# Some New Common Fixed Point Theorems under Strict Contractive Conditions inG-Metric Spaces

Authors: Zead Mustafa
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce some new types of pairs of mappings(f,g) on G-metric space called G-weakly commuting of type (Af) and G-R-weakly commuting of type (Af). 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.


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## Body

## 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]).
LetS 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 (xn) such that
(1.1)lim⁡n→∞Txn=lim⁡n→∞Sxn=t,forsomet∈X.In 2005 Zead Mustafa and Brailey Sims introduced the notion ofG-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-mapsf and g on G-metric space called G-weakly commuting of type Af and G-R-weakly commuting of type Af. 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]).
AG-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 functionG is called G-metric on X.EveryG-metric on X defines a metric dG on X by
(1.2)dG(x,y)=G(x,y,y)+G(y,x,x)∀x,y∈X.Example 1.3 (see [29]).
Let(X,d) be a metric space, and define Gs and Gm on X×X×X to R+ by
(1.3)Gs(x,y,z)=d(x,y)+d(y,z)+d(x,z),Gm(x,y,z)=max⁡{d(x,y),d(y,z),d(x,z)},
for all x,y,z∈X. Then (X,Gs) and (X,Gm) are G-metric spaces.Example 1.4 (see [29]).
LetX=R, and define G:X×X×X→R+, by
(1.4)G(x,y,z)={|x-y|+|y-z|+|x-z|,ifallx,y,andzarestrictlypositiveortheyareallstrictlynegativeorallx,y,andzarezero,1+|x-y|+|y-z|+|x-z|,otherwise,
then (X,G) is G-metric space.Definition 1.5 (see [29]).
A sequence(xn) in a G-metric space X is said to converge if there exists x∈X such that limn,m→∞G(x,xn,xm)=0, and one says that the sequence (xn) is G-convergent to x. We call x the limit of the sequence (xn) and write xn→x or limn→∞xn=x (through this paper we mean by N the set of all natural numbers).Proposition 1.6 (see [29]).
LetX be G-metric space. Then the following statements are equivalent:(1)
(xn) is G-convergent to x,(2)
G(xn,xn,x)→0, as n→∞,(3)
G(xn,x,x)→0, as n→∞,(4)
G(xm,xn,x)→0, as m,n→∞.Definition 1.7 (see [29]).
In aG-metric space X, a sequence (xn) is said to be G-Cauchy if given ε>0, there is N∈N such that G(xn,xm,xl)<ε, for all n,m,l≥N. That is G(xn,xm,xl)→0 as n,m,l→∞.Proposition 1.8 (see [29]).
In aG-metric space X, the following statements are equivalent:(1)
the sequence(xn) is G-Cauchy;(2)
for everyε>0,thereexistsN∈N such that G(xn,xm,xm)<ε, for all n,m≥N.Definition 1.9 (see [29]).
AG-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.
LetX=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]).
LetX 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]).
AG-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]).
Letf 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]).
Letf 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]).
LetX 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
(1.5)G(f(g(x)),f(g(x)),g(f(x)))≤RG(f(x),f(x),g(x))holdforeachx∈X.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 Gf and G-R-weakly commuting of type Gf, 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 Gf if
(1.6)G(fgx,gfx,ffx)≤G(fx,gx,fx),∀x∈X.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 Gf if there exists some positive real number R such that
(1.7)G(fgx,gfx,ffx)≤RG(fx,gx,fx),∀x∈X.Remark 1.18.
TheG-R-weakly commuting maps of type Gf 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 typeAf 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 Af if
(2.1)G(fgx,ggx,ffx)≤G(fx,gx,fx),∀x∈X.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 Af if there exists some positive real number R such that
(2.2)G(fgx,ggx,ffx)≤RG(fx,gx,fx),∀x∈X.Remark 2.3.
TheG-weakly commuting maps of type Af are G-R-weakly commuting of type Af. Reciprocally, if R≤1, then G-R-weakly commuting maps of type Af are G-weakly commuting of type Af.
If we interchangef and g in (2.1) and (2.2), then the pair of mappings (f,g) is called G-weakly commuting of type Ag and G-R-weakly commuting of type Ag, respectively.Example 2.4.
LetX=[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 Af and G-R-Weakly commuting of type Af.Example 2.5.
LetX=[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 Af or Ag, but it is G-R-weakly commuting of type Af (and Ag) for R≥4.The following examples show a pair of mappings(f,g) that G-weakly commuting of type Gf need not be G-weakly commuting of type Af.Example 2.6.
LetX=[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 Af, but it is G-weakly commuting of type Gf.The following example shows that(1)
a pair of mappings(f,g) that is G-weakly commuting of type Af need not be G-weakly commuting of type Ag;(2)
a pair of mappings(f,g) that is G-weakly commuting of type Af need not be G-weakly commuting of type Gf;(3)
a pair of mappings(f,g) that is G-R-weakly commuting of type Af need not be R-weakly commuting;Example 2.7.
LetX=[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
(2.3)f(x)={2ifx=2,6if2<x≤5,5ifx>5,g(x)={2ifx=2,9if2<x≤5,6ifx>5.
Then,(2.4)f(g(x))={2ifx=2,5ifx>2,g(f(x))={2ifx=2,6if2<x≤5,9ifx>5,g(g(x))={2ifx=2,6ifx>2,f(f(x))={2ifx=2,5if2<x≤5,6ifx>5.
Moreover,(2.5)|f(x)-g(x)|={0ifx=2,3if2<x≤5,1ifx>5.
Ifx=2, we have G(f(g(x)),g(g(x)),f(f(x)))=0=G(f(x),g(x),f(x)).
If2<x≤5, we have
(2.6)G(f(g(x)),f(f(x)),g(g(x)))=max⁡{0,1,1}≤3=G(f(x),g(x),f(x)).
Ifx>5, then
(2.7)G(f(g(x)),f(f(x)),g(g(x)))=max⁡{1,1,0}≤1=G(g(x),f(x),g(x)).
Thus,f and g are G weakly commuting of type Af, but for x=6, we have
(2.8)G(g(f(6)),f(f(6)),g(g(6)))=max⁡{3,3,0}≰1=G(g(6),f(6),g(6)).
Therefore, the pair(f,g) is not G-weakly commuting of type Ag, but it is G-weakly commuting of type Af.
Also forx=7, we have
(2.9)G(f(g(7)),g(f(7)),f(f(7)))=4≰1=G(g(6),f(6),g(6)).
Therefore, the pair(f,g) is not G-weakly commuting of type Gf.
AS an easy calculation one can see that(f,g) are G-R-weakly commuting of type Ag 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.
Iff and g are G-weakly commuting of type Af or G-R-weakly commuting of type Af, then f and g are weakly compatible.Proof.
Letx 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 Af, we have
(2.10)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.
It followsf(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 Gf, we have
(2.11)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.
Thusf(g(x))=g(f(x)); then the pair (f,g) is weakly compatible.The following example shows that(1)
the converse of Lemma2.8 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 Af,(3)
a pair of mappings(f,g) that is R-weakly commuting need not be G-R-weakly commuting of type Gf.Example 2.9.
LetX=[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 forx=3 we have,
(2.12)G(f(g(x)),g(g(x)),f(f(x)))=72≰4=G(f(x),g(x),f(x)).
Therefore, f and g are not G-weakly commuting of type Af.
Also, we see thatG(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 Af 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 Gf for R=2.Now, we rewrite Definition1.1 on G-metric spaces setting.Definition 2.10.
LetS 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 (xn) such that (Txn) and (Sxn)G-converge to t for some t∈X; that is, thanks to Proposition 1.6,
(2.13)lim⁡n⟶∞G(Txn,Txn,t)=lim⁡n⟶∞G(Sxn,Sxn,t)=0.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 iff and g satisfy the E.A. property, then the pair (f,g) need not be G-weakly commuting of type Af.Example 2.12.
We return to Example2.9. Let xn=1+(1/3n). We have limn→∞f(xn)=limn→∞(1+(2/3n))=1, and limn→∞g(xn)=limn→∞(1+(1/3n))2=1, therefore, limn→∞f(xn)=limn→∞g(xn)=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 Af.Following Matkowski (see [32]), let Φ be the set of all functions ϕ such that  ϕ:[0,∞)→[0,∞) be a nondecreasing function with limn→∞ϕn(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 Af,(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,where(2.14)M(x,y,z)=max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),f(x)),12G(g(x),f(y),f(y)),12G(g(x),f(z),f(z)),G(g(y),f(y),f(y)),G(g(y),f(x),f(x)),G(g(y),f(z),f(z)),G(g(z),f(z),f(z)),G(g(z),f(x),f(x)),G(g(z),f(y),f(y))}.   Then f and g have a unique common fixed point.Proof.
Letx0∈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 (yn)∈X as follows:
(2.15)yn=f(xn)=g(xn+1),n∈N∪{0}
We will show that the sequence (yn) is G-cauchy sequence:
(2.16)G(yn,yn+1,yn+1)=G(f(xn),f(xn+1),f(xn+1))≤ϕ(M(xn,xn+1,xn+1)),
where
(2.17)M(xn,xn+1,xn+1)=max⁡{G(g(xn),g(xn+1),g(xn+1)),G(g(xn),f(xn),f(xn)),12G(g(xn),f(xn+1),f(xn+1)),G(g(xn+1),f(xn),f(xn)),G(g(xn+1),f(xn+1),f(xn+1)),}=max⁡{G(yn-1,yn,yn),12G(yn-1,yn+1,yn+1),G(yn,yn,yn),G(yn,yn+1,yn+1),}.
We will have different cases.Case (1): ifM(xn,xn+1,xn+1)=G(yn,yn+1,yn+1), then G(yn,yn+1,yn+1)≤ϕ(G(yn,yn+1,yn+1)) <G(yn,yn+1,yn+1), which is contradiction.Case (2): ifM(xn,xn+1,xn+1)=(1/2)G(yn-1,yn+1,yn+1), then in this case we havemax{G(yn-1,yn,yn),G(yn,yn+1,yn+1)}<(1/2)G(yn-1,yn+1,yn+1),  which implies that(2.18)G(yn-1,yn,yn)+G(yn,yn+1,yn+1)<G(yn-1,yn+1,yn+1),
but fromG-metric property (G5) we have
(2.19)G(yn-1,yn+1,yn+1)≤G(yn-1,yn,yn)+G(yn,yn+1,yn+1).
Thus, from (2.18) and (2.19) we see that case (2) is impossible.
Then, we must have the case(2.20)M(xn,xn+1,xn+1)=G(yn-1,yn,yn).
Thus, forn∈N∪{0} and from (2.16) we have,
(2.21)G(yn,yn+1,yn+1)≤ϕ(G(yn-1,yn,yn))≤ϕ2(G(yn-2,yn-1,yn-1))⋮≤ϕn(G(y0,y1,y1))…(1).
Givenϵ>0, since limn→∞ϕn(G(y0,y1,y1))=0, and ϕ(ϵ)<ϵ, there is an integer no∈N, such that
(2.22)ϕn(G(y0,y1,y1))<ϵ-ϕ(ϵ),∀n≥n0.
Hence, we have(2.23)G(yn,yn+1,yn+1)≤ϕn(G(y0,y1,y1))<ϵ-ϕ(ϵ).
Now form,n∈N; m>n, we claim that
(2.24)G(yn,ym,ym)<ϵ,∀m≥n≥n0.
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
(2.25)G(yn,yk+1,yk+1)≤G(yn,yn+1,yn+1)+G(yn+1,yk+1,yk+1)<ϵ-ϕ(ϵ)+ϕ(G(yn,yk,yk))<ϵ-ϕ(ϵ)+ϕ(ϵ)=ϵ.
By induction onm, we conclude that (2.24) holds for all m≥n≥n0.
Hence, the sequence(yn)=g(xn+1) is G-cauchy sequence in g(X); since g(X) is G-complete, then there exists t∈g(X) such that limn→∞g(xn)=t=limn→∞f(xn).
Thus, there existsp∈X such that g(p)=t, also limn→∞f(xn)=g(p).
We will show thatf(p)=g(p). Supposing that f(p)≠g(p), then condition (4) implies that, G(f(p),f(p),f(xn))≤ϕ(M(p,p,xn)), where
(2.26)M(p,p,xn)=max⁡{G(g(p),g(p),g(xn)),G(g(p),f(p),f(p)),12G(g(p),f(p),f(p)),12G(g(p),f(xn),f(xn)),G(g(p),f(p),f(p)),G(g(p),f(p),f(p)),G(g(p),f(xn),f(xn)),G(g(xn),f(xn),f(xn)),G(g(xn),f(p),f(p)),G(g(xn),f(p),f(p))}.
Taking the limit asn→∞ and using the fact that the function G is continuous we get(2.27)G(f(p),f(p),g(p))≤ϕ(max⁡{G(g(p),f(p),f(p)),12G(g(p),f(p),f(p))})=ϕ(G(g(p),f(p),f(p))).
Therefore,
(2.28)G(f(p),f(p),g(p))≤ϕ(G(g(p),f(p),f(p)))<G(g(p),f(p),f(p)),
which is contradiction; hence fp=gp.
Sincef and g are G-weakly commuting of type Af, 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 thatt:=f(p) is common fixed point of f and g.
Supposing thatft≠t, then
(2.29)G(f(t),t,t)=G(f(t),f(p),f(p))≤ϕ(M(t,p,p)),
where
(2.30)M(t,p,p)=max⁡{G(g(t),g(p),g(p)),G(g(t),f(t),f(t)),12G(g(t),f(p),f(p)),12G(g(t),f(p),f(p)),G(g(p),f(p),f(p)),G(g(p),f(t),f(t)),G(g(p),f(p),f(p)),G(g(p),f(p),f(p)),G(g(p),f(t),f(t)),G(g(p),f(p),f(p))}.
Since g(t)=f(t), and g(p)=f(p), therefore (2.30) implies that
(2.31)M(t,p,p)=max⁡{G(f(t),t,t),12G(f(t),t,t),G(t,f(t),f(t))}.
Hence, (2.29) becomes
(2.32)G(f(t),t,t)≤ϕ(max⁡{G(f(t),t,t),G(t,f(t),f(t))})=ϕ(G(t,f(t),f(t)))<G(t,f(t),f(t)).
Similarly we get,
(2.33)G(t,f(t),f(t))<G(f(t),t,t).
So,
(2.34)G(f(t),t,t)<G(f(t),t,t),
a contradiction which implies that t=ft=gt. Then t is a common fixed point.
To prove uniqueness suppose we haveu and v such that u≠v, fu=gu=u and fv=gv=v; then condition (4) implies that
(2.35)G(u,v,v)≤ϕ(G(v,u,u)).
Therefore,
(2.36)G(u,v,v)≤ϕ(G(v,u,u))<G(v,u,u).
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.
LetX=[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 Af,(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 Af.
To show (d), forx,y,z∈X we have
(2.37)G(f(x),f(y),f(z))=18max⁡{|x3-y3|,|y3-z3|,|x3-z3|}≤13max⁡⁡{|x3-y3|,|y3-z3|,|x3-z3|}=23(12max⁡⁡{|x3-y3|,|y3-z3|,|x3-z3|})=ϕ(g(x),g(y),g(z))≤ϕ(M(x,y,z)).
Therefore, all hypotheses of Theorem2.13 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 Af,(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),where(2.38)M(x,y,z)=max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),f(x)),12G(g(x),f(y),f(y)),12G(g(x),f(z),f(z)),G(g(y),f(y),f(y))),G(g(y),f(x),f(x)),G(g(y),f(z),f(z)),G(g(z),f(z),f(z)),G(g(z),f(x),f(x)),G(g(z),f(y),f(y))}, for all x,y,z∈X, where k∈[0,1); then f and g have a unique common fixed point.Proof.
It suffices to takeϕ(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 Ag 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),where(2.39)M(x,y,z)=max⁡{[G(g(x),f(x),f(x))+G(g(y),f(y),f(y))+G(g(z),f(z),f(z))],[G(g(x),f(y),f(y))+G(g(y),f(x),f(x))+G(g(z),f(y),f(y))],[G(g(x),f(z),f(z))+G(g(y),f(z),f(z))+G(g(z),f(x),f(x))]} for all x,y,z∈X, where k∈[0,1/3); then f and g have a unique common fixed point.Proof.
Sincef and g satisfy E.A. property, there exists in X a sequence (xn) satisfying limn→∞f(xn)=limn→∞g(xn)=t for some t∈X.
Sinceg(X) is closed subspace of X and limn→∞g(xn)=t, there exists p∈X such that g(p)=t, alsolimn→∞f(xn)=g(p).
We will show thatf(p)=g(p) supposing that f(p)≠g(p), then condition (3) implies that
(2.40)G(f(p),f(p),f(xn))≤kM(p,p,xn),
where,
(2.41)M(p,p,xn)=max⁡{[G(g(p),f(p),f(p))+G(g(p),f(p),f(p))+G(g(xn),f(xn),f(xn))],[G(g(p),f(p),f(p))+G(g(p),f(p),f(p))+G(g(xn),f(p),f(p))],[G(g(p),f(xn),f(xn))+G(g(p),f(xn),f(xn))+G(g(xn),f(p),f(p))]}.
Taking the limit as n→∞ and using the fact that the function G is continuous, we get
(2.42)G(f(p),f(p),g(p))≤3kG(g(p),f(p),f(p)),
which is contradiction since k∈[0,1/3), so fp=gp. Since f and g are G-weakly commuting of type Ag, then
(2.43)G(gf(p),gg(p),ff(p))≤G(g(p),f(p),g(p))=0.
Therefore, f(g(p))=ff(p)=gf(p)=gg(p); then
(2.44)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 thatft≠t, then
(2.45)G(f(t),t,t)=G(f(t),f(p),f(p))≤kM(t,p,p),
where
(2.46)M(t,p,p)=max⁡{[G(g(t),f(t),f(t))+G(g(p),f(p),f(p))+G(g(p),f(p),f(p))],[G(g(t),f(p),f(p))+G(g(p),f(t),f(t))+G(g(p),f(p),f(p))],[G(g(t),f(p),f(p))+G(g(p),f(p),f(p))+G(g(p),f(t),f(t))]}.
But f(t)=g(t) and f(p)=g(p). Thus,
(2.47)G(f(t),t,t)≤kmax⁡{G(t,f(t),f(t)),[G(f(t),t,t)+G(t,f(t),f(t))]}<k{G(f(t),t,t)+G(t,f(t),f(t))}.
Hence,
(2.48)G(f(t),t,t)≤(k1-k)G(t,f(t),f(t)).
Adjusting similarly, we get(2.49)G(t,f(t),f(t))≤(k1-k)G(ft,t,t).
Therefore,
(2.50)G(ft,t,t)≤(k1-k)2G(ft,t,t),
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 haveu and v such that u≠v, fu=gu=u and fv=gv=v; then
(2.51)G(u,v,v)=G(f(u),f(v),f(v))≤k{G(u,v,v)+G(v,u,u)}.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.
LetX=[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 Ag,(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 thatforallx∈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 Ag.
To show (c), if we consider the sequence{xn}={1/2n}, then fxn→0 and gxn→0 as n→∞. Thus, f and g satisfy the E.A. property.
To show (d), forx,y,z∈X we have
(2.52)|x25-y25|≤x25+y25,|y25-z25|≤y25+z25,|x25-z25|≤x25+z25.
Then(2.53)G(f(x),f(y),f(z))=max⁡{|x25-y25|,|y25-z25|,|x25-z25|}≤x25+y25+z25=14(4x25+4y25+4z25)=14(G(g(x),f(x),f(x))+G(g(y),f(y),f(y))+G(g(z),f(z),f(z)))≤kM(x,y,z).
Therefore, all hypotheses of Theorem2.16 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 Af. 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,
(2.54)G(f(x),f(y),f(z))<max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),g(y)),G(g(z),f(z),f(x)),G(g(y),f(y),f(z))}, for all x,y,z∈X; then f and g have a unique common fixed point.Proof.
Letx0∈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 (yn)∈X as follows:
(2.55)yn=f(xn)=g(xn+1),n∈N∪{0}.
We will show that the sequence (yn) is G-cauchy sequence:
(2.56)G(g(xn),g(xn+1),g(xn+1))=G(g(xn),f(xn),f(xn))<ψ(g(xn))-ψ(f(xn))=ψ(g(xn))-ψ(g(xn+1)).
Consideran=ψ(g(xn)),n=1,2,3,4,…, then
(2.57)0≤G(g(xn),g(xn+1),g(xn+1))<an-an+1.
Thus, the sequence(an) is nonincreasing and bounded below by 0; hence (an)is convergent sequence.
On the other hand we have, from (G5) and (2.57), that for m,n∈N;m>n(2.58)G(g(xn),g(xn+m),g(xn+m))≤∑‍j=nn+m-1G(g(xj),g(xj+1),g(xj+1))<∑‍j=nn+m-1aj-aj+1(Telescopingsum)=an-an+m.
Therefore, the sequence(g(xn)) is G-cauchy sequence in g(X).
Sinceg(X) is G-complete subspace, then there exists t∈g(X) such that  limn→∞g(xn)=t; having t∈g(X) there exists p∈X such that g(p)=t,  also  limn→∞f(xn)=g(p)=t.
We will show thatf(p)=g(p); supposing that f(p)≠g(p), then condition (4) implies that
(2.59)G(f(p),f(p),f(xn))<max⁡{G(g(p),g(p),g(xn)),G(g(p),f(p),g(p)),G(g(p),f(p),f(xn)),G(g(xn),f(xn),f(p))}.
Taking the limit as n→∞, we get
(2.60)G(f(p),f(p),g(p))<max⁡{G(g(p),g(p),g(p)),G(g(p),f(p),g(p)),G(g(p),g(p),f(p))},
hence,
(2.61)G(f(p),f(p),g(p))<G(g(p),g(p),f(p)).
Adjusting similarly, we get
(2.62)G(g(p),g(p),f(p))<G(f(p),f(p),g(p))
Therefore,(2.63)G(f(p),f(p),g(p))<G(g(p),f(p),g(p))<G(f(p),f(p),g(p)).
Thus, a contradiction impliesfp=gp.
Sincef and g are G-weakly commuting of type Af, then
(2.64)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), then f(t)=fg(p)=gf(p)=g(t).
Finally, we will show thatt=f(p) is common fixed point of f and g.
Suppose thatft≠t, so
(2.65)G(f(t),t,t)=G(f(t),f(p),f(p))<max⁡{G(g(t),g(p),g(p)),G(g(t),f(t),g(p)),G(g(p),f(p),f(p))}.
Sinceg(p)=f(p) and g(t)=f(t), therefore (2.65) implies that
(2.66)G(f(t),t,t)<G(f(t),f(t),t).
Similarly, we haveG(f(t),f(t),t)<G(f(t),t,t).
A contradiction implies thatft=fp=t. Then t is a common fixed point.
To prove uniqueness suppose we haveu and v such that u≠v where fu=gu=u and fv=gv=v; then as an easy calculation one can get
(2.67)G(u,v,v)<G(v,u,u).
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.
LetX=[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 thatG(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 Af.
Also we see thatf(X)⊂g(X) and g(X) is G-complete subspace of X.
To prove (c),forallx∈X we see that
(2.68)G(gx,fx,fx)=x-1≤3x-3=ψ(g(x))-ψ(f(x)).
To prove (d), for allx,y,z∈X we have
(2.69)G(fx,fy,fz)=2max⁡{|x-y|,|y-z|,|x-z|}≤3max⁡{|x-y|,|y-z|,|x-z|}=G(gx,gy,gz)≤max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),g(y)),≤max⁡2G(g(z),f(z),f(x)),G(g(y),f(y),f(z))}.
Therefore, all hypotheses of the previous theorem are satisfied andx=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,
(2.70)G(f(3),f(1),f(1))=4>2k=kG(3,1,1)∀k∈[0,1).
Also, the Banach principle [34] is not applicable. Indeed, for d(x,y)=|x-y| for all x,y∈X we have for x≠y(2.71)d(f(x),f(y))=2|x-y|>k|x-y|∀k∈[0,1).Corollary 2.20.
Theorems2.13, 2.16, and 2.18 remain true if we replace, respectively, G-weakly commuting of type Af, G-weakly commuting of type Ag, weakly compatible and G-R-weakly commuting of type Af by any one of them (retaining the rest of hypothesis).Corollary 2.21.
Some corollaries could be derived from Theorems2.13, 2.16, and 2.18 by taking z=y or g=IdX.


## 2.1. New Concepts and Some Properties

In this section we introduce the concept of G-weakly commuting of typeAf 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 Af if
(2.1)G(fgx,ggx,ffx)≤G(fx,gx,fx),∀x∈X.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 Af if there exists some positive real number R such that
(2.2)G(fgx,ggx,ffx)≤RG(fx,gx,fx),∀x∈X.Remark 2.3.
TheG-weakly commuting maps of type Af are G-R-weakly commuting of type Af. Reciprocally, if R≤1, then G-R-weakly commuting maps of type Af are G-weakly commuting of type Af.
If we interchangef and g in (2.1) and (2.2), then the pair of mappings (f,g) is called G-weakly commuting of type Ag and G-R-weakly commuting of type Ag, respectively.Example 2.4.
LetX=[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 Af and G-R-Weakly commuting of type Af.Example 2.5.
LetX=[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 Af or Ag, but it is G-R-weakly commuting of type Af (and Ag) for R≥4.The following examples show a pair of mappings(f,g) that G-weakly commuting of type Gf need not be G-weakly commuting of type Af.Example 2.6.
LetX=[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 Af, but it is G-weakly commuting of type Gf.The following example shows that(1)
a pair of mappings(f,g) that is G-weakly commuting of type Af need not be G-weakly commuting of type Ag;(2)
a pair of mappings(f,g) that is G-weakly commuting of type Af need not be G-weakly commuting of type Gf;(3)
a pair of mappings(f,g) that is G-R-weakly commuting of type Af need not be R-weakly commuting;Example 2.7.
LetX=[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
(2.3)f(x)={2ifx=2,6if2<x≤5,5ifx>5,g(x)={2ifx=2,9if2<x≤5,6ifx>5.
Then,(2.4)f(g(x))={2ifx=2,5ifx>2,g(f(x))={2ifx=2,6if2<x≤5,9ifx>5,g(g(x))={2ifx=2,6ifx>2,f(f(x))={2ifx=2,5if2<x≤5,6ifx>5.
Moreover,(2.5)|f(x)-g(x)|={0ifx=2,3if2<x≤5,1ifx>5.
Ifx=2, we have G(f(g(x)),g(g(x)),f(f(x)))=0=G(f(x),g(x),f(x)).
If2<x≤5, we have
(2.6)G(f(g(x)),f(f(x)),g(g(x)))=max⁡{0,1,1}≤3=G(f(x),g(x),f(x)).
Ifx>5, then
(2.7)G(f(g(x)),f(f(x)),g(g(x)))=max⁡{1,1,0}≤1=G(g(x),f(x),g(x)).
Thus,f and g are G weakly commuting of type Af, but for x=6, we have
(2.8)G(g(f(6)),f(f(6)),g(g(6)))=max⁡{3,3,0}≰1=G(g(6),f(6),g(6)).
Therefore, the pair(f,g) is not G-weakly commuting of type Ag, but it is G-weakly commuting of type Af.
Also forx=7, we have
(2.9)G(f(g(7)),g(f(7)),f(f(7)))=4≰1=G(g(6),f(6),g(6)).
Therefore, the pair(f,g) is not G-weakly commuting of type Gf.
AS an easy calculation one can see that(f,g) are G-R-weakly commuting of type Ag 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.
Iff and g are G-weakly commuting of type Af or G-R-weakly commuting of type Af, then f and g are weakly compatible.Proof.
Letx 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 Af, we have
(2.10)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.
It followsf(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 Gf, we have
(2.11)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.
Thusf(g(x))=g(f(x)); then the pair (f,g) is weakly compatible.The following example shows that(1)
the converse of Lemma2.8 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 Af,(3)
a pair of mappings(f,g) that is R-weakly commuting need not be G-R-weakly commuting of type Gf.Example 2.9.
LetX=[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 forx=3 we have,
(2.12)G(f(g(x)),g(g(x)),f(f(x)))=72≰4=G(f(x),g(x),f(x)).
Therefore, f and g are not G-weakly commuting of type Af.
Also, we see thatG(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 Af 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 Gf for R=2.Now, we rewrite Definition1.1 on G-metric spaces setting.Definition 2.10.
LetS 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 (xn) such that (Txn) and (Sxn)G-converge to t for some t∈X; that is, thanks to Proposition 1.6,
(2.13)lim⁡n⟶∞G(Txn,Txn,t)=lim⁡n⟶∞G(Sxn,Sxn,t)=0.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 iff and g satisfy the E.A. property, then the pair (f,g) need not be G-weakly commuting of type Af.Example 2.12.
We return to Example2.9. Let xn=1+(1/3n). We have limn→∞f(xn)=limn→∞(1+(2/3n))=1, and limn→∞g(xn)=limn→∞(1+(1/3n))2=1, therefore, limn→∞f(xn)=limn→∞g(xn)=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 Af.Following Matkowski (see [32]), let Φ be the set of all functions ϕ such that  ϕ:[0,∞)→[0,∞) be a nondecreasing function with limn→∞ϕn(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 Af,(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,where(2.14)M(x,y,z)=max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),f(x)),12G(g(x),f(y),f(y)),12G(g(x),f(z),f(z)),G(g(y),f(y),f(y)),G(g(y),f(x),f(x)),G(g(y),f(z),f(z)),G(g(z),f(z),f(z)),G(g(z),f(x),f(x)),G(g(z),f(y),f(y))}.   Then f and g have a unique common fixed point.Proof.
Letx0∈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 (yn)∈X as follows:
(2.15)yn=f(xn)=g(xn+1),n∈N∪{0}
We will show that the sequence (yn) is G-cauchy sequence:
(2.16)G(yn,yn+1,yn+1)=G(f(xn),f(xn+1),f(xn+1))≤ϕ(M(xn,xn+1,xn+1)),
where
(2.17)M(xn,xn+1,xn+1)=max⁡{G(g(xn),g(xn+1),g(xn+1)),G(g(xn),f(xn),f(xn)),12G(g(xn),f(xn+1),f(xn+1)),G(g(xn+1),f(xn),f(xn)),G(g(xn+1),f(xn+1),f(xn+1)),}=max⁡{G(yn-1,yn,yn),12G(yn-1,yn+1,yn+1),G(yn,yn,yn),G(yn,yn+1,yn+1),}.
We will have different cases.Case (1): ifM(xn,xn+1,xn+1)=G(yn,yn+1,yn+1), then G(yn,yn+1,yn+1)≤ϕ(G(yn,yn+1,yn+1)) <G(yn,yn+1,yn+1), which is contradiction.Case (2): ifM(xn,xn+1,xn+1)=(1/2)G(yn-1,yn+1,yn+1), then in this case we havemax{G(yn-1,yn,yn),G(yn,yn+1,yn+1)}<(1/2)G(yn-1,yn+1,yn+1),  which implies that(2.18)G(yn-1,yn,yn)+G(yn,yn+1,yn+1)<G(yn-1,yn+1,yn+1),
but fromG-metric property (G5) we have
(2.19)G(yn-1,yn+1,yn+1)≤G(yn-1,yn,yn)+G(yn,yn+1,yn+1).
Thus, from (2.18) and (2.19) we see that case (2) is impossible.
Then, we must have the case(2.20)M(xn,xn+1,xn+1)=G(yn-1,yn,yn).
Thus, forn∈N∪{0} and from (2.16) we have,
(2.21)G(yn,yn+1,yn+1)≤ϕ(G(yn-1,yn,yn))≤ϕ2(G(yn-2,yn-1,yn-1))⋮≤ϕn(G(y0,y1,y1))…(1).
Givenϵ>0, since limn→∞ϕn(G(y0,y1,y1))=0, and ϕ(ϵ)<ϵ, there is an integer no∈N, such that
(2.22)ϕn(G(y0,y1,y1))<ϵ-ϕ(ϵ),∀n≥n0.
Hence, we have(2.23)G(yn,yn+1,yn+1)≤ϕn(G(y0,y1,y1))<ϵ-ϕ(ϵ).
Now form,n∈N; m>n, we claim that
(2.24)G(yn,ym,ym)<ϵ,∀m≥n≥n0.
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
(2.25)G(yn,yk+1,yk+1)≤G(yn,yn+1,yn+1)+G(yn+1,yk+1,yk+1)<ϵ-ϕ(ϵ)+ϕ(G(yn,yk,yk))<ϵ-ϕ(ϵ)+ϕ(ϵ)=ϵ.
By induction onm, we conclude that (2.24) holds for all m≥n≥n0.
Hence, the sequence(yn)=g(xn+1) is G-cauchy sequence in g(X); since g(X) is G-complete, then there exists t∈g(X) such that limn→∞g(xn)=t=limn→∞f(xn).
Thus, there existsp∈X such that g(p)=t, also limn→∞f(xn)=g(p).
We will show thatf(p)=g(p). Supposing that f(p)≠g(p), then condition (4) implies that, G(f(p),f(p),f(xn))≤ϕ(M(p,p,xn)), where
(2.26)M(p,p,xn)=max⁡{G(g(p),g(p),g(xn)),G(g(p),f(p),f(p)),12G(g(p),f(p),f(p)),12G(g(p),f(xn),f(xn)),G(g(p),f(p),f(p)),G(g(p),f(p),f(p)),G(g(p),f(xn),f(xn)),G(g(xn),f(xn),f(xn)),G(g(xn),f(p),f(p)),G(g(xn),f(p),f(p))}.
Taking the limit asn→∞ and using the fact that the function G is continuous we get(2.27)G(f(p),f(p),g(p))≤ϕ(max⁡{G(g(p),f(p),f(p)),12G(g(p),f(p),f(p))})=ϕ(G(g(p),f(p),f(p))).
Therefore,
(2.28)G(f(p),f(p),g(p))≤ϕ(G(g(p),f(p),f(p)))<G(g(p),f(p),f(p)),
which is contradiction; hence fp=gp.
Sincef and g are G-weakly commuting of type Af, 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 thatt:=f(p) is common fixed point of f and g.
Supposing thatft≠t, then
(2.29)G(f(t),t,t)=G(f(t),f(p),f(p))≤ϕ(M(t,p,p)),
where
(2.30)M(t,p,p)=max⁡{G(g(t),g(p),g(p)),G(g(t),f(t),f(t)),12G(g(t),f(p),f(p)),12G(g(t),f(p),f(p)),G(g(p),f(p),f(p)),G(g(p),f(t),f(t)),G(g(p),f(p),f(p)),G(g(p),f(p),f(p)),G(g(p),f(t),f(t)),G(g(p),f(p),f(p))}.
Since g(t)=f(t), and g(p)=f(p), therefore (2.30) implies that
(2.31)M(t,p,p)=max⁡{G(f(t),t,t),12G(f(t),t,t),G(t,f(t),f(t))}.
Hence, (2.29) becomes
(2.32)G(f(t),t,t)≤ϕ(max⁡{G(f(t),t,t),G(t,f(t),f(t))})=ϕ(G(t,f(t),f(t)))<G(t,f(t),f(t)).
Similarly we get,
(2.33)G(t,f(t),f(t))<G(f(t),t,t).
So,
(2.34)G(f(t),t,t)<G(f(t),t,t),
a contradiction which implies that t=ft=gt. Then t is a common fixed point.
To prove uniqueness suppose we haveu and v such that u≠v, fu=gu=u and fv=gv=v; then condition (4) implies that
(2.35)G(u,v,v)≤ϕ(G(v,u,u)).
Therefore,
(2.36)G(u,v,v)≤ϕ(G(v,u,u))<G(v,u,u).
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.
LetX=[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 Af,(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 Af.
To show (d), forx,y,z∈X we have
(2.37)G(f(x),f(y),f(z))=18max⁡{|x3-y3|,|y3-z3|,|x3-z3|}≤13max⁡⁡{|x3-y3|,|y3-z3|,|x3-z3|}=23(12max⁡⁡{|x3-y3|,|y3-z3|,|x3-z3|})=ϕ(g(x),g(y),g(z))≤ϕ(M(x,y,z)).
Therefore, all hypotheses of Theorem2.13 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 Af,(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),where(2.38)M(x,y,z)=max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),f(x)),12G(g(x),f(y),f(y)),12G(g(x),f(z),f(z)),G(g(y),f(y),f(y))),G(g(y),f(x),f(x)),G(g(y),f(z),f(z)),G(g(z),f(z),f(z)),G(g(z),f(x),f(x)),G(g(z),f(y),f(y))}, for all x,y,z∈X, where k∈[0,1); then f and g have a unique common fixed point.Proof.
It suffices to takeϕ(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 Ag 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),where(2.39)M(x,y,z)=max⁡{[G(g(x),f(x),f(x))+G(g(y),f(y),f(y))+G(g(z),f(z),f(z))],[G(g(x),f(y),f(y))+G(g(y),f(x),f(x))+G(g(z),f(y),f(y))],[G(g(x),f(z),f(z))+G(g(y),f(z),f(z))+G(g(z),f(x),f(x))]} for all x,y,z∈X, where k∈[0,1/3); then f and g have a unique common fixed point.Proof.
Sincef and g satisfy E.A. property, there exists in X a sequence (xn) satisfying limn→∞f(xn)=limn→∞g(xn)=t for some t∈X.
Sinceg(X) is closed subspace of X and limn→∞g(xn)=t, there exists p∈X such that g(p)=t, alsolimn→∞f(xn)=g(p).
We will show thatf(p)=g(p) supposing that f(p)≠g(p), then condition (3) implies that
(2.40)G(f(p),f(p),f(xn))≤kM(p,p,xn),
where,
(2.41)M(p,p,xn)=max⁡{[G(g(p),f(p),f(p))+G(g(p),f(p),f(p))+G(g(xn),f(xn),f(xn))],[G(g(p),f(p),f(p))+G(g(p),f(p),f(p))+G(g(xn),f(p),f(p))],[G(g(p),f(xn),f(xn))+G(g(p),f(xn),f(xn))+G(g(xn),f(p),f(p))]}.
Taking the limit as n→∞ and using the fact that the function G is continuous, we get
(2.42)G(f(p),f(p),g(p))≤3kG(g(p),f(p),f(p)),
which is contradiction since k∈[0,1/3), so fp=gp. Since f and g are G-weakly commuting of type Ag, then
(2.43)G(gf(p),gg(p),ff(p))≤G(g(p),f(p),g(p))=0.
Therefore, f(g(p))=ff(p)=gf(p)=gg(p); then
(2.44)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 thatft≠t, then
(2.45)G(f(t),t,t)=G(f(t),f(p),f(p))≤kM(t,p,p),
where
(2.46)M(t,p,p)=max⁡{[G(g(t),f(t),f(t))+G(g(p),f(p),f(p))+G(g(p),f(p),f(p))],[G(g(t),f(p),f(p))+G(g(p),f(t),f(t))+G(g(p),f(p),f(p))],[G(g(t),f(p),f(p))+G(g(p),f(p),f(p))+G(g(p),f(t),f(t))]}.
But f(t)=g(t) and f(p)=g(p). Thus,
(2.47)G(f(t),t,t)≤kmax⁡{G(t,f(t),f(t)),[G(f(t),t,t)+G(t,f(t),f(t))]}<k{G(f(t),t,t)+G(t,f(t),f(t))}.
Hence,
(2.48)G(f(t),t,t)≤(k1-k)G(t,f(t),f(t)).
Adjusting similarly, we get(2.49)G(t,f(t),f(t))≤(k1-k)G(ft,t,t).
Therefore,
(2.50)G(ft,t,t)≤(k1-k)2G(ft,t,t),
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 haveu and v such that u≠v, fu=gu=u and fv=gv=v; then
(2.51)G(u,v,v)=G(f(u),f(v),f(v))≤k{G(u,v,v)+G(v,u,u)}.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.
LetX=[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 Ag,(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 thatforallx∈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 Ag.
To show (c), if we consider the sequence{xn}={1/2n}, then fxn→0 and gxn→0 as n→∞. Thus, f and g satisfy the E.A. property.
To show (d), forx,y,z∈X we have
(2.52)|x25-y25|≤x25+y25,|y25-z25|≤y25+z25,|x25-z25|≤x25+z25.
Then(2.53)G(f(x),f(y),f(z))=max⁡{|x25-y25|,|y25-z25|,|x25-z25|}≤x25+y25+z25=14(4x25+4y25+4z25)=14(G(g(x),f(x),f(x))+G(g(y),f(y),f(y))+G(g(z),f(z),f(z)))≤kM(x,y,z).
Therefore, all hypotheses of Theorem2.16 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 Af. 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,
(2.54)G(f(x),f(y),f(z))<max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),g(y)),G(g(z),f(z),f(x)),G(g(y),f(y),f(z))}, for all x,y,z∈X; then f and g have a unique common fixed point.Proof.
Letx0∈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 (yn)∈X as follows:
(2.55)yn=f(xn)=g(xn+1),n∈N∪{0}.
We will show that the sequence (yn) is G-cauchy sequence:
(2.56)G(g(xn),g(xn+1),g(xn+1))=G(g(xn),f(xn),f(xn))<ψ(g(xn))-ψ(f(xn))=ψ(g(xn))-ψ(g(xn+1)).
Consideran=ψ(g(xn)),n=1,2,3,4,…, then
(2.57)0≤G(g(xn),g(xn+1),g(xn+1))<an-an+1.
Thus, the sequence(an) is nonincreasing and bounded below by 0; hence (an)is convergent sequence.
On the other hand we have, from (G5) and (2.57), that for m,n∈N;m>n(2.58)G(g(xn),g(xn+m),g(xn+m))≤∑‍j=nn+m-1G(g(xj),g(xj+1),g(xj+1))<∑‍j=nn+m-1aj-aj+1(Telescopingsum)=an-an+m.
Therefore, the sequence(g(xn)) is G-cauchy sequence in g(X).
Sinceg(X) is G-complete subspace, then there exists t∈g(X) such that  limn→∞g(xn)=t; having t∈g(X) there exists p∈X such that g(p)=t,  also  limn→∞f(xn)=g(p)=t.
We will show thatf(p)=g(p); supposing that f(p)≠g(p), then condition (4) implies that
(2.59)G(f(p),f(p),f(xn))<max⁡{G(g(p),g(p),g(xn)),G(g(p),f(p),g(p)),G(g(p),f(p),f(xn)),G(g(xn),f(xn),f(p))}.
Taking the limit as n→∞, we get
(2.60)G(f(p),f(p),g(p))<max⁡{G(g(p),g(p),g(p)),G(g(p),f(p),g(p)),G(g(p),g(p),f(p))},
hence,
(2.61)G(f(p),f(p),g(p))<G(g(p),g(p),f(p)).
Adjusting similarly, we get
(2.62)G(g(p),g(p),f(p))<G(f(p),f(p),g(p))
Therefore,(2.63)G(f(p),f(p),g(p))<G(g(p),f(p),g(p))<G(f(p),f(p),g(p)).
Thus, a contradiction impliesfp=gp.
Sincef and g are G-weakly commuting of type Af, then
(2.64)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), then f(t)=fg(p)=gf(p)=g(t).
Finally, we will show thatt=f(p) is common fixed point of f and g.
Suppose thatft≠t, so
(2.65)G(f(t),t,t)=G(f(t),f(p),f(p))<max⁡{G(g(t),g(p),g(p)),G(g(t),f(t),g(p)),G(g(p),f(p),f(p))}.
Sinceg(p)=f(p) and g(t)=f(t), therefore (2.65) implies that
(2.66)G(f(t),t,t)<G(f(t),f(t),t).
Similarly, we haveG(f(t),f(t),t)<G(f(t),t,t).
A contradiction implies thatft=fp=t. Then t is a common fixed point.
To prove uniqueness suppose we haveu and v such that u≠v where fu=gu=u and fv=gv=v; then as an easy calculation one can get
(2.67)G(u,v,v)<G(v,u,u).
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.
LetX=[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 thatG(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 Af.
Also we see thatf(X)⊂g(X) and g(X) is G-complete subspace of X.
To prove (c),forallx∈X we see that
(2.68)G(gx,fx,fx)=x-1≤3x-3=ψ(g(x))-ψ(f(x)).
To prove (d), for allx,y,z∈X we have
(2.69)G(fx,fy,fz)=2max⁡{|x-y|,|y-z|,|x-z|}≤3max⁡{|x-y|,|y-z|,|x-z|}=G(gx,gy,gz)≤max⁡{G(g(x),g(y),g(z)),G(g(x),f(x),g(y)),≤max⁡2G(g(z),f(z),f(x)),G(g(y),f(y),f(z))}.
Therefore, all hypotheses of the previous theorem are satisfied andx=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,
(2.70)G(f(3),f(1),f(1))=4>2k=kG(3,1,1)∀k∈[0,1).
Also, the Banach principle [34] is not applicable. Indeed, for d(x,y)=|x-y| for all x,y∈X we have for x≠y(2.71)d(f(x),f(y))=2|x-y|>k|x-y|∀k∈[0,1).Corollary 2.20.
Theorems2.13, 2.16, and 2.18 remain true if we replace, respectively, G-weakly commuting of type Af, G-weakly commuting of type Ag, weakly compatible and G-R-weakly commuting of type Af by any one of them (retaining the rest of hypothesis).Corollary 2.21.
Some corollaries could be derived from Theorems2.13, 2.16, and 2.18 by taking z=y or g=IdX.



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*Source: 248937-2012-09-13.xml*