R E S E A R C H
Open Access
Common fixed point theorems for
generalized multivalued contractions on
cone metric spaces over a non-normal solid
cone
Dragan Ðori´c
**Correspondence:
Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca 154, Beograd, Serbia
Abstract
In this paper we define a new class of multivalued generalized contractions on cone metric spaces. Then, by using a necessary new technique, we prove two common fixed point theorems for a pair of those mappings on complete cone metric spaces over solid, not necessarily normal cone. Our main theorems are generalizations of the theorem of Wardowski (Appl. Math. Lett. 24:275-278, 2011) and many existing theorems in the literature. By using our main theorems, we can obtained some important corollaries which are generalizations of the well-known metric fixed point theorems to setting of cone metric spaces over a solid non-normal cone.
MSC: 47H10; 54H25
Keywords: multivalued mapping; common fixed point; multivalued contraction; generalized multivalued contraction; cone metric space; non-normal cone
1 Introduction and preliminaries
There exist many generalizations of the concept of metric spaces in the literature. Fixed point theory in abstract (cone) metric, or inK-metric spaces over a Banach space, was developed in the mid-s. Huang and Zhang [] reintroduced cone metric spaces and defined the convergence via interior points of the cone which determines an order onE. Although they considered and proved several fixed point theorems only in cone metric spaces over a normal cone, their approach enables the investigation of cone metric spaces over a cone which is not necessarily normal. It is well known that many fixed point results in the setting of cone metric spaces can be obtained from the corresponding results in metric spaces (see [–]). The results in the setting of cone metric spaces are appropriate only if the underlying cone is not necessarily normal (see []).
Definition . LetE be a topological vector space andPbe a subset ofE. The setPis called a cone if
(P) Pis closed, nonempty, andP={θ}, whereθ is the zero vector ofE;
(P) a,b∈R,a,b≥,x,y∈P ⇒ ax+by∈P;
(P) P∩(–P) ={θ}.
A conePis called solid [] ifintP=∅, whereintPis the interior ofP.
Each conePofEdetermines a partial order onEbyx yif and only ify–x∈Pfor eachx,y∈X. We writex≺yifx ybutx=y, whilexywill denote thaty–x∈intP. This relation is compatible with the vector structure.
Definition . LetPbe a cone in a real Banach spaceE. The conePis called normal, if there exists a constantK> such that, for allx,y∈E,
θ x y implies x ≤Ky,
or, equivalently, if
xnynzn and lim
n→∞xn=nlim→∞zn=x, then nlim→∞yn=x. (.)
The least positive numberKsatisfying the above inequality is called the normal constant ofP.
The following example shows that there are non-normal cones.
Example . LetE=C
R([, ]) with the normf=f∞+f∞and consider the cone
P={f ∈E:f(t)≥}. For eachn≥, putf(x) =xandg(x) =xn. Thenθ g f,f=
andg= n+ . Since for eachK> there existsn∈Nsuch that n+ > K, we have g f, butgKffor anyK> . Therefore, the conePis non-normal.
Definition .([, ]) LetEbe a Banach space andθ be the zero vector ofE. LetPbe a cone inEwithint(P)=∅and let be a partial ordering with respect toP. A mapping d:X×X→Eis called a cone metric on the nonempty setXif the following axioms are satisfied:
(d) θ d(x,y)for allx,y∈Xandd(x,y) =θif and only ifx=y; (d) d(x,y) =d(y,x)for allx,y∈X;
(d) d(x,y) d(x,z) +d(z,y)for allx,y,z∈X.
The pair (X,d), whereXis a nonempty set anddis a cone metric, is called a cone metric space.
Example . LetE=R,P={(x,y)∈E:x,y≥},X=Randd:X×X→Edefined by
d(x,y) = (|x–y|,c|x–y|), wherec≥ is a constant. Then (X,d) is a cone metric space with the normal coneP, where the normal constantK= .
Definition .(Huang and Zhang []) Let (X,d) be a cone metric space. We say that a sequence{xn}inXis
(i) a convergent sequence if, for everycinEwithθc, there is anNsuch that
d(xn,x)cfor alln>Nand for some fixedxinX;
(ii) a Cauchy sequence if, for everycinEwithθc, there is anNsuch that
d(xn,xm)cfor alln,m>N.
In the following lemma we suppose that E is a Banach space, P is a cone inE with
int(P)=∅, without the assumption of normality of coneP.
Lemma . ([, ]) Let(X,d)be a cone metric space.Then the following properties are often used(particularly when dealing with cone metric spaces in which the cone need not be normal).
(p) Ifu vandvw,thenuw.
(p) Ifθ ucfor eachc∈intP,thenu=θ.
(p) Ifa b+cfor eachc∈intP,thena b.
(p) Ifθ x yanda≥,then ax ay.
(p) IfEis a real Banach space with a conePand ifa λa,wherea∈Pand <λ< ,then
a=θ.
(p) Ifc∈intP,θ anandan→θ,then there existsnsuch that,for alln>n,we have
anc.
From (p) it follows that the sequence{xn}converges tox∈Xifd(xn,x)→θ asn→ ∞
and{xn}is a Cauchy sequence ifd(xn,xm)→θ asn,m→ ∞. In the situation with a
non-normal cone we have only one part of Lemmas and from []. Also, in this case from xn→xandyn→yit need not follow thatd(xn,yn)→d(x,y), as well as fromxnynzn
andlimn→∞xn=limn→∞zn=xit need not follow thatlimn→∞yn=x.
Example . LetE=CR([, ]),P⊂Eand the norm · be as in Example .. Consider the sequencesxn(t) =tn/nandyn(t) = /n. Then xn≺ynandlimn→∞yn= , but
xn=max t∈[,]
tnn+max
t∈[,]
tn–= n+ > .
Therefore,{xn}does not converge to , although ≤xn(t) < /n. Thus it follows by (.)
thatPis a non-normal cone.
The study of fixed points of multivalued mappings satisfying certain contractive con-ditions has many applications and studied by many researchers (see [–]). An element x∈Xis said to be a fixed point of a multivalued mapT:X→Xifx∈Tx. Recently many
authors proved fixed point theorems for multivalued mappings on complete cone metric spaces assuming that the corresponding cone is regular or normal (see [–]). For a cone metric space (X,d) letA˜ be a family of subsets ofX. Wardowski [, Definition .] introduced a new cone metricH:A˜ × ˜A→E. Then he introduced the concept of set-valued contraction of Nadler type [] and proved a fixed point theorem by assumption that a conePofEis solid and normal. But, as noted in [], most of the fixed points results in cone metric spaces over a normal cone can be obtained as a consequences from the corresponding results in metric spaces. Very recently Arshad and Ahmad [] modified Wardowski’s [] idea ofH-cone metric. They introduced the following notion ofH-cone metric.
(H) θH(A,B)for allA,B∈ ˜AandH(A,B) =θ if and only ifA=B;
(H) H(A,B) =H(B,A)for allA,B∈ ˜A;
(H) H(A,B)H(A,C) +H(C,B)for allA,B,C∈ ˜A;
(H) IfA,B∈ ˜A,θ ≺∈EwithH(A,B)≺, then for eacha∈Athere existsb∈Bsuch
thatd(a,b)≺.
Example . Let (X,d) be a metric space and letA˜ be a family of all nonempty, closed, bounded subsets ofX. Then the mappingH:A˜× ˜A→R+given by the formula
H(A,B) =maxsup
x∈A
d(x,B),sup
y∈B
d(y,A), A,B∈ ˜A, (.)
which is called a Hausdorff metric induced by the metricd, is anH-cone metric induced byd.
Arshad and Ahmad [] extended the theorem of Wardowski [] to a complete cone metric space without the assumption that a conePis normal. They proved the following theorem.
Theorem .(Arshad and Ahmad []) Let(X,d)be a complete cone metric space.Let ˜
A be a collection of nonempty,closed,and bounded subsets of X and let H:A˜× ˜A→E be an H-cone metric induced by d.If for a map T:X→ ˜A there existsλ∈(, )such that,for all x,y∈X,
H(Tx,Ty)≤λ·d(x,y), (.)
then T has a fixed point.
Clearly, Theorem . is a generalization of the classical theorem of multivalued contrac-tive mappings (Nadler []). Recall that some of the initial generalizations of the theorem of Nadler are given in [] and in []. In Ćirić in [] introduced the concept of a generalized single-valued contraction, and then in in [] he used the following concept of a generalized multivalued contraction.
Definition .(Ćirić []) Let (X,d) be a metric space and letA˜be a family of nonempty, closed, and bounded subsets ofX. A mappingT:X→ ˜Ais said to be a generalized mul-tivalued contraction if and only if there existsλ∈[, ) such that, for allx,y∈X,
H(Tx,Ty)≤λ·max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
, (.)
whereH(A,B) forA,B∈ ˜Ais the Hausdorff metric (.) induced by metricd.
2 Main results
Inspired by Definition . of Ćirić we shall introduce the notion of the cone generalized multivalued contraction.
Definition . LetEbe a Banach space and let (X,d) be a cone metric space overE. Let ˜
Abe a family of nonempty, closed, and bounded subsets ofXand let there exists an H-cone metric H:A˜ × ˜A→E induced byd. A mappingT :X→ ˜Ais said to be a cone generalized multivalued contraction if and only if there existsλ∈[, ) such that, for all x,y∈X, a mappingT satisfies one of the following contractive conditions:
(D) H(Tx,Ty)λ·d(x,y);
(D) H(Tx,Ty)λ·d(x,u)for each fixedu∈Tx; (D) H(Tx,Ty)λ·d(y,v)for each fixedv∈Ty;
(D) H(Tx,Ty)λ·d(x,v)+d(y,u)for each fixedv∈Tyand each fixedu∈Tx.
It is easy to show that the generalized multivalued contraction defined in Definition . is an example of the cone generalized multivalued contraction defined in Definition ..
Example . LetX=Rand (X,d) be the usual metric space ordered by a usual
order-ing≤. LetA˜ be a family of all nonempty, closed, bounded subsets ofXandH:A˜× ˜A→E be a Hausdorff metric. Suppose that a mappingT:X→ ˜Ais a generalized multivalued contraction defined in Definition .. If we setE=R,θ = ,P={x∈E:x≥}=R+and forx,y∈E, we definex yif and only ifx≤y, then (X,d) is a cone metric space over cone PandT:X→ ˜Ais a cone generalized multivalued contraction.
Now we prove our main theorem.
Theorem . Let E be a Banach space,let P be a solid not necessarily normal cone of E and let(X,d)be a cone metric space over E.LetA be a family of nonempty,˜ closed,and bounded subsets of X and let there exists an H-cone metric H:A˜× ˜A→E induced by d. Suppose that T,S:X→ ˜A are two cone multivalued mappings and suppose that there is λ∈(, )such that,for all x,y∈X,at least one of the following conditions holds:
(C) H(Tx,Sy)λ·d(x,y);
(C) H(Tx,Sy)λ·d(x,u)for each fixedu∈Tx; (C) H(Tx,Sy)λ·d(y,v)for each fixedv∈Sy;
(C) H(Tx,Sy)λ·d(x,v)+d(y,u)for each fixedv∈Sy,u∈Tx.
Then T and S have a common fixed point.
Proof Letx∈Xandx∈Sxbe arbitrary. Consider the elementH(Tx,Sx)∈E. If each
right hand side of (C), (C), (C), and (C) withx=xandy=xisθinE, thend(x,x) =θ
and hence from the property (d) of the metricdit followsx=x. This andx∈Sximply
x∈Sx. Further,d(x,u) =d(x,u) =θfor each fixedu∈Tximpliesx=u∈Tx=Tx.
Hencex∈Tx. Therefore, in this casexis a common fixed point ofSandT and proof
is done.
Consider now the elementH(Tx,Sx)∈Ein the case that, in the one of the
=H(Tx,Sx) +λe. Then fromH(Tx,Sx)≺and from the property (H) of theH-cone
metric in Definition . we find, asx∈Sx, that there existsx∈Txsuch that
d(x,x)≺=H(Tx,Sx) +λe.
Consider now the element H(Sx,Tx). Clearly,H(Sx,Tx)≺H(Sx,Tx) +λe. Again
from (H) with=H(Sx,Tx) +λe, asx∈Tx, there existsx∈Sxsuch that
d(x,x)≺H(Sx,Tx) +λe.
Continuing this process we can construct a sequence{xn} inX such thatxn+ ∈Sxn,
xn+∈Txn+and
d(xn+,xn)≺H(Sxn,Txn–) +λne, (.)
d(xn+,xn+)≺H(Txn+,Sxn) +λn+e. (.)
According to the inequality (.) and the inequalities (C), (C), (C), and (C) with x=xn+andy=xn, we have to consider four cases.
() IfH(Txn+,Sxn) λ·d(xn+,xn), then from (.) we have
d(xn+,xn+)≺λ·d(n+,xn) +λn+e. (.)
() IfH(Txn+,Sxn) λ·d(xn+,u) for anyu∈Txn+, then we can takeu=xn+∈
Txn+. So, we obtainH(Txn+,Sxn) λ·d(xn+,xn+) and from (.) we get
d(xn+,xn+)≺λ·d(xn+,xn+) +λn+e.
Hence
d(xn+,xn+)λn+( –λ)–e. (.)
() IfH(Txn+,Sxn) λ·d(xn,v) for anyv∈Sxn, then we may takev=xn+∈Sxn
and we obtainH(Txn+,Sxn) λ·d(xn,xn+). Then from (.) we again have (.).
() IfH(Txn+,Sxn) λ·d(xn+,v)+d(xn,u) for anyv∈Sxnandu∈Txn+, then we may
takev=xn+∈Sxn,u=xn+∈Txn+. So we obtain
H(Txn+,Sxn) λ·
d(xn,xn+)
.
Then from (.) and by the triangle inequality we have
d(xn+,xn+)≺λ·
d(xn,xn+) +d(xn+,xn+)
+λ
n+e,
which implies that
d(xn+,xn+)≺
λ
–λ·d(xn,xn+) + λn+
Sinceλ< , we have /( –λ) < , and from (.) we obtain
d(xn+,xn+)≺λ·d(xn,xn+) +λn+e. (.)
It is easy see that from (.), (.), and (.) we get
d(xn+,xn+)≺λ·d(xn+,xn) +λn+( –λ)–e. (.)
Using similar arguments to (.) we can prove that
d(xn+,xn)≺λ·d(xn,xn–) +λn( –λ)–e. (.)
From (.) and (.) we conclude that
d(xn+,xn)λ·d(xn,xn–) +λn( –λ)–e (.)
for alln≥. From (.) we get
d(xn+,xn)λ
λ·d(xn–,xn–) +λn–( –λ)–e
+λn( –λ)–e =λ·d(xn–,xn–) + λn( –λ)–e.
Using mathematical induction it is easy to prove that
d(xn+,xn)λn·d(x,x) +nλn( –λ)–e. (.)
By the triangle inequality and (.) for anym>nwe have
d(xn,xm)d(xn,xn+) +d(xn+,xn+) +· · ·+d(xm–,xm)
λn·d(x,x) +nλn( –λ)–e
+λn+·d(x,x) + (n+ )λn+( –λ)–e
+· · ·
+λm–·d(x,x) + (m– )λm–( –λ)–e
.
Hence we get
d(xn,xm) λn+λn++· · ·+λm–
d(x,x) +Rn(λ)( –λ)–e
λn
–λ·d(x,x) +Rn(λ)( –λ)
–e, (.)
where Rn(λ) is the remainder of the convergent series
∞
n=n·λn. Since λn→ and
Rn(λ)→ asn→ ∞, we get
Letc∈Ewithθcbe arbitrary. From (.) and (p) in Lemma . it follows that we can
choose a natural numbernsuch that
λn( –λ)–·d(x,x) +Rn(λ)( –λ)–ec
for alln≥n. Thus, by (.),d(xn,xm)cfor allm>n≥n. Therefore, by (ii) in
Defini-tion ., we conclude that{xn}is Cauchy sequence. SinceXis complete, there existsz∈X
such thatlimn→∞d(xn,z) =θ.
Now we shall show thatzis a common fixed point ofT andS. Since
H(Txn+,Sz)≺H(Txn+,Sz) +λn+e,
from the property (H) of theH-cone metric in Definition . we see, asxn+∈Txn+,
that there existsyn+∈Szsuch that
d(xn+,yn+)≺H(Txn+,Sz) +λn+e. (.)
According to (.) and the inequalities (C), (C), (C), and (C) with x=xn+ and
y=zwe have to consider four cases.
() IfH(Txn+,Sz) λ·d(xn+,z), then from (.) we have
d(xn+,yn+)≺λ·d(xn+,z) +λn+e. (.)
() IfH(Txn+,Sz) λ·d(xn+,u) for any fixedu∈Txn+, then we can takeu=xn+∈
Txn+. Thus from (.) we get
d(xn+,yn+)≺λ·d(xn+,xn+) +λn+e. (.)
() IfH(Txn+,Sz) λ·d(z,v) for any fixedv∈Sz, then we can takev=yn+∈Sz. Thus
from (.) and by the triangle inequality we get
d(xn+,yn+)≺λ·d(z,xn+) +λ·d(xn+,yn+) +λn+e,
which implies that
d(xn+,yn+)≺
λ
–λ·d(z,xn+) + λn+
–λe. (.)
() IfH(Txn+,Sz) λ·d(xn+,v)+d(z,u) for anyv∈Szandu∈Txn+, then we can take
v=yn+∈Szandu=xn+∈Txn+. Thus from (.) and by the triangle inequality we
get
d(xn+,yn+)≺λ·
d(xn+,xn+) +d(xn+,yn+) +d(z,xn+)
+λ
n+e,
which implies that
d(xn+,yn+)≺λ·
d(xn+,xn+) +d(z,xn+)
–λ +
Since /( –λ) < forλ∈(, ), we have
d(xn+,yn+)≺λ·d(xn+,xn+) +λ·d(z,xn+) +λn+e. (.)
Thus, from (.), (.), (.), and (.) we have
d(xn+,yn+)≺λ
d(xn+,z) +d(xn+,xn+) +d(z,xn+)( –λ)–
+λn+( –λ)–e. (.)
By the triangle inequality and (.) we get
d(z,yn+)d(z,xn+) +d(xn+,yn+)
≺d(z,xn+) +λn+( –λ)–e
+ λ –λ·
d(xn+,z) +d(z,xn+) +d(xn+,xn+)
. (.)
Since{xn}converges tozand sinceλn+→ and by (.)d(xn+,xn+)→θasn→ ∞,
the right hand side of the inequality (.) converges toθasn→ ∞. Therefore, from (p)
in Lemma . and (.) we can choose a natural numbernsuch thatd(z,yn+)cfor all
n≥n, wherec∈Ewithθcis arbitrary. By (i) in Definition . we conclude that{yn}
converges toz. Sinceyn+∈SzandSzis closed, we getz∈Sz.
Analogously, we can get z∈Tz. So, we proved thatz is a common fixed point ofT
andS.
If we takeS=T in Theorem ., then we obtain the following fixed point theorem in complete non-normal cone metric spaces.
Theorem . Let(X,d)be a complete cone metric space over a solid non-normal cone and letA be a family of nonempty,˜ closed,and bounded subsets of X.Suppose that there exists an H-cone metric H:A˜× ˜A→E induced by d and suppose that T:X→ ˜A is a cone generalized multivalued contraction.Then T has a fixed point.
From Theorem . we can obtain Theorem . of Arshad and Ahmad [] and Theo-rem . of Wardowski [].
Now we shall present an example where Theorem . can be applied, but the theorem of Arshad and Ahmad [] (Theorem .) and the theorem of Wardowski [] cannot be applied.
Example . LetX= [, ] and letE=CR([, ]),P⊂Eand the norm · be as in Exam-ple .. Defined:X×X→Eby
d(x,y)(t) =|x–y| ·et,
where ≤t≤. Thendis a cone metric onX. LetA˜ be a family of all nonempty, closed, bounded subsets ofXand let the mappingT:X→ ˜Abe defined by
T(x) = ⎧ ⎨ ⎩
LetH:A˜× ˜A→Ebe defined by
H(A,B)(t) =H(A,B)·et forA,B∈ ˜A,
whereHis the usual Hausdorff metric onXinduced by the metricd(x,y) =|x–y|. Now we can show thatT satisfies all conditions of Theorem . withλ=
.
() If≤x,y< andx=y, then we have
H T(x),T(y)(t) =H ,x , ,y ·et=x
– y ·et<
|x–y| ·e
t=λd(x,y)(t).
ThusTsatisfies the contractive condition (D) in Definition . withλ= /. () If≤x< andy= , then we have
H T(x),T()(t) < –
·et=
·e
t<
· e
t≤λd(,u)(t)
for allu∈[, /] =T(). Therefore, in this caseT satisfies the contractive condition (D) in Definition . withλ= /.
From () and () we see that the mappingTsatisfies all of the conditions of Theorem . and has a fixed pointx= .
Now we shall show that in this example the theorems of Arshad and Ahmad [] and Wardowski [], as well as other theorems known in the literature, cannot be applied. Let y= and <x< . Then
H T(x),T()(t) =x –
·et>
–
·et> –
·et>| –x| ·et=d(x, )(t).
Clearly, there does not existλ< such thatH(T(x),T())(t)≤λ·d(x, )(t). Therefore, The-orem . of Arshad and Ahmad [] (TheThe-orem .) and TheThe-orem . of Wardowski [] cannot be applied in this example.
In a coneP of an ordered Hausdorff topological vector space (E,P), from a,b∈P it does not need to follow that (/)(a+b) anor (/)(a+b) b. Thus in addition to the conditions (C)-(C) of Theorem . we can consider the condition
(C) H(Tx,Sy)λ·d(x,u)+d(y,v) for each fixedu∈Txandv∈Sy.
The following theorem is a generalization of Theorem ..
Theorem . Let(X,d)be a complete cone metric space over a solid non-normal cone,let ˜
A be a family of non-empty,closed,and bounded subsets of X and let there exists an H-cone metric H:A˜× ˜A→E induced by d.Suppose that T,S:X→ ˜A are two cone multivalued mappings and suppose that there isλ∈(, )such that,for all x,y∈X,at least one of the conditions(C)-(C)holds.Then T and S have a common fixed point.
[] and Ćirić [] to non-normal cone metric spaces. For example, the following corol-lary is a cone multivalued version of Kannan’s fixed point theorem, and it easily follows from Theorem ..
Corollary . Let(X,d)be a complete cone metric space over a solid non-normal cone,let ˜
A be a family of non-empty,closed,and bounded subsets of X and let there exists an H-cone metric H:A˜× ˜A→E induced by d.Suppose that T,S:X→ ˜A are two cone multivalued mappings and suppose that there isγ∈(, /)such that,for all x,y∈X,the mappings T and S satisfy the condition
H(Tx,Sy)γ d(x,u) +d(y,v)
for each u∈Tx and for each v∈Sy.Then T and S have a common fixed point.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author is thankful to the Ministry of Education, Science and Technological Development of the Republic of Serbia. The author thanks the editor and the referees for their valuable comments and suggestions that helped to improve the text.
Received: 4 April 2014 Accepted: 5 June 2014 Published:22 Jul 2014
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