Some New Common Fixed Point Theorems under Strict Contractive Conditions in G-Metric Spaces

Abstract

We introduce some new types of pairs of mappings (𝑓,𝑔) on G-metric space called G-weakly commuting of type (𝐴_𝑓) and G-R-weakly commuting of type (𝐴_𝑓). We obtain also several common fixed point results for these mappings under certain contractive condition in G-metric space. Also some examples illustrated to support our results, and comparison between different types of pairs of mappings are studied.

1. Introduction and Preliminaries

The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of strong research activity and, being the area of the fixed point theory, has very important application in applied mathematics and sciences. In 1976 Jungck [1] proved a common fixed point theorem for commuting maps, but his results required the continuity of one of the maps.

Sessa [2] in 1982 first introduced a weaker version of commutativity for a pair of self-maps, and it is shown in Sessa [2] that weakly commuting pair of maps in metric pace is commuting, but the converse may not be true.

Later, Jungck [3] introduced the notion of compatible mappings in order to generalize the concepts of weak commutativity and showed that weak commuting map is compatible, but the reverse implication may not hold.

In 1996, Jungck [4] 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 commuting maps ⇒ compatible maps ⇒ weakly Compatible maps. Recently various authors have introduced coincidence points results for various classes of mappings on metric spaces for more detail of coincidence point theory and related results see [5–7].

However, the study of common fixed point of noncompatible mappings has recently been initiated by Pant (see [8, 9]).

In 2002 Amari and El Moutawakil [10] defined a new property called E.A. property which generalizes the concept of noncompatible mappings, and they proved some common fixed point theorem.

Definition 1.1 (see [10]). Let 𝑆 and 𝑇 be two self-mappings of a metric space (𝑋,𝑑). We say that 𝑇 and 𝑆 satisfy the E.A. property if there exists a sequence (𝑥_𝑛) such that lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑡,forsome𝑡∈𝑋.(1.1)

In 2005 Zead Mustafa and Brailey Sims introduced the notion of G-metric spaces as generalization of the concept of ordinary metric spaces. Based on the notion of G-metric space Mustafa et al. [11–15] obtained some fixed point results for mapping satisfying different contractive conditions on complete G-metric space, while in [16] the completeness property was omitted and replaced by sufficient conditions, where these conditions do not imply the completeness property.

Chugh et al. [17] obtained some fixed point results for maps satisfying property P in G-metric spaces. Saadati et al. [18] studied fixed point of contractive mappings in partially ordered G-metric spaces. Shatanawi obtained fixed points of 𝜙-maps in G-metric spaces [19] and a number of fixed point results for the two weakly increasing mappings with respect to partial ordering in G-metric spaces [20]. In [21, 22] authors established coupled fixed point theorems in a partially ordered G-metric spaces.

Abbas and rhoades [23] proved several common fixed points for noncommuting mappings without continuity in G-metric space, and they show that the results 2.3–2.6 generalize Theorems 2.1–2.4 of [11].

In [24] Abbas et al. proved several unique common fixed points for mappings satisfying E.A. property under generalized contraction condition and show that Corollary 3.1 extends the main result in [13] (Theorem 2.1) and Corollary 3.3 is G-version of Theorem 2 from [10] in the case of two self-mappings. Also this corollary is in relation with Theorem 2.5 of [23].

In [25] the authors proved some coupled coincidence and common coupled fixed point results for mappings defined on a set equipped with two G-metric spaces and these results do not rely on continuity of mappings involved therein as well as they show that Theorem 2.13 is an extension and generalization of (1) Theorem 2.2, Corollary 2.3, Theorem 2.6, Corollaries 2.7 and 2.8 in [26] and (2) Theorem 2.4 and Corollary 2.5 in [27].

Aydi et al. [28] established some common fixed point results for two mappings 𝑓 and 𝑔 on G-metric spaces with assumption that 𝑓 is a generalized weakly G-contraction mappings of type A and B with respect to 𝑔.

In this paper, we define new types of self-maps 𝑓 and 𝑔 on 𝐺-metric space called 𝐺-weakly commuting of type 𝐴_𝑓 and 𝐺-𝑅-weakly commuting of type 𝐴_𝑓. Also we obtain several common fixed point results for these mappings under certain contractive condition in 𝐺-metric space, and some examples are illustrated to support our results, and a comparison between different types of pairs of mappings are stated.

The following definitions and results will be needed in the sequel.

Definition 1.2 (see [29]). A 𝐺-metric space is a pair (𝑋,𝐺), where 𝑋 is a nonempty set, and 𝐺 is a nonnegative real-valued function defined on 𝑋×𝑋×𝑋 such that for all 𝑥,𝑦,𝑧,𝑎∈𝑋 we have(G1)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧,(G2)0<𝐺(𝑥,𝑥,𝑦);forall𝑥,𝑦∈𝑋,𝑤𝑖𝑡ℎ𝑥≠𝑦,(G3)𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧),forall𝑥,𝑦,𝑧∈𝑋,𝑤𝑖𝑡ℎ𝑧≠𝑦,(G4)𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)=⋯, (symmetry in all three variables),(G5)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧), for all 𝑥,𝑦,𝑧,𝑎∈𝑋, (rectangle inequality).
The function 𝐺 is called 𝐺-metric on 𝑋.

Every 𝐺-metric on 𝑋 defines a metric 𝑑_𝐺 on 𝑋 by 𝑑_𝐺(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦)+𝐺(𝑦,𝑥,𝑥)∀𝑥,𝑦∈𝑋.(1.2)

Example 1.3 (see [29]). Let (𝑋,𝑑) be a metric space, and define 𝐺_𝑠 and 𝐺_𝑚 on 𝑋×𝑋×𝑋 to 𝐑⁺ by 𝐺𝐺_𝑠(𝑥,𝑦,𝑧)=𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)+𝑑(𝑥,𝑧),_𝑚(𝑥,𝑦,𝑧)=max{𝑑(𝑥,𝑦),𝑑(𝑦,𝑧),𝑑(𝑥,𝑧)},(1.3) for all 𝑥,𝑦,𝑧∈𝑋. Then (𝑋,𝐺_𝑠) and (𝑋,𝐺_𝑚) are 𝐺-metric spaces.

Example 1.4 (see [29]). Let 𝑋=𝐑, and define 𝐺∶𝑋×𝑋×𝑋→𝐑⁺, by 𝐺(𝑥,𝑦,𝑧)=⎧⎪⎨⎪⎩||1𝑥+−||𝑥𝑦||−+𝑦||||𝑦+−||𝑧𝑦||−+𝑧|||𝑥+−|𝑥𝑧|−,𝑧|,ioooftrrhataelhllrlewy𝑥𝑥i,s,ae𝑦𝑦r,,e,aaannldlds𝑧𝑧traiacrretelyzesnrteorgi,cattliyvepositive(1.4) then (𝑋,𝐺) is G-metric space.

Definition 1.5 (see [29]). A sequence (𝑥_𝑛) in a 𝐺-metric space 𝑋 is said to converge if there exists 𝑥∈𝑋 such that lim_{𝑛,𝑚→∞}𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)=0, and one says that the sequence (𝑥_𝑛) is 𝐺-convergent to 𝑥. We call 𝑥 the limit of the sequence (𝑥_𝑛) and write 𝑥_𝑛→𝑥 or lim_𝑛→∞𝑥_𝑛=𝑥 (through this paper we mean by 𝐍 the set of all natural numbers).

Proposition 1.6 (see [29]). Let 𝑋 be 𝐺-metric space. Then the following statements are equivalent:(1)(𝑥_𝑛) is 𝐺-convergent to 𝑥,(2)𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)→0, as 𝑛→∞,(3)𝐺(𝑥_𝑛,𝑥,𝑥)→0, as 𝑛→∞,(4)𝐺(𝑥_𝑚,𝑥_𝑛,𝑥)→0, as 𝑚,𝑛→∞.

Definition 1.7 (see [29]). In a 𝐺-metric space 𝑋, a sequence (𝑥_𝑛) is said to be 𝐺-Cauchy if given 𝜀>0, there is 𝑁∈𝐍 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜀, for all 𝑛,𝑚,𝑙≥𝑁. That is 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→∞.

Proposition 1.8 (see [29]). In a 𝐺-metric space 𝑋, the following statements are equivalent:(1)the sequence (𝑥_𝑛) is 𝐺-Cauchy;(2)for every 𝜀>0,𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝑁∈𝐍 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜀, for all 𝑛,𝑚≥𝑁.

Definition 1.9 (see [29]). A 𝐺-metric space (𝑋,𝐺) is called symmetric 𝐺-metric space if 𝐺(𝑥,𝑦,𝑦)=𝐺(𝑦,𝑥,𝑥) for all 𝑥,𝑦∈𝑋 and called nonsymmetric if it is not symmetric.

Example 1.10. Let 𝑋=𝐍 be the set of all natural numbers, and define  𝐺∶𝑋×𝑋×𝑋→𝐑 such that for all 𝑥,𝑦,𝑧∈𝑋:  𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧, 𝐺(𝑥,𝑦,𝑦)=𝑥+𝑦, if 𝑥<𝑦, 𝐺(𝑥,𝑦,𝑦)=𝑥+𝑦+1/2, if 𝑥>𝑦, 𝐺(𝑥,𝑦,𝑧)=𝑥+𝑦+𝑧 if 𝑥≠𝑦≠𝑧 and symmetry in all three variables.
Then, (𝑋,𝐺) is G-metric space and nonsymmetric since if 𝑥<𝑦, we have 𝐺(𝑥,𝑦,𝑦)=𝑥+𝑦≠𝑥+𝑦+1/2=𝐺(𝑦,𝑥,𝑥).

Proposition 1.11 (see [29]). Let 𝑋 be a 𝐺-metric space; then the function 𝐺(𝑥,𝑦,𝑧) is jointly continuous in all three of its variables.

Definition 1.12 (see [29]). A 𝐺-metric space 𝑋 is said to be complete if every 𝐺-Cauchy sequence in 𝑋 is 𝐺-convergent in 𝑋.

Definition 1.13 (see [23]). Let 𝑓 and 𝑔 be self-maps of a set 𝑋. If 𝑤=𝑓𝑥=𝑔𝑥 for some 𝑥∈𝑋, then 𝑥 is called a coincidence point of 𝑓 and 𝑔, and 𝑤 is called a point of coincidence of 𝑓 and 𝑔.

Recall that a pair of self-mappings are called weakly compatible if they commute at their coincidence points.

Proposition 1.14 (see [23]). Let 𝑓 and 𝑔 be weakly compatible self-maps of a set 𝑋. If 𝑓 and 𝑔 have a unique point of coincidence 𝑤=𝑓𝑥=𝑔𝑥, then 𝑤 is the unique common fixed point of 𝑓 and 𝑔.

In 2001, Abbas et al. [30] introduce a new type of pairs of mappings (𝑓,𝑔) called R-weakly commuting and they proved a unique common fixed point of four R-weakly commuting, maps satisfying generalized contractive condition.

Definition 1.15 (see [30]). Let 𝑋 be a G-metric space, and let 𝑓 and 𝑔 be two self-mappings of 𝑋; then 𝑓 and 𝑔 are called R-weakly commuting if there exists a positive real number 𝑅 such that 𝐺(𝑓(𝑔(𝑥)),𝑓(𝑔(𝑥)),𝑔(𝑓(𝑥)))≤𝑅𝐺(𝑓(𝑥),𝑓(𝑥),𝑔(𝑥))holdforeach𝑥∈𝑋.(1.5)

Very recently, Mustafa et al. [31] introduce some new types of pairs of mappings (𝑓,𝑔) on G-metric space called G-weakly commuting of type 𝐺_𝑓 and G-R-weakly commuting of type 𝐺_𝑓, and they obtained several common fixed point results by using E.A. property.

Definition 1.16 (see [31]). A pair of self-mappings (𝑓,𝑔) of a G-metric space (𝑋,𝐺) is said to be 𝐺-weakly commuting of type 𝐺_𝑓 if 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑓𝑓𝑥)≤𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥),∀𝑥∈𝑋.(1.6)

Definition 1.17 (see [31]). A pair of self-mappings (𝑓,𝑔) of a 𝐺-metric space (𝑋,𝐺) is said to be 𝐺-𝑅-weakly commuting of type 𝐺_𝑓 if there exists some positive real number 𝑅 such that 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑓𝑓𝑥)≤𝑅𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥),∀𝑥∈𝑋.(1.7)

Remark 1.18. The 𝐺-𝑅-weakly commuting maps of type 𝐺_𝑓 are 𝑅-weakly commuting since 𝐺(𝑓𝑔𝑥,𝑓𝑔𝑥,𝑔𝑓𝑥)≤𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑓𝑓𝑥)≤𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥), but the converse need not be true.

2. Main Results

2.1. New Concepts and Some Properties

In this section we introduce the concept of G-weakly commuting of type 𝐴_𝑓 for pairs of mapping (𝑓,𝑔) and comparison between this concept and Definitions 1.15, 1.16, and 1.17 is studied as well as examples illustrated to show that these types of mappings are different.

First, we introduce the following concepts as follows.

Definition 2.1. A pair of self-mappings (𝑓,𝑔) of a G-metric space (𝑋,𝐺) is said to be G-weakly commuting of type 𝐴_𝑓 if 𝐺(𝑓𝑔𝑥,𝑔𝑔𝑥,𝑓𝑓𝑥)≤𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥),∀𝑥∈𝑋.(2.1)

Definition 2.2. A pair of self-mappings (f,g) of a G-metric space (𝑋,𝐺) is said to be G-R-weakly commuting of type 𝐴_𝑓 if there exists some positive real number 𝑅 such that 𝐺(𝑓𝑔𝑥,𝑔𝑔𝑥,𝑓𝑓𝑥)≤𝑅𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥),∀𝑥∈𝑋.(2.2)

Remark 2.3. The 𝐺-weakly commuting maps of type 𝐴_𝑓 are 𝐺-𝑅-weakly commuting of type 𝐴_𝑓. Reciprocally, if 𝑅≤1, then 𝐺-𝑅-weakly commuting maps of type 𝐴_𝑓 are 𝐺-weakly commuting of type 𝐴_𝑓.
If we interchange 𝑓 and 𝑔 in (2.1) and (2.2), then the pair of mappings (𝑓,𝑔) is called 𝐺-weakly commuting of type 𝐴_𝑔 and 𝐺-𝑅-weakly commuting of type 𝐴_𝑔, respectively.

Example 2.4. Let 𝑋=[0,3/4], with the G-metric 𝐺(𝑥,𝑦,𝑧)=|𝑥−𝑦|+|𝑦−𝑧|+|𝑥−𝑧|, for all 𝑥,𝑦,𝑧∈𝑋. Define 𝑓,𝑔∶𝑋→𝑋 by, 𝑓(𝑥)=(1/4)𝑥²,𝑔(𝑥)=𝑥²; then as an easy calculation one can show that 𝐺(𝑓𝑔𝑥,𝑔𝑔𝑥,𝑓𝑓𝑥)=(126/64)𝑥⁴≤𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥)=(6/4)𝑥²,forall𝑥∈𝑋. Then the pair (𝑓,𝑔) is G-Weakly commuting of type 𝐴_𝑓 and G-R-Weakly commuting of type 𝐴_𝑓.

Example 2.5. Let 𝑋=[2,∞], with the G-metric 𝐺(𝑥,𝑦,𝑧)=|𝑥−𝑦|+|𝑦−𝑧|+|𝑥−𝑧|, for all 𝑥,𝑦,𝑧∈𝑋. Define 𝑓,𝑔∶𝑋→𝑋 by, 𝑓(𝑥)=𝑥+1,𝑔(𝑥)=2𝑥+1, then for 𝑥=2 we see that 𝐺(𝑓𝑔𝑥,𝑔𝑔𝑥,𝑓𝑓𝑥)=𝐺(𝑔𝑓𝑥,𝑔𝑔𝑥,𝑓𝑓𝑥)=20 and 𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥)=𝐺(𝑔𝑥,𝑓𝑥,𝑔𝑥)=6. Therefore the pair (𝑓,𝑔) is not 𝐺-weakly commuting of type 𝐴_𝑓 or 𝐴_𝑔, but it is G-R-weakly commuting of type 𝐴_𝑓 (and 𝐴_𝑔) for 𝑅≥4.

The following examples show a pair of mappings (𝑓,𝑔) that 𝐺-weakly commuting of type 𝐺_𝑓 need not be 𝐺-weakly commuting of type 𝐴_𝑓.

Example 2.6. Let 𝑋=[0,89/100], with the G-metric 𝐺(𝑥,𝑦,𝑧)=max{|𝑥−𝑦|,|𝑦−𝑧|,|𝑥−𝑧|}, for all 𝑥,𝑦,𝑧∈𝑋. Define 𝑓(𝑥)=(1/4)𝑥²,𝑔(𝑥)=𝑥²; then we see that 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑓𝑓𝑥)=(15/16)𝑥⁴ and 𝐺(𝑓𝑥,𝑔𝑥,𝑓𝑥)=(3/4)𝑥², while as an easy calculation one can show that for 𝑥=88/100 we have 𝐺(𝑓(𝑔(𝑥),𝑔𝑔(𝑥),𝑓(𝑓(𝑥)))=(59/100)≰𝐺(𝑔(𝑥),𝑓(𝑥),𝑔(𝑥))=(58/100). Therefore the pair (𝑓,𝑔) is not 𝐺-weakly commuting of type 𝐴_𝑓, but it is G-weakly commuting of type 𝐺_𝑓.

The following example shows that(1)a pair of mappings (𝑓,𝑔) that is 𝐺-weakly commuting of type 𝐴_𝑓 need not be 𝐺-weakly commuting of type 𝐴_𝑔;(2)a pair of mappings (𝑓,𝑔) that is 𝐺-weakly commuting of type 𝐴_𝑓 need not be 𝐺-weakly commuting of type 𝐺_𝑓;(3)a pair of mappings (𝑓,𝑔) that is 𝐺-𝑅-weakly commuting of type 𝐴_𝑓 need not be 𝑅-weakly commuting;

Example 2.7. Let 𝑋=[2,9] and 𝐺(𝑥,𝑦,𝑧)=𝑚𝑎𝑥{|𝑥−𝑦|,|𝑦−𝑧|,|𝑥−𝑧|} for all 𝑥,𝑦,𝑧∈𝑋. Define the mappings 𝑓,𝑔∶𝑋→𝑋 by 𝑓(𝑥)=⎧⎪⎨⎪⎩265iiifff𝑥2𝑥<=>𝑥25,,≤5,𝑔(𝑥)=⎧⎪⎨⎪⎩296iiifff𝑥2𝑥<=>𝑥25,.≤5,(2.3)
Then, 𝑓𝑔((𝑔𝑔((𝑥𝑥))))==2526iiififff𝑥𝑥𝑥𝑥==>>2222,,,,𝑓𝑔((𝑓𝑓((𝑥𝑥))))==⎧⎪⎨⎪⎩⎧⎪⎨⎪⎩269256iiiiifffffif𝑥𝑥𝑥2𝑥2<=>=><𝑥2525𝑥,,,.≤≤55,,(2.4)
Moreover, ||𝑓(𝑥)−𝑔(𝑥)||=⎧⎪⎨⎪⎩031iiifff𝑥2𝑥<=>𝑥25,.≤5,(2.5)
If 𝑥=2, we have 𝐺(𝑓(𝑔(𝑥)),𝑔(𝑔(𝑥)),𝑓(𝑓(𝑥)))=0=𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥)).
If 2<𝑥≤5, we have 𝐺(𝑓(𝑔(𝑥)),𝑓(𝑓(𝑥)),𝑔(𝑔(𝑥)))=max{0,1,1}≤3=𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥)).(2.6)
If 𝑥>5, then 𝐺(𝑓(𝑔(𝑥)),𝑓(𝑓(𝑥)),𝑔(𝑔(𝑥)))=max{1,1,0}≤1=𝐺(𝑔(𝑥),𝑓(𝑥),𝑔(𝑥)).(2.7)
Thus, 𝑓 and 𝑔 are 𝐺 weakly commuting of type 𝐴_𝑓, but for 𝑥=6, we have 𝐺(𝑔(𝑓(6)),𝑓(𝑓(6)),𝑔(𝑔(6)))=max{3,3,0}≰1=𝐺(𝑔(6),𝑓(6),𝑔(6)).(2.8)
Therefore, the pair (𝑓,𝑔) is not G-weakly commuting of type 𝐴_𝑔, but it is G-weakly commuting of type 𝐴_𝑓.
Also for 𝑥=7, we have 𝐺(𝑓(𝑔(7)),𝑔(𝑓(7)),𝑓(𝑓(7)))=4≰1=𝐺(𝑔(6),𝑓(6),𝑔(6)).(2.9)
Therefore, the pair (𝑓,𝑔) is not G-weakly commuting of type 𝐺_𝑓.
AS an easy calculation one can see that (𝑓,𝑔) are 𝐺-𝑅-weakly commuting of type 𝐴_𝑔 for 𝑅=3; but for 𝑥=6 we have 𝐺(𝑓(𝑔(6)),𝑓(𝑔(6)),𝑔(𝑓(6)))=4≰3𝐺(𝑓(6),𝑓(6),𝑔(6))=3, hence (𝑓,𝑔) is NOT 𝑅-weakly commuting for 𝑅=3.

Lemma 2.8. If 𝑓 and 𝑔 are 𝐺-weakly commuting of type 𝐴_𝑓 or 𝐺-𝑅-weakly commuting of type 𝐴_𝑓, then 𝑓 and 𝑔 are weakly compatible.

Proof. Let 𝑥 be a coincidence point of 𝑓 and 𝑔, that is, 𝑓(𝑥)=𝑔(𝑥); then if the pair (𝑓,𝑔) is 𝐺-weakly commuting of type 𝐴_𝑓, we have 𝐺(𝑓(𝑔(𝑥)),𝑔(𝑓(𝑥)),𝑓(𝑔(𝑥)))=𝐺(𝑓(𝑔(𝑥)),𝑔(𝑔(𝑥)),𝑓(𝑓(𝑥)))≤𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥))=0.(2.10)
It follows 𝑓(𝑔(𝑥))=𝑔(𝑓(𝑥)); then they commute at their coincidence point.
Similarly, if the pair (𝑓,𝑔) is 𝐺-𝑅-weakly commuting of type 𝐺_𝑓, we have 𝐺(𝑓(𝑔(𝑥)),𝑔(𝑓(𝑥)),𝑓(𝑔(𝑥)))=𝐺(𝑓(𝑔(𝑥)),𝑔(𝑔(𝑥)),𝑓(𝑓(𝑥)))≤𝑅𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥))=0.(2.11)
Thus 𝑓(𝑔(𝑥))=𝑔(𝑓(𝑥)); then the pair (𝑓,𝑔) is weakly compatible.

The following example shows that(1)the converse of Lemma 2.8 fails (for the case of 𝐺-weakly commutativity),(2)a pair of mappings (𝑓,𝑔) that is 𝑅-weakly commuting need not be 𝐺-𝑅-weakly commuting of type 𝐴_𝑓,(3)a pair of mappings (𝑓,𝑔) that is 𝑅-weakly commuting need not be 𝐺-𝑅-weakly commuting of type 𝐺_𝑓.

Example 2.9. Let 𝑋=[1,+∞) and 𝐺(𝑥,𝑦,𝑧)=max{|𝑥−𝑦|,|𝑦−𝑧|,|𝑥−𝑧|}. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓(𝑥)=2𝑥−1 and 𝑔(𝑥)=𝑥². We see that 𝑥=1 is the only coincidence point and 𝑓(𝑔(1))=𝑓(1)=1 and 𝑔(𝑓(1))=𝑔(1)=1, so 𝑓 and 𝑔 are weakly compatible.
But, by an easy calculation, one can see that for 𝑥=3 we have, 𝐺(𝑓(𝑔(𝑥)),𝑔(𝑔(𝑥)),𝑓(𝑓(𝑥)))=72≰4=𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥)).(2.12) Therefore, 𝑓 and 𝑔 are not 𝐺-weakly commuting of type 𝐴_𝑓.
Also, we see that 𝐺(𝑓(𝑔(𝑥)),𝑓(𝑔(𝑥)),𝑔(𝑓(𝑥)))=2𝑥²−4𝑥+2≤2𝐺(𝑓(𝑥),𝑓(𝑥),𝑔(𝑥))=2(𝑥²−2𝑥+1); therefore the mappings (𝑓,𝑔) are 𝑅-weakly commuting for 𝑅=2, but for 𝑥=4 we have 𝐺(𝑓(𝑔(4)),𝑔(𝑔(4)),𝑓(𝑓(4)))=243≰2𝐺(𝑓(4),𝑔(4),𝑓(4))=18; hence (𝑓,𝑔) are not 𝐺-𝑅-weakly commuting of type 𝐴_𝑓 for 𝑅=2 and 𝐺(𝑓(𝑔(4)),𝑔(𝑓(4)),𝑓(𝑓(4)))=49≰2𝐺(𝑓(4),𝑔(4),𝑓(4))=18; hence (𝑓,𝑔) are not 𝐺-𝑅-weakly commuting of type 𝐺_𝑓 for 𝑅=2.

Now, we rewrite Definition 1.1 on 𝐺-metric spaces setting.

Definition 2.10. Let 𝑆 and 𝑇 be two self-mappings of a 𝐺-metric space (𝑋,𝐺). We say that 𝑇 and 𝑆 satisfy the E.A. property if there exists a sequence (𝑥_𝑛) such that (𝑇𝑥_𝑛) and (𝑆𝑥_𝑛)𝐺-converge to 𝑡 for some 𝑡∈𝑋; that is, thanks to Proposition 1.6, lim_𝑛⟶∞𝐺𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝑡=lim_𝑛⟶∞𝐺𝑆𝑥_𝑛,𝑆𝑥_𝑛,𝑡=0.(2.13)

Remark 2.11. In view of (1.2) and Example 1.3, Definition 1.1 is equivalent to Definition 2.10.

In the following example, we show that if 𝑓 and 𝑔 satisfy the E.A. property, then the pair (𝑓,𝑔) need not be 𝐺-weakly commuting of type 𝐴_𝑓.

Example 2.12. We return to Example 2.9. Let 𝑥_𝑛=1+(1/3𝑛). We have lim_𝑛→∞𝑓(𝑥_𝑛)=lim_𝑛→∞(1+(2/3𝑛))=1, and lim_𝑛→∞𝑔(𝑥_𝑛)=lim_𝑛→∞(1+(1/3𝑛))²=1, therefore, lim_𝑛→∞𝑓(𝑥_𝑛)=lim_𝑛→∞𝑔(𝑥_𝑛)=1∈[1,∞). Then 𝑓 and 𝑔 satisfy the E.A. property, but we know that the pair (𝑓,𝑔) is not 𝐺-weakly commuting of type 𝐴_𝑓.

Following Matkowski (see [32]), let Φ be the set of all functions 𝜙 such that 𝜙∶[0,∞)→[0,∞) be a nondecreasing function with lim_𝑛→∞𝜙^𝑛(𝑡)=0 for all 𝑡∈(0,+∞). If 𝜙∈Φ, then 𝜙 is called Φ-map. If 𝜙 is Φ-map, then it is easy to show that(1)𝜙(𝑡)<𝑡 for all 𝑡∈(0,+∞),(2)𝜙(0)=0.

2.2. Some Common Fixed Point Results

We start this section with the following theorem.

Theorem 2.13. Let (𝑋,𝐺) be a 𝐺-metric space; suppose mappings 𝑓,𝑔∶𝑋→𝑋 satisfy the following condition: (1)𝑓 and 𝑔 be 𝐺-weakly commuting of type 𝐴_𝑓, (2)𝑓(𝑋)⊆𝑔(𝑋),(3)𝑔(𝑋) is 𝐺-complete subspace of 𝑋, (4)𝐺(𝑓(𝑥),𝑓(𝑦),𝑓(𝑧))≤𝜙(𝑀(𝑥,𝑦,𝑧)),𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋,𝑤ℎ𝑒𝑟𝑒𝑀(𝑥,𝑦,𝑧)=max⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐺(𝑔(𝑥),𝑔(𝑦),𝑔(𝑧)),𝐺(𝑔(𝑥),𝑓(𝑥),𝑓(𝑥)),112𝐺(𝑔(𝑥),𝑓(𝑦),𝑓(𝑦)),2𝐺𝐺((𝑔𝑔((𝑦𝑥)),,𝑓𝑓(𝑧(𝑧)),,𝑓𝑓(𝑧(𝑧)))),,𝐺𝐺𝐺((𝑔(𝑔𝑔((𝑧(𝑧𝑦))),,,𝑓𝑓𝑓((𝑧(𝑦𝑦))),,,𝑓𝑓𝑓((𝑧(𝑦𝑦)))))),,𝐺𝐺(𝑔(𝑔(𝑧(𝑦)),,𝑓𝑓(𝑥(𝑥)),,𝑓𝑓(𝑥(𝑥)))),,⎫⎪⎪⎪⎬⎪⎪⎪⎭.(2.14) Then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Let 𝑥₀∈𝑋, and then choose 𝑥₁∈𝑋 such that 𝑓(𝑥₀)=𝑔(𝑥₁) and 𝑥₂∈𝑋 where 𝑓(𝑥₁)=𝑔(𝑥₂); then by induction we can define a sequence (𝑦_𝑛)∈𝑋 as follows: 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1,𝑛∈𝑁∪{0}(2.15) We will show that the sequence (𝑦_𝑛) is 𝐺-cauchy sequence: 𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1=𝐺𝑓𝑥_𝑛,𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+1≤𝜙𝑀𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,(2.16) where 𝑀𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=max⎧⎪⎪⎨⎪⎪⎩𝐺𝑔𝑥^𝑛1,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1,𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛,2𝐺𝑔𝑥^𝑛,𝑓𝐺𝑥𝑔_𝑛+1𝑥,𝑓𝑥_𝑛+1,𝐺𝑔𝑥_𝑛+1,𝑓𝑥_𝑛,𝑓𝑥_𝑛_𝑛+1,=max⎧⎪⎨⎪⎩𝐺𝑦_𝑛−1,𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+1,⎫⎪⎪⎬⎪⎪⎭,𝑦_𝑛,𝑦_𝑛,12𝐺𝑦^𝑛−1𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1,,𝑦_𝑛,𝑦_𝑛,𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1,⎫⎪⎬⎪⎭.(2.17)
We will have different cases. Case (1): if 𝑀(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)=𝐺(𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1), then 𝐺(𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1)≤𝜙(𝐺(𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1)) <𝐺(𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1), which is contradiction. Case (2): if 𝑀(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)=(1/2)𝐺(𝑦_𝑛−1,𝑦_𝑛+1,𝑦_𝑛+1), then in this case we havemax{𝐺(𝑦_𝑛−1,𝑦_𝑛,𝑦_𝑛),𝐺(𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1)}<(1/2)𝐺(𝑦_𝑛−1,𝑦_𝑛+1,𝑦_𝑛+1), which implies that 𝐺𝑦_𝑛−1,𝑦_𝑛,𝑦_𝑛+𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1<𝐺𝑦_𝑛−1,𝑦_𝑛+1,𝑦_𝑛+1,(2.18)
but from G-metric property (G5) we have 𝐺𝑦_𝑛−1,𝑦_𝑛+1,𝑦_𝑛+1≤𝐺𝑦_𝑛−1,𝑦_𝑛,𝑦_𝑛+𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1.(2.19)
Thus, from (2.18) and (2.19) we see that case (2) is impossible.
Then, we must have the case 𝑀𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝐺𝑦_𝑛−1,𝑦_𝑛,𝑦_𝑛.(2.20)
Thus, for 𝑛∈𝐍∪{0} and from (2.16) we have, 𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1≤≤𝜙𝜙₂𝐺𝑦_𝑛−1,𝑦_𝑛,𝑦_𝑛⋮≤𝜙_𝑛𝐺𝑦_𝑛−2,𝑦_𝑛−1,𝑦_𝑛−1𝐺𝑦₀,𝑦₁,𝑦₁…(1).(2.21)
Given𝜖>0, since lim_𝑛→∞𝜙^𝑛(𝐺(𝑦₀,𝑦₁,𝑦₁))=0, and 𝜙(𝜖)<𝜖, there is an integer 𝑛_𝑜∈𝐍, such that 𝜙^𝑛𝐺𝑦₀,𝑦₁,𝑦₁<𝜖−𝜙(𝜖),∀𝑛≥𝑛₀.(2.22)
Hence, we have 𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1≤𝜙^𝑛𝐺𝑦₀,𝑦₁,𝑦₁<𝜖−𝜙(𝜖).(2.23)
Now for 𝑚,𝑛∈𝐍; 𝑚>𝑛, we claim that 𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑚<𝜖,∀𝑚≥𝑛≥𝑛₀.(2.24)
We will prove (2.24) by induction on 𝑚.
Inequality (2.24) holds for 𝑚=𝑛+1, by using (2.23) and the fact that 𝜖−𝜙(𝜖)<𝜖.
Assume (2.24) holds for 𝑚=𝑘. For 𝑚=𝑘+1, we have 𝐺𝑦_𝑛,𝑦_𝑘+1,𝑦_𝑘+1≤<𝐺𝜖−𝑦𝜙_𝑛,(𝑦𝜖)+_𝑛+1,𝜙𝑦𝐺_𝑛+1+𝑦_𝑛𝐺𝑦_𝑛+1,𝑦_𝑘+1,𝑦_𝑘+1<𝜖−𝜙(𝜖)+𝜙(𝜖)=𝜖,.𝑦_𝑘,𝑦_𝑘(2.25)
By induction on 𝑚, we conclude that (2.24) holds for all 𝑚≥𝑛≥𝑛₀.
Hence, the sequence (𝑦_𝑛)=𝑔(𝑥_𝑛+1) is 𝐺-cauchy sequence in 𝑔(𝑋); since 𝑔(𝑋) is 𝐺-complete, then there exists 𝑡∈𝑔(𝑋) such that lim_𝑛→∞𝑔(𝑥_𝑛)=𝑡=lim_𝑛→∞𝑓(𝑥_𝑛).
Thus, there exists 𝑝∈𝑋 such that 𝑔(𝑝)=𝑡, also lim_𝑛→∞𝑓(𝑥_𝑛)=𝑔(𝑝).
We will show that 𝑓(𝑝)=𝑔(𝑝). Supposing that 𝑓(𝑝)≠𝑔(𝑝), then condition (4) implies that, 𝐺(𝑓(𝑝),𝑓(𝑝),𝑓(𝑥_𝑛))≤𝜙(𝑀(𝑝,𝑝,𝑥_𝑛)), where 𝑀𝑝=,𝑝m,𝑥ax_𝑛⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐺𝑔(𝑝),𝑔(𝑝),𝑔𝑥_𝑛1,𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)),12𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)),𝐺𝑔2𝐺(𝑝),𝑔𝑓(𝑝)𝑥,_𝑛𝑓𝑥^𝑛,𝑓𝑥_𝑛,𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)),𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)),,𝑓𝑥_𝑛,𝐺𝐺𝑔𝑔𝑥_𝑛𝑥,𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝐺𝑔𝑥_𝑛,𝑓(𝑝),𝑓(𝑝)_𝑛,,𝑓(𝑝),𝑓(𝑝)⎫⎪⎪⎪⎬⎪⎪⎪⎭.(2.26)
Taking the limit as 𝑛→∞ and using the fact that the function 𝐺 is continuous we get𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝))≤𝜙max𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)),1=𝜙(𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝))).2𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝))(2.27) Therefore, 𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝))≤𝜙(𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)))<𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)),(2.28) which is contradiction; hence 𝑓𝑝=𝑔𝑝.
Since 𝑓 and 𝑔 are 𝐺-weakly commuting of type 𝐴_𝑓, then (𝐺(𝑓(𝑔(𝑝)),𝑔(𝑔(𝑝)),𝑓(𝑓(𝑝)))  ≤𝐺(𝑓(𝑝),𝑔(𝑝),𝑓(𝑝))=0).
Thus, 𝑓𝑓(𝑝)=𝑓𝑔(𝑝)=𝑔𝑓(𝑝)=𝑔𝑔(𝑝); it follows that 𝑓(𝑡)=𝑓𝑔(𝑝)=𝑔𝑓(𝑝)=𝑔(𝑡).
Finally, we will show that 𝑡∶=𝑓(𝑝) is common fixed point of 𝑓 and 𝑔.
Supposing that 𝑓𝑡≠𝑡, then 𝐺(𝑓(𝑡),𝑡,𝑡)=𝐺(𝑓(𝑡),𝑓(𝑝),𝑓(𝑝))≤𝜙(𝑀(𝑡,𝑝,𝑝)),(2.29) where 𝑀(𝑡,𝑝,𝑝)=max⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐺(𝑔(𝑡),𝑔(𝑝),𝑔(𝑝)),𝐺(𝑔(𝑡),𝑓(𝑡),𝑓(𝑡)),112𝐺(𝑔(𝑡),𝑓(𝑝),𝑓(𝑝)),2𝐺𝐺(𝑔(𝑔(𝑝(𝑡)),,𝑓𝑓((𝑝𝑝)),,𝑓𝑓((𝑝𝑝)))),,𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑝𝑝𝑝))),,,𝑓𝑓𝑓(((𝑝𝑝𝑝))),,,𝑓𝑓𝑓(((𝑝𝑝𝑝)))))),,𝐺𝐺((𝑔𝑔((𝑝𝑝)),,𝑓𝑓((𝑡𝑡)),,𝑓𝑓((𝑡𝑡)))),,⎫⎪⎪⎪⎬⎪⎪⎪⎭.(2.30) Since 𝑔(𝑡)=𝑓(𝑡), and 𝑔(𝑝)=𝑓(𝑝), therefore (2.30) implies that 𝑀(𝑡,𝑝,𝑝)=max𝐺(𝑓(𝑡),𝑡,𝑡),12𝐺(𝑓(𝑡),𝑡,𝑡),𝐺(𝑡,𝑓(𝑡),𝑓(𝑡)).(2.31) Hence, (2.29) becomes 𝐺(𝑓(𝑡),𝑡,𝑡)≤=𝜙𝜙((m𝐺a(x𝑡,𝑓{𝐺(𝑡()𝑓,𝑓(𝑡()𝑡,)𝑡),)𝑡<),𝐺𝐺((𝑡𝑡,,𝑓𝑓((𝑡𝑡)),,𝑓𝑓((𝑡𝑡))))}.)(2.32) Similarly we get, 𝐺(𝑡,𝑓(𝑡),𝑓(𝑡))<𝐺(𝑓(𝑡),𝑡,𝑡).(2.33) So, 𝐺(𝑓(𝑡),𝑡,𝑡)<𝐺(𝑓(𝑡),𝑡,𝑡),(2.34) a contradiction which implies that 𝑡=𝑓𝑡=𝑔𝑡. Then 𝑡 is a common fixed point.
To prove uniqueness suppose we have 𝑢 and 𝑣 such that 𝑢≠𝑣, 𝑓𝑢=𝑔𝑢=𝑢 and 𝑓𝑣=𝑔𝑣=𝑣; then condition (4) implies that 𝐺(𝑢,𝑣,𝑣)≤𝜙(𝐺(𝑣,𝑢,𝑢)).(2.35) Therefore, 𝐺(𝑢,𝑣,𝑣)≤𝜙(𝐺(𝑣,𝑢,𝑢))<𝐺(𝑣,𝑢,𝑢).(2.36) Similarly, 𝐺(𝑣,𝑢,𝑢)<𝐺(𝑢,𝑣,𝑣); thus 𝐺(𝑢,𝑣,𝑣)<𝐺(𝑢,𝑣,𝑣), a contradiction which implies that 𝑢=𝑣. Then 𝑡 is a unique common fixed point of 𝑓 and 𝑔.

Now we give an example to support our result.

Example 2.14. Let 𝑋=[0,4/3], and define 𝐺∶𝑋×𝑋×𝑋→[0,∞) by 𝐺(𝑥,𝑦,𝑧)=max{|𝑥−𝑦|,|𝑦−𝑧|,|𝑥−𝑧|} and 𝑓,𝑔∶𝑋→𝑋 by 𝑓(𝑥)=𝑥³/8,𝑔(𝑥)=𝑥³/2 and 𝜙(𝑡)=(2/3)𝑡. Then,(a)𝑔(𝑋) is 𝐺-complete subspace of 𝑋,(b)𝑓(𝑋)⊂𝑔(𝑋),(c)𝑓 and 𝑔 are 𝐺-weakly commuting of type 𝐴_𝑓, (d)𝑓 and 𝑔 satisfy condition (4) of Theorem 2.13.
It is clear that (a) and (b) are satisfied.
To show (c), as an easy calculation one can show that ∀𝑥∈𝑋; we have 𝐺(𝑓(𝑔(𝑥)),𝑔(𝑔(𝑥)),𝑓(𝑓(𝑥)))=max{(3/64)𝑥⁹,(63/4096)𝑥⁹,(255/4096)𝑥⁹}≤(3/8)𝑥³=𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥)). Then 𝑓 and 𝑔 are G-weakly commuting of type 𝐴_𝑓.
To show (d), for 𝑥,𝑦,𝑧∈𝑋 we have 𝐺(𝑓(𝑥),𝑓(𝑦),𝑓(𝑧))=18max||𝑥³≤1−𝑦³||,||𝑦³−𝑧³||,||𝑥³−𝑧³||3max||𝑥³=2−𝑦³||,||𝑦³−𝑧³||,||𝑥³−𝑧³||312max||𝑥³=𝜙(𝑔(𝑥),𝑔(𝑦),𝑔−(𝑧𝑦)³)||,≤||𝑦𝜙³−(𝑀𝑧³(||𝑥,||,𝑥𝑦³,−𝑧))𝑧.³||(2.37)
Therefore, all hypotheses of Theorem 2.13 are satisfied and 𝑥=0 unique common fixed point of 𝑓 and 𝑔.

Corollary 2.15. Let (𝑋,𝐺) be a 𝐺-metric space, and suppose mappings 𝑓,𝑔∶𝑋→𝑋 satisfy the following conditions: (1)𝑓 and 𝑔 be G -weakly commuting of type 𝐴_𝑓, (2)𝑓(𝑋)⊆𝑔(𝑋), (3)𝑔(𝑋) is 𝐺-complete subspace of 𝑋, (4)𝐺(𝑓(𝑥),𝑓(𝑦),𝑓(𝑧))≤𝑘𝑀(𝑥,𝑦,𝑧),𝑤ℎ𝑒𝑟𝑒𝑀(𝑥,𝑦,𝑧)=max⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐺(𝑔(𝑥),𝑔(𝑦),𝑔(𝑧)),𝐺(𝑔(𝑥),𝑓(𝑥),𝑓(𝑥)),112𝐺(𝑔(𝑥),𝑓(𝑦),𝑓(𝑦)),2𝐺𝐺((𝑔𝑔((𝑦𝑥)),,𝑓𝑓((𝑧𝑧)),,𝑓𝑓((𝑧𝑧)))),,𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑧𝑦𝑧))),,,𝑓𝑓𝑓(((𝑧𝑦𝑦))),,,𝑓𝑓𝑓(((𝑧𝑦𝑦)))))),)𝐺,𝐺(𝑔(𝑔(𝑧()𝑦,)𝑓,𝑓(𝑥(𝑥),)𝑓,𝑓(𝑥(𝑥))),),⎫⎪⎪⎪⎬⎪⎪⎪⎭,(2.38) for all 𝑥,𝑦,𝑧∈𝑋, where 𝑘∈[0,1); then 𝑓 and 𝑔 have a unique common fixed point.

Proof. It suffices to take 𝜙(𝑡)=𝑘𝑡 in Theorem 2.13.

Theorem 2.16. Let (𝑋,𝐺) be a 𝐺-metric space. Suppose the mappings 𝑓,𝑔∶𝑋→𝑋 are 𝐺-weakly commuting of type 𝐴_𝑔 and satisfy the following condition: (1)𝑓 and 𝑔 satisfy E.A. property, (2)𝑔(𝑋) is closed subspace of 𝑋, (3)𝐺(𝑓(𝑥),𝑓(𝑦),𝑓(𝑧))≤𝑘𝑀(𝑥,𝑦,𝑧),𝑤ℎ𝑒𝑟𝑒𝑀(𝑥,𝑦,𝑧)=max⎧⎪⎪⎨⎪⎪⎩[[[𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑥𝑥𝑥))),,,𝑓𝑓𝑓(((𝑥𝑦𝑧))),,,𝑓𝑓𝑓(((𝑦𝑥𝑧))))))+++𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑦𝑦𝑦))),,,𝑓𝑓𝑓(((𝑥𝑦𝑧))),,,𝑓𝑓𝑓(((𝑥𝑦𝑧))))))+++𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑧𝑧𝑧))),,,𝑓𝑓𝑓(((𝑧𝑦𝑥))),,,𝑓𝑓𝑓(((𝑦𝑧𝑥))))))]]],,⎫⎪⎪⎬⎪⎪⎭(2.39) for all 𝑥,𝑦,𝑧∈𝑋, where 𝑘∈[0,1/3); then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Since 𝑓 and 𝑔 satisfy E.A. property, there exists in 𝑋 a sequence (𝑥_𝑛) satisfying lim_𝑛→∞𝑓(𝑥_𝑛)=lim_𝑛→∞𝑔(𝑥_𝑛)=𝑡 for some 𝑡∈𝑋.
Since 𝑔(𝑋) is closed subspace of 𝑋 and lim_𝑛→∞𝑔(𝑥_𝑛)=𝑡, there exists 𝑝∈𝑋 such that 𝑔(𝑝)=𝑡, alsolim_𝑛→∞𝑓(𝑥_𝑛)=𝑔(𝑝).
We will show that 𝑓(𝑝)=𝑔(𝑝) supposing that 𝑓(𝑝)≠𝑔(𝑝), then condition (3) implies that 𝐺𝑓(𝑝),𝑓(𝑝),𝑓𝑥_𝑛≤𝑘𝑀𝑝,𝑝,𝑥_𝑛,(2.40) where, 𝑀𝑝=,𝑝m,𝑥ax_𝑛⎧⎪⎪⎨⎪⎪⎩𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝))+𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝))+𝐺𝑔𝑥_𝑛𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝))+𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝))+𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝐺𝑔(𝑝),𝑓𝑥_𝑛,𝑓(𝑝),𝑓(𝑝),,𝑓𝑥_𝑛+𝐺𝑔(𝑝),𝑓𝑥_𝑛,𝑓𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑓(𝑝),𝑓(𝑝)⎫⎪⎪⎬⎪⎪⎭.(2.41) Taking the limit as 𝑛→∞ and using the fact that the function 𝐺 is continuous, we get 𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝))≤3𝑘𝐺(𝑔(𝑝),𝑓(𝑝),𝑓(𝑝)),(2.42) which is contradiction since 𝑘∈[0,1/3), so 𝑓𝑝=𝑔𝑝. Since 𝑓 and 𝑔 are 𝐺-weakly commuting of type 𝐴_𝑔, then 𝐺(𝑔𝑓(𝑝),𝑔𝑔(𝑝),𝑓𝑓(𝑝))≤𝐺(𝑔(𝑝),𝑓(𝑝),𝑔(𝑝))=0.(2.43) Therefore, 𝑓(𝑔(𝑝))=𝑓𝑓(𝑝)=𝑔𝑓(𝑝)=𝑔𝑔(𝑝); then 𝑓(𝑡)∶=𝑓𝑔(𝑝)=𝑔𝑓(𝑝)=𝑔(𝑡).(2.44) Finally, we will show that 𝑡=𝑓(𝑝) is common fixed point of 𝑓 and 𝑔.
Supposing that 𝑓𝑡≠𝑡, then 𝐺(𝑓(𝑡),𝑡,𝑡)=𝐺(𝑓(𝑡),𝑓(𝑝),𝑓(𝑝))≤𝑘𝑀(𝑡,𝑝,𝑝),(2.45) where 𝑀(𝑡,𝑝,𝑝)=max⎧⎪⎪⎨⎪⎪⎩[[[𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑡𝑡𝑡))),,,𝑓𝑓𝑓(((𝑡𝑝𝑝))),,,𝑓𝑓𝑓((𝑡(𝑝)𝑝))))+)++𝐺𝐺𝐺(𝑔((𝑔𝑔(𝑝((𝑝)𝑝),),𝑓,𝑓𝑓(𝑝((𝑡)𝑝),),𝑓,𝑓𝑓((𝑝𝑡())𝑝)))++)+𝐺𝐺𝐺((𝑔𝑔(𝑔((𝑝𝑝())𝑝,,)𝑓𝑓,𝑓((𝑝𝑝())𝑡),,,𝑓𝑓𝑓((𝑝𝑝(𝑡))))))]]],,⎫⎪⎪⎬⎪⎪⎭.(2.46) But 𝑓(𝑡)=𝑔(𝑡) and 𝑓(𝑝)=𝑔(𝑝). Thus, 𝐺(𝑓(𝑡),𝑡,<𝑡)𝑘≤{𝑘𝐺m(𝑓ax(𝑡{)𝐺,𝑡,(𝑡𝑡,)𝑓+(𝐺𝑡),(𝑡𝑓,𝑓(𝑡)()𝑡),,[𝑓𝐺((𝑡𝑓))(}𝑡).,𝑡,𝑡)+𝐺(𝑡,𝑓(𝑡),𝑓(𝑡))]}(2.47) Hence, 𝐺(𝑓(𝑡),𝑡,𝑡)≤𝑘1−𝑘𝐺(𝑡,𝑓(𝑡),𝑓(𝑡)).(2.48)
Adjusting similarly, we get 𝐺(𝑡,𝑓(𝑡),𝑓(𝑡))≤𝑘1−𝑘𝐺(𝑓𝑡,𝑡,𝑡).(2.49) Therefore, 𝐺(𝑓𝑡,𝑡,𝑡)≤𝑘1−𝑘²𝐺(𝑓𝑡,𝑡,𝑡),(2.50) a contradiction which implies that 𝑓𝑡=𝑡=𝑓𝑝, but 𝑔𝑡=𝑓𝑡=𝑡. Then 𝑡 is a common fixed point of 𝑓 and 𝑔.
To prove uniqueness, suppose we have 𝑢 and 𝑣 such that 𝑢≠𝑣, 𝑓𝑢=𝑔𝑢=𝑢 and 𝑓𝑣=𝑔𝑣=𝑣; then 𝐺(𝑢,𝑣,𝑣)=𝐺(𝑓(𝑢),𝑓(𝑣),𝑓(𝑣))≤𝑘{𝐺(𝑢,𝑣,𝑣)+𝐺(𝑣,𝑢,𝑢)}.(2.51) Hence, 𝐺(𝑢,𝑣,𝑣)≤(𝑘/(1−𝑘))𝐺(𝑣,𝑢,𝑢). Similarly, 𝐺(𝑣,𝑢,𝑢)≤(𝑘/(1−𝑘))𝐺(𝑢,𝑣,𝑣). Therefore, 𝐺(𝑢,𝑣,𝑣)≤(𝑘/(1−𝑘))²𝐺(𝑢,𝑣,𝑣)a contradiction which implies that 𝑢=𝑣. Then 𝑡 is a unique common fixed point of 𝑓 and 𝑔.
Now we give an example to support our result.

Example 2.17. Let 𝑋=[0,3/4], define 𝐺∶𝑋×𝑋×𝑋→[0,∞) by
𝐺(𝑥,𝑦,𝑧)=max{|𝑥−𝑦|,|𝑦−𝑧|,|𝑥−𝑧|} and let 𝑓,𝑔∶𝑋→𝑋 by 𝑓(𝑥)=𝑥²/5, 𝑔(𝑥)=𝑥².
Then,(a)𝑔(𝑋) is closed subspace of 𝑋,(b)𝑓 and 𝑔 are G-weakly commuting of type 𝐴_𝑔,(c)𝑓 and 𝑔 satisfy E.A. property. (d)𝑓 and 𝑔 satisfy condition (4) for 𝑘=(1/4).

Proof. (a) is obvious.
To show (b), as an easy calculation one can show that forall𝑥∈𝑋; we have 𝐺(𝑔(𝑓(𝑥)),𝑔(𝑔(𝑥)),𝑓(𝑓(𝑥)))=max{(4/125)𝑥⁴,(24/25)𝑥⁴,(124/125)𝑥⁴}≤(4/5)𝑥²=𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥)). Then 𝑓 and 𝑔 are 𝐺-weakly commuting of type 𝐴_𝑔.
To show (c), if we consider the sequence {𝑥_𝑛}={1/2𝑛}, then 𝑓𝑥_𝑛→0 and 𝑔𝑥_𝑛→0 as 𝑛→∞. Thus, 𝑓 and 𝑔 satisfy the E.A. property.
To show (d), for 𝑥,𝑦,𝑧∈𝑋 we have ||||𝑥₂||||𝑦5₂−𝑦5²||||≤𝑥5²+𝑦5²,||||𝑥5₂−𝑧5²||||≤𝑦5²+𝑧5²,5−𝑧5²||||≤𝑥5²+𝑧5².(2.52)
Then𝐺(𝑓(𝑥),𝑓(𝑦),𝑓(𝑧))=max||||𝑥²≤𝑥₂5−𝑦5²||||,||||𝑦5²−𝑧5²||||,||||𝑥5²−𝑧5²||||=15+𝑦5²+𝑧5²=1445𝑥²+45𝑦²+45𝑧²+4𝐺(𝐺(𝑔(𝑔(𝑧()𝑥,)𝑓,𝑓(𝑧()𝑥,)𝑓,𝑓(𝑧()𝑥))))≤+𝑘𝐺𝑀(𝑔(𝑥(𝑦,)𝑦,,𝑓𝑧)(.𝑦),𝑓(𝑦))(2.53)
Therefore, all hypotheses of Theorem 2.16 are satisfied for 𝑘=1/4 and 𝑥=0, a unique common fixed point of 𝑓 and 𝑔.

Theorem 2.18. Let (𝑋,𝐺) be a 𝐺-metric space, and suppose mappings 𝑓,𝑔∶𝑋→𝑋 be 𝐺-𝑅-weakly commuting of type 𝐴_𝑓. Suppose that there exists a mapping 𝜓∶𝑋→[0,∞) such that (1)𝑓(𝑋)⊂𝑔(𝑋), (2)𝑔(𝑋) is 𝐺-complete subspace of 𝑋, (3)𝐺(𝑔𝑥,𝑓𝑥,𝑓𝑥)<𝜓(𝑔(𝑥))−𝜓(𝑓(𝑥)),forall𝑥∈𝑋, 𝐺(𝑓(𝑥),𝑓(𝑦),𝑓(𝑧))<max⎧⎪⎨⎪⎩𝐺𝐺((𝑔𝑔((𝑥𝑧)),,𝑓𝑔((𝑦𝑧)),,𝑔𝑓((𝑧𝑥)))),,𝐺𝐺(𝑔(𝑔(𝑥(𝑦)),,𝑓𝑓((𝑥𝑦)),,𝑔𝑓(𝑦(𝑧)))),⎫⎪⎬⎪⎭,(2.54) for all 𝑥,𝑦,𝑧∈𝑋; then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Let 𝑥₀∈𝑋, and then choose 𝑥₁∈𝑋 such that 𝑓(𝑥₀)=𝑔(𝑥₁) and 𝑥₂∈𝑋 where 𝑓(𝑥₁)=𝑔(𝑥₂); then by induction we can define a sequence (𝑦_𝑛)∈𝑋 as follows: 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1,𝑛∈𝑁∪{0}.(2.55) We will show that the sequence (𝑦_𝑛) is 𝐺-cauchy sequence: 𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1=<𝐺𝜓𝑔𝑔𝑥𝑥^𝑛_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛=𝜓𝑔𝑥_𝑛−𝜓𝑓𝑥_𝑛−𝜓𝑔𝑥_𝑛+1.(2.56)
Consider 𝑎_𝑛=𝜓(𝑔(𝑥_𝑛)),𝑛=1,2,3,4,…, then 0≤𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1<𝑎_𝑛−𝑎_𝑛+1.(2.57)
Thus, the sequence (𝑎_𝑛) is nonincreasing and bounded below by 0; hence (𝑎_𝑛)is convergent sequence.
On the other hand we have, from (G5) and (2.57), that for 𝑚,𝑛∈𝐍;𝑚>𝑛𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛+𝑚,𝑔𝑥_𝑛+𝑚≤^{𝑛+𝑚−1}_𝑗=𝑛<𝐺𝑔𝑥_𝑗,𝑔𝑥_𝑗+1,𝑔𝑥_𝑗+1_{𝑛+𝑚−1}_𝑗=𝑛=𝑎_𝑛𝑎_𝑗−𝑎_𝑗+1(Telescopingsum)−𝑎_𝑛+𝑚.(2.58)
Therefore, the sequence (𝑔(𝑥_𝑛)) is G-cauchy sequence in 𝑔(𝑋).
Since 𝑔(𝑋) is 𝐺-complete subspace, then there exists 𝑡∈𝑔(𝑋) such that lim_𝑛→∞𝑔(𝑥_𝑛)=𝑡; having 𝑡∈𝑔(𝑋) there exists 𝑝∈𝑋 such that 𝑔(𝑝)=𝑡, also lim_𝑛→∞𝑓(𝑥_𝑛)=𝑔(𝑝)=𝑡.
We will show that 𝑓(𝑝)=𝑔(𝑝); supposing that 𝑓(𝑝)≠𝑔(𝑝), then condition (4) implies that 𝐺𝑓(𝑝),𝑓(𝑝),𝑓𝑥_𝑛<max⎧⎪⎨⎪⎩𝐺𝑔(𝑝),𝑔(𝑝),𝑔𝑥^𝑛𝐺𝑔(𝑝),𝑓(𝑝),𝑓𝑥_𝑛,𝐺(𝑔(𝑝),𝑓(𝑝),𝑔(𝑝)),,𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛,𝑓(𝑝)⎫⎪⎬⎪⎭.(2.59) Taking the limit as 𝑛→∞, we get 𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝))<max⎧⎪⎪⎨⎪⎪⎩𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑝𝑝𝑝))),,,𝑓𝑔𝑔(((𝑝𝑝𝑝))),,,𝑔𝑔𝑓(((𝑝𝑝𝑝)))))),,⎫⎪⎪⎬⎪⎪⎭,(2.60) hence, 𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝))<𝐺(𝑔(𝑝),𝑔(𝑝),𝑓(𝑝)).(2.61) Adjusting similarly, we get 𝐺(𝑔(𝑝),𝑔(𝑝),𝑓(𝑝))<𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝))(2.62)
Therefore, 𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝))<𝐺(𝑔(𝑝),𝑓(𝑝),𝑔(𝑝))<𝐺(𝑓(𝑝),𝑓(𝑝),𝑔(𝑝)).(2.63)
Thus, a contradiction implies 𝑓𝑝=𝑔𝑝.
Since 𝑓 and 𝑔 are 𝐺-weakly commuting of type 𝐴_𝑓, then 𝐺(𝑓(𝑔(𝑝)),𝑔(𝑔(𝑝)),𝑓(𝑓(𝑝)))≤𝐺(𝑓(𝑝),𝑔(𝑝),𝑓(𝑝))=0.(2.64)
Thus, 𝑓𝑓(𝑝)=𝑓𝑔(𝑝)=𝑔𝑓(𝑝)=𝑔𝑔(𝑝), then 𝑓(𝑡)=𝑓𝑔(𝑝)=𝑔𝑓(𝑝)=𝑔(𝑡).
Finally, we will show that 𝑡=𝑓(𝑝) is common fixed point of 𝑓 and 𝑔.
Suppose that 𝑓𝑡≠𝑡, so 𝐺(𝑓(𝑡),𝑡,𝑡)=𝐺(𝑓(𝑡),𝑓(𝑝),𝑓(𝑝))<max⎧⎪⎪⎨⎪⎪⎩𝐺𝐺𝐺(((𝑔𝑔𝑔(((𝑝𝑡𝑡))),,,𝑔𝑓𝑓(((𝑝𝑡𝑝))),,,𝑔𝑔𝑓((𝑝𝑝(𝑝)))))),,⎫⎪⎪⎬⎪⎪⎭.(2.65)
Since 𝑔(𝑝)=𝑓(𝑝) and 𝑔(𝑡)=𝑓(𝑡), therefore (2.65) implies that 𝐺(𝑓(𝑡),𝑡,𝑡)<𝐺(𝑓(𝑡),𝑓(𝑡),𝑡).(2.66)
Similarly, we have 𝐺(𝑓(𝑡),𝑓(𝑡),𝑡)<𝐺(𝑓(𝑡),𝑡,𝑡).
A contradiction implies that 𝑓𝑡=𝑓𝑝=𝑡. Then 𝑡 is a common fixed point.
To prove uniqueness suppose we have 𝑢 and 𝑣 such that 𝑢≠𝑣 where 𝑓𝑢=𝑔𝑢=𝑢 and 𝑓𝑣=𝑔𝑣=𝑣; then as an easy calculation one can get 𝐺(𝑢,𝑣,𝑣)<𝐺(𝑣,𝑢,𝑢).(2.67) Similarly, 𝐺(𝑣,𝑢,𝑢)<𝐺(𝑢,𝑣,𝑣), a contradiction which implies that 𝑢=𝑣. Then, 𝑡 is a unique common fixed point of 𝑓 and 𝑔.

Now we give an example to support our result.

Example 2.19. Let 𝑋=[1,∞), 𝜓∶𝑋→[0,∞) such that 𝜓(𝑡)=3𝑡,𝑡∈𝑋 and 𝐺(𝑥,𝑦,𝑧)=max{|𝑥−𝑦|,|𝑦−𝑧|,|𝑧−𝑥|}. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓(𝑥)=2𝑥−1 and 𝑔(𝑥)=3𝑥−2.
Then, (a)𝑓(𝑋)⊂𝑔(𝑋), (b)𝑔(𝑋) is 𝐺-complete subspace of 𝑋, (c)𝐺(𝑔𝑥,𝑓𝑥,𝑓𝑥)<𝜓(𝑔(𝑥))−𝜓(𝑓(𝑥)),forall𝑥∈𝑋,(d)𝑓 and 𝑔 satisfy condition (4) of Theorem 2.18.
Then as an easy calculation one can see that 𝐺(𝑓(𝑔(𝑥)),𝑔(𝑔(𝑥)),𝑓(𝑓(𝑥)))=max{2𝑥−2,5𝑥−5,3𝑥−3}=5𝑥−5≤𝑅(𝑥−1)=𝑅𝐺(𝑓(𝑥),𝑔(𝑥),𝑓(𝑥)), for 𝑅≥5, then 𝑓 and 𝑔 are G-R-weakly commuting of type 𝐴_𝑓.
Also we see that 𝑓(𝑋)⊂𝑔(𝑋) and 𝑔(𝑋) is G-complete subspace of 𝑋.
To prove (c), forall𝑥∈𝑋 we see that 𝐺(𝑔𝑥,𝑓𝑥,𝑓𝑥)=𝑥−1≤3𝑥−3=𝜓(𝑔(𝑥))−𝜓(𝑓(𝑥)).(2.68)
To prove (d), for all 𝑥,𝑦,𝑧∈𝑋 we have 𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)𝐺≤≤=(23m𝑔mma(𝑧xaa)xx{,𝐺𝑓||||𝑥𝑥((𝑧𝑔−−)(,𝑥𝑦𝑦𝑓)||||,,,(𝑔||||𝑥𝑦𝑦)()−−𝑦,)𝑧𝑧𝐺,||||𝑔,,(𝑔||(𝑥𝑥𝑧()−−𝑦)),𝑧𝑧,𝐺||𝑓((𝑔𝑦=()𝑥𝐺,)𝑓,(𝑔(𝑓𝑧𝑥)(,)𝑥𝑔})𝑦.,,𝑔𝑔𝑧(𝑦))),(2.69)
Therefore, all hypotheses of the previous theorem are satisfied and 𝑥=1 a unique common fixed point of 𝑓 and 𝑔.
Note that the main result of Mustafa [33] is not applicable in this case. Indeed, for 𝑦=𝑧=1 and 𝑥=3, 𝐺(𝑓(3),𝑓(1),𝑓(1))=4>2𝑘=𝑘𝐺(3,1,1)∀𝑘∈[0,1).(2.70)
Also, the Banach principle [34] is not applicable. Indeed, for 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋 we have for 𝑥≠𝑦𝑑(𝑓(𝑥),𝑓(𝑦))=2||𝑥−𝑦||>𝑘||𝑥−𝑦||∀𝑘∈[0,1).(2.71)

Corollary 2.20. Theorems 2.13, 2.16, and 2.18 remain true if we replace, respectively, 𝐺-weakly commuting of type 𝐴_𝑓, 𝐺-weakly commuting of type 𝐴_𝑔, weakly compatible and 𝐺-𝑅-weakly commuting of type 𝐴_𝑓 by any one of them (retaining the rest of hypothesis).

Corollary 2.21. Some corollaries could be derived from Theorems 2.13, 2.16, and 2.18 by taking 𝑧=𝑦 or 𝑔=𝐼𝑑_𝑋.