Some fixed point results on quasi b metric like spaces

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R E S E A R C H

Open Access

Some ﬁxed point results on

quasi-

b

-metric-like spaces

Marija Cvetkovi´c

1

_{, Erdal Karapınar}

2,3*

_{and Vladimir Rakocevi´c}

1

*_{Correspondence:}

erdalkarapinar@yahoo.com

2_{Department of Mathematics,}

Atilim University, ˙Incek, Ankara, 06836, Turkey

3_{Nonlinear Analysis and Applied}

Mathematics (NAAM) Research Group, King Abdulaziz University, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

In this paper, we investigate the existence and uniqueness of a ﬁxed point of certain operators in the setting of complete quasi-b-metric-like spaces via admissible mappings. Our results improve, extend, and unify several well-known existence results.

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

Keywords: b-metric like space; ﬁxed point;

α-admissible; contractive mapping

1 Introduction and preliminaries Throughout this paper, we denoteR+

= [, +∞) andN=N∪ {}, whereNis the set of all positive integers. First, we recall some basic concepts and notation.

The concept ofb-metric was introduced by Czerwik [] as a generalization of metric (see also Bakhtin [, ]) to extend the celebrated Banach contraction mapping principle. Following the initial paper of Czerwik [], a number of researchers in nonlinear analy-sis investigated the topology of the paper and proved several ﬁxed point theorems in the context of completeb-metric spaces (see [–] 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∈X, the following conditions are satisﬁed:

(b) d(x,y) = if and only ifx=y; (b) d(x,y) =d(y,x);

(b) d(x,z)≤s[d(x,y) +d(y,z)].

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

Deﬁnition .[] LetXbe a nonempty set, ands≥ be a given real number. A map-pingd:X×X→[, +∞) is said to be a quasi-b-metric if for allx,y,z∈X, the following conditions are satisﬁed:

(bm) d(x,y) = if and only ifx=y; (bm) d(x,z)≤s[d(x,y) +d(y,z)].

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

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Deﬁnition .[] LetXbe a nonempty set, ands≥ be a given real number. A mapping d:X×X→[, +∞) is said to be a quasi-b-metric-like if for allx,y,z∈X, the following conditions are satisﬁed:

(bM) d(x,y) = impliesx=y; (bM) d(x,z)≤s[d(x,y) +d(y,z)].

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

Example . LetX={,_,_} ∪[,∞), and letd:X×X→[, +∞) be deﬁned as

d(x,y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 ifx=y= ,  ifx=y=_,  ifx= ,y=_,



 ifx= ,y=  , 

 ifx=  ,y= ,

|x–y| otherwise.

It is clear that (X,d) is a quasi-b-metric-like space with constants= .

Deﬁnition .(seee.g.[]) Let (X,d) be a quasi-b-metric-like space. Then:

(i)a a sequence{xn}inXis called a left-Cauchy sequence if and only if for everyε> ,

there exists a positive integerN=N(ε)such thatd(xn,xm) <εfor alln>m>N;

(ii)b a sequence{xn}inXis called a right-Cauchy sequence if and only if for everyε> ,

there exists a positive integerN=N(ε)such thatd(xn,xm) <εfor allm>n>N;

(iii)a a quasi-partial metric space is said to be left-complete if every left-Cauchy sequence

{xn}inXconverges with respect todto a pointu∈Xsuch that

lim

n→∞d(xn,u) =d(u,u) =n,mlim→∞d(xm,xn) = , wherem≥n;

(iii)b a quasi-partial metric space is said to be right-complete if every left-Cauchy sequence

{xn}inXconverges with respect todto a pointu∈Xsuch that

lim

n→∞d(u,xn) =d(u,u) =n,mlim→∞d(xn,xm) = , wherem≥n.

Let (X,d) and (Y,α) be quasi-b-metric-like spaces, and letf :X→Y be a continuous mapping. Then

lim

n→∞xn=u ⇒ nlim→∞fxn=fu.

In , Sametet al.[] introduced the concept ofα-admissible mappings, and in , Karapınaret al.[] improved this notion as triangularα-admissible mappings.

Deﬁnition .[, ] Letα:X×X→[, +∞) be a function. A self-mappingf is called anα-admissible mapping if

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for allx,y∈X. If, further,f satisﬁes the condition

α(x,z)≥ and α(z,y)≥ ⇒ α(x,y)≥

for allx,y,z∈X, then it is called triangularα-admissible mapping.

Very recently, Popescu [] improved these notions as follows.

Deﬁnition .[] Letα:X×X→[,∞) be a function. Iff :X→Xsatisﬁes the condi-tion

(T) α(x,fx)≥ ⇒ αfx,fx≥

for allx∈X, then it is called a right-α-orbital admissible mapping. Iff satisﬁes the condi-tion

(T) α(fx,x)≥ ⇒ αfx,fx≥

for all x∈X, then it is called a left-α-orbital admissible mapping. Furthermore, iff is both right-α-orbital admissible and left-α-orbital admissible, thenf is called anα-orbital admissible mapping.

Triangularα-admissible mappings deﬁned by Popescu [] impose the following deﬁ-nitions.

Deﬁnition .[] Letf:X→Xbe a self-mapping, andα:X×X→[,∞) be a function. Thenf is said to be triangular right-α-orbital admissible iff is right-α-orbital admissible and

(T) α(x,y)≥ and α(y,fy)≥ ⇒ α(x,fy)≥

and is said to be triangular left-α-orbital admissible iff isα-orbital admissible and

(T) α(fx,x)≥ and α(x,y)≥ ⇒ α(fx,y)≥.

IfT satisﬁes both (T) and (T) , then it is called triangularα-orbital admissible.

It is easy to conclude that eachα-admissible mapping is anα-orbital admissible mapping and each triangularα-admissible mapping is a triangularα-orbital admissible mapping. However, the converses of the statements are false. In the following example, we see that a mapping that is triangularα-orbital admissible need not be triangularα-admissible.

Example . LetX={xi:i= , . . . ,n}for somen≥, andd:X×X→R+withd(x,y) =

|x–y|. We deﬁne a self-mappingf :X→Xsuch thatfxi=xifori= , ,fxi=xjfori,j∈

{, },i=j,fxi=xi+fori∈ {, . . . ,n– }, andfxn=fx. Moreover, letα:X×X→R+be such that

α(x,y) = ⎧ ⎪ ⎨ ⎪ ⎩

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Note that f is α-orbital admissible since α(x,fx) =α(x,x) =  and α(x,fx) = α(x,x) = . On the other hand, we haveα(x,x) =α(x,x) = , butα(x,x) = . Hence, T is not triangularα-admissible.

Deﬁnition . [] Let (X,d) be a quasi-b-metric-like space. Then X is said to be α-regular if for every sequence{xn}inXsuch thatα(xn,xn+)≥ for allnandxn→x∈X

asn→ ∞, there exists a subsequence{xn(k)}of{xn}such thatα(xn(k),x)≥ for allk.

2 Main result

The notion of (b)-comparison was introduced by Berinde [] in order to extend the notion of (c)-comparison.

Deﬁnition . [] Let s≥ be a real number. A mapping ψ :R+_ →R+_ is called a (b)-comparison function if the following conditions are fulﬁlled:

() ψis monotone increasing;

() there existk∈N,a∈(, ), and a convergent series of nonnegative terms ∞

k=vk

such thatsk+_ψk+_(t)_≤_ask_ψk_{(t) +}_v

kfor allk≥kandt∈[,∞).

The class of (b)-comparison functions will be denoted byb. Notice that the notion of

a (b)-comparison function reduces to the concept of a (c)-comparison function ifs= . The following lemma will be used in the proof of our main result.

Lemma .[, ] Let s≥be a real number.Ifψ:R+

→R+is a(b)-comparison func-tion,then:

() the series∞_k_=sk_ψk_(t)_{converges for any}_t_∈_R+ ; () the functionps: [,∞)→[,∞)deﬁned by

ps(t) =

∞

k=

skψk(t) for allt∈[,∞)

is increasing and continuous at.

Remark . It is easy to see that ifψ(t)∈b, thenψ(t) <tfor allt> . In fact, if there is a

t∗>  such thatψ(t∗)≥t∗, then we haveψ(t∗)≥ψ(t∗)≥t∗(sinceψ is increasing). Con-tinuing in the same manner, we getψn(t∗)≥t∗> ,n∈N. This contradicts Lemma ..

Deﬁnition . Let (X,d) be a complete quasi-b-metric-like space with a constants≥. A self-mappingf :X→Xis called (α,ψ)-contractive mapping if there exist two functions ψ∈bandα:X×X→[,∞) satisfying the following condition:

α(x,y)d(fx,fy)≤ψd(x,y) (.)

for allx,y∈X.

Theorem . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be an (α,ψ)-contractive mapping.Suppose also that

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(ii) there existsx∈Xsuch thatα(x,fx)≥andα(fx,x)≥; (iii) f is continuous.

Then f has a ﬁxed point u in X,and d(u,u) = .

Proof By (ii) there existsx∈Xsuch thatα(x,fx)≥ andα(fx,x)≥. Deﬁne the it-erative sequence{xn}inXbyxn+=fxnfor alln∈N. Note that if there existsn∈N such thatxn=xn+, thenxnbecomes a ﬁxed point, which completes the proof. Hence,

throughout the proof, we suppose thatxn=xn+for alln∈N. Regarding the fact thatf is α-orbital admissible, from (ii) we derive that

α(x,x) =α(x,fx)≥ ⇒ α(fx,fx) =α(x,x)≥.

Inductively, we get that

α(xn,xn+)≥ for alln∈N. (.)

Analogously, again by (ii) and the fact thatf isα-orbital admissible we ﬁnd that

α(x,x) =α(fx,x)≥ ⇒ α(fx,fx) =α(x,x)≥.

Consequently, we observe that

α(xn+,xn)≥ for alln∈N. (.)

From (.), by takingx=xnandy=xn–, we ﬁnd that

d(xn+,xn) =d(fxn,fxn–)

≤α(xn,xn–)d(fxn,fxn–)

≤ψd(xn,xn–)

.

In view of Remark ., we get that

d(xn+,xn)≤ψ

d(xn,xn–)

<d(xn,xn–) for alln∈N. (.)

By analogy, again by (.) and by substitutingx=xn–andy=xn, we have

d(xn,xn+) =d(fxn–,fxn)

≤α(xn–,xn)d(fxn–,fxn)

≤ψd(xn–,xn)

.

Consequently,

d(xn,xn+)≤ψ

d(xn–,xn)

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From (.) and (.) we derive that

d(xn,xn+)≤ψn

d(x,x)

and d(xn+,xn)≤ψn

d(x,x)

for alln∈N. (.)

By Lemma .() and lettingn→ ∞in (.), we havelimn→∞d(xn,xn+) =limn→∞d(xn+, xn) = .

We further prove that the sequence{xn}is right-Cauchy and left-Cauchy. For alln,p∈N,

we have

d(xn,xn+p)≤ p–

i=

sid(xn+i–,xn+i) +sp–d(xn+p–,xn+p)

<

p

i=

siψn+i–d(x,x)

=  sn–

n+p–

k=n

skψkd(x,x)

.

By lettingn,p→ ∞we get that

lim

n,p→∞d(xn,xn+p) = ,

that is, the sequence{xn}is right-Cauchy.

Analogously,

lim

n,p→∞d(xn+p,xn) = ,

that is, the sequence{xn}is left-Cauchy. As a result, the sequence{xn}is a Cauchy

se-quence. Since (X,d) is complete, there exists a pointu∈Xsuch that

lim

n→∞d(u,xn) =nlim→∞d(xn,u) =d(u,u) =n,mlim→∞d(xn,xm) =n,mlim→∞d(xm,xn) = . (.)

Sincef is continuous, we have

u= lim

n→∞xn+=nlim→∞fxn=fu.

Example . Let (X,d) be a quasib-metric like space deﬁned in Example ., and let the mappingf:X→Xbe deﬁned as

fx= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 

 ifx= , 

 ifx=  , 

 ifx=  , x+  ifx≥.

Letψ(t) =_t,t≥, and letα:X×X→[,∞) be deﬁned as

α(x,y) =

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Thenψ∈b, andf is an (α,ψ)-contractive mapping. Since the conditions of Theorem .

are satisﬁed, it follows thatf has a ﬁxed point inX.

It is possible to remove the heavy condition of continuity of the self-mappingf in Theo-rem .. For this purpose, we need the following result, which is inspired from the results in [].

Lemma . Let(X,d)be a quasi-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) = ,thenlimn→∞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). (.)

If d(x,x) = ,then



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

Theorem . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be an (α,ψ)-contractive mapping.Suppose also that

(i) f isα-orbital admissible;

(ii) there existsx∈Xsuch thatα(x,fx)≥andα(fx,x)≥; (iii) Xisα-regular.

Proof By verbatim of the proof of Theorem . we ﬁnd an iterative sequence{xn} that

converges to a pointu∈Xsuch that (.) holds. Sinced(u,u) = , by Lemma . we have



sd(u,fu)≤lim infn→∞ d(xn+,fu)

≤lim sup

n→∞ d(xn+,fu)

=lim sup

n→∞ d(fxn,fu) ≤lim sup

n→∞ α(xn,u)d(fxn,fu) ≤lim sup

n→∞ ψ

d(xn,u)

.

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It is natural to consider the uniqueness of a ﬁxed point of an (α,ψ)-contractive mapping. We notice that we need to add an additional condition to guarantee the uniqueness.

(U) For allx,y∈Fix(f), eitherα(x,y)≥orα(y,x)≥. Here,Fix(f) denotes the set of all ﬁxed points off.

Theorem . Adding condition(U) to hypotheses of Theorem. (or Theorem.),we obtain the uniqueness of a ﬁxed point of f.

Proof Suppose thatx∗andy∗are two distinct ﬁxed points off, so thatd(x∗,y∗) > . If, for example,α(x∗,y∗)≥, then

dx∗,y∗=dfx∗,fy∗

≤αx∗,y∗dfx∗,fy∗

≤ψdx∗,y∗

<dx∗,y∗,

which is a contradiction.

Deﬁnition . Let (X,d) be a complete quasi-b-metric-like space with a constants≥. A self-mappingf :X→Xis called a generalized (α,ψ)-contractive mapping of type (A) if there exist two functionsψ∈bandα:X×X→[,∞) satisfying the following

condi-tion:

α(x,y)d(fx,fy)≤ψM(x,y) (.)

for allx,y∈X, where

M(x,y) =maxd(x,y),d(x,fx),d(y,fy). (.)

Theorem . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a generalized(α,ψ)-contractive mapping of type(A).Assume that

Proof As in the proof of Theorem ., we construct an iterative sequencexn+=fxn,n∈

N, where the existence ofx∈Xis guaranteed by (ii). By the same reason as in the proof of Theorem ., we may assume thatxn=xn+for alln∈N, and we can conclude that

α(xn,xn+)≥ and α(xn+,xn)≥ for alln∈N. (.)

From (.) we have

d(xn,xn+) =d(fxn–,fxn)

≤α(xn–,xn)d(fxn–,fxn)

≤ψM(xn–,xn)

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for alln∈N, where

M(xn–,xn) =max

d(xn–,xn),d(xn,fxn),d(xn–,fxn–)

=maxd(xn–,xn),d(xn,xn+)

.

IfM(xn–,xn) =d(xn,xn+), then since we assumed thatxn=xn+,

d(xn,xn+)≤ψ

d(xn,xn+)

<d(xn,xn+),

which is a contradiction. It allows us to conclude thatM(xn–,xn) =d(xn–,xn),n∈N.

Thus,

d(xn,xn+)≤ψ

d(xn–,xn)

<d(xn–,xn) for alln∈N

and

d(xn,xn+)≤ψn

d(x,x)

for alln∈N. (.)

Analogously, lettingx=xnandy=xn–in (.), we get

d(xn+,xn) =d(fxn,fxn–)

≤α(xn,xn–)d(fxn,fxn–)

≤ψM(xn,xn–)

(.)

for alln∈N, where

M(xn,xn–) =max

d(xn,xn–),d(xn,fxn),d(xn–,fxn–)

=maxd(xn,xn–),d(xn,xn+),d(xn–,xn)

.

For the estimation ofd(xn+,xn), we will consider three diﬀerent cases.

Case. IfM(xn,xn–) =d(xn–,xn), then, by (.),

d(xn+,xn)≤ψ

d(xn–,xn)

. (.)

Case. IfM(xn,xn–) =d(xn,xn+), then

d(xn+,xn)≤ψ

d(xn,xn+)

.

By Remark . we ﬁnd that

d(xn+,xn)≤ψ

d(xn,xn+)

<ψn+d(x,x)

.

Case. Otherwise,M(xn,xn–) =d(xn,xn–) and

d(xn+,xn)≤ψ

d(xn,xn–)

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Observing (.) and (.), it follows that, for anyn∈N,

d(xn+,xn)≤max

ψnd(x,x)

,ψnd(x,x)

. (.)

Obviously, in all considered cases, we deduce that

lim

n→∞d(xn+,xn) =nlim→∞d(xn,xn+) = .

Sinceψis an increasing function, let

v=maxd(x,x),d(x,x)

.

Consequently, we have thatd(xn+,xn)≤ψn(v) andd(xn,xn+)≤ψn(v). By applying (bM) for anyn,p∈Nit follows that

d(xn,xn+p)≤ p–

i=

sid(xn+i–,xn+i) +sp–d(xn+p–,xn+p)

≤

p

i=

sid(xn+i–,xn+i)

≤

p

i=

siψn+i–(v)

=  sn–

p

i=

sn+i–ψn+i–(v).

Therefore,lim_n_,_p_→∞d(xn,xn+p) =  and, likewise,limn,p→∞d(xn+p,xn) = . Since,Xis

com-plete, there existsu∈Xsuch thatlimn→∞xn=uand

lim

n→∞d(u,xn) =nlim→∞d(xn,u) =d(u,u) = . (.)

Furthermore,f is a continuous mapping, and henceu=limn→∞xn=limn→∞fxn–=fu.

Theorem . Adding condition(U)to hypotheses of Theorem.,we obtain the unique-ness of a ﬁxed point of T.

Proof Suppose thatfx∗=x∗andfy∗=y∗. Then

dx∗,y∗=dfx∗,fy∗

≤αx∗,y∗ψdfx∗,fy∗

≤ψMx∗,y∗

=ψdx∗,y∗,

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In the following example, we show the existence of a function satisfying conditions of Theorem . but not satisfying conditions of Theorem ..

Example . Let (X,d) be a quasi-b-metric-like space described in Example ., and f :X→Xthe mapping

fx= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 

 ifx= , , ifx=_,



 ifx=  , x+  ifx≥.

Letψ(t) = t

,t≥, and letα:X×X→[,∞) be deﬁned as

α(x,y) =

 if (x,y)∈ {(,_), (,_), (_, ), (_,_), (_,_)},  otherwise.

Then (.) does not hold, for example, forx=  andy=_, but (.) holds,f has a unique ﬁxed pointu=_, andd(u,u) = .

Deﬁnition . Let (X,d) be a complete quasi-b-metric-like space with a constants≥. A self-mappingf :X→Xis called a generalized (α,ψ)-contractive mapping of type (B) if there exist two functionsψ∈bandα:X×X→[,∞) satisfying the following

condi-tion:

α(x,y)d(fx,fy)≤ψN(x,y) (.)

N(x,y) =max

d(x,y),d(x,fx) +d(y,fy) 

. (.)

The following theorem can be deduced from the inequalityN(x,y)≤M(x,y) for allx,y, together with the monotonicity ofψ.

Theorem . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a generalized(α,ψ)-contractive mapping of type(B).Assume that

Deﬁnition . Let (X,d) be a complete quasi-b-metric-like space with a constants≥. A self-mappingf :X→Xis called a generalized (α,ψ)-contractive mapping of type (C) if there exist two functionsψ∈bandα:X×X→[,∞) satisfying the following

condi-tion:

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M(x,y) =maxd(x,y),d(x,fx),d(y,fy). (.)

The following theorem is easily observed from Theorem . since inequality (.) can be easily derived from inequality (.).

Theorem . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a generalized(α,ψ)-contractive mapping of type(C).Assume that

In the next theorems, we establish a ﬁxed point result for a generalized (α,ψ )-contractive mapping of type (C) without any continuity assumption on the mappingf.

Theorem . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a generalized(α,ψ)-contractive mapping of type(C).Suppose that

Proof As in the proof of Theorem ., we consider an iterative sequence{xn}, and we

obtain the existence ofu∈Xsuch that (.) holds. By Lemma . we get

d(u,fu)≤slim inf

n→∞ d(xn+,fu) ≤slim sup

n→∞

d(xn+,fu)

≤slim sup

n→∞

α(xn,u)d(fxn,fu)

≤lim sup

n→∞ ψ

M(xn,u)

,

where

M(xn,u) =max

d(xn–,u),d(xn–,xn),d(u,fu)

.

According to (.) and the fact thatlim_n_→∞d(xn–,xn) = , it remains to discuss only

the caseM(xn,u) =d(u,fu) because otherwise it followsd(u,fu) = ⇒u=fu.

Notice that, under this assumption,d(u,fu)≤ψ(d(u,fu)) also impliesd(u,fu) =  since ψ(t) <tfor anyt> . Hence,uis a ﬁxed point of the mappingf.

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Example . Let (X,d) be a quasib-metric like space deﬁned in Example ., and let the mappingf:X→Xbe deﬁned as

fx= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

 ifx= , 

 ifx=  , 

 ifx=  , x+  ifx≥.

Letψ(t) = t

,t≥, andα:X×X→[,∞) be deﬁned as

α(x,y) = ⎧ ⎪ ⎨ ⎪ ⎩

 if (x,y)∈ {(_,_), (_,_)}, 

 if (x,y)∈ {,  ,  } × {,  ,  } \ {(  ,  ), (  ,  )},  otherwise.

Then ψ ∈b, andf is a generalized (α,ψ)-contractive mapping of type (C). Since the

conditions of Theorem . are satisﬁed, it follows thatf has a ﬁxed point inX.

Deﬁnition . Let (X,d) be a complete quasi-b-metric-like space with a constants≥. A self-mappingf :X→Xis called a generalized (α,ψ)-contractive mapping of type (D) if there exist two functionsψ∈bandα:X×X→[,∞) satisfying the following

condi-tion:

α(x,y)d(fx,fy)≤ψL(x,y) (.)

L(x,y) =max

d(x,y),d(x,fx) +d(y,fy) s

. (.)

Theorem . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a generalized(α,ψ)-contractive mapping of type(D).Suppose that

(i) f is triangularα-orbital admissible;

Then f has a ﬁxed point in X,that is,there exists u∈X such that fu=u and d(u,u) = .

Proof As in the proof of Theorem ., we consider an iterative sequence{xn}and obtain

the existence ofu∈Xsuch that (.) holds. By Lemma . we get



sd(u,fu)≤lim infn→∞ d(xn+,fu) ≤lim sup

n→∞ d(xn+,fu) ≤lim sup

n→∞

α(xn,u)d(fxn,fu)

≤lim sup

n→∞ ψ

N(xn,u)

,

where

N(xn,u) =max

d(xn–,u),

d(xn–,xn) +d(u,fu)

s

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IfN(xn,u) =d(xn–,u), then we conclude the result due to (.). Takinglimn→∞d(xn–, xn) =  into account, we deduce thatlimn→∞N(xn,u) =d(u_,_sfu). Notice that, under this

as-sumption,_sd(u,fu)≤ψ(d(u_,_sfu)) also impliesd(u,fu) =  sinceψ(t) <tfor anyt> . Hence,

uis a ﬁxed point of the mappingf.

Theorem . Adding condition(U)to hypotheses of Theorem. (and respectively, The-orem.),we obtain the uniqueness of a ﬁxed point of T.

3 Consequences

In this section, we will list some consequences of our main results.

3.1 For standard quasi-b-metric-like

Corollary . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a mapping such that

d(fx,fy)≤ψmaxd(x,y),d(x,fx),d(y,fy) (.)

for all x,y∈X,whereψ ∈b.If f is continuous,then f has a ﬁxed point u in X,and

d(u,u) = .

Proof The proof of Corollary . follows from Theorem . by takingα(x,y) =  for all x,y∈X, so (ii) is satisﬁed for anyx∈X,f is obviously anα-orbital admissible, and (U) holds. Inequality (.) allows us to conclude thatf is a generalized (α,ψ)-contractive

map-ping of type (A).

Corollary . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a continuous mapping such that

d(fx,fy)≤ψ

max

d(x,y),d(x,fx) +d(y,fy) 

(.)

for all x,y∈X,whereψ∈b.Then f has a ﬁxed point u in X,and d(u,u) = .

Proof The proof of Corollary . follows from Theorem . by takingα(x,y) =  for all x,y∈Xsince then (.) follows from (.).

Notice that the continuity condition off in Corollary . can be removed by adding an extra terms.

Corollary . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a mapping such that

d(fx,fy)≤sψmaxd(x,y),d(x,fx),d(y,fy) (.)

for all x,y∈X,whereψ∈b.Then f has a ﬁxed point u in X such that d(u,u) = .

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anyx∈X. Notice that sinceα(x,y) = , any constructive sequence turns to be regular, and

thusXisα-regular.

Corollary . Let(X,d)be a complete quasi-b-metric-like space,and let f :X→X be a mapping such that

d(fx,fy)≤ψd(x,y) (.)

for all x,y∈X,whereψ∈b.Then f has a ﬁxed point u in X such that d(u,u) = .

Proof The proof of Corollary . follows from Theorem . by takingα(x,y) =  for all x,y∈Xand observing thatXisα-regular and that (i) and (ii) hold.

3.2 For standard quasi-b-metric-like spaces with a partial order

In this section, we deduce various ﬁxed point results on a quasi-b-metric-like space en-dowed with a partial order. We, ﬁrst, recollect some basic notions and notation.

Deﬁnition . Let (X,) be a partially ordered set, andf :X→Xbe a given mapping. We say thatf is nondecreasing with respect toif for allx,y∈X,

xy ⇒ fxfy.

Deﬁnition . Let (X,) be a partially ordered set. A sequence{xn} ⊆Xis said to be

non-decreasing (respectively, nonincreasing) with respect toifxnxn+,n∈N(respectively, xn+xn,n∈N).

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,xxnk) 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 f :X→X be a nondecreasing mapping with respect to.Suppose that there existsψ∈bsuch that

d(fx,fy)≤ψM(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fxandfxx; (ii) f is continuous or

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

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Proof Deﬁne the mappingα:X×X→[,∞) by

α(x,y) =

 ifxyorxy,  otherwise.

Clearly,f satisﬁes (.), that is,

α(x,y)sd(fx,fy)≤ψM(x,y)

for allx,y∈X. From condition (i) we havexfxandfxx. Moreover, for allx,y∈X, from the monotone property off we have

α(x,y)≥ ⇒ xy or

xy ⇒ fxfy or

fxfy ⇒ α(fx,fy)≥.

Hence, the self-mapping f isα-admissible. Similarly, we can prove thatf is triangular α-admissible and so triangularα-orbital admissible. Now, iff is continuous, then the ex-istence of a ﬁxed point follows from Theorem ..

Suppose that (X,,d) is regular. Let{xn}be a sequence inXsuch thatα(xn,xn+)≥ for allnandxn→x∈Xasn→ ∞. By the regularity hypothesis, sinceXdoes not contain an

inﬁnite totally unordered subset, there exists a subsequence{xnk}of{xn}such thatxnkx

orxxnk for allk.

This implies from the deﬁnition ofαthatα(xnk,x)≥ for allk. In this case, the existence

of a ﬁxed point follows again from Theorem ..

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 f :X→X be a nondecreasing mapping with respect to.Suppose that there existsψ∈bsuch that

d(fx,fy)≤ψd(x,y) (.)

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

(ii) f is continuous.

Then T has a ﬁxed point u∈X with d(u,u) = .Moreover,if for all x,y∈X,there exists z∈X such that xz and yz,we have the uniqueness of a ﬁxed point.

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.

Author details

1_{Faculty of Sciences and Mathematics, University of Niš, Visegradska 33, Niš, 18000, Serbia.}2_{Department of Mathematics,}

Atilim University, ˙Incek, Ankara, 06836, Turkey. 3_{Nonlinear Analysis and Applied Mathematics (NAAM) Research Group,}

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Acknowledgements

The authors are grateful to the reviewers for their careful reviews and useful comments. The ﬁrst and third authors were supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

Received: 27 August 2015 Accepted: 17 November 2015 References

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