z) = max { Gb(fx, Sy, Tz), Gb(gy,Ry,Ry), Gb(fx, fx, hz), Gb(Tz, Tz, hz) +Gb(fx, Sx, Sx) 2s } . Then f, g, h,R, S and T have a unique common fixed point in X. By specifying ψ(t) = t and φ(t) = t k with k > 1 in Theorem 1, we get the following corollary. Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 103 Corollary 3. Let (X,Gb) be a complete Gb-metric space and let f, g, h,R, S, T : X → X are self mappings such that (1) (f, S) and (g,R) satisfy the (E.A) property; (2) f(X) ⊆ T (X), g(X) ⊆ S(X) and h(X) ⊆ R(X); (3) R(X) is a closed subspace of X; (4) (f, S), (g,R) and (h, T ) are weakly compatible pairs of mappings; (5) ψ ( s2Gb(fx, gy, hz) ) ≤ k−1 k M(x, y, z) for each x, y, z ∈ X where k is a positive integer and M(x, y, z) = max { Gb(fx, Sy, Tz), Gb(gy,Ry,Ry), Gb(fx, fx, hz), Gb(Tz, Tz, hz) +Gb(fx, Sx, Sx) 2s } . Then f, g, h,R, S and T have a unique common fixed point in X. By taking f = g and R = S in Theorem 1, we get the following result. Corollary 4. Let (X,Gb) be a complete Gb-metric space and let f, g, h,R, S, T : X → X be self mappings such that (1) (g, S) satisfies the (E.A) property; (2) g(X) ⊆ T (X), g(X) ⊆ S(X) and h(X) ⊆ S(X); (3) S(X) is a closed subspace of X; (4) (g, S) and (h, T ) are weakly compatible pairs of mappings; (5) ψ ( s2Gb(gx, gy, hz) ) ≤ ψ ( M(x, y, z) ) − φ ( M(x, y, z) ) for each x, y, z ∈ X where φ ∈ Ψ and M(x, y, z) = max { Gb(gx, Sy, Tz), Gb(gy, Sy, Sy), Gb(gx, gx, hz), Gb(Tz, Tz, hz) +Gb(gx, Sx, Sx) 2s } . Then g, h, S and T have a unique common fixed point in X. The following example is to illustrate Theorem 1. Example 5. Let X = [0,∞) and G : X × X × X → [0,∞) be the complete G-metric which is defined by G(x, y, z) = { 0, if x = y = z, max{x, y, z}, otherwise. Define the Gb metric by Gb(x, y, z) = (G(x, y, z))2. Then it is clear that (X,Gb) is a complete Gb-metric with s = 2. Also, define the mappings f, g, h,R, S and T by fx = x 32 , g(x) = x 36 , h(x) = x 48 , Z. Mustafa et al. / Eur. J. Pure Appl. Math, 11 (1) (2018), 90-109 104 and R(x) = 4x 9 , S(x) = x 2 , and T (x) = x 3 for all x ∈ X. Further, define ψ(t) = 4 √ t and φ(t) = √ t 3 for all t ∈ [0,∞). Then f, g, h,R, S and T have a unique common fixed point. Proof. (1) (f, S) and (g,R) satisfy the (E.A) property with xn = 1 n . (2) f(X) ⊆ T (X), g(X) ⊆ S(X) and h(X) ⊆ R(X). In fact, f(X) = g(x) = S(x) = R(x) = T (X) = [0,∞). (3) R(X) = [0,∞) is a closed subspace of X. (4) (f, S), (g,R) and (h, T ) are weakly compatible pairs of mappings. In fact, the only coincident point for f and R is 0 and f(R(0)) = R(f(0)) = 0. Similarly for the other two pairs. (5) We shall show that the above mappings satisfy the contractive condition (3). On one hand, we observe that ψ(s2Gb(fx, gy, hz)) = ψ(4(max{ x 32 , y 36 , z 48 })2) = ψ(4(max{( x 32 )2, ( y 36 )2, ( z 48 )2}) = ψ(max{(2x 32 )2, ( 2y 36 )2, ( 2z 48 )2}) = ψ(max{( x 16 )2, ( y 18 )2, ( z 24 )2}) = 4 max{( x 16 ), y 18 , z 24 } = max{x 4 , 2y 9 , z 6 }. (36) On the other hand, M(x, y, z) = max { max{( x32)2, (y2 )2, ( z3)2},max{( y36)2, (4y9 )2} max{( x32)2, ( z48)2}, max{( z 3 )2,( z 48 )2}+max{( x 32 )2,(x 2 )2} 4 } = max { max{( x32)2, (y2 )2, ( z3)2},max{( y36)2, (4y9 )2}, max{( x32)2, ( z48)2}, ( z 3 )2+(x 2 )2 4 . } = max { max{( x32)2, (y2 )2, ( z3)2}, (4y9 )2, ( z 3 )2+(x 2 )2 4 . } = max { ( x32)2, ( z3)2, (y2 )2, ( z 3 )2+(x 2 )2 4 . } = max { ( y 2 )2, ( z 3 )2, ( z3)2 + (x2 )2 4 } REFERENCES 105 = max { ( y 2 )2, ( z 3 )2, ( z 6 )2 + ( x 4 )2 } , and so, ψ(M(x, y, z))− φ(M(x, y, z)) = 4 max {y 2 , z 3 , √ ( z 6 )2 + ( x 4 )2 } − 1 3 max {y 2 , z 3 , √ ( z 6 )2 + ( x 4 )2 } = max {11y 6 , 11z 9 , 11 3 √ ( z 6 )2 + ( x 4 )2 } . (37) Combining (36) and (37), it is clear to see that ψ(s2Gb(fx, gy, hz)) = max{x 4 , 2y 9 , z 6 } ≤ max {11y 6 , 11z 9 , 11 3 √ ( z 6 )2 + ( x 4 )2 } = ψ(M(x, y, z))− φ(M(x, y, z)). 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