# Some Almost Generalized(ψ,ϕ)-Contractions in G-Metric Spaces

Authors: Hassen Aydi; Sana Hadj Amor; Erdal Karapinar
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In this paper, we introduce some almost generalized(ψ,ϕ)-contractions in the setting of G-metric spaces. We prove some fixed points results for such contractions. The presented theorems improve and extend some known results in the literature. An example is also presented.


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## Body

## 1. Introduction and Preliminaries

In 2006, a new structure of a generalized metric space was introduced by Mustafa and Sims [1] as an appropriate notion of a generalized metric space called a G-metric space. Fixed point theory in this space was initiated in [2]. Particularly, Banach contraction mapping principle was established in this work. Since then the fixed point theory in G-metric spaces has been studied and developed by many authors, see [1–29].The following definitions and results will be needed in the sequel.Definition 1 (see [1]).
LetX be a nonempty set, G:X×X×X→ℝ+ be a function satisfying the following properties:(G1)
G(x,y,z)=0 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 y≠z,(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 function G is called a generalized metric, or, more specially, a G-metric on X, and the pair (X,G) is called a G-metric space.EveryG-metric on X defines a metric dG on X by
(1)dG(x,y)=G(x,y,y)+G(y,x,x),∀x,y∈X.Example 2.
Let(X,d) be a metric space. The function G:X×X×X→[0,+∞), defined by
(2)G(x,y,z)=max{d(x,y),d(y,z),d(z,x)},
or
(3)G(x,y,z)=d(x,y)+d(y,z)+d(z,x),
for all x,y,z∈X, is a G-metric on X.Definition 3 (see [1]).
Let(X,G) be a G-metric space, and let {xn} be a sequence of points of X; therefore, we say that xn is G-convergent to x∈X if limn,m→+∞G(x,xn,xm)=0; that is, for any ɛ>0, there exists N∈ℕ such that G(x,xn,xm)<ɛ, for all n,m≥N. We call x the limit of the sequence and write xn→x or limn→+∞xn=x.Proposition 4 (see [1]).
Let(X,G) be a G-metric space. The following are equivalent:(1)
{xn}isG-convergent tox,(2)
G(xn,xn,x)→0asn→+∞,(3)
G(xn,x,x)→0asn→+∞.Definition 5 (see [1]).
Let(X,G) be a G-metric space. A sequence {xn} is called a G-Cauchy sequence if, for any ɛ>0, there is N∈ℕ such that G(xn,xm,xl)<ɛ for all m,n,l≥N; that is, G(xn,xm,xl)→0 as n,m,l→+∞.Proposition 6 (see [1]).
Let(X,G) be a G-metric space. Then the following are equivalent:(1)
the sequence{xn}isG-Cauchy,(2)
for anyɛ>0, there existsN∈ℕsuch thatG(xn,xm,xm)<ɛ, for allm,n≥N.Definition 7 (see [1]).
AG-metric space (X,G) is called G-complete if every G-Cauchy sequence is G-convergent in (X,G).Definition 8.
Let(X,G) be a G-metric space. A mapping F:X→X is said to be G-continuous if for any G-convergent sequence {xn} to x, then {F(xn)} is G-convergent to F(x).Now, letℱ denote the set of functions f:[0,+∞)→[0,+∞) such that f(t)=0 if and only if t=0. We denote by Ψ and Φ the subsets of ℱ such that
(4)Ψ={ψ∈ℱ:ψiscontinuousandnondecreasing},Φ={ϕ∈ℱ:ϕislowersemicontinuous}.
There are a lot of fixed point theorems for different type contractions in the literature. In particular, Berinde [30–32] introduced the concept of an almost contraction in metric spaces and studied many interesting fixed point theorems for a Ćirić strong almost contraction. For other fixed point results on generalized almost contractions, see [33–37]. In this paper, we introduce some almost generalized (ψ,ϕ)-contractions in the setting of G-metric spaces, and we establish some fixed points results for such contractions.


## 2. Main Results

Let(X,G) be a G-metric space. First, we consider the following expressions:
(5)M(x,y,z)=max{G(x,Tx,y),G(y,T2x,Ty),G(Tx,T2x,Ty),G(x,Tx,z),G(z,T2x,Tz),G(Tx,T2x,Tz),G(x,y,z)},N(x,y,z)=min{G(x,Tx,Tx),G(y,Ty,Ty),G(z,Tz,Tz),G(z,Tx,Tx),G(y,Tz,Tz)},
for all x,y,z∈X.Our first result for almost generalized(ψ,ϕ)-contractions is the following.Theorem 9.
Let(X,G) be a complete G-metric space. Let T:X→X be a self-mapping. Suppose there exist ψ∈Ψ, ϕ∈Φ, and L≥0 such that for all x,y,z∈X,
(6)ψ(G(Tx,Ty,Tz))≤ψ(M(x,y,z))-ϕ(M(x,y,z))+LN(x,y,z).
Then T has a unique fixed point; say u∈X.Proof.
Letx0∈X, and define a sequence {xn} in X such that
(7)xn+1=Txn,foranyn∈ℕ.
If for some n∈ℕ,xn+1=xn, then xn=Txn, and the proof is completed. Thus, we may assume that xn+1≠xn for all n∈ℕ. By (6) we have
(8)ψ(G(xn,xn+1,xn+1))=ψ(G(Txn-1,Txn,Txn))≤ψ(M(xn-1,xn,xn))-ϕ(M(xn-1,xn,xn))+LN(xn-1,xn,xn),
where
(9)M(xn-1,xn,xn)=max{G(xn-1,Txn-1,xn),G(xn,T2xn-1,Txn),G(Txn-1,T2xn-1,Txn),G(xn-1,Txn-1,xn),G(xn,T2xn-1,Txn),G(Txn-1,T2xn-1,Txn),G(xn-1,xn,xn)}=max{G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn,xn+1,xn+1),G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn,xn+1,xn+1),G(xn-1,xn,xn)}=max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)},N(xn-1,xn,xn)=min{G(xn-1,Txn-1,Txn-1),G(xn,Txn,Txn),G(xn,Txn,Txn),G(xn,Txn-1,Txn-1),G(xn,Txn,Txn)}=min{G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn,xn+1,xn+1),G(xn,xn,xn),G(xn,xn+1,xn+1)}=min{G(xn-1,xn,xn),G(xn,xn+1,xn+1),0}=0.
From (8) and (9), we get
(10)ψ(G(xn,xn+1,xn+1))≤ψ(max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)})-ϕ(max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)}).
If
(11)max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)}=G(xn,xn+1,xn+1),
then by (10) we have
(12)ψ(G(xn,xn+1,xn+1))≤ψ(G(xn,xn+1,xn+1))-ϕ(G(xn,xn+1,xn+1)).
Thus ϕ(G(xn,xn+1,xn+1))=0, and hence G(xn,xn+1,xn+1)=0. Therefore, xn=xn+1 which is a contradiction. So,
(13)max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)}=G(xn-1,xn,xn).
Therefore, we get
(14)G(xn,xn+1,xn+1)≤G(xn-1,xn,xn)∀n∈ℕ,
and (10) becomes
(15)ψ(G(xn,xn+1,xn+1))≤ψ(G(xn-1,xn,xn))-ϕ(G(xn-1,xn,xn))∀n∈ℕ.
Thus, by (14), the sequence {G(xn,xn+1,xn+1)} is monotone nonincreasing. It follows that G(xn,xn+1,xn+1)→α as n→+∞ for some α≥0. Next we claim that α=0. On taking limit as n→+∞ in (15), we obtain
(16)ψ(α)≤ψ(α)-ϕ(α).
Hence ϕ(α)=0 and we get α=0. Hence
(17)limn→+∞G(xn,xn+1,xn+1)=0.
Next, we show that {xn} is a G-Cauchy sequence. On contrary, assume that {xn} is not a G-Cauchy sequence. Then, there is an ɛ>0 for which we can find subsequences {xn(k)},{xm(k)} of {xn} with m(k)>n(k)≥k such that
(18)G(xm(k),xn(k),xn(k))≥ɛ,G(xm(k)-1,xn(k),xn(k))<ɛ.
Using (18) and (G5), we have
(19)ɛ≤G(xm(k),xn(k),xn(k))≤G(xm(k),xm(k)-1,xm(k)-1)+G(xm(k)-1,xn(k),xn(k))<ɛ+2G(xm(k)-1,xm(k),xm(k)).
Taking limit as k→+∞ and using (17), we have
(20)limk→+∞G(xm(k),xn(k),xn(k))=ɛ.
By (G5), we get
(21)G(xm(k),xn(k),xn(k))≤G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k)+1,xn(k)+1,xn(k)+1)+G(xn(k)+1,xn(k),xn(k))≤G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k)+1,xn(k)+1,xn(k)+1)+2G(xn(k),xn(k)+1,xn(k)+1)≤G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k)+1,xm(k),xm(k))+G(xm(k),xn(k),xn(k))+G(xn(k),xn(k)+1,xn(k)+1)+2G(xn(k),xn(k)+1,xn(k)+1)≤3G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k),xn(k),xn(k))+G(xn(k),xn(k)+1,xn(k)+1)+2G(xn(k),xn(k)+1,xn(k)+1).
On taking limit as k→+∞ in the above inequality and using (17) and (20), we obtain
(22)limk→+∞G(xm(k)+1,xn(k)+1,xn(k)+1)=ɛ.
Again, by (G5), we have
(23)G(xn(k),xm(k)+2,xn(k)+1)≤G(xn(k),xn(k)+1,xn(k)+1)+G(xn(k)+1,xn(k)+1,xm(k)+2)≤G(xn(k),xn(k)+1,xn(k)+1)+G(xm(k)+2,xm(k)+1,xm(k)+1)+G(xm(k)+1,xn(k)+1,xn(k)+1)≤G(xn(k),xn(k)+1,xn(k)+1)+2G(xm(k)+1,xm(k)+2,xm(k)+2)+G(xm(k)+1,xm(k)+2,xm(k)+2)+G(xm(k)+2,xn(k)+1,xn(k)+1)≤G(xn(k),xn(k)+1,xn(k)+1)+2G(xm(k)+1,xm(k)+2,xm(k)+2)+G(xm(k)+1,xm(k)+2,xm(k)+2)+G(xn(k)+1,xn(k),xn(k))+G(xn(k),xn(k)+1,xm(k)+2)≤G(xn(k),xn(k)+1,xn(k)+1)+2G(xm(k)+1,xm(k)+2,xm(k)+2)+G(xm(k)+1,xm(k)+2,xm(k)+2)+2G(xn(k),xn(k)+1,xn(k)+1)+G(xn(k),xn(k)+1,xm(k)+2).
On taking limit as k→+∞ in the above inequality and using (17) and (22), we obtain
(24)limk→+∞G(xn(k),xm(k)+2,xn(k)+1)=ɛ.
We have also, by (G5),
(25)G(xm(k),xm(k)+1,xn(k))≤G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k)+1,xm(k)+1,xn(k)),G(xm(k)+1,xm(k)+1,xn(k))≤G(xm(k)+1,xm(k),xm(k))+G(xm(k),xm(k)+1,xn(k)).
On taking limit as k→+∞ in the above inequality and using (17) and (G3), we obtain
(26)limk→+∞G(xm(k)+1,xm(k)+1,xn(k))=limk→+∞G(xm(k),xm(k)+1,xn(k)).
We have also, by (G3) and (G5),
(27)G(xn(k),xm(k)+1,xm(k)+1)≤G(xn(k),xm(k)+1,xn(k)+1)≤G(xn(k),xn(k)+1,xn(k)+1)+G(xn(k)+1,xn(k)+1,xm(k)+1),G(xm(k),xn(k),xn(k))≤G(xm(k),xn(k),xm(k)+1).
Now, on taking limit as k→+∞ in (27), and using (17), (20), and (26), we obtain
(28)limk→+∞G(xm(k)+1,xm(k)+1,xn(k))=limk→+∞G(xm(k),xm(k)+1,xn(k))=ɛ.
Furthermore, by (G3) and (G5), we get
(29)G(xm(k)+1,xn(k)+1,xn(k)+1)≤G(xm(k)+1,xm(k)+2,xn(k)+1)≤G(xm(k)+2,xm(k)+1,xm(k)+1)+G(xm(k)+1,xm(k)+1,xn(k)+1)≤2G(xm(k)+1,xm(k)+2,xm(k)+2)+G(xn(k)+1,xn(k),xn(k))+G(xn(k),xm(k)+1,xm(k)+1)≤2G(xm(k)+1,xm(k)+2,xm(k)+2)+2G(xn(k),xn(k)+1,xn(k)+1)+G(xn(k),xm(k)+1,xm(k)+1).
On taking limit as k→+∞ in the above inequality, and using (17), (20), (22), and (28), we obtain
(30)limk→+∞G(xm(k)+1,xm(k)+2,xn(k)+1)=ɛ.
Now, we have
(31)ψ(G(xm(k)+1,xn(k)+1,xn(k)+1))=ψ(G(Txm(k),Txn(k),Txn(k)))≤ψ(M(xm(k),xn(k),xn(k)))-ϕ(M(xm(k),xn(k),xn(k)))+LN(xm(k),xn(k),xn(k))≤ψ(max{G(xm(k),Txm(k),xn(k)),G(xn(k),T2xm(k),Txn(k)),G(Txm(k),T2xm(k),Txn(k)),G(xm(k),Txm(k),xn(k)),G(xn(k),T2xm(k),Txn(k)),G(Txm(k),T2xm(k),Txn(k)),G(xm(k),xn(k),xn(k))})-ϕ(max{G(xm(k),Txm(k),xn(k)),G(xn(k),T2xm(k),Txn(k)),G(Txm(k),T2xm(k),Txn(k)),G(xm(k),Txm(k),xn(k)),G(xn(k),T2xm(k),Txn(k)),G(Txm(k),T2xm(k),Txn(k)),G(xm(k),xn(k),xn(k))})+L(min{G(xm(k),Txm(k),Txm(k)),G(xn(k),Txn(k),Txn(k)),G(xn(k),Txn(k),Txn(k)),G(xn(k),Txm(k),Txm(k)),G(xn(k),Txn(k),Txn(k))})≤ψ(max{G(xm(k),xm(k)+1,xn(k)),G(xn(k),xm(k)+2,xn(k)+1),G(xm(k)+1,xm(k)+2,xn(k)+1),G(xm(k),xn(k),xn(k))})-ϕ(max{G(xm(k),xm(k)+1,xn(k)),G(xn(k),xm(k)+2,xn(k)+1),G(xm(k)+1,xm(k)+2,xn(k)+1),G(xm(k),xn(k),xn(k))})+L(min{G(xm(k),xm(k)+1,xm(k)+1),G(xn(k),xn(k)+1,xn(k)+1),G(xn(k),xm(k)+1,xm(k)+1)(xn(k),xn(k)+1,xn(k)+1)}).
Letting k→+∞, and using (17), (20), (22), (24), (28), and (30), and the properties of ϕ and ψ, we have
(32)ψ(ɛ)≤ψ(ɛ)-ϕ(ɛ).
Thus ϕ(ɛ)=0 and hence ɛ=0, a contradiction. Thus {xn} is a G-Cauchy sequence in X.
Now, since(X,G) is G-complete, there are x∈X such that {xn} is G-convergent to x; that is
(33)limn→+∞G(xn,xn,x)=limn→+∞G(xn,x,x)=0.
By (6), we get
(34)ψ(G(xn+1,xn+1,Tx))=ψ(G(Txn,Txn,Tx))≤ψ(M(xn,xn,x))-ϕ(M(xn,xn,x))+LN(xn,xn,x),
where
(35)M(xn,xn,x)=max{G(xn,Txn,xn),G(xn,T2xn,Txn),G(Txn,T2xn,Txn),G(xn,Txn,x),G(x,T2xn,Tx),G(Txn,T2xn,Tx),G(xn,xn,x)}=max{G(xn,xn+1,xn),G(xn,xn+2,xn+1),G(xn+1,xn+2,xn+1),G(xn,xn+1,x),G(x,xn+2,Tx),G(xn+1,xn+2,Tx),G(xn,xn,x)},N(xn,xn,x)=min{G(xn,Txn,Txn),G(xn,Txn,Txn),G(x,Tx,Tx),G(x,Txn,Txn),G(xn,Tx,Tx)}=min{G(xn,xn+1,xn+1),G(x,Tx,Tx),G(x,xn+1,xn+1),G(xn,Tx,Tx)}.
Letting n→+∞ in (35), we get
(36)limn→+∞M(xn,xn,x)=G(x,x,Tx),limn→+∞N(xn,xn,x)=0.
On letting n→+∞ in (34), and using the properties of ψ and ϕ and (36), we obtain
(37)ψ(G(x,x,Tx))≤ψ(G(x,x,Tx))-ϕ(G(x,x,Tx)).
Therefore G(x,x,Tx)=0 and hence x=Tx. Thus x is a fixed point of T.
Now our purpose is to check that such point is unique. Suppose that there are two fixed points ofT; say x,y∈X such that x≠y. By (6), we have
(38)ψ(G(x,x,y))=ψ(G(Tx,Tx,Ty))≤ψ(M(x,x,y))-ϕ(M(x,x,y))+LN(x,x,y),
where
(39)M(x,x,y)=max{G(x,Tx,x),G(x,T2x,Tx),G(Tx,T2x,Tx),G(x,Tx,y),G(y,T2x,Ty),G(Tx,T2x,Ty),G(x,x,y)}=max{G(x,x,y),G(y,x,y)},N(x,x,y)=min{G(x,Tx,Tx),G(x,Tx,Tx),G(y,Ty,Ty),G(y,Tx,Tx),G(x,Ty,Ty)}=0.
Similarly, we can prove that
(40)ψ(G(x,y,y))=ψ(G(Tx,Ty,Ty))≤ψ(M(x,y,y))-ϕ(M(x,y,y))+LN(x,y,y),
where
(41)M(x,y,y)=max{G(x,x,y),G(y,x,y)},N(x,y,y)=0.
If
(42)max{G(x,x,y),G(y,x,y)}=G(x,x,y).
By (38) and (39), we get
(43)ψ(G(x,x,y))≤ψ(G(x,x,y))-ϕ(G(x,x,y)),
a contradiction. Then
(44)max{G(x,x,y),G(y,x,y)}=G(x,y,y).
By (40) and (41), we get
(45)ψ(G(x,y,y))≤ψ(G(x,y,y))-ϕ(G(x,y,y)),
a contradiction. Thus, x=y, and hence the fixed point of T is unique.As consequence of Theorem9, we present the following corollaries.Corollary 10.
Let(X,G) be a complete G-metric space. Let T:X→X be a self mapping. Suppose there exist k∈[0,1) and L≥0 such that for all x,y,z∈X(46)G(Tx,Ty,Tz)≤kM(x,y,z)+LN(x,y,z).
Then T has a unique fixed point; say u∈X.Proof.
It suffices to takeψ(t)=t and ϕ(t)=(1-k)(t) in Theorem 9.Corollary 11.
Let(X,G) be a complete G-metric space. Let T:X→X be a self mapping. Suppose there exist k∈[0,1) and L≥0 such that for all x,y,z∈X(47)G(Tx,Ty,Tz)≤kG(x,y,z)+LN(x,y,z).
Then T has a unique fixed point; say u∈X.Proof.
Note thatG(x,y,z)≤M(x,y,z) in Corollary 10.Remark 12.
Corollary11 is a generalization of Mustafa’s result [2].Our second main result is given as follows.Theorem 13.
Let(X,G) be a complete G-metric space. Let T:X→X be a self mapping. Suppose there exist ψ∈Ψ,  ϕ∈Φ and L≥0 such that for all x,y∈X:
(48)ψ(G(Tx,Ty,T2x))≤ψ(M*(x,y,x))-ϕ(M*(x,y,x))+LN*(x,y,x),
where
(49)M*(x,y,x)=max{G(x,Tx,y),G(y,T2x,Ty),G(Tx,T2x,Ty),G(x,Tx,Tx),G(Tx,T2x,T2x),G(x,y,Tx)},N*(x,y,x)=min{G(x,Tx,Tx),G(y,Ty,Ty),G(y,Tx,Tx)}
for all x,y∈X. Then T has a unique fixed point; say u∈X.Proof.
Letx0 be an arbitrary point in X and define a sequence {xn} in X such that
(50)xn+1=Txn,foranyn∈ℕ.
If for some n∈ℕ,xn+1=xn, then xn=Txn and the proof is completed. Thus, we may assume that xn+1≠xn for all n∈ℕ. By (48), we have
(51)ψ(G(xn,xn+1,xn+1))=ψ(G(Txn-1,Txn,T2xn-1))≤ψ(M*(xn-1,xn,xn-1))-ϕ(M*(xn-1,xn,xn-1))+LN*(xn-1,xn,xn-1),
where
(52)M*(xn-1,xn,xn-1)=max{G(xn-1,Txn-1,xn),G(xn,T2xn-1,Txn),G(Txn-1,T2xn-1,Txn),G(xn-1,Txn-1,Txn-1),G(Txn-1,T2xn-1,T2xn-1),G(xn-1,xn,Txn-1)}=max{G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn,xn+1,xn+1),G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn-1,xn,xn)}=max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)},N*(xn-1,xn,xn-1)=min{G(xn-1,Txn-1,Txn-1),G(xn,Txn,Txn),G(xn,Txn-1,Txn-1)}=min{G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn,xn,xn)}=0.
From (51) and (52), we get
(53)ψ(G(xn,xn+1,xn+1))≤ψ(max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)})-ϕ(max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)}).
If
(54)max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)}=G(xn,xn+1,xn+1),
then by (53) we have
(55)ψ(G(xn,xn+1,xn+1))≤ψ(G(xn,xn+1,xn+1))-ϕ(G(xn,xn+1,xn+1)).
Thus ψ(G(xn,xn+1,xn+1))=0 and hence G(xn,xn+1,xn+1)=0. Therefore, xn=xn+1, which is a contradiction. So,
(56)max{G(xn-1,xn,xn),G(xn,xn+1,xn+1)}=G(xn-1,xn,xn).
Therefore, we get
(57)G(xn,xn+1,xn+1)≤G(xn-1,xn,xn)∀n∈ℕ,
and (53) becomes
(58)ψ(G(xn,xn+1,xn+1))≤ψ(G(xn-1,xn,xn))-ϕ(G(xn-1,xn,xn))∀n∈ℕ.
Thus, by (57), the sequence {G(xn,xn+1,xn+1)} is monotone nonincreasing. It follows that G(xn,xn+1,xn+1)→g as n→+∞ for some g≥0. Next we claim that g=0. On taking limit as n→+∞ in (58), we obtain
(59)ψ(g)≤ψ(g)-ϕ(g).
Hence ϕ(g)=0 and we get g=0. Then
(60)limn→+∞G(xn,xn+1,xn+1)=0.
Next, we show that {xn} is a G-Cauchy sequence. On contrary, assume that {xn} is not a G-Cauchy sequence. Then, there is an ɛ>0 for which we can find subsequences {xn(k)},{xm(k)} of {xn} with m(k)>n(k)≥k such that
(61)G(xm(k),xn(k),xn(k))≥ɛ,G(xm(k)-1,xn(k),xn(k))<ɛ.
Using (61) and (G5), we have
(62)ɛ≤G(xm(k),xn(k),xn(k))≤G(xm(k),xm(k)-1,xm(k)-1)+G(xm(k)-1,xn(k),xn(k))<ɛ+2G(xm(k)-1,xm(k),xm(k)).
Taking limit as k→+∞ and using (60), we have
(63)limk→+∞G(xm(k),xn(k),xn(k))=ɛ.
By (G5), and using (60) and (63), we obtain, as stated in the proof of Theorem 9,
(64)limk→+∞G(xm(k)+1,xn(k)+1,xn(k)+1)=ɛ.
Again, by (G5), and using (60) and (64), similarly as in the proof of Theorem 9, we obtain
(65)limk→+∞G(xn(k),xm(k)+2,xn(k)+1)=ɛ.
Furthermore, by (G3) and (G5), we get
(66)G(xm(k)+1,xn(k)+1,xn(k)+1)≤G(xm(k)+1,xm(k)+2,xn(k)+1)≤G(xm(k)+2,xm(k)+1,xm(k)+1)+G(xm(k)+1,xm(k)+1,xn(k)+1)≤2G(xm(k)+1,xm(k)+2,xm(k)+2)+G(xn(k)+1,xn(k),xn(k))+G(xn(k),xm(k)+1,xm(k)+1)≤2G(xm(k)+1,xm(k)+2,xm(k)+2)+2G(xn(k),xn(k)+1,xn(k)+1)+G(xn(k),xm(k)+1,xm(k)+1).
By the proof of Theorem 9, we have stated that
(67)limk→+∞G(xm(k)+1,xm(k)+1,xn(k))=ɛ.
So, on taking limit as k→+∞ in (66), and using (60), (64), and (67), we obtain
(68)limk→+∞G(xm(k)+1,xm(k)+2,xn(k)+1)=ɛ.
Furthermore, by (G3) and (G5), we get
(69)G(xn(k)-1,xn(k),xm(k)+1)≤G(xn(k)-1,xn(k),xn(k))+G(xn(k),xn(k),xm(k)+1)≤G(xn(k)-1,xn(k),xn(k))+G(xm(k)+1,xm(k),xm(k))+G(xm(k),xn(k),xn(k))≤G(xn(k)-1,xn(k),xn(k))+2G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k),xn(k),xn(k)),G(xm(k),xn(k),xn(k))≤G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k)+1,xn(k),xn(k))≤G(xm(k),xm(k)+1,xm(k)+1)+G(xn(k),xn(k)-1,xn(k)-1)+G(xn(k)-1,xn(k),xm(k)+1).
On taking limit as k→+∞ in above inequalities, and using (60) and (63), we have
(70)limk→+∞G(xn(k)-1,xn(k),xm(k)+1)=ɛ.
Also, by (G3) and (G5), we get
(71)G(xm(k)+1,xn(k),xn(k))≤G(xm(k)+1,xm(k),xm(k))+G(xm(k),xn(k),xn(k))≤2G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k),xn(k),xn(k)),G(xm(k),xn(k),xn(k))≤G(xm(k),xm(k)+1,xm(k)+1)+G(xm(k)+1,xn(k),xn(k)).
On taking limit as k→+∞ in above inequalities, and using (60), (63), and (70), we obtain
(72)limk→+∞G(xm(k)+1,xn(k),xn(k))=ɛ.
Now, we have
(73)ψ(G(xn(k),xm(k)+2,xn(k)+1))=ψ(G(Txn(k)-1,Txm(k)+1,T2xn(k)-1))≤ψ(M*(xn(k)-1,xm(k)+1,xn(k)-1))-ϕ(M*(xn(k)-1,xm(k)+1,xn(k)-1))+LN*(xn(k)-1,xm(k)+1,xn(k)-1)≤ψ(max{G(xn(k)-1,Txn(k)-1,xm(k)+1),G(xm(k)+1,T2xn(k)-1,Txm(k)+1),G(Txn(k)-1,T2xn(k)-1,Txm(k)+1),G(xn(k)-1,Txn(k)-1,Txn(k)-1),G(Txn(k)-1,T2xn(k)-1,T2xn(k)-1),G(xn(k)-1,xm(k)+1,Txn(k)-1)})-ϕ(max{G(xn(k)-1,Txn(k)-1,xm(k)+1),G(xm(k)+1,T2xn(k)-1,Txm(k)+1),G(Txn(k)-1,T2xn(k)-1,Txm(k)+1),G(xn(k)-1,Txn(k)-1,Txn(k)-1),G(Txn(k)-1,T2xn(k)-1,T2xn(k)-1),G(xn(k)-1,xm(k)+1,Txn(k)-1)})+Lmin{G(xn(k)-1,Txn(k)-1,Txn(k)-1),G(xm(k)+1,Txm(k)+1,Txm(k)+1),G(xm(k)+1,Txn(k)-1,Txn(k)-1)}≤ψ(max{G(xn(k)-1,xn(k),xm(k)+1),G(xm(k)+1,xn(k)+1,xm(k)+2),G(xn(k),xn(k)+1,xm(k)+2),G(xn(k)-1,xn(k),xn(k)),G(xn(k),xn(k)+1,xn(k)+1)})-ϕ(max{G(xn(k)-1,xn(k),xm(k)+1),G(xm(k)+1,xn(k)+1,xm(k)+2),G(xn(k),xn(k)+1,xm(k)+2),G(xn(k)-1,xn(k),xn(k)),G(xn(k),xn(k)+1,xn(k)+1)})+Lmin{G(xn(k)-1,xn(k),xn(k)),G(xm(k)+1,xm(k)+2,xm(k)+2),G(xm(k)+1,xn(k),xn(k))}.
Letting k→+∞, and using (60), (65), (68), (70), and (72), and the properties of ϕ and ψ, we have
(74)ψ(ɛ)≤ψ(ɛ)-ϕ(ɛ).
Thus ϕ(ɛ)=0 and hence ɛ=0, a contradiction. Thus {xn} is a G-Cauchy sequence in X.
Now, since(X,G) is G-complete, there is x∈X such that {xn} is G-convergent to x; that is
(75)limn→+∞G(xn,xn,x)=limn→+∞G(xn,x,x)=0.
By (48), we get
(76)ψ(G(xn,Tx,xn+1))=ψ(G(Txn-1,Tx,T2xn-1))≤ψ(M*(xn-1,x,xn-1))-ϕ(M*(xn-1,x,xn-1))+LN*(xn-1,x,xn-1),
where
(77)M*(xn-1,x,xn-1)=max{G(xn-1,Txn-1,x),G(x,T2xn-1,Tx),G(Txn-1,T2xn-1,Tx),G(xn-1,Txn-1,Txn-1),G(Txn-1,T2xn-1,T2xn-1),G(xn-1,x,Txn-1)}=max{G(xn-1,xn,x),G(x,xn+1,Tx),G(xn,xn+1,Tx),G(xn-1,xn,xn),G(xn,xn+1,xn+1),G(xn-1,x,xn)},N*(xn-1,x,xn-1)=min{G(xn-1,Txn-1,Txn-1),G(x,Tx,Tx),G(x,Txn-1,Txn-1)}=min{G(xn-1,xn,xn),G(x,Tx,Tx),G(x,xn,xn)}.
Letting n→+∞ in (77), we get
(78)limn→+∞M*(xn-1,x,xn-1)=G(x,x,Tx),limn→+∞N*(xn-1,x,xn-1)=0.
On letting n→+∞ in (76), and using the properties of ψ and ϕ and (78), we obtain
(79)ψ(G(x,x,Tx))≤ψ(G(x,x,Tx))-ϕ(G(x,x,Tx)).
Therefore G(x,x,Tx)=0 and hence x=Tx. Thus x is a fixed point of T.
Now our purpose is to check that such point is unique. Suppose that there are two fixed points ofT; say x,y∈X such that x≠y. By (48), we have
(80)ψ(G(x,y,x))=ψ(G(Tx,Ty,T2x))ψ(G(x,y,x))≤ψ(M*(x,y,x))-ϕ(M*(x,y,x))ψ(G(x,y,x))+LN*(x,y,x),
where
(81)M*(x,y,x)=max{G(x,Tx,y),G(y,T2x,Ty),G(Tx,T2x,Ty),G(x,Tx,Tx),G(Tx,T2x,T2x),G(x,y,Tx)}=max{G(x,x,y),G(y,x,y)},N*(x,y,x)=min{G(x,Tx,Tx),G(y,Ty,Ty),G(y,Tx,Tx)}=0.
Similarly, we can prove that
(82)ψ(G(y,x,y))=ψ(G(Ty,Tx,T2y))≤ψ(M*(y,x,y))-ϕ(M*(y,x,y))+LN*(y,x,y),
where
(83)M*(y,x,y)=max{G(x,x,y),G(y,x,y)},N*(y,x,y)=0.
If
(84)max{G(x,x,y),G(y,x,y)}=G(x,x,y).
By (80) and (81), we get
(85)ψ(G(x,x,y))≤ψ(G(x,x,y))-ϕ(G(x,x,y)),
a contradiction. Then
(86)max{G(x,x,y),G(y,x,y)}=G(x,y,y).
By (82) and (83), we get
(87)ψ(G(x,y,y))≤ψ(G(x,y,y))-ϕ(G(x,y,y)),
a contradiction. Thus, x=y, and hence the fixed point of T is unique.Finally, we provide the following example.Example 14.
LetX=[0,1] and G:X×X×X→ℝ be defined by
(88)G(x,y,z)={0ifx=y=z,max{x,y,z}otherwise.
Then (X,G) is a G-complete G-metric space. Let T:X→X be defined by
(89)Tx={14xif0≤x<13,18x4if13≤x≤1.
Take ψ(t)=(1/2)t and ϕ(t)=(1/4)t for all t∈[0,+∞[. We examine the following cases.
(i) Let0≤x,y<1/3. Then
(90)G(Tx,T2x,Ty)=max{14x,116x,14y}=14max{x,14x,y}≤14max{x,y}=ψ(M*(x,y,x))-ϕ(M*(x,y,x)),
where M*(x,y,x)=max{x,y}.
(ii) Let1/3≤x,y<1. Then
(91)G(Tx,T2x,Ty)=max{18x4,164x16,18y4}≤14max{x,18x4,y}≤14max{x,y}=ψ(M*(x,y,x))-ϕ(M*(x,y,x)),
where M*(x,y,z)=max{x,y}.
(iii) Let0≤x<1/3≤y<1. Then
(92)G(Tx,T2x,Ty)=max{14x,116x,18y4}≤14max{x,14x,y}≤14y=ψ(M*(x,y,x))-ϕ(M*(x,y,x)),
where M*(x,y,x)=y.
(iv) Let0≤y<1/3≤x<1. Then
(93)G(Tx,T2x,Ty)=max{18x4,164x16,14y}≤14max{x,18x4,y}≤14x=ψ(M*(x,y,x))-ϕ(M*(x,y,x)),
where M*(x,y,x)=x. Then
(94)G(Tx,T2x,Ty)≤ψ(M*(x,y,x))-ϕ(M*(x,y,x))+LN*(x,y,x)
for all L≥0. Then the conditions of Theorem 13, hold and T has a unique fixed point. Notice that u=0 is the desired fixed point of T.



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*Source: 165420-2013-10-22.xml*