Some extensions for Geragthy type contractive mappings

(1)

R E S E A R C H

Open Access

Some extensions for Geragthy type

contractive mappings

Erdal Karapınar

1,2*

_{, Hamed H Alsulami}

2

_{and Maha Noorwali}

2

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} Atilim University, ˙Incek, Ankara, 06836, Turkey

2_{Nonlinear Analysis and Applied} Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia

Abstract

In this paper, we establish some ﬁxed point theorems on some extensions of Geragthy contractive type mappings in the context ofb-metric-like spaces.

MSC: 46T99; 46N40; 47H10; 54H25

Keywords: b-metric-like space; ﬁxed point;

α-admissible; contractive mapping

1 Introduction and preliminaries

One of the interesting extensions of the notion of a metric space is the dislocated space, introduced by Hitzler []. This notion was rediscovered by Amini-Harandi [] and given the name of a metric-like space.

Deﬁnition . On a nonempty setXwe deﬁne a functionσ:X×X→[,∞) such that for allx,y,z∈X:

(σ) ifσ(x,y) = thenx=y; (σ) σ(x,y) =σ(y,x);

(σ) σ(x,y)≤σ(x,z) +σ(z,y);

and the pair (X,σ) is called a dislocated (metric-like) space.

Throughout this paper, we suppose that N=N∪ {}whereNdenotes the set of all

positive integers. Further, the symbolsR+_and_R+

denotes the set of positive reals and the

set of non-negative reals. First, we recall some basic concepts and notations.

The concept of ab-metric was introduced by Czerwik [] as a generalization of the met-ric (see also Bakhtin [] and Bourbaki []) to extend the celebrated Banach contraction mapping principle. Following this initial paper of Czerwik [], a number of researchers in nonlinear analysis investigated the topology of the paper and proved several ﬁxed point theorems in the context of completeb-metric spaces (seee.g.[–] and references therein).

Deﬁnition .[] LetXbe a nonempty set ands≥ be a given real number. A mapping

d: X×X→[,∞) is said to be ab-metric if for allx,y,z∈Xthe following conditions are satisﬁed:

(bM) d(x,y) = if and only ifx=y;

(bM) d(x,y) =d(y,x);

(2)

(bM) d(x,z)≤s[d(x,y) +d(y,z)].

In this case, the pair (X,d) is called ab-metric space (with constants).

In what follows, we recall the notion ofb-metric-like space which is an interesting gen-eralization of bothb-metric space and metric-like space.

Deﬁnition .[] LetXbe a nonempty set ands≥ be a given real number. A mapping

d: X×X→[,∞) is said to beb-metric-like if for allx,y,z∈Xthe following conditions are satisﬁed:

(bML) ifd(x,y) = thenx=y;

(bML) d(x,y) =d(y,x);

(bML) d(x,z)≤s[d(x,y) +d(y,z)].

In this case, the pair (X,d) is called ab-metric-like space (with constants).

Example . LetX=C([,T]) be the set of all real continuous functions on the closed interval [,T]. Letd:X×X→R+_be deﬁned

d(f,g) =maxf(t) –g(t)p+a,

for allf,g∈X,a∈R+

, andp> . It is easy to see that (X,d) is a completeb-metric-like

space withs= p–_{. For more examples, see}_e.g._[].

Remark . Let (X,d) be ab-metric-like space with constants≥. Then it is clear that

ds₍_x_,_y_{) =}_|__d₍_x_,_y_{) –}_d₍_x_,_x_{) –}_d₍_y_,_y₎_|_{satisﬁes the following:} (S) ds₍_x_,_x_{) = }_{for all}_x_∈_X_.

Deﬁnition .[] Let (X,d) be ab-metric-like space. Then: () a sequence{xn}inXis called convergent tox∈Xif and only if

lim_n→∞d(xn,x) =d(x,x);

() a sequence{xn}inXis called Cauchy sequence if and only iflimn,m→∞d(xn,xm) exists and ﬁnite;

() (X,d)is complete if and only if every Cauchy sequence{xn}inXconverges tox∈X so that

lim

n→∞d(xn,x) =d(x,x) =m,limn→∞d(xn,xm).

Proposition .[] Let(X,d)be a b-metric-like space with constant s and let{xn}be a

sequence in X such thatlim_n→∞d(xn,x) = .Then: () xis unique.

() _sd(x,y)≤limn→∞d(xn,y)≤sd(x,y)for ally∈X.

Lemma .[] Let(X,d)be a b-metric-like space with constant s and{xn}a sequence in

X such that

d(xn+,xn+)≤kd(xn,xn+), n= , , . . . ,

(3)

Lemma .[] Let(X,d)be a b-metric-like space with constant s and assume that{xn}

and{yn}are sequences in X converging to x and y,respectively.Then



sd(x,y) –



sd(x,x) –d(y,y)≤lim infn→∞ d(xn,yn)≤lim sup_n→∞ d(xn,yn)

≤sd(x,x) +sd(y,y) +sd(x,y).

In particular,if d(x,y) = thenlim_n→∞d(xn,yn) = .

Moreover,for each z∈X we have



sd(x,z) –d(x,x)≤lim infn→∞ d(xn,z)≤lim sup_n→∞ d(xn,z)≤sd(x,z) +sd(x,x). ()

In particular,if d(x,x) = ,then



sd(x,z)≤lim infn→∞ d(xn,z)≤lim sup_n→∞ d(xn,z)≤sd(x,z).

Notice that, in general, ab-metric-like mapping does not need to be continuous. The notion ofα-admissible and triangularα-admissible mappings were introduced by Sametet al.[] and Karapınaret al.[], respectively.

Deﬁnition . LetT:X→Xbe a mapping andα:X×X→[,∞) be a function. We say thatT is anα-admissible mapping if

x,y∈X, α(x,y)≥ ⇒ α(Tx,Ty)≥.

Moreover, a self-mappingTis called triangularα-admissible ifTisα-admissible and

x,y,z∈X, α(x,z)≥ andα(z,y)≥ ⇒ α(x,y)≥.

For more details onα-admissible and triangularα-admissible mappings, seee.g.[–]. Very recently, Popescu [] reﬁned the notion of triangularα-orbital admissible as fol-lows.

Deﬁnition .[] LetT:X→Xbe a mapping andα:X×X→[,∞) be a function. We say thatT isα-orbital admissible if

α(x,Tx)≥ ⇒ αTx,Tx≥.

Furthermore,Tis called triangularα-orbital admissible ifTisα-orbital admissible and

α(x,y)≥ andα(y,Ty)≥ ⇒ α(x,Ty)≥.

(4)

Example . LetX={a,b,c,d,e,f,g,h}. We deﬁne a self-mappingT:X→Xsuch that

Tx=x, forx=a,dand

Tx=y for (x,y)∈(b,c), (c,b), (e,f), (f,e), (g,h), (h,g).

Moreover, we deﬁneα:X×X→R+

, such that

α(x,y) =

 if (x,y)∈ {(a,b), (a,c), (b,b), (c,c), (b,c), (c,b), (b,d), (c,d), (d,e)},  otherwise.

Note thatT isα-orbital admissible, sinceα(b,Tb) =α(b,c) =  andα(c,Tc) =α(c,b) = . On the other hand, we have α(d,e) = , but α(Td,Te) =α(d,f) = . Hence,T is notα -admissible.

Lemma .[] Let T:X→X be a triangularα-orbital admissible mapping.Assume that there exists x∈X such thatα(x,Tx)≥.Deﬁne a sequence{xn}by xn+=Txnfor

each n∈N.Then we haveα(xn,xm)≥for all m,n∈Nwith n<m.

Lemma . Let T:X→X be a triangularα-orbital admissible mapping.Assume that there exists x∈X such thatα(Tx,x)≥.Deﬁne a sequence{xn}by xn+=Txnfor each

n∈N.Then we haveα(xm,xn)≥for all m,n∈Nwith n<m.

We characterize the notion ofα-regular in the setting of ab-metric-like space.

Deﬁnition .(cf.[]) Let (X,d) be ab-metric-like space,Xis said to beα-regular, if for every sequence{xn}inXsuch thatα(xn,xn+)≥ (respectively,α(xn+,xn)≥) for alln

andxn→x∈Xasn→ ∞, there exists a subsequence{xnk}of{xn}such thatα(xnk,x)≥ (respectively,α(x,xnk)≥) for allk.

In this paper, we shall prove the existence and uniqueness of a ﬁxed point for certain operators in the setting ofb-metric-like spaces. The presented results improve, extend, and unify a number of existing results in the literature.

2 Main result forb-metric-like spaces

In this section, we shall state and prove our main results. First, we recall the following classes of auxiliary functions. Letbe the set of all increasing and continuous functions

ψ: [,∞)→[,∞) withψ–({}) ={}. LetFsbe the family of all functionsβ: [,∞)→ [,_s) which satisfy the condition

lim

n→∞β(tn) = 

s implies n→∞lim tn= , ()

for somes≥.

Deﬁnition . Let (X,d) be ab-metric-like space with constants≥, andT:X→Xbe a map. We say thatTis a generalized almostα-ψ-φ-Geraghty contractive type mapping if there exist a functionα:X×X→[,∞),ψ,φ∈,β∈Fs, and someL≥ such that

(5)

for allx,y∈X, where

M(x,y) =max

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

s and ()

N(x,y) =minds(x,Ty),ds(y,Tx),d(x,Tx),d(y,Ty). ()

Remark . Since the functions belonging toFsare strictly smaller than_s, for somes≥, the expressionβ(ψ(M(x,y))) in () can be estimated from above as follows:

βψM(x,y)<

s for anyx,y∈X.

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥ and T :

X→X be a generalized almostα-ψ-φ-Geraghty contractive type mapping.We suppose also that

(i) Tis triangularα-orbital admissible; (ii) there existsx∈Xsuch thatα(x,Tx)≥;

(iii) Tis continuous.

Then T has a ﬁxed point,u∈X with d(u,u) = .

Proof By (ii) there existsx∈Xsuch thatα(x,Tx)≥. Deﬁne a sequence{xn} ⊂Xby

xn+=Txnfor alln∈N. AsT is triangularα-orbital admissible, by Lemma . we have α(xn,xn+)≥ for alln∈N. Throughout the proof, we suppose thatxn=xn+for alln∈

N. Indeed, if there existsnsuch thatxn=xn+, thenxnbecomes the ﬁxed point ofT, which completes the proof.

SinceTis a generalized almostα-ψ-φ-Geraghty contractive type mapping we have

ψsd(xn+,xn+)

≤α(xn,xn+)ψ

sd(Txn,Txn+)

≤βψM(xn,xn+)

ψM(xn,xn+)

+LφN(xn,xn+)

. ()

Thus, we have

<

sψ

M(xn,xn+)

+LφN(xn,xn+)

, ()

whereN(xn,xn+) =min{ds(xn,xn+),ds(xn+,xn+),d(xn,xn+),d(xn+,xn+)}= , and

M(xn,xn+) =max

d(xn,xn+),d(xn,xn+),d(xn+,xn+),

d(xn,xn+) +d(xn+,xn+)

s .

Note that

d(xn,xn+) +d(xn+,xn+)

s ≤

s[d(xn,xn+) + d(xn+,xn+)]

s

=[d(xn,xn+) + d(xn+,xn+)] 

≤maxd(xn,xn+),d(xn+,xn+)

(6)

Consequently, we have

M(xn,xn+) =max

d(xn,xn+),d(xn+,xn+)

.

IfM(xn,xn+) =d(xn+,xn+), then from () we have

<

sψ

d(xn+,xn+)

≤ψd(xn+,xn+)

.

Sinceψis increasing, we derive thatsd(xn+,xn+) <d(xn+,xn+), which is a contradiction

ass≥. Thus,M(xn,xn+) =d(xn,xn+). Again by (), we ﬁnd

<

sψ

d(xn,xn+)

≤ψd(xn,xn+)

.

Hence, we get

sd(xn+,xn+)≤d(xn,xn+) equivalently d(xn+,xn+)≤



sd(xn,xn+). ()

Case (i):s> . Since _s >  ands_s =_s< , by Lemma ., the sequence{xn}is Cauchy and

lim

n,m→∞d(xn,xm) = . ()

Case (ii):s= . From (), we haved(xn+,xn+)≤d(xn,xn+) for alln. Thus, we conclude

that

lim

n→∞d(xn,xn+) =r, ()

for somer≥. We shall prove thatr= . Suppose, on the contrary, thatr> . Note that, fors= , the inequality () turns into

ψd(xn+,xn+)

<βψM(xn,xn+)

ψM(xn,xn+)

, ()

whereN(xn,xn+) =  andM(xn,xn+) =d(xn,xn+) as evaluated above. Thus, () yields

ψ(d(xn+,xn+)) ψ(d(xn,xn+)) ≤

βψd(xn,xn+)

< . ()

By taking the limit asn→ ∞in () and regarding the continuity ofψ, we get

lim

n→∞β

ψd(xn,xn+)

= .

Hence, we have

lim

n→∞ψ

d(xn,xn+)

=  and so lim

(7)

Consequentlyr= . In what follows, we shall prove that{xn}is a Cauchy sequence. Indeed we will prove thatlimm,n→∞d(xn,xm) = . Suppose, on the contrary, that there existε>  and corresponding subsequences{nk}and{mk}ofNsatisfyingnk>mk>kfor which

d(xmk,xnk)≥ε, ()

wherenk,mkare chosen as the smallest integers satisfying (), that is,

d(xmk,xnk–) <ε. ()

By (), (), and the triangle inequality, we easily derive that

ε≤d(xmk,xnk)≤d(xmk,xnk–) +d(xnk–,xnk) <ε+d(xnk–,xnk). ()

Using () and the squeeze theorem we get

lim

k→∞d(xmk,xnk) =ε. ()

In a similar way, we can prove thatlim_k→∞d(xmk,xnk+) = ,limk→∞d(xnk,xmk+) = .

Regarding thatTis a generalized almostα-ψ-φ-Geraghty contractive type mapping, we have

ψd(xmk+,xnk+)

≤α(xmk,xnk)ψ

d(Txmk,Txnk)

≤βψM(xmk,xnk)

ψM(xmk,xnk)

+LφN(xmk,xnk)

, ()

M(xmk,xnk) = max

d(xmk,xnk),d(xmk,xmk+),d(xnk,xnk+),

d(xmk,xnk+) +d(xnk,xmk+)

 , ()

and

N(xmk,xnk) =min

ds(xmk,xnk+),ds(xnk,xmk+),d(xmk,xmk+),d(xnk,xnk+)

.

By taking the limit ask→ ∞in () and taking (), () into account, we get

ψ(ε)≤ lim

k→∞β

ψM(xmk,xnk)

ψ(ε). ()

Sinceβis a Geraghty function, we derive thatψ(M(xmk,xnk))→. Consequently, we have

d(xmk,xnk)→, which is a contradiction. Hence, we conclude thatlimm,n→∞d(xn,xm) = , and the sequence{xn}is Cauchy for anys≥.

By completeness of (X,d), there existsu∈Xsuch that

lim

(8)

SinceTis continuous,

Tu=T

lim

n→∞xn

= lim

n→∞Txn=n→∞limxn+=u,

anduis a ﬁxed point forT.

In what follows, we replace the condition of continuity of the operator by the condition ofα-regularity of the space.

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥ and T :

X→X be a generalized almostα-ψ-φ-Geraghty contractive type mapping.We suppose also that:

(iii) Xisα-regular anddis continuous.

Proof Following the lines of the proof of Theorem ., we conclude that there existsu∈X

such that

lim

n→∞d(xn,u) =n,m→∞lim d(xn,xm) =d(u,u) = .

SinceXisα-regular,α(xn,xn+)≥ for alln. Due to the fact thatlimn→∞xn=u, there exists a subsequence{xnk}of{xn}such thatα(xnk,u)≥ for allk. To prove thatuis a ﬁxed point forT, suppose on the contrary thatd(u,Tu) > .

Now, by using the properties ofψ and asT is a generalized almost α-ψ-φ-Geraghty contractive type mapping we have

ψd(xnk+,Tu)

≤α(xnk,u)ψ

s_d₍_Tx

nk,Tu)

≤βψM(xnk,u)

ψM(xnk,u)

+LφN(xnk,u)

.

Thus we have

ψd(xnk+,Tu)

<

sψ

M(xnk,u)

+LφN(xnk,u)

, ()

where N(xnk,u) = min{d s₍_x

nk,Tu),d s₍_u_,_x

nk+),d(xnk,xnk+),d(u,Tu)}, and note that

limk→∞N(xnk,u) = . Moreover,

M(xnk,u) = max

d(xnk,u),d(xnk,xnk+),d(u,Tu),

d(xnk,Tu) +d(u,xnk+) s

≤max

d(xnk,u),d(xnk,xnk+),d(u,Tu),

d(xnk,u) +d(u,Tu) +d(u,xnk) +d(xnk,xnk+)

(9)

Hence

lim

k→∞M(xnk,u)≤max

,d(u,Tu),d(u,Tu)

 =d(u,Tu),

and by the deﬁnition ofM(xnk,u) we havelimk→∞M(xnk,u) =d(u,Tu).

By the continuity ofψand theb-metric-liked, taking the limit askgoes to∞on both sides of () we have

ψd(u,Tu)≤

sψ

d(u,Tu).

Thus  = ψ_ψ(₍_dd(₍u_u_,,Tu_Tu))₎₎≤ _s, which is a contradiction in the cases> . Henced(u,Tu) = ; thereforeTu=u. In the cases=  we take the limit askgoes to∞on both sides of

ψd(xnk+,Tu)

≤βψM(xnk,u)

ψM(xnk,u)

and getlim_k→∞β(ψ(M(xnk,u))) =  and asβ∈F so we havelimk→∞ψ(M(xnk,u)) = .

Thus we haved(u,Tu) = ; thereforeTu=u.

For the uniqueness of a ﬁxed point of a generalizedα-ψ-φcontractive mapping, we will consider the following hypothesis.

(H) For allx,y∈Fix(T), eitherα(x,y)≥orα(y,x)≥. Here,Fix(T) denotes the set of ﬁxed points ofT.

Theorem . Adding condition(H)to the hypotheses of Theorem. (or Theorem.),we obtain the uniqueness of the ﬁxed point of T.

Proof Suppose thatx∗andy∗are two ﬁxed points ofT. Then it is obvious thatM(x∗,y∗) =

d(x∗,y∗) andN(x∗,y∗) = . So, we have

ψdx∗,y∗≤ψsdTx∗,Ty∗

≤αx∗,y∗ψsdTx∗,Ty∗

≤βψdx∗,y∗ψdx∗,y∗

<

sψ

dx∗,y∗,

which is a contradiction.

Deﬁnition . Let (X,d) be ab-metric-like space with constants≥,T :X→Xbe a map, we say thatTis a generalized rationalα-ψ-φ-Geraghty contractive mapping of type (I) if there exist a functionα:X×X→[,∞),ψ,φ∈,β∈Fs, and someL≥ such that

α(x,y)ψsd(Tx,Ty)≤βψK(x,y)ψK(x,y)+LφN(x,y), ()

for allx,y∈X, whereN(x,y) is deﬁned as in () and

K(x,y) =max

d(x,y),d(x,Tx)d(y,Ty)  +d(x,y) ,

d(x,Tx)d(y,Ty)

(10)

Deﬁnition . Let (X,d) be ab-metric-like space with constants≥,T:X→Xbe a map, we say thatT is a generalized rationalα-ψ-φ-Geraghty contractive of type (II) mapping if there exist a functionα:X×X→[,∞),ψ,φ∈,β∈Fsand someL≥ such that

α(x,y)ψsd(Tx,Ty)≤βψQ(x,y)ψQ(x,y)+LφN(x,y), ()

for allx,y∈X, whereN(x,y) is deﬁned as in () and

Q(x,y) =max

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

s₍_x_,_Ty₎_ds₍_y_,_Tx₎

 +s(d(x,Tx) +d(y,Ty)) . ()

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥and T:X→ X be a generalized rationalα-ψ-φ-Geraghty contractive mapping of type(I)such that:

Proof We shall use the same techniques as in the proof of Theorem .. First of all, we shall construct a sequence{xn} ⊂Xwherexn+=Txnfor whichα(xn,xn+)≥ andxn=xn+for

alln∈N.

SinceTis generalized rationalα-ψ-φ-Geraghty contractive of type (I) we have

≤α(xn,xn+)ψ

sd(Txn,Txn+)

≤βψK(xn,xn+)

ψK(xn,xn+)

. ()

SinceN(xn,xn+) = , the above inequality implies that

≤βψK(xn,xn+)

ψK(xn,xn+)

, ()

where

K(xn,xn+) =max

d(xn,xn+),

d(xn,xn+)d(xn+,xn+)

 +d(xn,xn+)

,d(xn,xn+)d(xn+,xn+)  +d(xn+,xn+)

.

On the other hand, we have

 +d(xn+,xn+) ≤

d(xn+,xn+)

=d(xn,xn+)

and

d(xn,xn+)d(xn+,xn+)

 +d(xn,xn+) ≤

d(xn,xn+)d(xn+,xn+)

d(xn,xn+)

=d(xn+,xn+).

Consequently, we getK(xn,xn+)≤max{d(xn,xn+),d(xn+,xn+)}.

Ifmax{d(xn,xn+),d(xn+,xn+)}=d(xn+,xn+), then from () together with Remark .,

we have

<

sψ

d(xn+,xn+)

(11)

This is a contradiction sinceψis increasing. Thus, we havemax{d(xn,xn+),d(xn+,xn+)}=

d(xn,xn+) and by the deﬁnition ofK(xn,xn+) we shall haveK(xn,xn+) =d(xn,xn+).

Con-sequently, the inequality () turns into

ψd(xn+,xn+)

≤ψsd(xn+,xn+)

≤βψK(xn,xn+)

ψd(xn,xn+)

. ()

By Remark ., we get

<ψd(xn,xn+)

and hence,d(xn+,xn+) <



sd(xn,xn+). ()

Case (i):s> . Since _s >  ands_s =_s< , by Lemma ., the sequence{xn}is Cauchy and

lim

n,m→∞d(xn,xm) = . ()

Case (ii):s= . Since{d(xn,xn+)}is a decreasing sequence, there existsr≥ such that

lim

n→∞d(xn,xn+) =r, ()

for somer≥. We shall prove thatr= . Suppose, on the contrary, thatr> . By letting

n→ ∞in () we ﬁnd

ψ(r)≤ lim

n→∞β

ψK(xn,xn+)

ψ(r).

It yields  =limn→∞β(ψ(K(xn,xn+))). Sinceβ∈F, we getψ(K(xn,xn+))→, which

im-plies thatd(xn,xn+)→, that is,r= .

In what follows, we shall prove that{xn}is a Cauchy sequence. Indeed we will prove that

limm,n→∞d(xn,xm) = . Suppose, on the contrary, that there existε>  and corresponding

subsequences{nk}and{mk}ofNsatisfyingnk>mk>kfor which

d(xmk,xnk)≥ε, ()

wherenk,mkare chosen as the smallest integers satisfying (), that is,

d(xmk,xnk–) <ε. ()

By (), (), and the triangle inequality, we easily derive that

ε≤d(xmk,xnk)≤d(xmk,xnk–) +d(xnk–,xnk)

<ε+d(xnk–,xnk). ()

Using () and the squeeze theorem we get

lim

(12)

Regarding that T is generalized rational α-ψ-φ-Geraghty contractive mapping of type (I), we have

d(Txmk,Txnk)

≤βψK(xmk,xnk)

ψK(xmk,xnk)

+LφN(xmk,xnk)

, ()

K(xmk,xnk) =max

d(xmk,xnk),

d(xmk,xmk+)d(xnk,xnk+)

 +d(xmk,xnk) ,

d(xmk,xmk+)d(xnk,xnk+)

 +d(xmk+,xnk+)

()

and

N(xmk,xnk) =min

ds(xmk,xnk+),ds(xnk,xmk+),d(xmk,xmk+),d(xnk,xnk+)

.

It is clear that

lim

k→∞K(xmk,xnk) =ε and k→∞limN(xmk,xnk) = .

By taking the limit ask→ ∞in () and taking (), () into account, we get

ψ(ε)≤ lim

k→∞β

ψK(xmk,xnk)

lim

k→∞ψ

K(xmk,xnk)

≤ lim

k→∞β

ψK(xmk,xnk)

ψ(ε). ()

Sinceβis a Geraghty function, we derive thatψ(K(xmk,xnk))→. Consequently, we have

d(xmk,xnk)→, which is a contradiction. Hencelimm,n→∞d(xn,xm) = , and the sequence {xn}is Cauchy for anys≥.

By completeness of (X,d), there existsu∈Xsuch that

lim

n→∞d(xn,u) =d(u,u) =n,m→∞lim d(xn,xm) = .

Now, ifTis continuous, then

Tu=T

lim

n→∞xn

= lim

n→∞Txn=n→∞limxn+=u,

anduis a ﬁxed point forT.

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥and T:X→ X be a generalized rationalα-ψ-φ-Geraghty contractive of mapping type(I)such that:

(13)

Proof Verbatim of the proof of Theorem ., we conclude that the iterative sequence{xn} is Cauchy and converges tou∈X. SinceXisα-regular, then, as in the proof of Theo-rem ., there exists a subsequence{xnk}of{xn}such that

ψd(xnk+,Tu)

<

sψ

K(xnk,u)

+LφN(xnk,u)

, ()

whereK(xnk,u) =max{d(xnk,u),

d(x_nk,x_nk+)d(u,Tu)

+d(x_nk,u) ,

d(x_nk,x_nk+)d(u,Tu)

+d(x_nk+,Tu) }.

Hencelimk→∞K(xnk,u) =  and as in the proof of Theorem .limk→∞N(xnk,u) = . Thus taking the limit ask→ ∞on both sides of () and keeping in mind thatψandd

are continuous we haveψ(d(u,Tu))≤. Henced(u,Tu) = ; thereforeTu=u.

Theorem . Adding condition(H)to the hypotheses of Theorem. (or Theorem.),

we obtain uniqueness of the ﬁxed point of T.

Proof As in the proof of Theorem ., we suppose thatx∗andy∗are two ﬁxed points ofT. Then, clearly, we haveK(x∗,y∗) =d(x∗,y∗) andN(x∗,y∗) = . So, we have

ψdx∗,y∗≤ψsdTx∗,Ty∗

≤αx∗,y∗ψsdTx∗,Ty∗

≤βψdx∗,y∗ψdx∗,y∗

<

sψ

dx∗,y∗,

which is a contradiction.

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥and T :

X→X be a generalized rationalα-ψ-φ-Geraghty contractive mapping of type(II)such that:

Proof Verbatim of the lines in the proof of Theorem ., we construct a sequence{xn} ⊂X wherexn+=Txnfor whichα(xn,xn+)≥ andxn=xn+for alln∈N. Moreover, by using

the fact thatT is a generalized rationalα-ψ-φ-Geraghty contractive mapping of type (II) and the property ofψwe have

≤α(xn,xn+)ψ

sd(Txn,Txn+)

≤βψQ(xn,xn+)

ψQ(xn,xn+)

. ()

By using the same arguments as in the proof of Theorem ., we derive that

<

sψ

Q(xn,xn+)

(14)

where

Q(xn,xn+) =max

d(xn,xn+),

d(xn,xn+)d(xn+,xn+) +ds(xn,xn+)ds(xn+,xn+)

 +s[d(xn,xn+) +d(xn+,xn+)]

.

Sinced(xn,xn+)≤s[d(xn,xn+) +d(xn+,xn+)], we have

d(xn,xn+)≥

d(xn,xn+)d(xn,xn+)

 +d(xn,xn+)

≥ d(xn,xn+)d(xn,xn+)

 +s[d(xn,xn+) +d(xn+,xn+)]

.

Hence, we getQ(xn,xn+) =d(xn,xn+).

By using () we get ψ(s_d₍_x

n+,xn+))≤ ψ(d(xn,xn+)). Since ψ is increasing we

haved(xn+,xn+) < _sd(xn,xn+). Ifs>  then, as in the proof of Theorem ., by using

Lemma ., we conclude that{xn} is a Cauchy sequence andlimn,m→∞d(xn,xm) = . If

s= , by verbatim of the proof of Theorem ., we deduce that{xn}is a Cauchy sequence. Since (X,d) is complete, there exists u ∈ X such that  = limn,m→∞d(xn,xm) =

limn→∞d(xn,u) =d(u,u). Now, sinceTis continuous,Tu=T(limn→∞xn) =limn→∞Txn=

limn→∞xn+=uanduis a ﬁxed point forT.

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥and T:

X→X be a generalized rationalα-ψ-φ-Geraghty contractive mapping of type(II)such that:

Proof By following the proof of Theorem . line by line, we see that{xn}converges to

u∈X. Due to the fact thatXisα-regular and by following the lines of the proof of Theo-rem . there exists a subsequence{xnk}of{xn}such that

ψd(xnk+,Tu)

<

sψ

Q(xnk,u)

+LφN(xnk,u)

, ()

where

Q(xnk,u) =max

d(xnk,u),

d(xnk,xnk+)d(u,Tu) +ds(xnk,Tu)ds(u,xnk+)

 +s[d(xnk,xnk+) +d(u,Tu)]

.

Note thatlim_k→∞Q(xnk,u) =  andlimk→∞N(xnk,u) = . Thus taking the limit asn→ ∞ on both sides of () and keeping in mind that ψ and d are continuous we have

ψ(d(u,Tu)) = , and sod(u,Tu) = . Thusu=Tu.

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥and T:

X→X be a mapping. Suppose that there exist a functionα:X×X→[,∞),ψ ∈,

β∈Fssuch that

(15)

for all x,y∈X.Suppose also that:

Proof By (ii) there existsx∈Xsuch thatα(x,Tx)≥. Deﬁne a sequence{xn} ⊂Xby

xn+=Txnfor alln∈N. AsT is triangularα-orbital admissible, by Lemma . we have α(xn,xn+)≥ for alln∈N. Notice that if there exists a natural numbernsuch that

xn=xn+, then the proof is complete. To avoid this trivial case, from now on, we assume thatxn=xn+for alln∈N.

SinceTsatisﬁes () we have

≤α(xn,xn+)ψ

sd(Txn,Txn+)

≤βψd(xn,xn+)

ψd(xn,xn+)

.

Thus, we have

<

sψ

d(xn,xn+)

<ψd(xn,xn+)

. ()

Sinceψis increasing, we haved(xn+,xn+) <_sd(xn,xn+).

Case (i): Ifs> , then, since _s >  ands_s =_s < , by Lemma .,{xn}is a Cauchy se-quence andlimn,m→∞d(xn,xm) = .

Case (ii): If s = , then as d(xn+,xn+) <d(xn,xn+), there exists r≥  such that

limn→∞d(xn,xn+) =r. Ifr>  then taking the limit asn→ ∞on both sides of

ψd(xn+,xn+)

≤βψd(xn,xn+)

ψd(xn,xn+)

we getlim_n→∞β(ψ(d(xn,xn+))) =  and asβis a Geraghty function, we derive that

lim

n→∞d(xn,xn+) = . ()

That is,r= . In what follows we shall prove that{xn}is a Cauchy sequence. Indeed we will prove thatlimm,n→∞d(xn,xm) = . Suppose, on the contrary, that there existε>  and

corresponding subsequences{nk}and{mk}ofNsatisfyingnk>mk>kfor which

d(xmk,xnk)≥ε, ()

wherenk,mkare chosen as the smallest integers satisfying (), that is,

d(xmk,xnk–) <ε. ()

By (), (), and the triangle inequality, we easily derive that

(16)

Using () and the squeeze theorem we get

lim

k→∞d(xmk,xnk) =ε. ()

AsTsatisﬁes (), we have

d(Txmk,Txnk)

≤βψd(xmk,xnk)

ψd(xmk,xnk)

. ()

Taking the limit ask→ ∞for () we getlimk→∞β(ψ(d(xmk,xnk))) = . Thus

lim

k→∞d(xmk,xnk) = .

Therefore,{xn}is a Cauchy sequence fors≥. By completeness of (X,d), there existsu∈X such that  =limn,m→∞d(xn,xm) =limn→∞d(xn,u) =d(u,u).

SinceT is continuous,Tu=T(lim_n→∞xn) =lim_n→∞Txn=limn→∞xn+=uanduis a

ﬁxed point forT.

Theorem . Let(X,d)be a complete b-metric-like space with constant s≥and T :

X→X be a mapping.Suppose that there exist a functionα:X×X→[,∞),ψ∈,and β∈Fssuch that

α(x,y)ψsd(Tx,Ty)≤βψd(x,y)ψd(x,y), ()

for all x,y∈X.Suppose also that:

Theorem . Adding condition(H)to the hypotheses of Theorem. (or Theorem.),

we obtain the uniqueness of the ﬁxed point of T.

Remark . Notice that we get several corollaries by replacing the auxiliary functions

ψ andβin a proper way. In particular, by takingψ(t) =twe ﬁnd the extended version of several existing results.

3 Expected consequences

In this section, we shall consider some immediate consequences of our main results. The following result is obtained by lettingL=  in Theorem . or ..

Corollary . Let(X,d)be a complete b-metric-like space with constant s≥and T:X→ X be a mapping.Suppose that there existα:X×X→[,∞),ψ∈,β∈Fssuch that

(17)

for all x,y∈X,where

M(x,y) =max

s . ()

Suppose also that:

(iii) Tis continuous or(iii)Xisα-regular anddis continuous.

Then T has a ﬁxed point.

Adding condition (H) to the hypothesis of Corollary ., we guarantee the uniqueness of the ﬁxed point.

Again by lettingL=  in Theorem . and Theorem . we get two more corollaries as Corollary .. We skip the details regarding the volume of the paper.

Corollary . Let(X,d)be a complete b-metric-like space with constant s≥,T:X→X

be a map andα:X×X→[,∞)be a function.Suppose that T satisﬁes at least one of the following conditions:

(a) α(x,y)d(Tx,Ty)≤__sM(x,y); (b) α(x,y)d(Tx,Ty)≤ 

sK(x,y);

where M(x,y),K(x,y)are deﬁned as in(), ().Suppose also that: (i) Tis triangularα-orbital admissible;

(ii) there existsx∈Xsuch thatα(x,Tx)≥;

Then T has a unique ﬁxed point u∈X with d(u,u) = .

Proof It is suﬃcient to takeL= ,ψ(t) =t, andβ(t) =__sin Theorem . and Theorem . (and thus, Theorem . or Theorem ., Theorem . or Theorem ., respectively).

Adding condition (H) to the hypothesis of Corollary ., we guarantee the uniqueness of the ﬁxed point.

Corollary . Let(X,d)be a complete b-metric-like space with constant s≥,T:X→X be a map,andα:X×X→[,∞)be a function.Suppose that T satisﬁes at least one of the following conditions:

(c) α(x,y)d(Tx,Ty)≤__sQ(x,y),

where M(x,y),K(x,y),Q(x,y)are deﬁned as in().Suppose also that: (i) Tis triangularα-orbital admissible;

(ii) there existsx∈Xsuch thatα(x,Tx)≥;

Then T has a ﬁxed point u∈X with d(u,u) = .

Proof It is suﬃcient to takeL= ,ψ(t) =t, andβ(t) = 

sin Theorem . or Theorem .,

(18)

3.1 For standardb-metric-like spaces

If we setα(x,y) =  for allx,y∈Xin Theorem ., then we derive the following results.

Corollary . Let(X,d)be a complete b-metric-like space with constant s≥such that d is continuous and T:X→X be a mapping.Suppose that there existψ,φ∈,β∈Fs,and

some L≥such that

ψsd(Tx,Ty)≤βψM(x,y)ψM(x,y)+LφN(x,y), ()

M(x,y) =max

s and ()

N(x,y) =minds(x,Ty),ds(y,Tx),d(x,Tx),d(y,Ty). ()

If we setα(x,y) =  for allx,y∈Xin Theorem ., then we derive the following results.

Corollary . Let(X,d)be a complete b-metric-like space with constant s≥such that d is continuous and T:X→X be a mapping.Suppose that there existψ,φ∈,β∈Fs,and

ψsd(Tx,Ty)≤βψK(x,y)ψK(x,y)+LφN(x,y), ()

K(x,y) =max

d(x,y),d(x,Tx)d(y,Ty)  +d(x,y) ,

d(x,Tx)d(y,Ty)

 +d(Tx,Ty) and ()

N(x,y) =minds(x,Ty),ds(y,Tx),d(x,Tx),d(y,Ty). ()

If we setα(x,y) =  for allx,y∈Xin Theorem ., then we derive the following results.

Corollary . Let(X,d)be a complete b-metric-like space with constant s≥such that d is continuous and T:X→X be a mapping.Suppose that there existψ,φ∈,β∈Fs,and

ψsd(Tx,Ty)≤βψQ(x,y)ψQ(x,y)+LφN(x,y), ()

Q(x,y) =max

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

s₍_x_,_Ty₎_ds₍_y_,_Tx₎

 +s(d(x,Tx) +d(y,Ty)) and ()

N(x,y) =minds(x,Ty),ds(y,Tx),d(x,Tx),d(y,Ty). ()

(19)

If takeL=  in Corollaries .-., we get three more consequences. Regarding the vol-ume of the paper, we skip the details.

Corollary . Let(X,d)be a complete b-metric-like space with constant s≥such that d

is continuous and T:X→X be mapping.Suppose that there existψ∈andβ∈Fssuch

that

ψsd(Tx,Ty)≤βψd(x,y)ψd(x,y), ()

for all x,y∈X.Then T has a unique ﬁxed point u∈X with d(u,u) = .

Proof It follows from Theorem . byα(x,y) =  for allx,y∈X.

3.2 Forb-metric-like spaces endowed with a partial order

In this section, from our main results, we shall derive easily various ﬁxed point results on ab-metric-like space endowed with a partial order. We, ﬁrst, recall some notions.

Deﬁnition . Let (X,) be a partially ordered set andT:X→Xbe a given mapping. We say thatT is nondecreasing with respect toif

x,y∈X, xy ⇒ TxTy.

Deﬁnition . Let (X,) be a partially ordered set. A sequence{xn} ⊂Xis said to be nondecreasing (respectively, nonincreasing) with respect toifxnxn+(respectively,

xn+xnfor alln).

Deﬁnition . Let (X,) be a partially ordered set anddbe ab-metric-like onX. We say that (X,,d) is regular if for every nondecreasing (respectively, nonincreasing) sequence {xn} ⊂Xsuch thatxn→x∈Xasn→ ∞, there exists a subsequence{xnk}of{xn}such thatxnkx(respectively,xnkx) for allk.

We have the following result.

Corollary . Let(X,)be a partially ordered set(which does not contain an inﬁnite totally unordered subset)and d be a b-metric-like on X with constant s≥such that(X,d)

is complete.Let T:X→X be a nondecreasing mapping with respect to.Suppose that

there existψ∈,β∈Fssuch that

ψs_d₍_Tx_,_Ty₎_≤_β_ψ_M₍_x_,_y₎_ψ_M₍_x_,_y₎_, _()

for all x,y∈X with xy or yx where M(x,y)is deﬁned as in().Suppose also that the following conditions hold:

(i) there existsx∈Xsuch thatxTx;

(ii) T is continuous or(ii)(X,,d)is regular anddis continuous.

(20)

Proof Deﬁne the mappingα:X×X→[,∞) by

α(x,y) =

 ifxyorxy,  otherwise.

Clearly,Tsatisﬁes (), that is,

α(x,y)ψsd(Tx,Ty)≤βψM(x,y)ψM(x,y),

for allx,y∈X. From condition (i), we haveα(x,Tx)≥. Moreover, for allx,y∈X, from

the monotone property ofT, we have

α(x,y)≥ ⇒ xyorxy ⇒ TxTyorTxTy ⇒ α(Tx,Ty)≥.

Hence, the self-mappingTisα-admissible. Similarly, we can prove thatTis triangularα -admissible and so triangularα-orbital admissible. Now, ifT is continuous, the existence of a fixed point follows from Corollary .. Suppose now that (X,,d) is regular. Let{xn} be a sequence inXsuch thatα(xn,xn+)≥ for allnandxn→x∈Xasn→ ∞. From the regularity hypothesis and asXdoes not contain an infinite totally unordered subset, there exists a subsequence{xnk}of{xn}such thatxnkxorxxnkfor allk. This implies from the definition ofαthatα(xnk,x)≥ for allk. In this case, the existence of a fixed point

follows again from Corollary ..

In an analogous way, we derive the following results from Theorem . and Theo-rem ., respectively.

Corollary . Let(X,)be a partially ordered set(which does not contain an inﬁnite totally unordered subset)and d be a b-metric-like on X with constant s≥such that(X,d)

ψsd(Tx,Ty)≤βψK(x,y)ψK(x,y), ()

for all x,y∈X with xy or yx where M(x,y)is deﬁned as in().Suppose also that the following conditions hold:

Corollary . Let(X,)be a partially ordered set(which does not contain an inﬁnite totally unordered subset)and d be a b-metric-like on X with constant s≥such that(X,d)

ψsd(Tx,Ty)≤βψQ(x,y)ψQ(x,y), ()

(21)

Corollary . Let(X,)be a partially ordered set(which does not contain an inﬁnite totally unordered subset)and d be b-metric-like on X with constant s≥such that(X,d)

ψsd(Tx,Ty)≤βψd(x,y)ψd(x,y), ()

for all x,y∈X with xy or yx.Suppose also that the following conditions hold: (i) there existsx∈Xsuch thatxTx;

Example . LetX= [,∞) deﬁned:X×X→[,∞) byd(x,y) =|x–y|_{. Then (}_X_,_d₎

is completeb-metric (sob-metric-like) space with constants= . DeﬁneT:X→Xand

α(x,y) :X×X→[,∞) as follows:

Tx=

x

 ifx∈[, ],

x

x+ ifx∈(,∞),

and

α(x,y) =

 ifx,y∈[, ],  otherwise.

It is clear thatTis triangularα-orbital admissible and we haveα(,T)≥. Moreover,X

isα-regular anddis continuous.

Letψ(t) =t,β(t) =_ then clearlyψ∈andβ∈F. Moreover,T satisﬁes () for the following reason: ifx,y∈[, ], then

α(x,y)ψd(Tx,Ty)= 

x

–

y





=(x–y)



 =β

ψd(x,y)ψ(d(x,y).

Otherwise,

α(x,y)ψd(Tx,Ty)= ≤βψd(x,y)ψ(d(x,y).

Therefore, by Theorem .,Thas a ﬁxed pointx= .

Example . LetX={, , }deﬁned:X×X→[,∞) byd(x,y) = (max{x,y})_{. Then} (X,d) is completeb-metric-like space with constants= –_{= } _{such that}_d_is continu-ous. DeﬁneT:X→XbyT={(, ), (, ), (, )}.

Letψ(t) =t,β(t) = 

e

–t _or _β₍_t_{) =} 

+t, then clearly

ψ ∈ andβ∈F

. Note that

(22)

4 Conclusion

It is clear that we can list several more results by replacing theb-metric-like space, with some other abstract space, such as ab-metric space, a metric space, a metric-like space, a partial metric space, and so on.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Received: 16 July 2015 Accepted: 15 September 2015

References

1. Hitzler, P: Generalized metrics and topology in logic programming semantics. Ph.D. thesis, School of Mathematics, Applied Mathematics and Statistics, National University Ireland, University College Cork (2001)

2. Amini-Harandi, A: Metric-like spaces, partial metric spaces and ﬁxed points. Fixed Point Theory Appl.2012, Article ID 204 (2012)

3. Czerwik, S: Contraction mappings inb-metric spaces. Acta Math. Inform. Univ. Ostrav.1(1), 5-11 (1993) 4. Bakhtin, IA: The contraction mapping principle in quasimetric spaces. In: Functional Analysis, vol. 30, pp. 26-37.

Ul’yanovsk. Gos. Ped. Inst., Ul’yanovsk (1989) 5. Bourbaki, N: General Topology. Hermann, Paris (1974)

6. Aydi, H, Bota, M-F, Karapınar, E, Moradi, S: A common ﬁxed point for weakφ-contractions onb-metric spaces. Fixed Point Theory13(2), 337-346 (2012)

7. Bota, M-F, Chifu, C, Karapınar, E: Fixed point theorems for generalized (α∗-ψ)- ´Ciri´c-type contractive multivalued operators inb-metric spaces. Abstr. Appl. Anal.2014, Article ID 246806 (2014)

8. Bota, M-F, Karapınar, E, Mlesnite, O: Ulam-Hyers stability results for ﬁxed point problems via alpha-psi-contractive mapping inb-metric space. Abstr. Appl. Anal.2013, Article ID 825293 (2013)

9. Bota, M-F, Karapınar, E: A note on ‘Some results on multi-valued weakly Jungck mappings inb-metric space’. Cent. Eur. J. Math.11(9), 1711-1712 (2013)

10. Kutbi, MA, Karapınar, E, Ahmed, J, Azam, A: Some ﬁxed point results for multi-valued mappings inb-metric spaces. J. Inequal. Appl.2014, Article ID 126 (2014)

11. Alghamdi, MA, Hussain, N, Salimi, P: Fixed point and coupled ﬁxed point theorems onb-metric-like spaces. J. Inequal. Appl.2013, Article ID 402 (2013)

12. Hussain, N, Roshan, JR, Parvaneh, V, Kadelburg, Z: Fixed points of contractive mappings inb-metric-like spaces. Sci. World J.2014, Article ID 471827 (2014)

13. Samet, B, Vetro, C, Vetro, P:α-ψ-Contractive type mappings. Nonlinear Anal.75, 2154-2165 (2012)

14. Karapınar, E, Kuman, P, Salimi, P: Onα-ψ-Meir-Keeler contractive mappings. Fixed Point Theory Appl.2013, Article ID 94 (2013)

15. Karapınar, E, Samet, B: Generalized (α-ψ) contractive type mappings and related ﬁxed point theorems with applications. Abstr. Appl. Anal.2012, Article ID 793486 (2012)

16. Karapınar, E:α-ψ-Geraghty contraction type mappings and some related ﬁxed point results. Filomat28(1), 37-48 (2014)

17. Karapınar, E: A discussion on ‘α-ψ-Geraghty contraction type mappings’. Filomat28(4), 761-766 (2014) 18. Popescu, O: Some new ﬁxed point theorems forα-Geraghty contractive type maps in metric spaces. Fixed Point