On some fixed point theorems in generalized metric spaces

R E S E A R C H

Open Access

On some fixed point theorems in

generalized metric spaces

Youness ElKouch and El Miloudi Marhrani

*_{Correspondence:} [email protected]

Laboratory of Algebra, Analysis and Applications (L3A), Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, Avenue Commandant Harti, B.P 7955, Sidi Othmane, Casablanca, Morocco

Abstract

In this paper, we obtain some generalizations of fixed point results for Kannan, Chatterjea and Hardy-Rogers contraction mappings in a new class of generalized metric spaces introduced recently by Jleli and Samet (Fixed Point Theory Appl. 2015:33, 2015).

MSC: Primary 47H10; secondary 54H25

Keywords: fixed point theorem; Kannan contraction; Chatterjea contraction; Hardy-Rogers contraction; T-contraction

1 Introduction

The concept of a metric space is a very important tool in many scientific fields and partic-ulary in the fixed point theory.

In recent years, this notion has been generalized in several directions and many no-tions of a metric-type space was introduced (b-metric, dislocated space, generalized met-ric space, quasi-metmet-ric space, symmetmet-ric space, etc.).

In , Jleli and Samet [] introduced a very interesting concept of a generalized met-ric space, which covers different well-known metmet-ric structures including classical metmet-ric spaces, b-metric spaces, dislocated metric spaces, modular spaces, and so on.

In this paper, we establish and generalize some well-known fixed point results for non-linear contractions in this new class of generalized metric spaces.

Let us recall that a mappingTon a metric space (X,d) is called a Kannan contraction if there existsα∈[,

[ such that

d(Tx,Ty)≤αd(x,Tx) +αd(y,Ty)

for allx,y∈X.

Using this contraction notion, Kannan [] proved the following result.

Theorem .([]) Let(X,d)be a complete metric space,λ∈[,_[,and T a self-mapping on X such that

d(Tx,Ty)≤λd(x,Tx) +d(y,Ty) ()

for all x,y∈X.Then T has a unique fixed point.

In , Chatterjea [] obtained a similar result by considering a constantλ∈[,_[ and a mappingT:X−→Xsuch that

d(Tx,Ty)≤λd(x,Ty) +d(y,Tx) ()

for allx,y∈X.

In this paper, we are interested by Kannan, Chatterjea, and Hardy-Rogers contraction types (see [–] and []); we establish some results on fixed points in generalized metric spaces. We also give some examples to show the effectiveness of the obtained results.

Our results generalize and improve many fixed point theorems existing in the literature in various metric-type spaces.

2 Definitions and preliminaries

LetXbe a nonempty set, and letD:X×X−→[, +∞] be a given mapping. For every

x∈X, let us define the set

C(D,X,x) ={xn} ⊂X: lim

n→∞D(xn,x) = 

Definition .([]) Dis called a generalized metric onXif it satisfies the following con-ditions:

(D) For every(x,y)∈X×X, we have

D(x,y) =  ⇒ x=y.

(D) For all(x,y)∈X×X, we have

D(x,y) =D(y,x).

(D) There exists a real constantC> such that, for all(x,y)∈X×Xand{xn} ∈C(D,X,x), we have

D(x,y)≤Clim sup n→∞

D(xn,y).

The pair(X,D)is called a generalized metric space.

Remark . If the setC(D,X,x) is empty for every x∈X, then (X,D) is a generalized metric space if and only if (D) and (D) are satisfied.

Definition . Let (X,D) be a generalized metric space, let{xn}be a sequence inX, and letx∈X. We say that{xn}D-converges toxinXif{xn} ∈C(D,X,x).

Remark . Let{xn}be the sequence defined byxn=xfor alln∈N. If it D-converges to

x, thenD(x,x) = .

Definition . Let (X,D) be a generalized metric space. A sequence{xn}inXis called a D-Cauchy sequence iflimm,n→∞D(xn,xn+m) = .

In the sequel, we use the following definition of a Cauchy sequence.

Definition . Let (X,D) be a generalized metric space, and let{xn}be a sequence inX. We say that{xn}is a D-Cauchy sequence iflimm,n→∞D(xn,xm) = .

Proposition . C(D,X,x)is a nonempty set if and only if D(x,x) = .

Proof IfC(D,X,x)=∅, then there exists a sequence{xn} ⊂Xsuch thatlimn→∞D(xn,x) = . Using property (D), we obtain

D(x,x)≤Clim sup

n→∞ D(xn,x),

and thusD(x,x) = .

Assume now that D(x,x) = . The sequence{xn} ⊂X defined byxn=xfor alln∈N

converges tox, which ends the proof.

3 Main results

Proposition . Let(X,D)be a generalized metric space,and let f :X−→X be a mapping satisfying inequality()for someλ∈[,_).Then any fixed pointω∈X of f satisfies

D(ω,ω) <∞ ⇒ D(ω,ω) = .

Proof Letω∈Xbe a fixed point off such thatD(ω,ω) <∞. Using (), we obtain

D(ω,ω) =D(fω,fω)

≤λD(ω,fω) +D(ω,fω) ≤λD(ω,ω).

Since λ∈[, [, we obtainD(ω,ω) = .

For everyx∈X, we define

δ(D,f,x) =supDfix,fjx:i,j∈N .

Theorem . Let(X,D)be a D-complete generalized metric space, and let f be a self-mapping on X satisfying()for some constantλ∈[,_[such that Cλ< .

If there exists an element x∈X such thatδ(D,f,x) <∞,then the sequence{fnx}

con-verges to someω∈X.Moreover,if D(ω,fω) <∞,thenωis a fixed point of f.Moreover,for each fixed pointωof f in X such that D(ω,ω) <∞,we haveω=ω.

Proof Letn∈N(n≥). For alli,j∈N, we have

Dfn+ix,fn+jx

≤λDfn+i–x,fn+ix

+Dfn+j–x,fn+jx

and then

Dfn+ix,fn+jx

≤λδD,f,fn–x

which gives

δD,f,fnx

≤λδD,f,fn–x

.

Consequently, we obtain

δD,f,fn_x 

≤(λ)n_δ(D,f,x )

and

Dfnx,fmx

≤δD,f,fnx

≤(λ)nδ(D,f,x) ()

for all integermsuch thatm>n.

Sinceδ(D,f,x) <∞and λ∈[, [, we obtain

lim m,n→∞D

fnx,fmx

= .

It follows that{fnx}is a D-Cauchy sequence, and thus there existsω∈Xsuch that

lim n→∞D

fnx,ω= 

and

D(fω,ω)≤Clim sup n→∞ D

fω,fn+x

. ()

By () we have

Dfn+x,fω≤λDfn+x,fnx

+D(ω,fω). ()

By () and () we obtain

lim sup n→∞ D

fω,fn+x

≤λD(ω,fω).

Using (), we obtain

D(ω,fω)≤CλD(ω,fω).

SinceCλ<  andD(ω,fω) <∞, we deduce thatD(ω,fω) = , which implies thatfω=ω. Ifωis any fixed point off such thatD(ω,ω) <∞, we obtain

Dω,ω=Dfω,fω

≤λD(fω,ω) +Dfω,ω ≤λD(ω,ω) +Dω,ω ≤,

Example . LetX= [, ], and letD:X×X→[,∞[ be the mapping defined by

⎧ ⎨ ⎩

D(x,y) =x+y ifx=  andy= ,

D(,x) =D(x, ) =_x for allx∈X.

Conditions (D) and (D) are trivially satisfied. By Proposition . we need to verify con-dition (D) only for elementsxofXsuch thatD(x,x) = , which implies thatx= .

Let (xn)⊂Xbe a sequence such thatlimn→∞D(xn, ) = . For alln∈Nandy∈X, we have:

D(xn,y) =

⎧ ⎨ ⎩

xn+y ifxn= , y

 ifxn= .

Then

y

≤D(xn,y),

which implies that

D(,y) =y

≤lim supn→∞ D(xn,y).

It follows that (X,D) is a generalized metric space that is not a standard metric space since the triangular inequality does not hold: Ifx,y∈X–{}, then we haveD(x,y) =x+yand

D(x, ) +D(,y) =x+_y, and thus

D(x,y) >D(x, ) +D(,y).

Note that (X,D) is D-complete. Define the mappingT onXby

T(x) = x

x+  for allx∈X.

For anyx∈X, we have:

DT(x),T()=D

x x+ , 

= x

(x+ )

and

DT(x),x+D,T()=D

x x+ ,x

+D(, ) = x

x+ +x.

Then

DT(x),T()≤ 

Forx,y∈X–{}, we have

DT(x),T(y)=D

x x+ ,

y y+ 

= x

(x+ )+

y

(y+ )

and

DT(x),x+Dy,T(y)=D

x x+ ,x

+D

y, y

y+ 

= x

x+ +

y

y+ +x+y.

Then

DT(x),T(y)≤  

DT(x),x+Dy,T(y).

The hypotheses of Theorem . are satisfied. ThereforeThas a unique fixed point since

Dis bounded; note thatT() = .

Example . LetX={a,b,c}and defineTonXbyT(a) =a,T(b) =b, andT(c) =a. There is no metric for whichTis a Kannan contraction onX.

DefineD:X×X−→[, +∞] by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D(a,a) =D(b,b) =  and D(b,b) = +∞,

D(a,b) =D(b,a) = ,

D(a,c) =D(c,a) = ,

D(b,c) =D(c,b) = .

Then (X,D) is a complete generalized metric space,T is a Kannan contraction on (X,D) for anyλ∈],_[, and we can apply Theorem ..

Theorem . generalizes well-known results for b-metric and metric spaces.

Corollary . Let(X,d)be a complete b-metric space with constant s≥,and let f :X→ X a mapping for which there existsλ∈[, 

s+[such that

d(fx,fy)≤λd(x,fx) +d(y,fy)

for all x,y∈X.Then f has a unique fixed point.

Proof Letx∈X. For alln∈N, we have

dfnx,fn+x

≤λdfnx,fn–x

+dfn+x,fnx

,

which implies

dfnx,fn+x

≤ λ  –λd

fnx,fn–x

.

Letk=_–λ_λ. By induction we obtain

dfnx,fn+x

and then

dfn_x ,fmx

≤sdfn_x ,fn+x

+· · ·+sm–n_dfm–_x ,fmx

≤s(k)nd(x,fx) +· · ·+sm–n(k)m–d(x,fx)

≤s(k)n – (sk) m–n

 –sk d(x,fx)≤ s

 –skd(x,fx)

for alln,m∈Nsuch thatm>n.

This implies thatδ(D,f,x) <∞. Then by Theorem . we conclude thatf has a unique

fixed point.

Corollary . (Kannan fixed point theorem []) Let(X,d)be a complete metric space,

and let f :X→X a mapping for which there existsλ∈[,

[such that

d(fx,fy)≤λd(x,fx) +d(y,fy)

for all x,y∈X.Then f has a unique fixed point.

Corollary . Let(X,d)be a dislocated metric space,and let f :X→X be a mapping for which there existsλ∈[,

[such that

d(fx,fy)≤λd(x,fx) +d(y,fy) ()

for all x,y∈X.Then f has a unique fixed point.

In the following, we need the basic lemma.

Lemma .([]) Letλis a real number such that≤λ< ,and let{bn}be a sequence

of positives reals numbers such that lim_n→∞bn= . Then,for any sequence of positives

numbers{an}satisfying

an+≤λan+bn for all n∈N,

we havelimn→∞an= .

Theorem . Let(X,D)be a D-complete generalized metric space,λ∈[,_[,and let f be a self-mapping on X such that

D(fx,fy)≤λD(y,fx) +D(x,fy) ()

for all x,y∈X.If there exists a point x∈X such thatδ(D,f,x) <∞,then the sequence {fn_x

}converges to someω∈X.Moreover,if D(x,fω) <∞,thenωis a fixed point of f,and

for any fixed pointωof f such that D(ω,ω) <∞,we haveω=ω.

Proof Letn∈N(n≥). For all integersi,j, we have

Dfn+ix,fn+jx

≤λDfn+ix,fn+j–x

+Dfn+i–x,fn+jx

which implies that

Dfn+ix,fn+jx

≤λδD,f,fn–x

.

Hence

δD,f,fnx

≤λδD,f,fn–x

,

and consequently

δD,f,fnx

≤(λ)nδ(D,f,x).

This inequality implies that

Dfnx,fmx

≤δD,f,fnx

≤(λ)nδ(D,f,x)

for all integersn,msuch thatm>n. Sinceδ(D,f,x) <∞and λ∈[, ), we obtain

lim m,n→∞D

fnx,fmx

= .

It follows that{fn_x

}is a D-Cauchy sequence, and thus there existsω∈Xsuch that

lim n→∞D

fnx,ω= .

By (D) we have

Dfnx,ω

≤Clim sup m→∞ D

fnx,fmx

≤(λ)nCδ(D,f,x)≤Cδ(D,f,x).

Then

Dfnx,ω<∞ for alln∈N.

By () we have

Dfn+x,fω≤λDfn+x,ω+Dfnx,fω.

SinceD(x,fω) <∞, we haveD(fn_{x,fω) <∞for alln∈N. By Lemma . we obtain}

lim n→∞D

fnx,fω

= .

It follows thatfω=ω.

Letωbe any fixed point ofX. We have

Dω,ω=Dfω,fω

SinceD(ω,ω) <∞, we obtainD(ω,ω) = , which ends the proof.

Example . LetX= [, ], and letD:X×X→[,∞] be defined by

⎧ ⎨ ⎩

D(x, ) =D(,x) =∞ for allx∈[, ],

D(x,y) =x+y ifx=  andy= .

It is easy to see that (X,D) is a D-complete generalized metric space withC= . Consider the functionT: [, ]→[, ] given by

T(x) = 

x ifx∈[, [,

T() = .

The functionT is a Chatterjea contraction withλ=_ in (X,D). By Theorem .,Thas a fixed pointω∈X.

Note that the mappingThas two different fixed points, so we cannot apply the classical fixed point theorems for Banach, Kannan, and Chatterjea contractions since they give the uniqueness of the fixed point.

Definition . Let (X,D) be a generalized metric space. A self-mappingf onXis called

a Hardy-Rogers contraction if there exists nonnegative real constantsλifori= , , , ,  such thatλ=i_i=_=λi∈[, [ and

D(fx,fy)≤λD(x,y) +λD(x,fx) +λD(y,fy) +λD(y,fx) +λD(x,fy) ()

for allx,y∈X.

Proposition . Let(X,D)be a generalized metric space,and let f :X−→X be a

Hardy-Rogers contraction.Then any fixed pointω∈X of f satisfies

D(ω,ω) <∞ ⇒ D(ω,ω) = .

Proof Letω∈Xbe a fixed point off such thatD(ω,ω) <∞. We have

D(ω,ω) =D(fω,fω)

≤λD(ω,ω) +λD(ω,fω) +λD(ω,fω) +λD(ω,fω) +λD(ω,fω) ≤λD(ω,ω).

Sinceλ∈[, [, we haveD(ω,ω) = .

Lemma .([]) Let(an)be a sequence of nonnegative real numbers,and let(λn)be a

real sequence in[, ]such that

∞

n= λn=∞.

If,for a givenε> ,there exists a positive integer nsuch that

an+≤( –λn)an+ελn for all n≥n,

then≤lim sup_n→∞an≤ε.

Theorem . Let(X,D)be a D-complete generalized metric space,and let f be a self-mapping on X satisfying().

Assume that Cλ+λ< and that there exists a point x∈X such thatδ(D,f,x) <∞.

Then the sequence{fn_x

}converges to someω∈X.If D(x,fω) <∞,thenωis a fixed point

of f.Moreover,ifω∈X is another fixed point of f such that D(ω,ω) <∞and D(ω,ω) <∞,

thenω=ω.

Proof Letn∈N(n≥). For alli,j∈N, we have

Dfn+ix,fn+jx

≤λδD,f,fn–x

.

We obtain

δD,f,fnx

≤λδD,f,fn–x

and

δD,f,fnx

≤λnδ(D,f,x),

which leads to

Dfnx,fmx

≤δD,f,fnx

≤λnδ(D,f,x). ()

Sinceδ(D,f,x) <∞andλ∈[, ), we obtain

lim m,n→∞D

fnx,fmx

= .

It follows that{fnx}is a D-Cauchy sequence, and thus there existsω∈Xsuch that

lim n→∞D

fnx,ω= .

From () we have

Dfn+x,fω≤λD

fnx,ω+λD

fn+x,fnx

+λD(fω,ω)

+λD

ω,fn+x

+λD

Let ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

an=D(fnx,fω),

bn=λD(fnx,ω) +λD(fn+x,fnx) +λD(ω,fn+x),

K=λD(fω,ω).

()

By () we obtain

an+≤λan+bn+K.

Sincelimn→∞bn= , for every>  such that>_–Kλ, there existsNsuch that

bn≤ε( –λ) –K for alln≥Nε.

Then

an+≤λan+bn+K≤λan+ε( –λ).

By Lemma . we have

≤lim sup

n→∞ an≤ε for allε>

K

 –λ .

Then

≤lim sup n→∞

an≤

K

 –λ

. ()

By (D) we obtain

D(fω,ω)≤Clim sup n→∞ D

fω,fn+x

≤Clim sup n→∞ an≤C

K

 –λ

, ()

and by () we have

D(fω,ω)≤CλD(fω,ω)  –λ

.

SinceCλ+λ< , we have_–Cλλ < . ThenD(fω,ω) = , which impliesfω=ω.

Ifωis any fixed point off such thatD(ω,ω) <∞andD(ω,ω) <∞, then () implies

Dω,ω=Dfω,fω

≤λD

ω,ω+λD(ω,fω) +λD

ω,fω+λD

ω,fω+λD

ω,fω

≤λD

ω,ω+λD(ω,ω) +λD

ω,ω+λD

ω,ω+λD

ω,ω.

Then we obtain

( –λ–λ–λ)D

ω,ω≤λD(ω,ω) +λD

From Proposition . we haveD(ω,ω) =D(ω,ω) = , and thenD(ω,ω) = , which ends

the proof.

By the symmetry of the generalized metricDthe theorem is true if eitherCλ+λ<  orCλ+λ< .

4 Fixed point results for T-contractions

Beiranvand, Moradi, Omid, and Pazandeh [] introduced the notion of a T-contraction and established a version of the Banach contraction principle.

Let us introduce the following definitions.

Definition . Let (X,D) be a metric space, and letT be a self-mapping onX. We say that

(a) Tis continuous if

lim

n→∞D(xn,x) =  ⇒ nlim→∞D

T(xn),T(x)

= 

for allx∈X;

(b) Tis sequentially convergent if for every sequence{xn}such that{T(xn)}converges, {xn}converges;

(c) Tis subsequentially convergent if for every sequence{xn}such that{T(xn)} converges,{xn}has a convergent subsequence.

Definition . Let (X,D) be a metric space, and letTandf be two self-mappings onX. We say that

(a) f is a T-Banach contraction if there existsk∈], [such that

DTf(x),Tf(y)≤kD(Tx,Ty)

for allx,y∈X;

(b) f is a T-Kannan contraction if there existsk∈],_[such that

DTf(x),Tf(y)≤kDTx,Tf(x)+DTy,Tf(y)

for allx,y∈X;

(c) f is a T-Chatterjea contraction if there existsk∈],_[such that

DTf(x),Tf(y)≤kDTx,Tf(y)+DTy,Tf(x)

for allx,y∈X;

(d) f is a T-Hardy-Rogers contraction if

D(Tfx,Tfy)≤λD(Tx,Ty) +λD(Tx,Tfx) +λD(Ty,Tfy) +λD(Ty,Tfx) +λD(Tx,Tfy)

To show new results for T-contractions on a complete generalized metric space (X,D), we consider the mappingDT:X×X→[,∞] defined by

DT(x,y) =D(Tx,Ty) for allx,y∈X,

whereTis continuous, sequentially convergent, and one-to-one.

Proposition . We have:

. For every sequencexninX,

lim

n D(xn,x) =  ⇐⇒ limn DT(xn,x) = ;

. DTis a generalized metric onX;

. If(X,D)is complete,then(X,DT)is complete.

Proof . Let{xn}be a sequence such that

lim

n→∞DT(xn,x) = .

By continuity we have

lim

n→∞D(Txn,Tx) = .

Assume thatlimn→∞D(Txn,Tx) = . SinceTis sequentially convergent, there existsu∈X such that

lim

n D(xn,u) = ,

and by the continuity ofT we havelimnD(Txn,Tu) = . It follows thatTu=Tx, and since

T is one-to-one, we haveu=x.

. Letx,y∈Xbe such thatDT(x,y) = . ThenD(Tx,Ty) = . SinceTis one-to-one, we obtainx=yby (D).

The symmetry is obvious. Let nowx,y∈X, and let{xn}be a sequence that converges to

xin (X,DT). Then{Txn}converges toTxin (X,D), and by (D) we have

D(Tx,Ty)≤Clim sup n

D(Txn,Ty),

which is equivalent toDT(x,y)≤Clim supnDT(xn,y). Hence (X,DT) is a generalized met-ric space with the same constantC.

. Iflim_n,m→∞DT(xn,xm) = , thenlimn,m→∞D(Txn,Txm) = . So{Txn}is a Cauchy se-quence in (X,D), which is complete. It follows that there existsu∈Xsuch that

lim

n→∞D(Txn,u) = .

SinceTis sequentially convergent, there existsx∈Xsuch thatlim_nD(xn,x) = , which is

Remark . If a mapping f is a T-Banach (resp. T-Kannan, T-Chatterjea, T-Hardy-Rogers) contraction in (X,D), thenf is a Banach (resp. Kannan, Chatterjea, Hardy-Rogers) contraction in (X,DT) with the same constants.

For everyx∈X, we define

δT(D,f,x) =sup

DTfix,Tfjx:i,j∈N .

From Remark . and Theorem . in [] we can deduce the following corollaries.

Corollary . Let(X,D)be a complete metric space,and let T,f :X→X be two mappings such that T is continuous,one-to-one,and sequentially convergent.Assume that f is a T-Banach contraction.If there exists x∈X such thatδT(D,f,x) <∞,then{fnx}converges

to a fixed pointωof f.Moreover,ifωis any fixed point of f such that D(Tω,Tω) <∞,

thenω=ω.

Corollary . Let(X,D)be a complete generalized metric space,and let T,f :X→X be two mappings such that T is continuous,one-to-one,and sequentially convergent.Assume that f is a T-Kannan contraction with constant k> such that Ck< .If there exists x∈X

such thatδT(D,f,x) <∞,then{fnx}converges to someω∈X.Moreover,if D(Tx,Tfω) < ∞,thenωis a fixed point of f,and for every fixed pointωof f such that D(Tω,Tω) <∞,

we haveω=ω.

Corollary . Let(X,D)be a complete generalized metric space,and let T,f :X→X be two mappings such that T is continuous,one-to-one,sequentially convergent.Assume that f is a T-Chatterjea contraction and that there exists x∈X such thatδT(D,f,x) <∞.

Then{fn_x

}converges to someω∈X,and if D(Tx,Tfω) <∞,thenωis a fixed point of f.

Moreover,ifω∈X is another fixed point of f such that D(Tω,Tω) <∞,thenω=ω.

Corollary . Let(X,D)be a complete generalized metric space, and let T,f :X→X be two mappings such that T is continuous,one-to-one,and sequentially convergent. As-sume that f is a T-Hardy-Rogers contraction with nonnegative constantsλi,i= , , , , ,

such thatλ=i_i=_=λi∈[, [and Cλ+λ< .Assume that there exists x∈X such that δT(D,f,x) <∞.Then {fnx}converges to someω∈X.If D(Tx,Tfω) <∞,thenωis a

fixed point of f.Moreover,ifω∈X is another fixed point of f such that D(Tω,Tω) <∞

and D(Tω,Tω) <∞,thenω=ω.

5 Some remarks on Senapati et al. results

In this section, we discuss some results by Senapati et al. [] on extensions of Ciric and Wardowski-type fixed point theorems in generalized metric spaces. The authors intro-duced the following definition.

Definition .([], Definition .) Let (X,D) be a generalized metric space, and letTbe a self-mapping onX. ThenT is said to be aD-admissible mapping if for allx,y∈X,

They proved the following lemma.

Lemma .([], Lemma .) Let(X,D)be a generalized metric space,and let T be a D-admissible mapping on X.Then,for every sequence{xn}converging to a point a point w∈X,

we have D(w,Tw) <∞.

The proof of this lemma uses the following implication:

D(Txn,Tw) <∞ (∀n≥n) ⇒ lim sup n

D(Txn,Tw) <∞.

Example . We consider Example . in [] and the mappingT:X−→Xdefined by

Tx=

⎧ ⎨ ⎩

 ifx= , 

x otherwise.

For allx,y∈X, we haveTx+Ty≤D(Tx,Ty)≤Tx+Ty+ . ThenD(Tx,Ty) <∞, which implies thatTis admissible.

Forxn=_n, we havelimnD(xn, ) =  butlim supnD(Txn,T) =∞.

Example . LetX= [, +∞[, and letDbe defined by

D(x, ) =D(,x) =  and D(x,y) = 

x+



y forx= ,y= .

LetTbe the mapping defined by

⎧ ⎨ ⎩

T = ,

Tx=_xx_+ forx= .

ThenTis admissible. If the sequence (xn)nis defined byx=  andxn+=Txnfor alln≥, then we have

lim

n D(xn, ) = , D(Txn,T) =n+  <∞, and lim sup_n D(Txn,T) =∞.

The lemma was used to prove Theorems . and . of [], and they deduced the fol-lowing corollary:

Corollary .([], Corollary .) Let T:X−→X be a D-admissible self-mapping,and let

(X,D)be a complete D-generalized metric space.Suppose the following conditions hold:

(i) for allx,y∈X,there existsk∈], [such that

D(Tx,Ty)≤kmax

D(x,y),D(x,Tx),D(y,Ty),D(x,Ty) +D(y,Tx) 

;

(ii) there existsx∈Xsuch thatδ(D,T,x) <∞.

Then(Tn(x))nconverges to some w∈X,and this w is a fixed point of T.Moreover,if wis

The inequalities

D(x,Tx)

 +D(x,Tx)≤D(x,Tx),

D(x,Ty) +D(y,Tx)

 ≤max

D(x,Ty),D(y,Tx)

give the following implications:

Tsatisfies (i) ⇒ Tis ak-quasi-contraction

and

Tsatisfies rational inequality ⇒ Tsatisfies (i).

These implications show that in [], Theorem . is a consequence of Corollary .. Finally, in the proof of Theorem . in [], the authors use theF-contraction defined as follows.

Definition .([], Definition .) A self-mappingT defined onX is said to be anF -contraction if, for allx,y∈X,

D(x,y) >  and D(Tx,Ty) >  ⇒ τ+FD(Tx,Ty)≤F(D(x,y)

for someτ > .

We suspect that that existence of somex∈Xsuch thatδ(D,T,x) =cdoes not give

D(Tn+i_x,Tn+j_{x) >  for all integersn,i,jas the proofs of Theorem . and Corollary .} of [] blame.

6 Conclusion

In this paper, we gave a generalized version of Kannan, Chatterjea, and Hardy-Rogers con-traction fixed point theorems and some fixed point results for T-concon-tractions in a gener-alized metric space.

Our examples show that the results can be applied to prove the existence of fixed points in generalized metric spaces, whereas its classical counterpart fails to give positive an-swers.

Acknowledgements

The authors are thankful to the editors and the anonymous referees for their valuable comments, which reasonably improved the presentation of the manuscript.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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References

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