A new type of contraction in a complete $G$-metric space

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Available online at www.ispacs.com/jnaa

Volume 2015, Issue 2, Year 2015 Article ID jnaa-00283, 6 Pages doi:10.5899/2015/jnaa-00283

Research Article

A new type of contraction in a complete

G

-metric space

Nidhi Malhotra1∗, Bindu Bansal1

(1)Department of Mathematics, Hindu College, University of Delhi, Delhi, India

Copyright 2015 c⃝Nidhi Malhotra and Bindu Bansal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we extend and generalize the concept ofF-contraction toF-weak contraction and prove a fixed point theorem forF-weak contraction in a completeG-metric space. The article includes a nontrivial example which verify the effectiveness and applicability of our main result.

Keywords:fixed point,F-contraction,F-weak contraction, completeG-metric spaces.

1 Introduction

The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions (see [1], [2], [3], [4], [7], [9], [10], [11] and [12]) and the reference therein. In [5], Dhage introducedD-metric space as a generalization of metric space and proved many results for this metric. But in 2005, Mustafa and Sims [8] proved that these results are not true in topological structure and hence they introducedG-metric space as a generalized form of metric space. Since then, many fixed point results have been developed by different authors inG-metric spaces. In 2012 ,Wardowski [13] introduced a new concept ofF-contraction and proved a fixed point theorem for such a map on a complete metric space which generalizes Banach contraction principle in a different direction. Recently in 2014, Wardowski and Van Dung [14] defined the notion ofF-weak contraction in metric spaces and generalized the theorem of Wardowski [13]. Also, Gupta [6] in 2014, introduced the notion ofF-contraction inG-metric space and proved a fixed point theorem concerningF-contraction.

In this paper, we introduce the notion ofF-weak contraction in a completeG-metric space which is a generalization of the concept ofF-weak contraction due to Wardowski and Van Dung [14]. We also extend and generalize the fixed point theorem due to Gupta.

2 Preliminaries and notations

Definition 2.1. [8]Let X be a nonempty set, G:X×X×X→ℜ+be a function satisfying the following properties: (G1) G(x,y,z) =0if 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 x̸=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 function G is called a generalized metric or a G-metric on X , and the pair(X,G)is called a G-metric space.

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Definition 2.2. [8] Let(X,G)be a G-metric space.

(1) A sequence{xn}in X , is said to be G-convergent to a point x∈X if for eachε>0, there exists n0∈Nsuch that, for all m,n≥n0,

G(xm,xn,x)<ε;

(2) A sequence{xn}in X , is said to be G-Cauchy sequence if for eachε>0, there exists n0∈Nsuch that, for all m,n,l≥n0,

G(xm,xn,xl)<ε.

Proposition 2.1. [8] Let(X,G)be a G-metric space. Then the following are equivalent:

(1) The sequence{xn}is G-Cauchy.

(2) For everyε>0, there is k∈Nsuch that G(xn,xm,xm)<ε, for all n,m≥k.

Proposition 2.2. [8] Let(X,G)be a G-metric space. Then the function G(x,y,z)is jointly continuous in all three of its variables.

Definition 2.3. [13]Let F:R+_→Rbe a mapping satisfying

(F1) F is strictly increasing. That is,α<β⇒F(α)<F(β)for allα,β ∈R+.

(F2) For every sequence{αn}inR+we have lim

n→∞αn=0if and only ifnlim→∞F(αn) =−∞.

(F3) There exists a number k∈(0,1)such that lim

α→0+α

k_F₍_α_{) =}₀_.

Definition 2.4. [6]Let(X,G)be a G-metric space. A mapping T:X→X is said to be an F-contraction if there exists a numberτ>0such that

G(T x,Ty,T z)>0⇒τ+F(G(T x,Ty,T z))≤F(G(x,y,z))for all x,y,z∈X.

Remark 2.1. Clearly Definition 2.4 and (F1) implies that G(T x,Ty,T z)<G(x,y,z)for all x,y,z∈X with T x̸=Ty̸=T z. Hence every F- contraction mapping is continuous.

We now introduce the notion of anF-weak contraction in aG-metric space and prove a fixed point theorem forF -weak contractions, which generalizes some results known from the literature. Examples are given to show that our result is a proper extension of [[6], Theorem 2.9].

Definition 2.5.Let(X,G)be a G-metric space. Let F be a mapping as defined in Definition 2.3. A mapping T:X→X said to be an F-weak contraction on(X,G)if there exists a numberτ>0 such that for all x,y,z∈X satisfying G(T x,Ty,T z)>0, the following holds:

τ+F(G(T x,Ty,T z))

≤F(max{G(x,y,z),G(x,T x,T x),G(y,Ty,Ty),G(z,T z,T z)}) (2.1)

Remark 2.2. (1) Every F-contraction is an F-weak contraction. The following example shows that the converse in not true.

Example 2.1. Let X= [0,1]and G(x,y,z) =1

3(|x−y|+|y−z|+|x−z|)for all x,y,z∈X. Then(X,G)is a complete G-metric space. Define a mapping T:X→X by

T x=

{₁

2, if0≤x<1 1

4, if x=1

Since T is not continuous, therefore it is not an F-contraction for any mapping F as described in Definition 2.3. However, for x,y∈[0,1),z=1, we have G(T x,Ty,T1) =G(1

2, 1 2,

1 4) =

1

6>0and max{G(x,y,1),G(x,T x,T x),G(y,Ty,Ty),G(1,T1,T1)} ≥G(1,T1,T1) =1

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Remark 2.3. Let T be an F-weak contraction. From (2.1) we have, for all x,y,z∈X with G(T x,Ty,T z)>0, F(G(T x,Ty,T z))<τ+F(G(T x,Ty,T z))

≤F(max{G(x,y,z),G(x,T x,T x),G(y,Ty,Ty),G(z,T z,T z)}.

Then by (F1), we get

G(T x,Ty,T z))<max{G(x,y,z),G(x,T x,T x),G(y,Ty,Ty),G(z,T z,T z)}for all x,y,z∈X with G(T x,Ty,T z)>0.

3 Main section

Theorem 3.1. Let(X,G)be a complete G-metric space . Let T :X→X be an F-weak contraction. If T or F is continuous, then T has a unique fixed point x∗in X and for every x0∈X , there is a sequence {Tnx0} in X that converges to x∗.

Proof. Letx0∈X be arbitrary. We define a sequence{xn}inX given byxn=T xn−1for alln∈N. If there exists n0∈Nfor whichxn0+1=xn0, thenT xn0=xn0.

This shows thatxn0 is a fixed point ofT.

Therefore, we assume thatxn+1̸=xnfor everyn∈N∪ {0}.

Letpn=G(xn,xn+1,xn+2)>0 for alln∈N_. It follows from (2.1) that for eachn∈N,

F(pn) =F(G(T xn−1,T xn,T xn+1))

≤F(max{G(xn−1,xn,xn+1),G(xn−1,T xn−1,T xn−1),G(xn,T xn,T xn), G(xn+1,T xn+1,T xn+1)} −τ

=F(max{pn−1,G(xn−1,xn,xn),G(xn,xn+1,xn+1),G(xn+1,xn+2,xn+2)} −τ

=F(pn−1)−τ.

By successive application, we get for alln∈N

F(pn)≤F(pn−1)−τ≤F(pn−2)−2τ≤...≤F(p0)−nτ. (3.2) Taking the limit asn→∞in ( 3.2) we get lim

n→∞F(pn) =−∞and then by (F2) of Definition 2.3 we have

lim

n→∞pn=0. (3.3)

Now, by (F3) of Definition 2.3, there existsk∈(0,1)such that

lim

n→∞p

k

nF(pn) =0. (3.4)

By (3.2), following holds for alln∈N.

pk_nF(pn)−pk_nF(p0) =p_nk(F(pn)−F(p0))≤npk_nτ. (3.5) Lettingn→∞in (3.5) and using (3.3) and (3.4) we have

lim

n→∞np

k

n=0. (3.6)

Then there exists a positive integern1such thatnpkn<1 for alln≥n1. Consequently

pn<

1

n1k

∀n≥n1. (3.7)

Further we show that{xn}is a Cauchy sequence. Now for allm>n≥n1using (3.7), we have G(xn,xm,xm)≤pn+pn+1+...+pm<Σ∞i=npi≤Σ∞i=n

1

i1k

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Ask∈(0,1), the seriesΣ∞_n₌₁ 1 n1k

is convergent and therefore from (3.8),{xn}is a Cauchy sequence inX. SinceX is

complete, there existsx∗∈X such that lim

n→∞xn=x ∗_.

Now we prove thatx∗is a fixed point ofT by considering the following two cases:

Case 1 IfT is continuous. We have

G(T x∗,x∗,x∗) =lim

n→∞G(T xn,xn,xn) =nlim→∞G(xn+1,xn,xn) =0.

This proves thatx∗is a fixed point ofT.

Case 2 IfFis continuous.

We consider the following two subcases:

Subcase 1 : There existsn0∈Nsuch thatxn+1̸=T x∗for alln≥n0.

That is,G(T xn,T x∗,T x∗)>0 for alln≥n0.

It follows from (2.1) that

τ+F(G(xn+1,T x∗,T x∗)) =τ+F(G(T xn,T x∗,T x∗))

≤F(max{G(xn,x∗,x∗),G(xn,T xn,T xn),G(x∗,T x∗,T x∗)}

=F(max{G(xn,x∗,x∗),G(xn,xn+1,xn+1),G(x∗,T x∗,T x∗)}.

IfG(x∗,T x∗,T x∗)>0, then using the fact

lim

n→∞G(xn,x

∗_,_x∗_{) =} _lim

n→∞G(xn,xn+1,xn+1) =0, there existsn1∈Nsuch that for alln≥n1,we have

max{G(xn,x∗,x∗),G(xn,xn+1,xn+1),G(x∗,T x∗,T x∗)}=G(X∗,T x∗,T x∗).

Therefore, we get

τ+F(G(xn+1,T x∗,T x∗))≤F(G(x∗,T x∗,T x∗)) ∀n≥max{n0,n1}. (3.9)

SinceFis continuous, taking limit asn→∞in (3.9), we obtain

τ+F(G(x∗,T x∗,T x∗))≤F(G(x∗,T x∗,T x∗))which is a contradiction. ThereforeG(x∗,T x∗,T x∗) =0 . Thusx∗is a fixed point ofT.

Subcase 2 :There exists a subsequence{xnk}of{xn}such thatxnk+1=T x

∗_{for all}_k_∈_N_._{Then we have}

x∗=lim

k→∞xnk+1=klim→∞T x

∗₌_{T x}∗_.

This shows thatx∗is a fixed point ofT.

Combining above two cases, we get thatT has a fixed pointx∗inX. Now we show the uniqueness.

Letx∗andy∗be two fixed points ofT. Suppose thatx∗̸=y∗. ThenT x∗̸=Ty∗.

It follows from (2.1) that

τ+F(G(x∗,y∗,y∗)) =τ+F(G(T x∗,Ty∗,Ty∗))

≤F(max{G(x∗,y∗,y∗),G(x∗,T x∗,T x∗),G(y∗,Ty∗,Ty∗)}

=F(max{G(x∗,y∗,y∗),G(x∗,x∗,x∗),G(y∗,y∗,y∗)}

=F(G(x∗,y∗,y∗)which is a contradiction. Thus,G(x∗,y∗,y∗) =0. That is,x∗=y∗.

Also we have seen above thatx∗=lim

n→∞xn=nlim→∞T

n₍_x_0).

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Corollary 3.1. Let(X,G)be a complete G-metric space . Let T:X→X satisfies:

τ+F(G(T x,Ty,T z))≤F(aG(x,y,z) +bG(x,T x,T x) +cG(y,Ty,Ty) +dG(z,T z,T z)) (3.10)

for all x,y,z∈X with G(T x,Ty,T z)>0and a,b,c,d≥0such that a+b+c+d<1.If T or F is continuous, then T has a unique fixed point x∗in X and for every x0∈X , there is a sequence{Tnx0}in X that converges to x∗.

Proof. For allx,y,z∈X, we have

aG(x,y,z) +bG(x,T x,T x) +cG(y,Ty,Ty) +dG(z,T z,T z)

≤(a+b+c+d)max{G(x,y,z),G(x,T x,T x),G(y,Ty,Ty),G(z,T z,T z)}

<max{G(x,y,z),G(x,T x,T x),G(y,Ty,Ty),G(z,T z,T z)}

Then by (F1), we observe that (2.1) is a consequence of (3.10). Hence the corollary is proved.

Remark 3.1. If we take a=1,b=c=d=0in above corollary, we get Theorem 2.9 [6] by Gupta.

The following example shows that our theorem is a proper extension of Theorem 2.9 by Gupta.

Example 3.1. Consider the mapping T defined in Example 2.1. Since T is not an F-contraction for any F, Theorem 2.9 [6] is not applicable to T . However T is an F-weak contraction for F=lnα,α∈(0,∞).Therefore Theorem 3.1 can be applied to T . We note that x=1

2is the unique fixed point of T . References

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