# On Generalized WeaklyG-Contractive Mappings in Partially Ordered G-Metric Spaces

Authors: A. Razani; V. Parvaneh
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The aim of this paper is to present some coincidence and common fixed point results for generalized weaklyG-contractive mappings in the setup of partially ordered G-metric space. We also provide an example to illustrate the results presented herein. As an application of our results, periodic points of weakly G-contractive mappings are obtained.


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## Body

## 1. Introduction and Mathematical Preliminaries

The concept of a generalized metric space, or aG-metric space, was introduced by Mustafa et al. [1]. In recent years, many authors have obtained different fixed point theorems for mappings satisfying various contractive conditions on G-metric spaces.  For a survey of fixed point theory, its applications, comparison of different contractive conditions, and related topics in G-metric spaces we refer the reader to [1–14] and the references mentioned therein.Definition 1.1 (G-metric space [1]).
LetX be a nonempty set and G:X×X×X→R+ be a function satisfying the following properties:(G1)
G(x,y,z)=0 if and only if  x=y=z;(G2)
0<G(x,x,y), for all x,y∈X with x≠y;(G3)
G(x,x,y)≤G(x,y,z), for all x,y,z∈X with z≠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).
Then, the functionG is called a G-metric on X and the pair (X,G) is called a G-metric space.Definition 1.2 (see [1]).
Let(X,G) be a G-metric space and let {xn} be a sequence of points of X. A point x∈X is said to be the limit of the sequence {xn} if limn,m→∞G(x,xn,xm)=0 and one says that the sequence {xn} is G-convergent to x. Thus, if xn→x in a G-metric space (X,G), then for any ε>0, there exists a positive integer N such that G(x,xn,xm)<ε, for all n,m≥N.Definition 1.3 (see [1]).
Let(X,G) be a G-metric space. A sequence {xn} is called G-Cauchy if for every ε>0, there is a positive integer N such that G(xn,xm,xl)<ε, for all n,m,l≥N, that is, if G(xn,xm,xl)→0, as n,m,l→∞.Lemma 1.4 (see [1]).
Let(X,G) be a G-metric space. Then, the following 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→∞.Lemma 1.5 (see [15]).
If(X,G) is a G-metric space, then {xn} is a G-Cauchy sequence if and only if for every ε>0, there exists a positive integer N such that G(xn,xm,xm)<ε, for all m>n≥N.Definition 1.6 (see [1]).
AG-metric space (X,G) is said to be G-complete (or complete G-metric space) if every G-Cauchy sequence in (X,G) is convergent in X.Definition 1.7 (see [1]).
Let(X,G) and (X',G') be two G-metric spaces. Then a function f:X→X' is G-continuous at a point x∈X if and only if it is G-sequentially continuous at x, that is, whenever {xn} is G-convergent to x, {f(xn)} is G-convergent to f(x).The concept of an altering distance function was introduced by Khan et al. [16] as follows.Definition 1.8.
The functionψ:[0,∞)→[0,∞) is called an altering distance function, if the following properties are satisfied.(1)
ψ is continuous and nondecreasing.(2)
ψ(t)=0 if and only if t=0.In [5], Aydi et al. established some common fixed point results for two self-mappings f and g on a generalized metric space X. They presented the following definitions.Definition 1.9 (see [5]).
Let(X,G) be a G-metric space and f,g:X→X be two mappings. We say that f is a generalized weakly G-contraction mapping of type A with respect to g if for all x,y,z∈X, the following inequality holds:
(1.1)ψ(G(fx,fy,fz))≤ψ(G(gx,fy,fy)+G(gy,fz,fz)+G(gz,fx,fx)3)-φ(G(gx,fy,fy),G(gy,fz,fz),G(gz,fx,fx)),
where(1)
ψ is an altering distance function;(2)
φ:[0,∞)3→[0,∞) is a continuous function with φ(t,s,u)=0 if and only if t=s=u=0.Definition 1.10 (see [5]).
Let(X,G) be a G-metric space and f,g:X→X be given mappings. We say that f is a generalized weakly G-contraction mapping of type B with respect to g if for all x,y,z∈X, the following inequality holds:
(1.2)ψ(G(fx,fy,fz))≤ψ(G(gx,gx,fy)+G(gy,gy,fz)+G(gz,gz,fx)3)-φ(G(gx,gx,fy),G(gy,gy,fz),G(gz,gz,fx)),
where(1)
ψ is an altering distance function;(2)
φ:[0,∞)3→[0,∞) is a continuous function with φ(t,s,u)=0 if and only if t=s=u=0.
Note that the concept of a generalized weaklyG-contraction is the extension of the concept of weakly C-contraction which has been defined by Choudhury in [17]. For more details on weakly C-contractive mappings we refer the reader to [18, 19].Definition 1.11 (see [20]).
Let(X,⪯) be a partially ordered set. A mapping f is called a dominating map on X if x⪯fx for each x in X.Example 1.12 (see [20]).
LetX=[0,1] be endowed with the usual ordering. Let f:X→X be defined by fx=x1/3. Then, x≤x1/3=fx for all x∈X. Thus, f is a dominating map.Example 1.13 (see [20]).
LetX=[0,∞) be endowed with the usual ordering. Let f:X→X be defined by fx=xn for x∈[0,1) and fx=xn for x∈[1,∞), for any n∈ℕ. Then, for all x∈X,x≤fx; that is, f is a dominating map.A subsetW of a partially ordered set X is said to be well ordered if every two elements of W be comparable [20].The following definition is Definition 2.5 of [21], but in the setup of partially ordered G-metric spaces.Definition 1.14.
Let(X,⪯,G) be a partially ordered G-metric space. We say that X is regular if and only if the following hypothesis holds.
For any nondecreasing sequence{xn} in X such that xn→z as n→∞, it follows that xn⪯z for all n∈ℕ.Jungck in [22] introduced the following definition.Definition 1.15 (see [22]).
Let(X,d) be a metric space and f,g:X→X. The pair (f,g) is said to be compatible if and only if limn→∞d(fgxn,gfxn)=0, whenever {xn} is a sequence in X such that limn→∞fxn=limn→∞gxn=t for some t∈X.LetX be a nonempty set and f:X→X be a given mapping. For every x∈X, let f-1(x)={u∈X∣fu=x}.Definition 1.16 (see [21]).
Let(X,⪯) be a partially ordered set and f,g,h:X→X are given mappings such that fX⊆hX and gX⊆hX. We say that f and g are weakly increasing with respect to h if and only if for all x∈X, we have
(1.3)fx⪯gy,∀y∈h-1(fx),gx⪯fy,∀y∈h-1(gx).
Iff=g, we say that f is weakly increasing with respect to h.Ifh=I (the identity mapping on X), then the above definition reduces to the weakly increasing mapping [23] (also see [21, 24]).Definition 1.17.
Let(X,G) be a G-metric space and f,g:X→X. The pair (f,g) is said to be compatible if and only if limn→∞G(fgxn,fgxn,gfxn)=0, whenever {xn} is a sequence in X such that limn→∞fxn=limn→∞gxn=t for some t∈X.Note that the concept of compatibility in aG-metric space has been defined by Kumar in [25] (Definition 2.1). In the above definition we only modify his definition, using the fact that G(x,y,y)≤2G(x,x,y), for all x,y∈X.The aim of this paper is to prove some coincidence and common fixed point theorems for nonlinear weaklyG-contractive mappings in partially ordered G-metric spaces.


## 2. Main Results

From now, we assume(2.1)Φ={φ∣φ:[0,∞)3→[0,∞)isacontinuousfunctionsuchthatφ(x,y,z)=0⟺x=y=z=0[0,∞)3}.Our first result is the following.Theorem 2.1.
Let(X,⪯,G) be a partially ordered complete G-metric space. Let f,g:X→X be two mappings such that f(X)⊆g(X); f is weakly increasing with respect to g and
(2.2)ψ(G(fx,fy,fz))≤ψ(G(gx,fy,fy)+G(gy,fz,fz)+G(gz,fx,fx)3)-φ(G(gx,fy,fy),G(gy,fz,fz),G(gz,fx,fx))
for every x,y,z∈X such that gx⪯gy⪯gz, where ψ:[0,∞)→[0,∞) is an altering distance function and φ∈Φ. Then f and g have a coincidence point in X provided that f and g are continuous and the pair (f,g) is compatible.Proof.
Letx0∈X be an arbitrary point. Since f(X)⊆g(X), we can construct a sequence {zn} defined by: zn=gxn=fxn-1, for all n≥0.
Now, sincex1∈g-1(fx0) and x2∈g-1(fx1), as f is weakly increasing with respect to g, we obtain
(2.3)gx1=fx0⪯fx1=gx2⪯fx2=gx3.
Continuing this process, we get:(2.4)gx1⪯gx2⪯gx3⪯⋯⪯gxn⪯gxn+1⪯⋯.
We complete the proof in three steps.
Step I. We will prove that limn→∞G(zn,zn+1,zn+1)=0.
Sincegxn-1⪯gxn, using (2.2) we obtain that
(2.5)ψ(G(zn,zn+1,zn+1))=ψ(G(fxn-1,fxn,fxn))≤ψ(G(gxn-1,fxn,fxn)+G(gxn,fxn,fxn)+G(gxn,fxn-1,fxn-1)3)-φ(G(gxn-1,fxn,fxn),G(gxn,fxn,fxn),G(gxn,fxn-1,fxn-1))=ψ(G(zn-1,zn+1,zn+1)+G(zn,zn+1,zn+1)+G(zn,zn,zn)3)-φ(G(zn-1,zn+1,zn+1),G(zn,zn+1,zn+1),G(zn,zn,zn))≤ψ(G(zn-1,zn,zn)+2G(zn,zn+1,zn+1)3)-φ(G(zn-1,zn+1,zn+1),G(zn,zn+1,zn+1),G(zn,zn,zn))≤ψ(G(zn-1,zn,zn)+2G(zn,zn+1,zn+1)3).
Sinceψ is a nondecreasing function, from (2.5), we have
(2.6)G(zn,zn+1,zn+1)≤G(zn-1,zn+1,zn+1)+G(zn,zn+1,zn+1)3≤G(zn-1,zn,zn)+2G(zn,zn+1,zn+1)3.
Hence, we conclude that{G(zn,zn+1,zn+1)} is a nondecreasing sequence of nonnegative real numbers. Thus, there is an r≥0 such that
(2.7)limn→∞G(zn,zn+1,zn+1)=r.
Lettingn→∞ in (2.6), we get that
(2.8)r≤limn→∞G(zn-1,zn+1,zn+1)+r3≤r,
that is,
(2.9)limn→∞G(zn-1,zn+1,zn+1)=2r.
Again, from (2.5) we have
(2.10)ψ(G(zn,zn+1,zn+1))=ψ(G(zn-1,zn+1,zn+1)+G(zn,zn+1,zn+1)+G(zn,zn,zn)3)-φ(G(zn-1,zn+1,zn+1),G(zn,zn+1,zn+1),G(zn,zn,zn)).
Lettingn→∞ and using (2.7), (2.9), and the continuities of ψ and φ, we get ψ(r)≤ψ((2r+r+0)/3)-φ(2r,r,0), and hence φ(2r,r,0)=0. This gives us that
(2.11)limn→∞G(zn,zn+1,zn+1)=0,
from our assumptions about φ.
Step II. We will show that {zn} is a G-Cauchy sequences in X. So, we will show that for every ε>0, there exists k∈ℕ such that for all m,n≥k,
(2.12)G(zm,zn,zn)<ε.
Suppose the above statement is false. Then, there existsε>0 for which we can find subsequences {zm(k)} and {zn(k)} of {zn} such that n(k)>m(k)>k and
(2.13)G(zm(k),zn(k),zn(k))≥ε,
where n(k) is the smallest index with this property, that is,
(2.14)G(zm(k),zn(k)-1,zn(k)-1)<ε.
From rectangle inequality,(2.15)G(zm(k),zn(k),zn(k))≤G(zm(k),zn(k)-1,zn(k)-1)+G(zn(k)-1,zn(k),zn(k)).
Makingk→∞ in (2.15), from (2.11), (2.13), and (2.14) we conclude that
(2.16)limk→∞G(zm(k),zn(k),zn(k))=ε.
Again, from rectangle inequality,(2.17)G(zm(k),zn(k),zn(k)+1)≤G(zm(k),zn(k),zn(k))+G(zn(k),zn(k),zn(k)+1)≤G(zm(k),zn(k),zn(k))+2G(zn(k),zn(k)+1,zn(k)+1),G(zn(k),zn(k),zm(k))≤G(zn(k),zm(k),zn(k)+1).
Hence in (2.17), if k→∞, using (2.11), and (2.16), we have
(2.18)limk→∞G(zm(k),zn(k),zn(k)+1)=ε.
On the other hand,(2.19)G(zm(k),zn(k)+1,zn(k)+1)≤G(zm(k),zn(k),zn(k))+G(zn(k),zn(k)+1,zn(k)+1),
and
(2.20)G(zn(k),zn(k)+1,zm(k))≤G(zn(k),zn(k)+1,zn(k)+1)+G(zn(k)+1,zn(k)+1,zm(k)).
Hence in (2.19) and (2.20), if k→∞, from (2.11), (2.16) and (2.18) we have
(2.21)limk→∞G(zm(k),zn(k)+1,zn(k)+1)=ε.
In a similar way, we have(2.22)G(zm(k)+1,zn(k),zn(k)+1)≤G(zm(k)+1,zm(k),zm(k))+G(zm(k),zn(k),zn(k)+1)≤2G(zm(k),zm(k)+1,zm(k)+1)+G(zm(k),zn(k),zn(k)+1),G(zm(k),zn(k),zn(k)+1)≤G(zm(k),zm(k)+1,zm(k)+1)+G(zm(k)+1,zn(k),zn(k)+1),
and therefore, from (2.22) by taking limit when k→∞, using (2.11) and (2.18), we get that
(2.23)limk→∞G(zm(k)+1,zn(k),zn(k)+1)=ε.
Also,(2.24)G(zm(k)+1,zn(k)+1,zn(k)+1)≤G(zm(k)+1,zn(k)+1,zn(k)),G(zm(k)+1,zn(k),zn(k)+1)≤G(zm(k)+1,zn(k)+1,zn(k)+1)+G(zn(k)+1,zn(k)+1,zn(k)).
So, from (2.11), (2.23), and (2.24), we have
(2.25)limk→∞G(zm(k)+1,zn(k)+1,zn(k)+1)=ε.
Finally,(2.26)G(zn(k),zm(k)+1,zm(k)+1)≤G(zn(k),zn(k)+1,zn(k)+1)+G(zn(k)+1,zm(k)+1,zm(k)+1),G(zn(k)+1,zm(k)+1,zm(k)+1)≤G(zn(k)+1,zn(k),zn(k))+G(zn(k),zm(k)+1,zm(k)+1)≤G(zn(k),zn(k)+1,zn(k)+1)+G(zn(k),zm(k)+1,zm(k)+1).
Hence in (2.26), if k→∞ and using (2.11) and (2.25), we have
(2.27)limk→∞G(zm(k),zn(k)+1,zn(k)+1)=ε.
Sincegxm(k)⪯gxn(k)⪯gxn(k), putting x=xm(k), y=xn(k), and z=xn(k) in (2.2), for all k≥0, we have
(2.28)ψ(G(zm(k)+1,zn(k)+1,zn(k)+1))=ψ(G(fxm(k),fxn(k),fxn(k)))≤ψ(G(gxm(k),fxn(k),fxn(k))+G(gxn(k),fxn(k),fxn(k))+G(gxn(k),fxm(k),fxm(k))3)-φ(G(gxm(k),fxn(k),fxn(k)),G(gxn(k),fxn(k),fxn(k)),G(gxn(k),fxm(k),fxm(k)))≤ψ(G(zm(k),zn(k)+1,zn(k)+1)+G(zn(k),zn(k)+1,zn(k)+1)+G(zn(k),zm(k)+1,zm(k)+1)3)-φ(G(zm(k),zn(k)+1,zn(k)+1),G(zn(k),zn(k)+1,zn(k)+1),G(zn(k),zm(k)+1,zm(k)+1)).
Now, ifk→∞ in (2.28), from (2.11), (2.21), (2.25), and (2.27), we have
(2.29)ψ(ε)≤ψ(2ε3)-φ(ε,0,ε).
Hence,ε=0 which is a contradiction. Consequently, {zn} is G-Cauchy.
Step III. We will show that f and g have a coincidence point.
Since{gxn} is a G-Cauchy sequence in the complete G-metric space X, there exists z∈X such that
(2.30)limn→∞G(zn,zn,z)=limn→∞G(gxn,gxn,z)=limn→∞G(fxn,fxn,z)=0.
From (2.30) and the continuity of g, we get
(2.31)limn→∞G(gzn,gzn,gz)=limn→∞G(g(gxn),g(gxn),gz)=0.
By the rectangle inequality, we have(2.32)G(gz,fz,fz)≤G(gz,ggxn+1,ggxn+1)+G(gfxn,fz,fz)≤G(gz,ggxn+1,ggxn+1)+G(gfxn,fgxn,fgxn)+G(fgxn,fz,fz).
From (2.30), as n→∞, we have
(2.33)gxn→z,fxn→z.
Since the pair(f,g) is compatible, this implies that
(2.34)limn→∞G(gfxn,fgxn,fgxn)=0.
Now, from the continuity off and (2.30), we have
(2.35)limn→∞G(fzn,fz,fz)=0.
Combining (2.31), (2.32), and (2.34) and letting n→∞ in (2.35), we obtain
(2.36)G(gz,fz,fz)≤0,
which implies that fz=gz, that is, z is a coincidence point of f and g.In the following theorem, we will omit the continuity off and g, and the compatibility of the pair (f,g).Theorem 2.2.
Let(X,⪯,G) be a partially ordered G-metric space. Let f,g:X→X be two mappings such that f(X)⊆g(X); f is weakly increasing with respect to g and
(2.37)ψ(G(fx,fy,fz))≤ψ(G(gx,fy,fy)+G(gy,fz,fz)+G(gz,fx,fx)3)-φ(G(gx,fy,fy),G(gy,fz,fz),G(gz,fx,fx)),
for every x,y,z∈X such that gx⪯gy⪯gz, where ψ:[0,∞)→[0,∞) is an altering distance function and φ∈Φ. Then, f and g have a coincidence point in X if X is regular and g(X) is a G-complete subset of (X,G).Proof.
Following the proof of Theorem2.1, there exists z∈X such that
(2.38)limn→∞G(zn,zn,z)=limn→∞G(gxn,gxn,z)=limn→∞G(fxn,fxn,z)=0.
Sinceg(X) is G-complete and {zn}⊆g(X), we have z∈g(X) and hence there exists u∈X such that z=gu and
(2.39)limn→∞G(zn,zn,gu)=limn→∞G(gxn,gxn,gu)=limn→∞G(fxn,fxn,gu)=0.
Now, we will prove thatu is a coincidence point of f and g.
We know that{gxn} is a nondecreasing sequence in X. Regularity of X yields that gxn⪯z=gu. So, from (2.2) we have(2.40)ψ(G(zn+1,zn+1,fu))=ψ(G(fxn,fxn,fu))≤ψ(G(gxn,fxn,fxn)+G(gxn,fu,fu)+G(gu,fxn,fxn)3)-φ(G(gxn,fxn,fxn),G(gxn,fu,fu),G(gu,fxn,fxn))=ψ(G(zn,zn+1,zn+1)+G(zn,fu,fu)+G(gu,zn+1,zn+1)3)-φ(G(zn,zn+1,zn+1)+G(zn,fu,fu)+G(gu,zn+1,zn+1)).
Lettingn→∞ in (2.40), from the continuity of ψ and φ, we get
(2.41)ψ(G(z,z,fu))≤ψ(G(z,fu,fu)3)-φ(0,G(z,fu,fu),0).
AsG(z,fu,fu)≤2G(z,z,fu), we have
(2.42)ψ(G(z,z,fu))≤ψ(2G(z,z,fu)3)-φ(0,G(z,fu,fu),0).
Hence,φ(0,G(z,fu,fu),0)≤ψ(2G(z,z,fu)/3)-ψ(G(z,z,fu))≤0. So, G(z,fu,fu)=0 and hence, gu=z=fu. This means that g and f have a coincidence point.Takingg=IX (the identity mapping on X) and ψ=I[0,∞) in the above theorems, we obtain the following fixed point result.Corollary 2.3.
Let(X,⪯,G) be a partially ordered complete G-metric space. Let f:X→X be a mapping such that fx⪯f(fx), for all x∈X and
(2.43)G(fx,fy,fz)≤G(x,fy,fy)+G(y,fz,fz)+G(z,fx,fx)3-φ(G(x,fy,fy),G(y,fz,fz),G(z,fx,fx)),
for every x,y,z∈X such that x⪯y⪯z, where ψ:[0,∞)→[0,∞) is an altering distance function and φ∈Φ. Then, f has a fixed point in X provided that one of the following two conditions is satisfied:(a)
f is continuous, or,(b)
X is regular.Takingφ(x,y,z)=(1/3-α)(x+y+z), where α∈[0,1/3), in the above corollary, we obtain the following result.Corollary 2.4.
Let(X,⪯,G) be a partially ordered complete G-metric space. Let f:X→X be a mapping such that fx⪯f(fx), for all x∈X and
(2.44)G(fx,fy,fz)≤α(G(x,fx,fx)+G(y,fy,fy)+G(z,fz,fz)),
for every x,y,z∈X such that x⪯y⪯z, where α∈[0,1/3). Then, f has a fixed point in X if one of the following two conditions is satisfied:(a)
f is continuous, or,(b)
X is regular.Theorem 2.5.
Under the hypotheses of Theorem2.1, f and g have a common fixed point in X if g is a nondecreasing dominating map.
Moreover, the set of common fixed points off and g is well ordered if and only if f and g have one and only one common fixed point.Proof.
Following the proof of the Theorem2.1 we obtain that the sequence {zn} is G-convergent to z and fz=gz. Since f and g are weakly compatible (since the pair (f,g) is compatible), we have fgz=gfz. Let w=gz=fz. Therefore, we have
(2.45)fw=gw.
As g is a nondecreasing dominating map,
(2.46)z⪯gz⪯ggz=gw.
Ifz=w, then z is a common fixed point. If z≠w, then, since from (2.46) gz⪯gw, from (2.2) we have
(2.47)ψ(G(fz,fz,fw))≤ψ(G(gz,fz,fz)+G(gz,fw,fw)+G(gw,fz,fz)3)-φ(G(gz,fz,fz),G(gz,fw,fw),G(gw,fz,fz))≤ψ(G(fz,fz,fz)+G(fz,fw,fw)+G(fw,fz,fz)3)-φ(G(fz,fz,fz),G(fz,fw,fw),G(fw,fz,fz))≤ψ(2G(fz,fz,fw)+G(fz,fz,fw)3)-φ(0,G(fz,fw,fw),G(fw,fz,fz)).
Therefore,φ(0,G(fz,fw,fw),G(fw,fz,fz))=0. So, fz=fw. Now, since w=gz=fz and fw=gw, we have w=gw=fw. This completes the proof.
Suppose that the set of common fixed points off and g is well ordered. We claim that common fixed point of f and g is unique. Assume on contrary that, fu=gu=u and fv=gv=v, and u≠v. Without any loss of generality, we may assume that gu=u⪯v=gv. Using (2.2), we obtain
(2.48)ψ(G(u,u,v))=ψ(G(fu,fu,fv))≤ψ(G(gu,fu,fu)+G(gu,fv,fv)+G(gv,fu,fu)3)-φ(G(gu,fu,fu),G(gu,fv,fv),G(gv,fu,fu))≤ψ(2G(v,u,u)+G(v,u,u)3)-φ(0,G(u,v,v),G(v,u,u)).
Therefore,u=v, a contradiction. Conversely, if f and g have only one common fixed point then, clearly, the set of common fixed points of f and g is well ordered.Theorem 2.6.
Under the hypotheses of Theorem2.2, f and g have a common fixed point in X provided that f and g are weakly compatible and g is a nondecreasing dominating map.
Moreover, the set of common fixed points off and g is well ordered if and only if f and g have one and only one common fixed point.Proof.
The proof is done as in Theorem2.5.Following arguments similar to those given in the proof of Theorems2.1 and 2.2, we have the following results for a generalized weakly G-contractive mapping of type B.Theorem 2.7.
Let(X,⪯,G) be a partially ordered complete G-metric space. Let f,g:X→X be two mappings such that f(X)⊆g(X)f is weakly increasing with respect to g and
(2.49)ψ(G(fx,fy,fz))≤ψ(G(gx,gx,fy)+G(gy,gy,fz)+G(gz,gz,fx)3)-φ(G(gx,gx,fy),G(gy,gy,fz),G(gz,gz,fx)),
for every x,y,z∈X such that gx⪯gy⪯gz, where ψ:[0,∞)→[0,∞) is an altering distance function and φ∈Φ. Then f and g have a coincidence point in X provided that f and g are continuous and the pair (f,g) is compatible.
Moreover,f and g have a common fixed point in X if g is a nondecreasing dominating map.
Also, the set of common fixed points off and g is well ordered if and only if f and g have one and only one common fixed point.Theorem 2.8.
Let(X,⪯,G) be a partially ordered G-metric space. Let f,g:X→X be two mappings such that f(X)⊆g(X)f is weakly increasing with respect to g and
(2.50)ψ(G(fx,fy,fz))≤ψ(G(gx,gx,fy)+G(gy,gy,fz)+G(gz,gz,fx)3)-φ(G(gx,gx,fy),G(gy,gy,fz),G(gz,gz,fx)),
for every x,y,z∈X such that gx⪯gy⪯gz, where ψ:[0,∞)→[0,∞) is an altering distance function and φ∈Φ. Then f and g have a coincidence point in X provided that X is regular and g(X) is a G-complete subset of (X,G).
Moreover,f and g have a common fixed point in X if f and g are weakly compatible and g is a nondecreasing dominating map.
Also, the set of common fixed points off and g is well ordered if and only if f and g have one and only one common fixed point.The following corollary is an immediate consequence of the above theorems.Corollary 2.9.
Let(X,⪯,G) be a partially ordered complete G-metric space. Let f:X→X be a mapping such that fx⪯f(fx), for all x∈X and
(2.51)G(fx,fy,fz)≤G(x,x,fy)+G(y,y,fz)+G(z,z,fx)3-φ(G(x,x,fy),G(y,y,fz),G(z,z,fx)),
for every x,y,z∈X such that x⪯y⪯z, where ψ:[0,∞)→[0,∞) is an altering distance function and φ∈Φ. Then f has a fixed point in X provided that one of the following two conditions is satisfied:(a)
f is continuous, or,(b)
X is regular.Example 2.10.
LetX=[0,∞) be endowed with the usual order in ℝ and G on X be given as
(2.52)G(x,y,z)=max{|x-y|,|y-z|,|z-x|}.
Definef,g:X→X as
(2.53)f(x)=1,g(x)={2-x2,if0≤x≤20,ifx>2,
for all x∈X.
Defineψ:[0,∞)→[0,∞) by ψ(t)=(1/4)t2 and φ:[0,∞)3→[0,∞) by φ(s,t,u)=(1/100)(s+t+u)2.
Let0≤x≤y≤z≤2. Now, we have
(2.54)ψ(G(fx,fy,fz))=0≤14(|x2-1|+|y2-1|+|z2-1|3)2-1100(|x2-1|+|y2-1|+|z2-1|)2≤14(3-x2-y2-z23)2-1100(3-x2-y2-z2)2=ψ(13(G(gx,fx,fx)+G(gy,fy,fy)+G(gz,fz,fz)))-φ(G(gx,fx,fx),G(gy,fy,fy),G(gz,fz,fz)).
There are other 3 cases as follows:(1)
0≤x≤y≤1 and 2<z.(2)
0≤x≤2 and 2<y≤z.(3)
2<x≤y≤z.
By a careful calculation for the remained cases above, we see that all the conditions of Theorems2.1 and 2.5 are satisfied. Moreover, (1) is the unique common fixed point of f and g.Denote byΛ the set of all functions μ:[0,+∞)→[0,+∞) verifying the following conditions:(I)
μ is a positive Lebesgue integrable mapping on each compact subset of [0,+∞).(II)
for allε>0,  ∫0ε‍μ(t)dt>0.Other consequences of the main theorems are the following results for mappings satisfying a contraction of integral type.Corollary 2.11.
Replace the contractive condition (2.2) of Theorem 2.1 by the following condition.
There exists aμ∈Λ such that
(2.55)∫0ψ(G(fx,fy,fz))‍μ(t)dt≤∫0ψ((G(gx,fy,fy)+G(gy,fz,fz)+G(gz,fx,fx))/3)‍μ(t)dt-∫0φ(G(gx,fy,fy),G(gy,fz,fz),G(gz,fx,fx))‍μ(t)dt.
Then,f and g have a coincidence point, if the other conditions of Theorem 2.1 are satisfied.Proof.
Consider the functionΓ(x)=∫0x‍μ(t)dt. Then, (2.55) becomes
(2.56)Γ(ψ(G(fx,fy,fz)))≤Γ(ψ(G(gx,fy,fy)+G(gy,fz,fz)+G(gz,fx,fx)3))-Γ(φ(G(gx,fy,fy),G(gy,fz,fz),G(gz,fx,fx))).
Takingψ1=Γ∘ψ and φ1=Γ∘φ and applying Theorem 2.1, we obtain the proof (it is easy to verify that ψ1 is an altering distance function and φ1∈Φ).Similar to [21], let N∈N* be fixed. Let {μi}1≤i≤N be a family of N functions which belong to Λ. For all t≥0, we define
(2.57)I1(t)=∫0t‍μ1(s)ds,I2(t)=∫0I1t‍μ2(s)ds=∫0∫0t‍μ1(s)ds‍μ2(s)ds,I3(t)=∫0I2t‍μ3(s)ds=∫0∫0∫0t‍μ1(s)ds‍μ2(s)ds‍μ3(s)ds,∣∣⋮IN(t)=∫0I(N-1)t‍μN(s)ds.
We have the following result.Corollary 2.12.
Replace the inequality (2.2) of Theorem 2.1 by the following condition:
(2.58)IN(ψ(G(fx,fy,fz)))≤IN(ψ(G(gx,fy,fy)+G(gy,fz,fz)+G(gz,fx,fx)3))-IN(φ(G(gx,fy,fy),G(gy,fz,fz),G(gz,fx,fx))).
Then,f and g have a coincidence point if the other conditions of Theorem 2.1 are satisfied.Proof.
ConsiderΨ^=IN∘ψ and Φ^=IN∘φ.


## 3. Periodic Point Results

LetF(f)={x∈X:fx=x}, the fixed point set of f.Clearly, a fixed point off is also a fixed point of fn for every n∈ℕ; that is, F(f)⊂F(fn). However, the converse is false. For example, the mapping f:ℕ→ℕ, defined by fx=1/2-x has the unique fixed point 1/4, but every x∈ℕ is a fixed point of f2. If F(f)=F(fn) for every n∈ℕ, then f is said to have property P. For more details, we refer the reader to [6, 26–28] and the references mentioned therein.Theorem 3.1.
LetX and f be as in Corollary 2.3. If f is a dominating map on X, then f has property P.Proof.
From Corollary2.3, F(f)≠∅. Let u∈F(fn) for some n>1. We will show that u=fu. Since f is dominating on X, we have u⪯fu, which implies that fn-1u⪯fnu, as f is nondecreasing. Using (2.2), we obtain that
(3.1)G(u,fu,fu)=G(fnu,fn+1u,fn+1u)=G(ffn-1u,ffnu,ffnu)≤13(G(fn-1u,fn+1u,fn+1u)+G(fnu,fn+1u,fn+1u)+G(fnu,fnu,fnu))-φ(G(fn-1u,fn+1u,fn+1u),G(fnu,fn+1u,fn+1u),G(fnu,fnu,fnu))≤13(G(fn-1u,fnu,fnu)+2G(fnu,fn+1u,fn+1u)+0)-φ(G(fn-1u,fn+1u,fn+1u),G(fnu,fn+1u,fn+1u),0),
that is,
(3.2)G(u,fu,fu)=G(fnu,fn+1u,fn+1u)≤G(fn-1u,fnu,fnu)-3φ(G(fn-1u,fn+1u,fn+1u),G(fnu,fn+1u,fn+1u),0).
Repeating the above process, we get(3.3)G(fn-(i)u,fn-(i-1)u,fn-(i-1)u)≤G(fn-(i+1)u,fn-(i)u,fn-(i)u)-3φ(G(fn-(i+1)u,fn-(i-1)u,fn-(i-1)u)G(fn-(i)u,fn-(i-1)u,fn-(i-1)u),0).
From the above inequalities, we have(3.4)G(u,fu,fu)≤G(u,fu,fu)-3∑i=0n-1‍φ(G(fn-(i+1)u,fn-(i-1)u,fn-(i-1)u),G(fn-(i)u,fn-(i-1)u,fn-(i-1)u),0).
Therefore,(3.5)∑i=0n-1‍φ(G(fn-(i+1)u,fn-(i-1)u,fn-(i-1)u),G(fn-(i)u,fn-(i-1)u,fn-(i-1)u),0)=0,
which from our assumptions about φ implies that
(3.6)G(fn-(i+1)u,fn-(i-1)u,fn-(i-1)u)=G(fn-(i)u,fn-(i-1)u,fn-(i-1)u)=0
for all 0≤i≤n-1. Now, taking i=n-1, we have u=fu.Analogously, we have the following theorem.Theorem 3.2.
LetX and f be as in Corollary 2.12. If f is a dominating map on X, then f has property P.



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*Source: 701910-2012-10-03.xml*