Istratescu-Suzuki-Ćirić-type fixed points results in the framework of G-metric spaces

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1 Available online at J. Nonlinear Sci. Appl ), Research Article Istratescu-Suzuki-Ćirić-type fixed points results in the framework of G-metric spaces Mujahid Abbas a,b, Azhar Hussain c, Branislav Popović d,, Stojan Radenović e a Department of Mathematics, University of Management and Technology, C-II, Johar Town, Lahore, Pakistan. b Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood Road, Pretoria 000, South Africa. c Department of Mathematics, University of Sargodha, Sargodha-40100, Pakistan. d University of Kragujevac, Faculty of Science, Radoja Domanovića 1, Kragujevac, Serbia. e University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 1110 Beograd, Serbia. Communicated by R. Saadati Abstract The aim of this paper is to present fixed point results of convex contraction, convex contraction of order, weakly Zamfirescu and Ćirić strong almost contraction mappings in the framework of G-metric spaces. Some examples are presented to support the results proved herein. As an application, we derive Suzuki type fixed point in G-metric spaces. Our results generalize and extend various results in the existing literature. We also present some examples to illustrate our new theoretical results. c 016 All rights reserved. Keywords: Coincidence point, convex contraction, convex contraction of order, Ćirić strong almost contraction. 010 MSC: 47H10, 54H5, 54E Introduction and preliminaries Over the past two decades, the development of fixed point theory in metric spaces has attracted a considerable attention due to numerous applications in areas such as variational inequalities, optimization, and approximation theory. Mustafa and Sims [5] generalized the concept of a metric in which to every triplet of points of an abstract set, a real number is assigned. Based on the notion of generalized metric Corresponding author addresses: abbas.mujahid@gmail.com Mujahid Abbas), hafiziqbal30@yahoo.com Azhar Hussain), bpopovic@kg.ac.rs Branislav Popović), radens@beotel.rs Stojan Radenović) Received

2 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), spaces, Mustafa et al. [, 4, 6] obtained several fixed point theorems for mappings satisfying different contractive conditions. On the other hand, Mustafa et al. [18 4, 6] also obtained some interesting fixed point results for mappings satisfying new different contractive conditions. Chugh et al. [13] obtained some fixed point results for maps satisfying property P in G-metric spaces. Saadati et al. [30] studied fixed point of contractive mappings in partially ordered G-metric spaces. Shatanawi [33] obtained fixed points of Φ-maps in G-metric spaces. For more details, we refer to [1 7, 9, 11, 1, 7 9, 34, 35]. Jleli and Samet [16] see also, [3]) observed that some fixed point results in the context of a G-metric space can be deduced by some existing results in the setting of a quasi-) metric space. In fact, if the contraction condition of the fixed point theorem on a G-metric space can be reduced to two variables instead of three variables, then one can construct an equivalent fixed point theorem in the setting of a usual metric space. More precisely, they noticed that dx, y) = Gx, y, y) forms a quasi-metric. Therefore, if one can transform the contraction condition of existence results in a G-metric space in terms such as Gx, y, y), then the related fixed point results become the known fixed point results in the context of a quasi-metric space. On the other hands, Istratescu [15] introduced the notion of a convex contraction mapping. Recently, Miandaragh et al. [17] proved some fixed point results for generalized convex contractions on complete metric space. The aim of this paper is to study the notion of convex contraction, convex contraction of order, weakly Zamfirescu mappings and Ćirić strong almost contraction in the setup of G-metric spaces. We obtain several fixed point results for such mappings in the setting of generalized metric spaces. As an application, Suzuki type fixed point result is also derived. Some examples are provided to support the results proved herein. Our results extend and generalize various existing results in the literature. Consistent with Mustafa and Sims [5], the following definitions and results will be needed in the sequel. Definition 1.1. Let X be a nonempty set. A mapping G : X X X R + is said to be a G-metric on X, if for any x, y, z X, the following conditions hold: a) Gx, y, z) = 0, if x = y = z; b) 0 < Gx, y, z), for all x, y X with x y; c) Gx, x, y) Gx, y, z), for all x, y, z X, with y z; d) Gx, y, z) = Gpx, y, z), where p is a permutation of x, y, z symmetry); e) Gx, y, z) Gx, a, a) + Ga, y, z), for all x, y, z, a X. The pair X, G) is called a G-metric space [5]. Definition 1.. A sequence x n in a G-metric space X is called: i) a G-Cauchy sequence if for any ε > 0, there is an n 0 N the set of natural numbers) such that for all n, m, l n 0, Gx n, x m, x l ) < ε; ii) a G-convergent sequence if for any ε > 0, there is an x X and an n 0 N such that for all n, m n 0, Gx, x n, x m ) < ε. A G-metric space is said to be G-complete, if every G-Cauchy sequence in X is G-convergent in X. It is known that a sequence x n converges to x X if and only if Gx m, x n, x) 0 as n, m [5]. Proposition 1.3 [5]). Let X be a G-metric space. Then following are equivalent: 1. x n is G-convergent to x.. Gx n, x n, x) 0 as n.

3 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Gx n, x, x) 0 as n. 4. Gx n, x m, x) 0 as n, m. Definition 1.4. A G-metric on X is said to be symmetric if Gx, y, y) = Gy, x, x) for all x, y X. Proposition 1.5. Every G-metric on X will define a metric d G on X given by For a symmetric G-metric, we have d G x, y) = Gx, y, y) + Gy, x, x), for all x, y X. d G x, y) = Gx, y, y), for all x, y X. However, if G is not symmetric, then the following inequality holds: 3 Gx, y, y) d Gx, y) 3Gx, y, y), for all x, y X. Definition 1.6 [31]). Let ϕ be the collection of all mappings ψ : [0, ) [0, ) that satisfy the following conditions: ψ n t) < for each t > 0, where ψ n is the n-th iterate of ψ; n=1 ψ is nondecreasing. Definition 1.7 [8]). Let X be a nonempty set and α: X X X [0, ). A self mapping T on X is said to be α-admissible, if for any x, y, z X αx, y, z) 1 implies that αt x, T y, T z) 1. Example 1.8 [8]). Let X = [0, ) and T : X X by ln x, if x 0; T x) = e, otherwise. Define α : X X X [0, ) by Then the mapping T is α-admissible. αx, y, z) = e, x y z; 0, x < y < z.. Fixed point results for convex contractions Definition.1. Let X be a G-metric space, T a self-map on X and ɛ > 0 a given number. A point x in X is called a) an ɛ-fixed point of T, if Gx, T x, T x) < ɛ; b) approximate fixed point of T, if T has an ɛ-fixed point for all ɛ > 0. Definition.. Let X be a G-metric space. A self-mapping T on X is called asymptotic regular if for any x in X, we have GT n x, T n+1 x, T n+ x) 0 as n. Now, we have the following simple lemma.

4 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Lemma.3. Let X be a G-metric space and T an asymptotic regular map on X. Then T has an approximate fixed point. Proof. Let x 0 be a given point in X and ɛ > 0. Since T is an asymptotic regular map on X, we can choose n 0 ɛ) N such that GT n x 0, T n+1 x 0, T n+ x 0 ) < ɛ for all n n 0 ɛ). That is, GT n x 0, T T n x 0 ), T T n x 0 )) < ɛ for all n n 0 ɛ). If we put T n x 0 ) = y 0 then Gy 0, T y 0 ), T y 0 )) < ɛ implies that y 0 = T n x 0 ) is an ɛ-fixed point of T in X. As ɛ > 0 is an arbitrary number, so T has an approximate fixed point. Definition.4. Let X be a nonempty set and α, η : X X X [0, ) two mappings. A self-mapping T on X is said to be α-admissible with respect to η if for any x, y, z X αx, y, z) ηx, y, z) implies that αt x, T y, T z) ηt x, T y, T z). Example.5. Let X = [0, ) and T : X X by ln x, if x 0; T x) = e, otherwise. Define α, η : X X X [0, ) by αx, y, z) = e, x y z;, x < y < z, and ηx, y, z) = 1. Then the mapping T is α-admissible with respect to η. Definition.6. Let X be a nonempty set and α, η : X X X [0, ) two mappings. A self-mapping T on X is said to be convex contraction, if for any x, y, z in X ηx, T x, T y) αx, y, z) implies that GT x, T y, T z, ) agt x, T y, T z) + bgx, y, z),.1) where a, b 0 with a + b < 1. Definition.7. Let X be a nonempty set and α, η : X X X [0, ) two mappings. A self-mapping T on X is said to be convex contraction if for any x, y, z in X implies that ηx, T x, T y) αx, y, z), GT x, T y, T z, ) a 1 Gx, T x, T x) + a GT x, T x, T x) + b 1 Gy, T y, T y) where a 1, a, b 1, b, c 1, c 0 with a 1 + a + b 1 + b + c 1 + c < 1. + b GT y, T y, T y) + c 1 Gz, T z, T z) + c GT z, T z, T z),.) Theorem.8. Let X be a complete G-metric space and T an α-admissible convex contraction with respect to η. If αx, T x, T x) ηx, T x, T x) for any x X, then T has an approximate fixed point. Proof. Let x 0 be a given point in X. Since αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ) and T is an α-admissible mapping with respect to η, we have αt x 0, T x 0, T 3 x 0 ) ηt x 0, T x 0, T 3 x 0 ). By continuing this way, we obtain that αt n x 0, T n+1 x 0, T n+ x 0 ) ηt n x 0, T n+1 x 0, T n+ x 0 ), for all n N 0. Put ϑ = GT 3 x 0, T x 0, T x 0 )+GT x 0, T x 0, x 0 ) and r = a+b. Obviously, GT 3 x 0, T x 0, T x 0 ), GT x 0, T x 0, x 0 ) ϑ. By using x = x 0, y = T x 0 and z = T x 0 in.1), we have GT x 0, T 3 x 0, T 4 x 0, ) agt x 0, T x 0, T 3 x 0 ) + bgx 0, T x 0, T x 0 ) rϑ.

5 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Similarly, we have GT 3 x 0, T 4 x 0, T 5 x 0, ) agt x 0, T 3 x 0, T 4 x 0 ) + bgt x 0, T x 0, T 3 x 0 ) r ϑ. By continuing this process, we arrive at GT m x 0, T m+1 x 0, T m+ x 0 ) r l ϑ, where m = l or m = l + 1. On taking the limit as m on both sides of above inequality, we have GT m x 0, T m+1 x 0, T m+ x 0 ) 0, for any x 0 X. By Lemma.3, T has an approximate fixed point. Let T be a self-mapping on a nonempty set X and α, η : X X X [0, ). We say that the set X has H property if for any x, y F ixt ) with αx, y, y) < ηx, T x, T x), there exists z X such that αx, z, z) ηx, z, z) and αy, z, z) ηy, z, z). Also for any x, y X, we have ηx, T x, T x) ηx, y, z). Theorem.9. Let X be a complete G-metric space and T a continuous convex contraction and α-admissible mapping with respect to η. Suppose that there exists a point x 0 in X such that αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ). Then T has a fixed point. Moreover, T has a unique fixed point provided that X has H property. Proof. Define a sequence x n in X by x n = T n x 0, for all n N. Since T is an α-admissible mapping with respect to η and αx 0, x 1, x 1 ) = αx 0, T x 0,, T x 0 ) ηx 0, T x 0, T x 0 ), we have αt x 0, T x 0, T x 0 ) = αx 1, x, x ) ηt x 0, T x 0, T x 0 )=ηx 1, x, x ). By continuing this way, we obtain that αx n, x n+1, z n+1 ) ηx n, x n+1, x n+1 ) = ηx n, T x n, T x n ), for all n N 0. Also, from.1), we have Gx n+, x n+3, x n+4 ) = GT n+ x 0, T n+3 x 0, T n+4 x 0 ) = GT T n x 0 ), T T n+1 x 0 ), T T n+ x 0 )) agt T n x 0 ), T T n+1 x 0 ), T T n+ x 0 )) + bgt n x 0, T n+1 x 0, T n+ x 0 ) = agx n+1, x n+, x n+3 ) + bgx n, x n+1, x n+ ). We set ϑ = Gx 3, x, x 1 ) + Gx, x 1, x 0 ) and r = a + b. Then Gx m, x m+1, x m+ ) r l ϑ, where m = l or m = l + 1. Suppose that m = l. Then for n, k = p with p >, l 1 and m < n, k we have Gx m, x n, x k ) Gx m, x m+1, x m+1 ) + Gx m+1, x m+, x m+ ) + Gx m+, x m+3, x m+3 ) + + Gx n, x n 1, x n l ) + Gx n 1, x n, x k ) = Gx l, x l+1, x l+1 ) + Gx l+1, x l+, x l+ ) + Gx l+, x l+3, x l+3 ) + + Gx p, x p 1, x p l ) + Gx p 1, x p, x p ) r l ϑ + r l ϑ + r l+1 ϑ + + r p 1 ϑ = r l ϑ + r l+1 ϑ + r l+ ϑ + + r p 1 ϑ rl 1 r ϑ. Similarly, for n, k = p + 1 with p 1, l 1 and m < n, k we have Gx m, x n, x k ) rl 1 r ϑ.

6 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Now, assume that m = l + 1. Then for n = p with p, l 1 and m < n, we have Gx m, x n, x k ) Gx m, x m+1, x m+1 ) + Gx m+1, x m+, x m+ ) + Gx m+, x m+3, x m+3 ) + + Gx n, x n 1, x n 1 ) + Gx n 1, x n, x k ) = Gx l+1, x l+, x l+ ) + Gx l+, x l+3, x l+3 ) + Gx l+3, x l+4, x l+4 ) + + Gx p, x p 1, x p l ) + Gx p 1, x p, x p ) r l ϑ + r l+1 ϑ + r l+1 ϑ + + r p ϑ r l ϑ + r l+1 ϑ + r l+ ϑ + + r p ϑ rl 1 r ϑ. Similarly, for n, k = p + 1 with p 1, l 1 and m < n, k we obtain that Gx m, x n, x k ) Hence, for all m, n, k N with m < n, k, we have Gx m, x n, x k ) rl 1 r ϑ. rl 1 r ϑ. On taking the limit as l on both sides of above inequality, we have Gx m, x n, x k ) 0 which implies that x n is a Cauchy sequence. By completeness of X, there exists z X such that x n z and n and hence T z = z as T is a continuous mapping. Let x, y F ixt ) where x y. To prove the uniqueness, we consider the following cases. i) If αx, y, y) ηx, T x, T x). As T is a convex contraction, so we have a contraction. ii) If αx, y, y) < ηx, T x, T x). Gx, y, y) = GT x, T y, T y) agt x, T y, T y) + bgx, y, y) = agx, y, y) + bgx, y, y) = a + b)gx, y, y) < Gx, y, y), Since X has H property, there exists z X such that αx, z) ηx, z) and αy, z, z) ηy, z, z). Also, T is an α-admissible mapping with respect to η, we have αx, T n z, T n z) ηx, T n z, T n z) ηx, T x, T x) and αy, T n z, T n z) ηy, T n z, T n z) ηy, T y, T y). If αx, T n z, T n z) ηx, T x, T x), then we have Gx, T n+ z, T n+ z) agx, T n+1 z, T n+1 z) + bgx, T n z, T n z). By taking ϑ = Gx, T z, T z) + Gx, z, z) and r = a + b < 1, we have Gx, T m z, T m z) r l ϑ, where m = l or m = l+1 which on taking the limit as m implies that T m z x. Similarly, T m z y as m. Hence x = y, a contradiction. Thus the result follows. Example.10. Let X = 0, 1, be a set. Let G : X X X [0, ) be defined by

7 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), x, y, z) Gx, y, z) 0,0,0), 1,1,1),,,) 0 0,1,1), 1,0,1), 1,1,0) 1 0,0,1), 0,1,0), 1,0,0) 1,,),,1,),,,1) 0,0,), 0,,0),,0,0) 3 0,,),,0,),,,0) 3 1,1,), 1,,1),,1,1) 4 0,1,), 0,,1), 1,0,) 4 1,,0),,0,1),,1,0) 4 It is clear that G is a non-symmetric G-metric as G0, 0, 1) G0, 1, 1). Let T : X X be defined by Now, x 0 1 T x) 1 0 T x, T y, T z) GT x, T y, T z) T x, T y, T z) GT x, T y, T z) 1,1,1), 0,0,0),,,) 0 0,0,0), 1,1,1),,,) 0 1,0,0), 0,1,0), 0,0,1) 0,1,1), 1,0,1), 1,1,0) 1 1,1,0), 1,0,1), 0,1,1) 1 0,0,1), 0,1,0), 1,0,0) 0,,),,0,),,,0) 3 1,,),,1,),,,1) 1,1,), 1,,1),,1,1) 4 0,0,), 0,,0),,0,0) 3 1,,),,1,),,,1) 0,,),,0,),,,0) 3 0,0,), 0,,0),,0,0) 3 1,1,), 1,,1),,1,1) 4 1,0,), 1,,0), 0,1,) 4 0,1,), 0,,1), 1,0,) 4 0,,1),,1,0),,0,1) 4 1,,0),,0,1),,1,0) 4 Define α, η : X X X [0, ) by αx, y, z) = 4 + xyz and ηx, y, z) = xyz. For x y z we consider the following cases to check that T is convex contraction. Case-I: For x = 0, y = 1 and z =. Case-II: For x = 0, y = and z = 1. Case-III: For x = 1, y = 0 and z =. GT x, T y, T z) = 4 a4) + b4) = ag0, 1, ) + bg0, 1, ) GT x, T y, T z) = 4 a4) + b4) = agt x, T y, T z) + bgx, y, z). = ag0,, 1) + bg0,, 1) GT x, T y, T z) = 4 a4) + b4) = agt x, T y, T z) + bgx, y, z). = ag1, 0, ) + bg1, 0, ) = agt x, T y, T z) + bgx, y, z).

8 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Case-IV: For x = 1, y = and z = 0. GT x, T y, T z) = 4 a4) + b4) = ag1,, 0) + bg1,, 0) = agt x, T y, T z) + bgx, y, z). Case-V: For x =, y = 0 and z = 1. GT x, T y, T z) = 4 a4) + b4) = ag, 0, 1) + bg, 0, 1) = agt x, T y, T z) + bgx, y, z). Case-VI: For x =, y = 1 and z = 0. GT x, T y, T z) = 4 a4) + b4) = ag, 1, 0) + bg, 1, 0) = agt x, T y, T z) + bgx, y, z). Thus, in all cases T is convex contraction with a, b 1.8 are satisfied and is a fixed point of T. with a b. Hence all the conditions of Theorem Remark.11. A G-metric naturally induces a metric d G given by d G x, y) = Gx, y, y) + Gx, x, y). If the G-metric is not symmetric, the inequalities.1) do not reduce to any metric inequality with the metric d G. Hence our results do not reduce to fixed point problems in the corresponding metric space X, d G ). For instance, if we take x = 0 and y = in above example, we obtain d G T x, T y) =, d G T x, T y) = 1, d G x, y) =, so there does not exist any a, b 0 with a + b < 1 such that d G T x, T y) ad G T x, T y) + bd G x, y) holds. So, we can not apply the result of [14] to obtain fixed point of T. Theorem.1. Let X be a complete G-metric space, T a convex contraction of order α-admissible with respect to η and αx, T x, T x) ηx, T x, T x) for all x X. Then T has an approximate fixed point. Proof. Let x 0 be a given point in X. The following arguments similar to those in the proof of Theorem.8 we obtain that αt n x 0, T n+1 x 0, T n+ x 0 ) ηt n x 0, T n+1 x 0, T n+ x 0 ), for all n N. We set r = a 1 +a +b 1 +c 1, = 1 b c and ϑ = GT x 0, T x 0, T x 0 ) + GT x 0, T x 0, x 0 ). From.) with x = x 0 and y = z = T x 0 we have GT x 0, T 3 x 0, T 3 x 0 ) a 1 Gx 0, T x 0, T x 0 ) + a GT x 0, T x 0, T x 0 ) + b 1 GT x 0, T x 0, T x 0 ) which implies that + b GT x 0, T 3 x 0, T 3 x 0 ) + c 1 GT x 0, T x 0, T x 0 ) + c GT x 0, T 3 x 0, T 3 x 0 ), GT x 0, T 3 x 0, T 3 x 0 ) = 1 b c )GT x 0, T 3 x 0, T 3 x 0 ) Thus GT x 0, T 3 x 0, T 3 x 0 ) r a 1 Gx 0, T x 0, T x 0 ) + a + b 1 + c 1 )GT x 0, T x 0, T x 0 ) rϑ. ) ϑ. Again from.) with x = T x 0 and y = z = T x 0, we have GT 3 x 0, T 4 x 0, T 4 x 0 ) a 1 GT x 0, T x 0, T x 0 ) + a GT x 0, T 3 x 0, T 3 x 0 ) + b 1 GT x 0, T 3 x 0, T 3 x 0 ) + b GT 3 x 0, T 4 x 0, T 4 x 0 ) + c 1 GT x 0, T 3 x 0, T 3 x 0 ) + c GT 3 x 0, T 4 x 0, T 4 x 0 ),

9 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), which implies that GT 3 x 0, T 4 x 0, T 4 x 0 ) rϑ, ) r GT 3 x 0, T 4 x 0, T 4 x 0 ) ϑ, GT 3 x 0, T 4 x 0, T 4 x 0 ) = 1 b c )GT 3 x 0, T 4 x 0, T 4 x 0 ) rϑ. ) Hence GT 3 x 0, T 4 x 0, T 4 x 0 ) r ϑ. Similarly, we have a 1 GT x 0, T x 0, T x 0 ) + a + b 1 + c 1 )GT x 0, T 3 x 0, T 3 x 0 ) ) r GT 4 x 0, T 5 x 0, T 5 x 0 ) ϑ, ) r GT 5 x 0, T 6 x 0, T 6 x 0 ) ϑ. By continuing this way, we can obtain that GT m x 0, T m+1 x 0, T m+1 x 0 ) r ) l ϑ, where m = l or m = l + 1. Hence GT m x 0, T m+1 x 0, T m+1 x 0 ) 0 as m, which by Lemma.3 implies that T has an approximate fixed point. Theorem.13. Let X be a complete G-metric space, T a continuous convex contraction of order and α- admissible mapping with respect to η. Suppose that there exists a point x 0 in X such that αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ). Then T has a fixed point. Moreover, T has a unique fixed point provided that X has H property. Proof. Define a sequence x n in X by x n = T n x 0, for all n N. We set r = a 1 +a +b 1 +c 1, = 1 b c and ϑ = GT x 0, T x 0, T x 0 ) + GT x 0, T x 0, x 0 ). From.) with x = x 0 and y = z = T x 0, we have GT x 0, T 3 x 0, T 3 x 0 ) a 1 Gx 0, T x 0, T x 0 ) + a GT x 0, T x 0, T x 0 ) + b 1 GT x 0, T x 0, T x 0 ) which implies that + b GT x 0, T 3 x 0, T 3 x 0 ) + c 1 GT x 0, T x 0, T x 0 ) + c GT x 0, T 3 x 0, T 3 x 0 ) rϑ, GT x 0, T 3 x 0, T 3 x 0 ) = 1 b c )GT x 0, T 3 x 0, T 3 x 0 ) rϑ. ) Thus, GT x 0, T 3 x 0, T 3 x 0 ) r ϑ. Again from.) with x = T x 0 and y = z = T x 0, we have a 1 Gx 0, T x 0, T x 0 ) + a + b 1 + c 1 )GT x 0, T x 0, T x 0 ) GT 3 x 0, T 4 x 0, T 4 x 0 ) a 1 GT x 0, T x 0, T x 0 ) + a GT x 0, T 3 x 0, T 3 x 0 ) + b 1 GT x 0, T 3 x 0, T 3 x 0 ) which implies that + b GT 3 x 0, T 4 x 0, T 4 x 0 ) + c 1 GT x 0, T 3 x 0, T 3 x 0 ) + c GT 3 x 0, T 4 x 0, T 4 x 0 ), GT 3 x 0, T 4 x 0, T 4 x 0 ) = 1 b c )GT 3 x 0, T 4 x 0, T 4 x 0 ) a 1 GT x 0, T x 0, T x 0 ) + a + b 1 + c 1 )GT x 0, T 3 x 0, T 3 x 0 ) rϑ.

10 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), ) Hence GT 3 x 0, T 4 x 0, T 4 x 0 ) r ϑ. Similarly, we have ) r GT 4 x 0, T 5 x 0, T 5 x 0 ) ϑ, ) r GT 5 x 0, T 6 x 0, T 6 x 0 ) ϑ. By continuing this way, we obtain that GT m x 0, T m+1 x 0, T m+1 x 0 ) r ) l ϑ, where m = l or m = l+1. Now for m = l, n, k = p with p >,l 1 and m < n, k, we have Gx m, x n, x k ) Gx m, x m+1, x m+1 ) + Gx m+1, x m+, x m+ ) + Gx m+, x m+3, x m+3 ) + + Gx n, x n 1, x n l ) + Gx n 1, x n, x k ) = Gx l, x l+1, x l+1 ) + Gx l+1, x l+, x l+ ) + Gx l+, x l+3, x l+3 ) + + Gx p, x p 1, x p l ) + Gx p 1, x p, x p ) ) r l ) r l ) r l+1 ) r p 1 ϑ + ϑ + ϑ + + ϑ ) r l ) r l+1 ) r l+ ) r p 1 = ϑ + ϑ + ϑ + + ϑ ) l r )ϑ. 1 r Similarly, for m = l and n, k = p + 1 with p 1, l 1 and m < n, k we get Gx m, x n, x k ) r ) l 1 r )ϑ. If m = l + 1, then for n = p with p, l 1 and m < n we have Gx m, x n, x k ) Gx m, x m+1, x m+1 ) + Gx m+1, x m+, x m+ ) + Gx m+, x m+3, x m+3 ) + + Gx n, x n 1, x n l ) + Gx n 1, x n, x k ) = Gx l+1, x l+, x l+ ) + Gx l+, x l+3, x l+3 ) + Gx l+3, x l+4, x l+4 ) + + Gx p, x p 1, x p l ) + Gx p 1, x p, x p ) ) r l ) r l+1 ) r l+1 ) r p ϑ + ϑ + ϑ + + ϑ ) r l ) r l+1 ) r l+ ) r p ϑ + ϑ + ϑ + + ϑ ) l r )ϑ. 1 r Similarly, for m = l + 1 and n, k = p + 1 with p 1, l 1 and m < n, k, we have Gx m, x n, x k ) r ) l 1 r )ϑ.

11 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Hence, for all m, n, k N with m < n, k we obtain that Gx m, x n, x k ) r ) l 1 r )ϑ. By taking the limit as l in the above inequality we get Gx m, x n, x k ) converges to 0. Since X, G) is a complete G-metric space, we have x n z and n for some z X. By continuity of T, T z = z. By following arguments similar to those in proof of Theorem.9, we obtain the uniqueness of fixed point of T provided that X has H -property. 3. α-η-weakly Zamfirescu mappings In this section we obtain fixed point results of α-η-weakly Zamfirescu mapping in the framework of G-metric spaces. Definition 3.1. Let T be a self-mapping on a G-metric space X and a, b R + with 0 < a b. If there exists a mapping γ : X X X [0, 1] with θa, b) := supγx, y, z) : a Gx, y, z) b < 1 such that for any x, y, z X ηx, T x, T x) αx, y, z), implies that Gx, T x, T x) + Gy, T y, T y) + Gz, T z, T z) GT x, T y, T z) γx, y, z) max Gx, y, z),, Gx, T y, T z) + Gy, T z, T x) + Gz, T x, T y), then T is α-η-weakly Zamfirescu mapping. Theorem 3.. Let X be a G-metric space and T a self-mapping on X. If T is an α-η-weakly Zamfirescu mapping and α-admissible with respect to η with αx, T x, T x) ηx, T x, T x) for any x X, then T has an approximate fixed point. Proof. If G is symmetric, then we have and 3.1) becomes d G T x, T y) γx, y) max d G x, y) = Gx, y, y), 3.1) d G x, y), d Gx, T x) + d G y, T y), d Gx, T y) + d G y, T x) by taking γ : X X [0, 1] instead of γ : X X X [0, 1]. The result then follows from Theorem 0 in [14]. Suppose that G is non-symmetric. We proceed as follows. Let x 0 be a given point in X. We define a sequence x n by x n = T n x 0. By following arguments similar to those in the proof of Theorem.8 we obtain that αt n x 0, T n+1 x 0, T n+1 x 0 ) ηt n x 0, T n+1 x 0, T n+1 x 0 ) for all n N. Then we have Gx n, x n+1, x n+1 ) = GT T n 1 x 0, T T n x 0, T T n x 0 ) γt n 1 x 0, T n x 0, T n x 0 ) max GT n 1 x 0, T n x 0, T n x 0 ), GT n 1 x 0, T T n 1 x 0, T T n 1 x 0 ) + GT n x 0, T T n x 0, T T n x 0 ),,

12 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), If then we have If then gives which implies that GT n 1 x 0, T T n x 0, T T n x 0 ) + GT n x 0, T T n 1 x 0, T T n 1 x 0 ) = γx n 1, x n, x n ) max Gx n 1, x n, x n ), Gx n 1, x n, x n ) + Gx n, x n+1, x n+1 ), Gx n 1, x n, x n ) + Gx n, x n, x n ) γx n 1, x n, x n ) max Gx n 1, x n, x n ), Gx n 1, x n, x n ) + Gx n, x n+1, x n+1 ). max Gx n 1, x n, x n ), Gx n 1, x n, x n ) + Gx n, x n+1, x n+1 ) = Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ) γx n 1, x n, x n )Gx n 1, x n, x n ). max Gx n 1, x n+1, x n+1 ), Gx n 1, x n, x n ) + Gx n, x n+1, x n+1 ) = Gx n 1, x n, x n ) + Gx n, x n+1, x n+1 ), [ ] Gxn 1, x n, x n ) + Gx n, x n+1, x n+1 ) Gx n, x n+1, x n+1 ) γx n 1, x n, x n ), γ)x n 1, x n, x n ))Gx n, x n+1, x n+1 ) γx n 1, x n, x n )Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ) γx n 1, x n, x n ) γx n 1, x n, x n ) Gx n 1, x n, x n ) γx n 1, x n, x n )Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ) γx n 1, x n, x n )Gx n 1, x n, x n ). Hence Gx n 1, x n, x n ) is a non-increasing sequence which converges to a real number s = inf n 1 Gx n 1, x n, x n ). Assume that s > 0. Since 0 < s Gx n, x n+1, x n+1 ) Gx 0, x 1, x 1 ) and γx n 1, x n, x n ) θ for all n N 0, where θ = θs, Gx 0, x 1, x 1 )), we obtain that and Gx n, x n+1, x n+1 ) θgx n 1, x n, x n ), s Gx n 1, x n, x n ) θ n Gx 0, x 1, x 1 ). This implies s = 0 on taking the limit as n ), a contradiction. Therefore, Now, lim Gx n 1, x n, x n ) = lim GT n 1 x 0, T n x 0, T n x 0 ) = 0. n n GT n x 0, T n+1 x 0, T n+ x 0 ) = Gx n, x n+1, x n+ )

13 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), gives GT n x 0, T n+1 x 0, T n+ x 0 ) 0, as n. point. Gx n, x n+1, x n+1 ) + Gx n+1, x n+1, x n+ ) Gx n, x n+1, x n+1 ) + Gx n+1, x n+, x n+ ) + Gx n+, x n+1, x n+ ) = Gx n, x n+1, x n+1 ) + Gx n+1, x n+, x n+ ), Hence, by Lemma.3, T has an approximate fixed Theorem 3.3. Let X be a complete G-metric space and T a continuous, α-η-weakly Zamfirescu and α- admissible mapping with respect to η. If there exists x 0 X such that αx 0, T x 0 ) ηx 0, T x 0 ), then T has a fixed point. Proof. Let x 0 X be such that αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ). Define a sequence x n as in Theorem.8. By following arguments similar to those in proof of Theorem 3., we obtain that Gx n, x n+1, x n+1 ) γx n 1, x n, x n )Gx n 1, x n, x n ) for all n N 0. Also, we deduce that x n is a Cauchy sequence. Since X is a complete G-metric space, there exists z X such that x n z. The result follows by the continuity of T. Example 3.4. Let X = [0, ) and Gx, y, z) = max x y, y z, z x be a G-metric on X. Define T : X X and α, η : X X X [0, ) by x, if x [0, ], 5 3x + x x+1 )4 x) + x 1, if x, 4], T x = 60 x+1 30 x) 16x + 1) x 4), if x 4, 0), 16 5x, if x [0, ), and 15, if x, y, z [0, 1], αx, y, z) = 1, otherwise, ηx, y, z) = 1. Let γ : X X X [0, 1] be a given function. If αx, y, z) 1 then x, y, z [0, 1]. Therefore, GT x, T y, T y) = 1 Gx, y, y) max Gx, y, y), Take γx, y, z) = 1 3 and so, GT x, T y, T y) = 1 Gx, y, y) 5 γx, y, z) max Gx, y, z), Gx, T x, T x) + Gy, T y, T y), Gx, T x, T x) + Gy, T y, T y), Gx, T y, T y) + Gy, T x, T x). Gx, T y, T y) + Gy, T x, T x). That is, there exists γ : X X X [0, 1] with θa, b) := supγx, y, z) : a Gx, y, z) b < 1 for all 0 < a b, such that ηx, T x, T x) αx, y, y), Gx, T x, T x) + Gy, T y, T y) Gx, T y, T y) + Gy, T x, T x) GT x, T y, T y) γx, y, y) max Gx, y, y),, holds for all x, y X. Then T is α-η-weakly Zamfirescu mapping. Thus, T has a fixed point by Theorem 3.3.

14 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), From α-η-ćirić strong almost contraction to Suzuki type contraction Definition 4.1. Let X be a G-metric space and α, η : X X X [0, ). A mapping T : X X is called an α-η-ćirić strong almost contraction, if there exists a constant r [0, 1) such that for any x, y, z X, ηx, T x, T x) αx, y, z) implies that GT x, T y, T z) rmx, y, z) + LGy, T z, T x), 4.1) where L 0 and Mx, y, z) = max Gx, y, z), Gx, T x, T x), Gy, T y, T y), Gz, T z, T z) Gx, T y, T z) + Gy, T z, T x) + Gz, T x, T y). Theorem 4.. Let X be a G-metric space and T be a continuous α-η-ćirić strong almost contraction on X. Also suppose that, T is an α-admissible mapping with respect to η. If there exists an x 0 X such that αx 0, T x 0 ) ηx 0, T x 0 ), then T has a fixed point. Proof. If G is symmetric, then we have and 4.1) becomes d G x, y) = Gx, y, y), d G T x, T y) rmx, y) + Ld G y, T x). The result then follows from Theorem 5 in [10]. Suppose that G is non-symmetric. Let x 0 X be such that αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ). Define a sequence x n by x n = T n x 0 = T x n 1. As T is an α-admissible mapping with respect to η, so we have αx 0, x 1, x 1 ) = αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ) = ηx 0, x 1, x 1 ). By continuing this process, we have for all n N. From given assumption we have ηx n 1, T x n 1, T x n 1 ) = ηx n 1, x n, x n ) αx n 1, x n, x n ) Gx n, x n+1, x n+1 ) = GT x n 1, T x n, T x n ) rmx n 1, x n, x n ) + LGx n, T x n 1, T x n 1 ) = rmx n 1, x n, x n ), where Mx n 1, x n, x n ) = max Gx n 1, x n, x n ), Gx n 1, T x n 1, T x n 1 ), Gx n, T x n, T x n ), Gx n 1, T x n, T x n ) + Gx n, T x n, T x n 1 ) + Gx n, T x n 1, T x n ) = max Gx n 1, x n, x n ), Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ), GT x n, T x n, T x n ) + Gx n, x n+1, x n ) + Gx n, x n, x n+1 ) = max Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ), Gx n, x n+1, x n ) + Gx n, x n, x n+1 ) = maxgx n 1, x n, x n ), Gx n, x n+1, x n+1, Gx n, x n, x n+1 )).

15 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Thus, If for some n, then Gx n, x n+1, x n+1 ) max Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ), Gx n, x n, x n+1 ). max Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ), Gx n, x n, x n+1 ) = Gx n, x n+1, x n+1 ), gives a contradiction. Similarly, Gx n, x n+1, x n+1 ) rgx n, x n+1, x n+1 ) < Gx n, x n+1, x n+1 ), max Gx n 1, x n, x n ), Gx n, x n+1, x n+1 ), Gx n, x n, x n+1 ) = Gx n, x n, x n+1 ), leads to a contradiction. Hence, which further implies that for all n N. Now for m < n Gx n, x n+1, x n+1 ) rgx n 1, x n, x n ), Gx n, x n+1, x n+1 ) rgx n 1, x n, x n ) r n Gx 0, x 1, x 1 ), Gx m, x n, x n ) Gx m, x m+1, x m+1 ) + Gx m+1, x m+, x m+ ) + + Gx n 1, x n, x n ) r m Gx 0, x 1, x 1 ) + r m+1 Gx 0, x 1, x 1 ) + + r n 1 Gx 0, x 1, x 1 ) = r m + r m r n 1 )Gx 0, x 1, x 1 ) rm 1 r Gx 0, x 1, x 1 ). By taking the limit as m, n, we get that x n is a Cauchy sequence. By completeness of X, there exists z X such that x n z, as n. The result follows by the continuity of T. Theorem 4.3. Let X be a G-metric space, α, η : X X X [0, ), T an α-admissible with respect to η and α-η-ćirić strong almost contraction on X. If there exists an x 0 X such that αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ) and for any sequence x n in X such that with x n x as n, then either αx n, x n+1, x n+1 ) ηx n, x n+1, x n+1 ), ηt x n, T x n, T x n ) αt x n, x, x), or ηt x n, T 3 x n, T 3 x n ) αt x n, x, x), holds for all n N. Then T has a fixed point. Proof. Let x 0 be a given point in X such that αx 0, T x 0, T x 0 ) ηx 0, T x 0, T x 0 ). Define a sequence x n by x n = T n x 0 = T x n 1. As in proof of Theorem 3.3, we obtain that αx n, x n+1, x n+1 ) ηx n, x n+1, x n+1 ) for all n N. Also, there exists z X such that, x n z as n. If Gz, T z, T z) 0, then ηt x n 1, T x n 1, T x n 1 ) αt x n 1, z, z), or ηt x n 1, T 3 x n 1, T 3 x n 1 ) αt x n 1, z, z), holds for all n N. Thus ηx n, T x n, T x n ) αx n, z, z), or ηt x n, T x n+1, T x n+1 ) αx n+1, z, z), holds for all n N. Suppose that ηx n, T x n, T x n ) αx n, z, z) holds for all n N. By given assumption we have

16 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), where Gx n+1, T z, T z) = GT x n, T z, T z) rmx n, z, z) + LGz, T x n, T x n ) = rmx n, z, z) + LGz, x n+1, x n+1 ), Mx n, z, z) = max Gx n, z, z), Gx n, T x n, T x n ), Gz, T z, T z), Gx n, T z, T z) + Gz, T z, T x n ) + Gz, T x n, T z) = max Gx n, z, z), Gx n, x n+1, x n+1 ), Gz, T z, T z), Gx n, T z, T z) + Gz, T z, x n+1 ) + Gz, x n+1, T z) By using 4.3) in 4.) and taking the limit as n, we have Gz, T z, T z) rgz, T z, T z) < Gz, T z, T z), 4.). 4.3) a contradiction. Hence Gz, T z, T z) = 0. By following arguments similar to those given above, we obtain that T z = z, if ηx n+1, T x n+1, T x n+1 ) αx n+1, z, z) holds for all n N. Example 4.4. Let X = [0, + ) and Gx, y, z) = max x y, y z, z x. Define T : X X and α, η : X X X [0, ) by x, if x [0, 1], 4 T x = x 3 + x + 1), if x 1, ], x + 1 3x, if x [, + ), and 1, if x, y, z [0, 1], αx, y, z) = 1 ηx, y, z) = 1 8, otherwise, 4. Let αx, y, z) ηx, y, z), then x, y, z [0, 1]. Also, T w [0, 1] for all w [0, 1]. Then αt x, T y, T z) ηt x, T y, T z). This shows T is α-admissible mapping with respect to η. Let x n be a sequence in X such that x n x as n and that αx n, x n+1, x n+1 ) ηx n, x n+1, x n+1 ). Then T x n, T x n, T 3 x n [0, 1] for all n N. That is, ηt x n, T x n, T x n ) αt x n, x, x), and ηt x n, T 3 x n, T 3 x n ) αt x n, x, x) hold for all n N. Clearly, α0, T 0, T 0) η0, T 0, T 0). Let αx, y, z) ηx, T x, T x). Now, if x / [0, 1], then, which is not possible. So, x, y, z [0, 1]. Therefore, GT x, T y, T z) = 1 4 max x y, y z, z x = 1 max x y x + y, y z y + z, z x z + x 4 1 max x y, y z, z x 1 Mx, y, z) + LGy, T z, T x).

17 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), Therefore, T is an α-η-ćirić strong almost contraction. Hence all the conditions for Theorem 4.3 are satisfied. Hence T has a fixed point. As an application of the above result, we obtain the following Suzuki type fixed point theorem in the setup of G-metric spaces. Theorem 4.5. Let X be a complete G-metric space and T a self-mapping on X. If there exists r [0, 1) such that for any x, y in X 1 Gx, T x, T x) Gx, y, y) implies that GT x, T y, T y) rmx, y, y) + LGy, T x, T x), 4.4) 1 + r where Mx, y, y) = max Gx, y, y), Gx, T x, T x), Gy, T y, T y), then T has a fixed point. Proof. Define α, η : X X X [0, ) by Gx, T y, T y) + Gy, T y, T x) + Gy, T x, T y), αx, y, y) = Gx, y, y) and ηx, y, y) = λr)gx, y, y), where 0 r < 1 and λr) = 1 1+r. As for any x, y X, λr)gx, y, y) Gx, y, y) so we have ηx, y, y) αx, y, y). Thus, conditions i) and ii) of Theorem 4.3 hold. Let x n be a sequence in X with x n x as n. If GT x n, T x n, T x n ) = 0 for some n. Then T x n = T x n implies that T x n is a fixed point of T and the result follows. Suppose that T x n T x n for all n N. As λr)gt x n, T x n, T x n ) GT x n, T x n, T x n ) for all n N, so from 4.4) we get, where, If GT x n, T 3 x n, T 3 x n ) rmt x n, T x n, T x n ) + LGT x n, T x n, T x n ) = rmt x n, T x n, T x n ), 4.5) MT x n, T x n, T x n ) = max GT x n, T x n, T x n ), GT x n, T x n, T x n ), GT x n, T 3 x n, T 3 x n ), GT x n, T 3 x n, T 3 x n ) + GT x n, T x n, T x n ) max GT x n, T x n, T x n ), GT x n, T 3 x n, T 3 x n ), GT x n, T x n, T x n ) + GT x n, T 3 x n, T 3 x n ) for some n, then 4.5) becomes = maxgt x n, T x n, T x n ), GT x n, T 3 x n, T 3 x n ). maxgt x n, T x n, T x n ), GT x n, T 3 x n, T 3 x n ) = GT x n, T 3 x n, T 3 x n ), GT x n, T 3 x n, T 3 x n ) rgt x n, T 3 x n, T 3 x n ), a contradiction. Hence GT x n, T 3 x n, T 3 x n ) rgt x n, T x n, T x n ). 4.6)

18 M. Abbas, A. Hussain, B. Popović, S. Radenović, J. Nonlinear Sci. Appl ), If there exists n 0 N such that ηt x n0, T x n0, T x n0 ) > αt x n0, x, x), and ηt x n0, T 3 x n0, T 3 x n0 ) > αt x n0, x, x), then we have and Thus from 4.6) we obtain that λr)gt x n0, T x n0, T x n 0 ) > GT x n0, x, x), λr)gt x n0, T 3 x n0, T 3 x n0 ) > GT x n0, T x n0, x 0 ). GT x n0, T x n0, T x n0 ) GT x n0, x, x) + Gx, T x n0, T x n0 ) a contradiction. Hence, either < λr)gt x n0, T x n0, T x n0 ) + λr)gt x n0, T 3 x n0, T 3 x n0 ) λr)gt x n0, T x n0, T x n0 ) + rλr)gt x n0, T x n0, T x n0 ) = λr)1 + r)gt x n0, T x n0, T x n0 ) = GT x n0, T x n0, T x n0 ), ηt x n, T x n, T x n ) αt x n, x, x), or ηt x n, T 3 x n, T 3 x n ) αt x n, x, x) holds for all n N and the condition iv) of Theorem 4.3 holds. Now ηx, T x, T x) αx, y, y) gives that λr)gx, T x, T x) Gx, y, y). Thus from 4.4) we obtain GT x, T y, T y) rmx, y, y) + LGy, T x, T x). Hence all the conditions of Theorem 4.3 are satisfied and the result follows. Acknowledgment The third author was supported in part by the Serbian Ministry of Science and Technological Developments Project: Methods of Numerical and Nonlinear Analysis with Applications, grant number #17400). References [1] M. Abbas, S. H. Khan, T. Nazir, Common fixed points of R-weakly commuting maps in generalized metric spaces, Fixed Point Theory Appl., ), 11 pages. 1 [] M. Abbas, A. R. Khan, T. Nazir, Coupled common fixed point results in two generalized metric spaces, Appl. Math. Comput., ), [3] M. Abbas, T. Nazir, S. Radenović, Some periodic point results in generalized metric spaces, Appl. Math. Comput., ), [4] M. Abbas, T. Nazir, S. Radenović, Common fixed point of generalized weakly contractive maps in partially ordered G-metric spaces, Appl. Math. Comput., 18 01), [5] M. Abbas, B. E. Rhoades, Common fixed point results for noncommuting mappings without continuity in generalized metric spaces, Appl. Math. Comput., ), [6] R. P. Agarwal, Z. Kadelburg, S. Radenović, On coupled fixed point results in asymmetric G-metric spaces, J. Inequal. Appl., ), 1 pages. [7] M. A. Alghamdi, S. H. Alnafei, S. Radenović, N. Shahzad, Fixed point theorems for convex contraction mappings on cone metric spaces, Math. Comput. Modelling, ), [8] M. A. Alghamdi, E. Karapınar, G--ψ-contractive type mappings in G-metric spaces, Fixed Point Theory Appl., ), 17 pages. 1.7, 1.8 [9] H. Aydi, B. Damjanović, B. Samet, W. Shatanawi, Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces, Math. Comput. Modelling, ), [10] H. Aydi, M. Postolache, W. Shatanawi, Coupled fixed point results for ψ, φ)-weakly contractive mappings in ordered G-metric spaces, Comput. Math. Appl., 63 01), [11] H. Aydi, W. Shatanawi, C. Vetro, On generalized weak G-contraction mapping in G-metric spaces, Comput. Math. Appl., 6 011),

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