Fixed point result and applications on a b-metric space endowed with an arbitrary binary relation

(1)

R E S E A R C H

Open Access

Fixed point result and applications on a

b-metric space endowed with an arbitrary

binary relation

Wutiphol Sintunavarat

1*

_{, Somyot Plubtieng}

2

_{and Phayap Katchang}

3*

*_{Correspondence:}

[email protected]; [email protected]; [email protected]

1_{Department of Mathematics and}

Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand

3_{Division of Mathematics, Faculty of}

Science and Agricultural

Technology, Rajamangala University of Technology Lanna Tak, Tak, 63000, Thailand

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce the concept ofq-set-valued

α-quasi-contraction mapping

and establish the existence of a ﬁxed point theorem for this mapping inb-metric spaces. Our results are generalizations and extensions of the result of Aydiet al.(Fixed Point Theory Appl. 2012:88, 2012) and some recent results. We also state some illustrative examples to claim that our results properly generalize some results in the literature. Further, by applying the main results, we investigate a ﬁxed point theorem in ab-metric space endowed with an arbitrary binary relation. At the end of this paper, we give open problems for further investigation.

MSC: 47H10; 54H25

Keywords:

α-admissible mapping; binary relation; ﬁxed point;

b-metric space; q-set-valued

α-quasi-contraction

1 Introduction

Fixed point theory is one of the cornerstones in the development of mathematics since it plays a basic role in applications of many branches of mathematics. The famous Banach contraction principle is one of the most efficient power tools to study in this field since it can be observed easily and comfortably. In , Bakhtin [] introduced the concept ofb-metric space and presented the contraction mapping inb-metric spaces that is gen-eralization of the Banach contraction principle in metric spaces (see also Czerwik []). Subsequently, several researchers studied fixed point theory or the variational principle for single-valued and set-valued mappings inb-metric spaces (see [–] and references therein).

Recently, Aydiet al.[] established theq-set-valued quasi-contraction mapping which is a generalization of theq-set-valued quasi-contraction mapping due to Amini-Harandi [] in . They also established a ﬁxed point theorem for such a mapping inb-metric spaces. This theorem extends, uniﬁes and generalizes several well-known comparable re-sults in the existing literature.

On the other hand, Sametet al.[] introduced the concept ofα-admissible mapping and using this concept proved a fixed point theorem for a single-valued mapping. They also showed that these results can be utilized to derive fixed point theorems in partially ordered spaces and coupled fixed point theorems. Moreover, they applied the main re-sults to ordinary differential equations. Recently, Mohammadiet al.[] introduced the

(2)

concept ofα-admissible for a set-valued mapping which is diﬀerent from the notion of

α_∗-admissible which was provided in []. Subsequently, there are a number of results via the concept ofα-admissible mapping in many spaces (see [–] and references therein). Inspired and motivated by Aydiet al.[] and Mohammadiet al.[], we introduce the class ofq-set-valuedα-quasi-contraction mappings and give a ﬁxed point theorem for such mappings via the idea ofα-admissible mapping. Our result improves and com-plements the main result of Aydiet al.[] and many results in the literature. We also provide some examples to show the generality of our result. The applications for ﬁxed point theorems in ab-metric space endowed with an arbitrary binary relation are also de-rived from our results. Furthermore, at the end of this paper, we give open problems for further investigation.

2 Auxiliary notions

Throughout this paper, the standard notations and terminologies in nonlinear analysis are used. For the convenience of the reader, we recall some of them. In the sequel,R,R+andN

denote the set of real numbers, the set of nonnegative real numbers and the set of positive integers, respectively.

Deﬁnition .([, ]) LetXbe a nonempty set ands≥ be a given real number. A func-tionald:X×X→R+is called ab-metric if for allx,y,z∈Xthe following conditions are

satisﬁed:

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

() d(x,z)≤s[d(x,y) +d(y,z)].

A pair (X,d) is called ab-metric space (with constants).

It is easy to see that any metric space is ab-metric space withs= . Therefore, the class ofb-metric spaces is larger than the class of metric spaces.

Some known examples ofb-metric, which show that ab-metric space is real generaliza-tion of a metric space, are the following.

Example . The set of real numbers together with the functional

d(x,y) :=|x–y|

for allx,y∈Ris ab-metric space with constants= . Also, we obtain thatdis not a metric onX.

Example . The setlp(R) with  <p< , wherelp(R) :={{xn} ⊆R|

∞

n=|xn|p <∞},

together with the functionald:lp(R)×lp(R)→R,

d(x,y) :=

_∞

n=

|xn–yn|p



p

(3)

Example . Letpbe a given real number in the interval (, ). The spaceLp[, ] of all real functionsx(t),t∈[, ] such that_|x(t)|p_dt_{< , together with the functional}

d(x,y) :=





x(t) –y(t) pdt /p

for eachx,y∈Lp[, ],

is ab-metric space with constants= p_.

Example . LetX={, , }and a functionald:X×X→R+be deﬁned by

d(, ) =d(, ) =d(, ) = ,

d(, ) =d(, ) =d(, ) =d(, ) = 

and

d(, ) =d(, ) =m,

wheremis a given real number such thatm≥. It is easy to see that

d(x,y)≤m 

d(x,z) +d(z,y)

for allx,y,z∈X. Therefore, (X,d) is ab-metric space with constants=m/. However, if

m> , the ordinary triangle inequality does not hold and thus (X,d) is not a metric space.

Next, we recall the notions of convergence, compactness, closedness and completeness in ab-metric space (see in []).

Deﬁnition .([]) Let (X,d) be ab-metric space. The sequence{xn}inXis called:

() convergentif and only if there existsx∈Xsuch thatd(xn,x)→asn→ ∞. In this case, we writelim_n_→∞d(xn,x) = .

() Cauchyif and only ifd(xn,xm)→asm,n→ ∞.

Remark . In ab-metric space (X,d), the following assertions hold:

() a convergent sequence has a unique limit; () each convergent sequence is Cauchy; () in general, ab-metric is not continuous.

Deﬁnition .([]) LetYbe a nonempty subset of ab-metric space (X,d). The closure

Y ofYis the set of limits of all convergent sequences of points inY,i.e.,

Y:=

x∈X: there exists a sequence{xn}inXsuch that lim

n→∞xn=x

.

Deﬁnition .([]) Let (X,d) be ab-metric space. A subsetY⊆Xis called:

(4)

() compactif and only if for every sequence of element inYthere exists a subsequence that converges to an element inY;

() boundedif and only ifδ(Y) :=sup{d(a,b)|a,b∈Y}<∞.

Deﬁnition . Theb-metric space (X,d) iscompleteif every Cauchy sequence inX con-verges.

Now, we consider the following notations of a collection of subsets of ab-metric space (X,d):

P(X) :={Y|Y⊆X};

P(X) :=Y∈P(X)|Y =∅;

Pb(X) :=Y∈P(X)|Yis bounded;

Pcp(X) :=Y∈P(X)|Yis compact;

Pcl(X) :=Y∈P(X)|Yis closed;

Pb,cl(X) :=Pb(X)∩Pcl(X).

Here, we give the concept of generalized functionals on ab-metric space (X,d).

Deﬁnition . Let (X,d) be ab-metric space.

() The functionalD:P(X)×P(X)→R∪ {+∞}is said to be gap functional if and only if it is deﬁned by

D(A,B) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

inf{d(a,b)|a∈A,b∈B}, A =∅ =B,

, A=∅=B,

+∞, otherwise.

In particular, ifx∈X, thend(x,B) :=D({x},B).

() The functionalρ:P(X)×P(X)→R∪ {+∞}is said to be excess generalized functional if and only if it is deﬁned by

ρ(A,B) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sup{d(a,B)|a∈A}, A=∅ =B,

, A=∅,

+∞, otherwise.

() The functionalH:P(X)×P(X)→R∪ {+∞}is said to be Pompeiu-Hausdorﬀ generalized functional if and only if it is deﬁned by

H(A,B) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

max{ρ(A,B),ρ(B,A)}, A =∅ =B,

, A=∅,

+∞, otherwise.

(5)

Remark . For ab-metric space (X,d), we have

d(x,B)≤ρ(A,B)≤H(A,B)

for allA,B∈P(X) andx∈A. We also obtain that forB∈P(X), we get

d(x,B)≤d(x,b)

for allb∈B.

The following lemmas from Czerwik [] are useful for some of the proofs in the main result.

Lemma .([]) Let(X,d)be a b-metric space.Then

d(x,A)≤sd(x,B) +H(B,A)

for all x∈X and A,B∈P(X).In particular,we have

d(x,A)≤sd(x,y) +d(y,A)

for all x,y∈X and A∈P(X).

Lemma .([]) Let(X,d)be a b-metric space.Then

H(A,C)≤sH(A,B) +H(B,C)

for all A,B,C∈P(X).

Lemma .([]) Let(X,d)be a b-metric space and A,B∈Pb,cl(X).Then,for each>  and for all b∈B,there exists a∈A such that d(a,b)≤H(A,B) +.

Lemma .([]) Let(X,d)be a b-metric space.For A∈Pb,cl(X)and x∈X,we have

d(x,A) =  ⇒ x∈A.

Lemma .([]) Let(X,d)be a b-metric space(with constant s≥),and let{xn}be a sequence in X such that

d(xn+,xn+)≤γd(xn,xn+),

for all n∈N,where≤γ < .Then the sequence{xn}is a Cauchy sequence in X provided that sγ < .

Deﬁnition .([]) LetXbe a nonempty set,t:X→Xandα:X×X→[,∞). We say thattisα-admissible if

(6)

Deﬁnition .([, ]) LetXbe a nonempty set,T:X→X_{, where }X_{is a collection} of subsets ofXandα:X×X→[,∞). We say that

() Tisα∗-admissible if

forx,y∈Xfor whichα(x,y)≥ ⇒ α_∗(Tx,Ty)≥,

whereα∗(Tx,Ty) :=inf{α(a,b)|a∈Tx,b∈Ty}.

() Tisα-admissible whenever for eachx∈Xandy∈Txwithα(x,y)≥, we have

α(y,z)≥for allz∈Ty.

Remark . It is easy to prove that the set-valued mappingT isα∗-admissible implies thatTisα-admissible mapping.

3 The existence of ﬁxed point theorems for a set-valued mapping inb-metric spaces

In this section, we introduce theq-set-valuedα-quasi-contraction mapping and obtain the existence of a ﬁxed point theorem for such a mapping inb-metric spaces.

Deﬁnition . Let (X,d) be ab-metric space andα:X×X→[,∞) be a mapping. The set-valued mappingT:X→Pb,cl(X) is said to be aq-set-valuedα-quasi-contraction if

α(x,y)H(Tx,Ty)≤qM(x,y), (.) for allx,y∈X, where ≤q<  and

M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

Next, we give the main result in this paper.

Theorem . Let(X,d)be a complete b-metric space(with constant s≥)such that the b-metric is a continuous functional on X×X,letα:X×X→[,∞)be a mapping,and let T:X→Pb,cl(X)be a q-set-valuedα-quasi-contraction.Suppose that the following con-ditions hold:

(i) Tisα-admissible;

(ii) there existx∈Xandx∈Txsuch thatα(x,x)≥;

(iii) if{xn}is a sequence inXsuch thatα(xn,xn+)≥for alln∈Nandxn→xas

n→ ∞for somex∈X,thenα(xn,x)≥.

If we set q<_s₊_s,then T has a ﬁxed point in X,that is,there exists u∈X such that u∈Tu.

Proof We obtain thatM(x,y) =  if and only ifx=yis a ﬁxed point ofT. Therefore, we suppose thatM(x,y) >  for allx,y∈X.

Next, we set

ε:=  



s₊_s–q

and β:=q+ε= 



s₊_s+q

.

(7)

Starting from (ii), we can choose x ∈X and x ∈Tx such that α(x,x)≥. By Lemma ., there existsx∈Txsuch that

d(x,x)≤H(Tx,Tx) +εM(x,x) ≤α(x,x)H(Tx,Tx) +εM(x,x)

≤qM(x,x) +εM(x,x)

=βM(x,x).

Sincex∈Xandx∈Txsuch that α(x,x)≥ and fromT isα-admissible, we have

α(x,x)≥. Using Lemma ., there existsx∈Txsuch that

d(x,x)≤H(Tx,Tx) +εM(x,x) ≤α(x,x)H(Tx,Tx) +εM(x,x) ≤qM(x,x) +εM(x,x)

=βM(x,x).

By induction, we can construct a sequence{xn}inXsuch that, for eachn∈N, we have

xn∈Txn–, α(xn–,xn)≥

and

d(xn,xn+)≤βM(xn–,xn). (.)

Assume thatxn–=xnfor somen∈N, then we havexn∈Txn, so the proof is completed.

For the rest, assume that for eachn∈N,d(xn–,xn) > . For eachn∈N, we get

d(xn,xn+)≤βM(xn–,xn)

=βmaxd(xn–,xn),d(xn–,Txn–),d(xn,Txn),d(xn–,Txn),d(xn,Txn–)

≤βmaxd(xn–,xn),d(xn–,xn),d(xn,xn+),d(xn–,xn+),d(xn,xn)

≤βmaxd(xn–,xn),d(xn,xn+),s

d(xn–,xn) +d(xn,xn+)

≤βsd(xn–,xn) +d(xn,xn+)

.

This implies that

d(xn,xn+)≤γd(xn–,xn) (.) for alln∈N, whereγ :=_–β_βs_s.

Sinces≥,β=_(_s₊_s+q) andq<_s₊_s, we have

(8)

By (.), (.) and Lemma ., the sequence{xn}is Cauchy in (X,d). It follows from the completeness ofXthat there existsu∈Xsuch that

lim

n→∞xn=u. (.)

Next, we show thatu∈Tu. By condition (iii), we haveα(xn,u)≥. From Lemma . and (.), forn∈N, we have

d(u,Tu)≤sd(u,xn+) +d(xn+,Tu)

≤sd(u,xn+) +H(Txn,Tu)

≤sd(u,xn+) +α(xn,u)H(Txn,Tu)

≤sd(u,xn+) +qM(xn,u)

=sd(u,xn+) +qmax

d(xn,u),d(xn,Txn),d(u,Tu),d(xn,Tu),d(u,Txn)

≤sd(u,xn+) +qmax

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

≤sd(u,xn+) +qmax

d(xn,u),sd(xn,u) +d(u,xn+)

,d(u,Tu),

sd(xn,u) +d(u,Tu)

,d(u,xn+)

.

Lettingn→ ∞in the above inequality, we get

d(u,Tu)≤qsd(u,Tu). (.)

Sinceq< _s₊_s, we getqs< . From (.), we haved(u,Tu) = . By Lemma ., we get

u∈Tu, that is,uis a ﬁxed point ofT. This completes the proof.

Theorem . Let(X,d)be a complete b-metric space(with constant s≥)such that the b-metric is a continuous functional on X×X,letα:X×X→[,∞)be a mapping,and let T:X→Pb,cl(X)be a q-set-valuedα-quasi-contraction.Suppose that the following con-ditions hold:

(i) Tisα_∗-admissible;

(ii) there existx∈Xandx∈Txsuch thatα(x,x)≥;

(iii) if{xn}is a sequence inXsuch thatα(xn,xn+)≥for alln∈Nandxn→xas

n→ ∞for somex∈X,thenα(xn,x)≥.

If we set q< 

s₊_s,then T has a ﬁxed point in X,that is,there exists u∈X such that u∈Tu.

Proof We can prove this result by using Theorem . and Remark ..

Corollary .(Theorem . in []) Let(X,d)be a complete b-metric space(with constant s≥)such that the b-metric is a continuous functional on X×X,and let T:X→Pb,cl(X)

be a q-set-valued quasi-contraction.If we set q<_s₊_s,then T has a ﬁxed point in X,that

is,there exists u∈X such that u∈Tu.

Proof Settingα(x,y) =  for allx,y∈X. By Theorem . (or Theorem .), we obtain the

(9)

Remark . The condition ofqin Theorem . becomesq< _ if we takes=  (it corre-sponds to the case of metric spaces). Therefore, Theorems . and . are generalization of many known results in metric spaces.

Remark . Theorem . is an extension of Theorem . of Aydiet al.[], which itself extends and improves the results of Amini-Harandi [], Daﬀer and Kaneko [], Rouhani and Moradi [] and Singhet al.[].

The following example shows that Theorem . properly generalizes the main result, Theorem ., of Aydiet al.[].

Example . LetX=Rwith the functionald:X×X→R+be deﬁned byd(x,y) :=|x– y|_{. Clearly, (}_X_,_d_{) is a complete}_b_{-metric space with constant}_s_{= . Deﬁne the set-valued}

mappingT:X→Pb,cl(X) by

Tx=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

[x,max{x, }], x> ,

[,x

], ≤x≤,

[min{x, –},x], x< ,

andα:X×X→[,∞) by

α(x,y) =

⎧ ⎨ ⎩

, x,y∈[, ],

, otherwise.

We obtain thatH(T,T) =  andM(, ) = . Therefore,

H(T,T) >qM(, )

for all ≤q< . This implies that the contraction condition of Theorem . of Aydiet al.

[] is not true for this case. Therefore, Theorem . cannot be used to claim the existence of a ﬁxed point ofT.

Next, we show that our result can be applied to this case. First of all, we show thatTis a

q-set-valuedα-quasi-contraction mapping, whereq=_. We need only to show the case ofx,y∈[, ] since the other case is trivial. Forx,y∈[, ], we have

α(x,y)H(Tx,Ty) = x –

y





= |x–y|



 =qd(x,y)

≤qM(x,y).

This shows thatTis aq-set-valuedα-quasi-contraction mapping. Also, we have

q=  <

 =



(10)

It is easy to check thatTis anα-admissible mapping. Forx=  andx= ∈Tx, we have

α(x,x)≥. Further, for any sequence{xn}inXwithxn→xasn→ ∞for somex∈X andα(xn,xn+)≥ for alln∈N, we obtain thatα(xn,x)≥ for alln∈N.

Therefore, all hypotheses of Theorem . are satisfied, and soThas infinitely many fixed points.

Next, we give the result for a single-valued mapping which is an extension of Corol-lary . of Aydiet al.[] and the result of Ćirić [].

Corollary . Let(X,d)be a complete b-metric space(with constant s≥)such that the b-metric is a continuous functional on X×X,letα:X×X→[,∞)be a mapping,and let t:X→X be a q-single-valuedα-quasi-contraction,that is,

α(x,y)d(tx,ty)≤qm(x,y), (.)

for all x,y∈X,where≤q< and

m(x,y) =maxd(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx).

Suppose that the following conditions hold:

(i) tisα-admissible;

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

(iii) if{xn}is a sequence inXsuch thatα(xn,xn+)≥for alln∈Nandxn→xas

n→ ∞for somex∈X,thenα(xn,x)≥.

If we set q<_s₊_s,then t has a ﬁxed point in X,that is,there exists u∈X such that u=tu.

Proof It follows by applying Theorem . or ..

4 Applications on ab-metric space endowed with an arbitrary binary relation In this section, we give the existence of ﬁxed point theorems on ab-metric space endowed with an arbitrary binary relation.

Before presenting our results, we give the following deﬁnitions.

Deﬁnition . Let (X,d) be ab-metric space andRbe a binary relation overX. We say thatT:X→Pb,cl(X) is a preserving mapping if for eachx∈Xandy∈TxwithxRy, we haveyRzfor allz∈Ty.

Deﬁnition . Let (X,d) be ab-metric space andRbe a binary relation over X. The set-valued mappingT:X→Pb,cl(X) is said to be aq-set-valued quasi-contraction with

respect toRif

H(Tx,Ty)≤qM(x,y) (.)

for allx,y∈Xfor whichxRy, where ≤q<  and

(11)

Theorem . Let(X,d)be a complete b-metric space(with constant s≥)such that the b-metric is a continuous functional on X×X,Rbe a binary relation over X and T:X→ Pb,cl(X)be a q-set-valued quasi-contraction with respect toR.Suppose that the following conditions hold:

(i) Tis a preserving mapping;

(ii) there existx∈Xandx∈Txsuch thatxRx;

(iii) if{xn}is a sequence inXsuch thatxnRxn+for alln∈Nandxn→xasn→ ∞for

somex∈X,thenxnRx.

Proof Consider the mappingα:X×X→[,∞) deﬁned by

α(x,y) =

⎧ ⎨ ⎩

 ifxRy,

 otherwise. (.)

From condition (ii), we getα(x,x)≥. It follows fromTis a preserving mapping thatT

is anα-admissible mapping. SinceT is aq-set-valued quasi-contraction with respect to

R, we have, for allx,y∈X,

α(x,y)H(Tx,Ty)≤qM(x,y). (.) This implies thatTis aq-set-valuedα-quasi-contraction mapping. Now all the hypotheses of Theorem . are satisﬁed and so the existence of the ﬁxed point ofT follows from

Theorem ..

Next, we give some special case of Theorem . in partially orderedb-metric spaces. Before studying next results, we give the following deﬁnitions.

Deﬁnition . LetXbe a nonempty set. Then (X,d,) is called a partially orderedb -metric space if (X,d) is ab-metric space and (X,) is a partially ordered space.

Deﬁnition . Let (X,d,) be a partially orderedb-metric space. We say thatT:X→ Pb,cl(X) is a preserving mapping withif for eachx∈Xandy∈Txwithxy, we have

yzfor allz∈Ty.

Deﬁnition . Let (X,d,) be a partially orderedb-metric space. The set-valued map-pingT:X→Pb,cl(X) is said to be aq-set-valued quasi-contraction with respect toif

H(Tx,Ty)≤qM(x,y), (.)

for allx,y∈Xfor whichxy, where ≤q<  and

M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

(12)

(i) Tis a preserving mapping with;

(ii) there existx∈Xandx∈Txsuch thatxx;

(iii) if{xn}is a sequence inXsuch thatxnxn+for alln∈Nandxn→xasn→ ∞for

somex∈X,thenxnx.

Proof The result follows from Theorem . by considering the binary relation.

Open question  In Theorems . and ., can we find the sufficient condition to prove the uniqueness of a fixed point for a set-valued mapping?

Open question  Is the ﬁxed point problem for aq-set-valuedα-contraction mapping well posed?

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center,}

Pathumthani, 12121, Thailand.2_{Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000,}

Thailand.3_{Division of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of}

Technology Lanna Tak, Tak, 63000, Thailand.

Acknowledgements

This research was supported by the Commission on Higher Education, the Thailand Research Fund and the Rajamangala University of Technology Lanna Tak (Grant No. MRG5580233).

Received: 7 June 2013 Accepted: 1 October 2013 Published:11 Nov 2013 References

1. Bakhtin, IA: The contraction mapping principle in quasimetric spaces. Funct. Anal., Unianowsk Gos. Ped. Inst.30, 26-37 (1989)

2. Czerwik, S: Contraction mappings inb-metric spaces. Acta Math. Inform. Univ. Ostrav.1, 5-11 (1993)

3. Abusalim, SM, Md Noorani, MS: Fixed point and common ﬁxed point theorems on ordered coneb-metric spaces. Abstr. Appl. Anal.2013, Article ID 815289 (2013)

4. Boriceanu, M: Fixed point theory for multivalued generalized contraction on a set with twob-metrics. Stud. Univ. Babe¸s-Bolyai, Math.LIV(3), 3-14 (2009)

5. Boriceanu, M, Bota, M, Petrusel, A: Multivalued fractals inb-metric spaces. Cent. Eur. J. Math.8(2), 367-377 (2010) 6. Czerwik, S, Dlutek, K, Singh, SL: Round-oﬀ stability of iteration procedures for operators inb-metric spaces. J. Natur.

Phys. Sci.11, 87-94 (1997)

7. Czerwik, S: Nonlinear set-valued contraction mappings inb-metric spaces. Atti Semin. Mat. Fis. Univ. Modena46(2), 263-276 (1998)

8. Hussain, N, Shah, MH, Harandi, AA, Akhtar, Z: Common ﬁxed point theorems for generalized contractive mappings with applications. Fixed Point Theory Appl.2013, 169 (2013)

9. Parvaneh, V, Roshan, JR, Radenovic, S: Existence of tripled coincidence point in orderedb-metric spaces and application to a system of integral equations. Fixed Point Theory Appl.2013, 130 (2013)

10. Aydi, H, Bota, M-F, Karapinar, E, Mitrovi´c, S: A ﬁxed point theorem for set-valued quasicontractions inb-metric spaces. Fixed Point Theory Appl.2012, 88 (2012)

11. Amini-Harandi, A: Fixed point theory for set-valued quasi-contraction maps in metric spaces. Appl. Math. Lett.24(11), 1791-1794 (2011)

12. Samet, B, Vetro, C, Vetro, P: Fixed point theorems forα-ψ-contractive type mappings. Nonlinear Anal.75, 2154-2165 (2012)

13. Mohammadi, B, Rezapour, S, Shahzad, N: Some results on ﬁxed points ofα-ψ-Ciric generalized multifunctions. Fixed Point Theory Appl.2013, 24 (2013)

14. Asl, JH, Rezapour, S, Shahzad, N: On ﬁxed points ofα-ψ-contractive multifunctions. Fixed Point Theory Appl.2012, 212 (2012)

15. Agarwal, RP, Sintunavarat, W, Kumam, P: PPF dependent ﬁxed point theorems for anαc-admissible non-self mapping

in the Razumikhin class. Fixed Point Theory Appl. (to appear)

(13)

17. Hussain, N, Karapinar, E, Salimi, P, Vetro, P: Fixed point results forGm_{-Meir-Keeler contractive and}_G_-(_α_,_ψ_{)-Meir-Keeler}

contractive mappings. Fixed Point Theory Appl.2013, 34 (2013)

18. Kutbi, MA, Sintunavarat, W: The existence of ﬁxed point theorems viaw-distance andα-admissible mappings and applications. Abstr. Appl. Anal.2013, Article ID 165434 (2013)

19. Hussain, N, Salimi, P, Latif, A: Fixed point results for single and set-valuedα-η-ψ-contractive mappings. Fixed Point Theory Appl.2013, 212 (2013)

20. Salimi, P, Latif, A, Hussain, N: Modiﬁedα-ψ-contractive mappings with applications. Fixed Point Theory Appl.2013, 151 (2013)

21. Singh, SL, Czerwik, S, Krol, K, Singh, A: Coincidences and ﬁxed points of hybrid contractions. Tamsui Oxford Univ. J. Math. Sci.24(4), 401-416 (2008)

22. Daﬀer, PZ, Kaneko, H: Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl.192, 655-666 (1995)

23. Djafari Rouhani, B, Moradi, S: Common ﬁxed point of multivalued generalized (ψ,φ)-weak contractive mappings. Fixed Point Theory Appl.2010, Article ID 708984 (2010)

24. ´Ciri´c, LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc.45, 267-273 (1974) 10.1186/1687-1812-2013-296