R E S E A R C H
Open Access
Fixed-point theorems for mappings satisfying
the ordered contractive condition on
noncommutative spaces
Qiaoling Xin and Lining Jiang
**Correspondence:
[email protected] School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China
Abstract
In the paper, we introduce noncommutative Banach spaces which generalize the concept of Banach spaces, and thek-ordered contractive condition; we then discuss an ordered structure and several properties on noncommutative Banach spaces. Moreover, some fixed-point theorems for mappings with thek-ordered contractive condition on noncommutative Banach spaces are presented. In addition, we investigate the existence and uniqueness of fixed points for an integral equation of Fredholm type.
MSC: 47H10
Keywords: noncommutative Banach space; ordered contractive condition; fixed-point theorem
1 Introduction
The well-known fixed-point theorem of Banach [] is a very important tool for solving existence problems in many branches of mathematics and physics. There are a large num-ber of generalizations of the Banach contraction principle in the literature (see [–] and others). The theorem has been generalized in two directions. On the one side, the usual contractive condition is replaced by weakly contractive conditions. On the other side, the action spaces are replaced by metric spaces endowed with an ordered or partially ordered structure. In particular, there is much interest in obtaining the existence and uniqueness of fixed points for self-maps by altering the action spaces. In this direction, Dhageet al.[] addressed a new category of fixed-point problems for a self-map with the help of ordered Banach spaces. Further improvements in those spaces were found in []. In recent years, Ran and Reurings [], O’Regan and Petruşel [] and others started the investigations concerning a fixed-point theory in ordered metric spaces. Later, many authors followed this concept by introducing and investigating the different types of contractive mappings,
e.g., in [] Caballeroet al.considered contractive-like mappings in ordered metric spaces and applied their results in ordinary differential equations. Some interesting fixed-point theorems concerning partially ordered metric spaces can also be found in [, ].
The results obtained by Huang and Zhang [] have become of interest for many scholars. They reconsidered the Banach contraction principle by initiating a new concept of cone metric spaces. Recently, also, the existence of fixed points for the given contractive type mappings in partially ordered cone metric spaces was investigated (see [, ]).
The purpose of this paper is to present some fixed-point theorems for mappings satis-fying the ordered contractive condition in the context of noncommutative metric spaces which are noncommutative sense of those in [].
The paper is organized as follows: we firstly introduce a noncommutative Banach space and thek-ordered contractive condition, and then discuss the ordered structure and sev-eral properties on noncommutative Banach spaces. Moreover, some fixed-point theorems on this space are established. Finally, we investigate the existence and uniqueness of fixed points for integral equation of Fredholm type.
Throughout this paper, the letters R, R+, Nwill denote the sets of all real numbers, nonnegative real numbers and natural numbers, respectively.
To begin with, we introduce some definitions and properties which will be used later.
Definition . LetEbe a group with a uniteand suppose that there exists a metricdon
E such that (E,d) is a complete metric space.Eis said to be a noncommutative Banach space if the following conditions hold:
() for anyx,y,z∈E, we haved(xz,yz) =d(x,y); () there exists a binary continuous operation
F:R×E→E, (α,x)→xα
such thatF(–,x) =x–is exactly the inverse ofxin the groupEandF(,x) =x=e
is the unit in the groupE, and that
F(mn,x) =Fm,F(n,x), F(m+n,x) =F(m,x)F(n,x)
form,n∈R,x∈E;
() for anyx∈E, there exists a constantMx> such that
dxα,e≤Mx|α|, ∀α∈R.
In particular, if there exists a constantM> such thatd(xα,e)≤M|α|forx∈E,α∈R, thenEis said to be uniformly bounded.
LetEbe a uniformly bounded noncommutative Banach space. Takingα= , we conclude thatd(x,e)≤M, which together with the triangular inequality yieldsd(x,y)≤M. This showsEis bounded.
Example . All Banach spaces are noncommutative Banach spaces. Let (X, · ) be a Banach space, then (X, +) is a group with a unitθ, and there exists a metricdinduced by the norm · such that (X,d) is a complete metric space. Firstly, the metricdsatisfies
d(x+z,y+z) =d(x,y) forx,y,z∈X. Secondly, there exists a binary continuous mapping
F:R×X→X, (k,x)→kxsatisfying the condition () in Definition .. Finally, for any
Example . LetRnbe the standardn-dimensional vector space overR. Define
d(x,y) =
n
i= i
|xi–yi| +|xi–yi|
forx= (x,x, . . . ,xn),y= (y,y, . . . ,yn)∈Rn. Clearly, Rn is a complete metric space. In
order to verify thatRnis a noncommutative Banach space. It suffices to show that for any x= (x,x, . . . ,xn)∈Rn, there exists a constantMx> such thatd(kx,θ)≤Mx|k|, fork∈R,
where θ is unit inRn. Indeed, chooseMx={,|x
|,|x|, . . . ,|xn|}, and we getd(kx,θ)≤
Mx|k|.
Example . Suppose thatHis a Hilbert space andU(H) is the unitary group ofH. Put
d(S,T) =ST––I=S–T ∀S,T∈U(H),
thenU(H) is a complete metric space as a subset ofB(H), whereB(H) denotes the algebra of all bounded linear operators onH. Moreover, for anyT∈U(H) andα∈R, set
Tα= π
eiαθdE θ,
whereEθmeans the spectral measure associated with the operatorT. Notice that
dTα,I= π
eiαθ– dEθ ≤ sup
θ∈[,π]
eiαθ– ≤π|α|
forα∈R. ThenU(H) is a uniformly bounded noncommutative Banach space.
Definition . LetEbe a noncommutative Banach space.Pis a subset ofEsatisfying the
following conditions:
() Pis nonempty, closed, andP={e}; () x,y∈Pandα,β∈R+impliesxαyβ∈P;
() P∩P–={e}whereP–={x–:x∈P}. ThenPis called a cone inE.
Given a conePin a noncommutative Banach spaceE, a relation can be introduced as follows:
xy ⇐⇒ yβx–β∈P for allβ∈[, ].
One can show that ‘’ is a partial ordering inEwith respect toP. In fact:
(i) Forx∈E,xβx–β=xβ–β=x=e∈Pfor allβ∈[, ]. This implies thatxx.
(ii) Ifxyandyx, thenyβx–β∈Pand(yβx–β)–=xβy–β∈Pfor allβ∈[, ]. By
P∩P–={e}, we getyβ=xβ, which implies thaty=x.
(iii) Ifxyandyz, thenyβx–β∈Pandzβy–β∈Pfor allβ∈[, ], which together
Definition . A coneP⊆Eis called normal if there is a numberN> such that
exy ⇒ d(x,e)≤Nd(y,e), ∀x,y∈E.
The least positive numberNsatisfying the above is called the normal constant ofP. It is clear thatN≥.
Remark . Letx∈P,α∈R, then the following relation holds:
⎧ ⎨ ⎩
xxα, α≥,
xαx, α< .
Indeed, for anyβ∈[, ], ifα≥, then (xα)βx–β=x(α–)β∈PsincePis a cone, thus we can getxxα; ifα< , thenxβ(xα)–β=x(–α)β∈P, which meansxαx.
Definition . Foru,v∈E, if eitheruvorvuholds, we say thatuandvare compa-rable, denoted
∨(u,v) = ⎧ ⎨ ⎩
u, vu,
v, uv.
From the above definitions, we have the following properties.
Lemma . Suppose that P is a cone in E.For u,v∈E,we have:
() Setuv,thenuαvαholds for any≤α≤.
() Ifuandvare comparable,thenuv–andvu–are comparable,and furthermore
e∨(uv–,vu–).
() Ifuandvare comparable,thend(∨(uv–,vu–),e) =d(u,v).
() (compatibility)Let{un},{vn} ⊂E,unandvnbe comparable for alln∈N.Ifun→u,
vn→v,thenuandvare comparable.
Proof () Letuv, we havevβu–β∈Pfor allβ∈[, ]. Sinceαβ∈[, ] for anyα∈[, ], we seevαβu–αβ∈P, which implies thatuαvα.
() Without loss of generality, one can suppose thatuv, which meansvu–∈P. Using Remark ., one can seeuv–vu–. Furthermore (vu–)βe–β= (vu–)β∈Pfor allβ∈[, ], which impliesevu–, and thereforee∨(uv–,vu–).
() Assume that uv, then∨(uv–,vu–) =vu–. It follows immediately from Defini-tion . that
d∨uv–,vu–,e=dvu–,e=d(v,u) =d(u,v).
() Sinceunandvnare comparable for alln, we can suppose that there exist two subse-quencesunk andvnk such thatunkvnkfor allk. Note that for allβ∈[, ]
dvβnku–nkβ,vβu–β≤dvβnkun–kβ,vβu–nkβ+dvβu–nkβ,vβu–β
It follows directly that
dvβnku–nkβ,vβu –β
→.
AsPis closed, one obtainsvβu–β∈P. This says thatuv.
2 Fixed-point theorems on noncommutative Banach spaces
From now on, we always suppose thatEis a noncommutative Banach space with a partial orderinginduced by a normal conePwith the normal constantN. And some fixed-point theorems for mappings onE satisfying the ordered contractive condition will be presented. Let us begin with the following theorem.
Theorem . Let A:E→E be a continuous mapping and suppose that the following two assertions hold.
() There exists a constantk∈(, )such that for allu,v∈E,ifuandvare comparable,
thenAuandAvare comparable and furthermore
∨Av(Au)–,Au(Av)–∨vu–,uv–k.
In this case,we sayAsatisfies thek-ordered contractive condition. () There existsx∈Esuch thatxandAxare comparable.
Then A has a fixed point x∗which is unique in the comparable sense.Namely,if Ay∗=y∗ and y∗and x∗are comparable,then y∗=x∗.Moreover,
dx∗,x
≤
–kNM(Ax)x–.
Proof Define a sequence{xn}n∈N⊆Eby the formulaxn=Anx,n∈N. The proof can be divided into three steps.
Step I.{xn}is a Cauchy sequence.
Now, sincexandAx=x are comparable, andAsatisfies thek-ordered contractive condition, we obtainxandx,xandx, . . . ,xnandxn+are comparable. Notice
∨xnx–n+,xn+x–n
∨xn–x–n,xnx–n– k
.
Inductively the following holds:
∨xx– ,xx–
∨xx– ,xx– k
.
By the above for alln∈Nand Lemma .(), we obtain
e∨xn+x–n,xnx–n+
∨xnx–n–,xn–x–n
k
· · ·∨xx–,xx– kn
.
From the above it is easy to conclude that
d∨xn+x–n,xnx–n+
,e≤Nd∨xx–,xx– kn
which together with Definition . yields
d∨xn+x–n,xnx–n+
,e≤NMxx– k
n.
Finally, by Lemma .(), this implies
d(xn,xn+)≤NMxx–kn→,
which shows{xn}is a Cauchy sequence. The completeness ofEimplies that there exists
x∗∈Esuch thatlimn→∞xn=x∗. Step II.x∗is a fixed point ofA.
Suppose thatlimn→∞xn=x∗, which together withAis continuous, we get
Ax∗=A lim
n→∞xn=nlim→∞Axn=x ∗,
which showsx∗is a fixed point ofA.
Step III. The uniqueness of the fixed point ofAin the comparable sense.
Let us considerAy∗=y∗, and letx∗andy∗be comparable. Without loss of generality, set
y∗x∗, which showsx∗y∗–∈P. By the condition (),
∨Ay∗Ax∗–,Ax∗Ay∗–∨y∗x∗–,x∗y∗–k.
That is
∨y∗x∗–,x∗y∗–∨y∗x∗–,x∗y∗–k.
In addition, we know ∨(y∗x∗–,x∗y∗–) =x∗y∗–. Thenx∗y∗–(x∗y∗–)k, which implies
(x∗y∗–)k–∈P. On the other hand, we have (x∗y∗–)–k∈P. Now, from the definition of a
cone, we havex∗y∗–=e, and theny∗=x∗. Furthermore,
dx∗,x
≤ ∞
n=
d(xn,xn+)≤
∞
n=
NM(Ax)x– k
n=
–kNM(Ax)x–.
Remark . Assume in addition thatEsatisfies the condition:
() Foru,v∈E, if they are not comparable, then there existsw∈Esuch thatuandw,v andware comparable, respectively.
ThenAhas a unique fixed point. Note that condition () is always valid ifEis a lattice. It suffices to show the uniqueness of the fixed point ofA. Suppose thatx∗,y∗are fixed points ofA. Claim thatx∗andy∗are comparable. If not, there existsz∈Esuch thatx∗and
z,y∗andzare comparable, respectively. SinceAsatisfies thek-ordered contractive condi-tion, thenAn+x∗andAn+z,An+y∗andAn+zare comparable for anyn∈N, respectively. Also,
and
e∨An+y∗An+z–,An+zAn+y∗–∨y∗z–,zy∗–kn.
Then
dAn+x∗,An+z=d∨An+x∗An+z–,An+zAn+x∗–,e
≤Nd∨x∗z–,zx∗–kn,e ≤NknM
x∗z–,
and similarly
dAn+y∗,An+z≤NknMy∗z–.
The triangular inequality tells us that
dx∗,y∗=dAn+x∗,An+y∗
≤dAn+x∗,An+z+dAn+y∗,An+z ≤Nkn(Mx∗z–+My∗z–)→,
asn→ ∞, which impliesx∗=y∗. This is a contradiction. Therefore,x∗andy∗are compa-rable. By Theorem .,x∗=y∗.
Corollary . Let E be a uniformly bounded noncommutative Banach space,P a normal cone with the normal constant N.For c∈R+,x
∈E,set B(x,c) ={x∈E:d(x,x)≤c}.
Suppose that a continuous mapping A:E→E satisfies the k-ordered contractive condition and d(Ax,x)≤c–kNM.Also,xand x∈B(x,c)are comparable.Then there exists a
unique fixed point in B(x,c)in the comparable sense.
Proof It suffices to show thatAx∈B(x,c) for anyx∈B(x,c). For anyx∈B(x,c), the triangular inequality gives
d(x,Ax)≤d(x,Ax) +d(Ax,Ax).
By Lemma .(),
d(Ax,Ax) =d
∨Ax(Ax)–,Ax(Ax)–
,e.
SinceAsatisfies thek-ordered contractive condition, we have
∨Ax(Ax)–,Ax(Ax)–
∨xx–,xx– k
,
and then
d∨Ax(Ax)–,Ax(Ax)–
,e≤Nd∨xx–,xx– k
Hence we get
d(x,Ax)≤d(x,Ax) +d(Ax,Ax)≤c–kNM+kNM=c,
from which one deduces that
Ax∈B(x,c).
Corollary . Let A:E→E be a continuous mapping and suppose that the following two assertions hold:
() there existk∈(, )andn∈Nsuch that for allu,v∈E,ifuandvare comparable, thenAuandAvare comparable and furthermore
∨AnvAnu–,AnuAnv–∨vu–,uv–k;
() there existsx∈Esuch thatxandAxare comparable.
Then A admits a unique fixed point x∗in the comparable sense.
Proof By Theorem ., we knowAnhas a unique fixed pointx∗in the comparable sense. Notice that
AnAx∗=AAnx∗=Ax∗
which meansAx∗is also a fixed point ofAn. Again, sincex
andAxare comparable, then
x∗andAx∗are comparable, which impliesAx∗=x∗. Since the fixed point ofAis also the fixed point ofAn, the fixed point ofAis unique in the comparable sense.
Theorem . Let A:E→E be a mapping and suppose that the following two assertions hold:
() there exists a constantk∈(,)such that for allβ∈[, ],ifuandvare
comparable,thenAuandAv,Auandu,Avandvare comparable,and furthermore
∨Av(Au)–,Au(Av)–β∨Auu–,u(Au)–kβ∨Avv–,v(Av)–kβ;
() Ais continuous.
Then A admits a unique fixed point x∗ in the comparable sense, and for each x∈E, limn→∞Anx=x∗.Moreover,
dx∗,x
≤ –k
– kNM(Ax)x– .
Proof By the reflexivity of the partial ordering ‘’ inE,xx. And sinceAsatisfies the condition (), thenxandAx,AxandAx, . . . ,AnxandAn+xare comparable. Put
xn=Anx. Then for each integern≥, from the condition (), we get
∨xnx–n+,xn+x–n
β
=∨Axn–(Axn)–,Axn(Axn–)– β
∨Axn–x–n–,xn–(Axn–)– kβ
∨Axnx–n,xn(Axn)–
kβ
=∨xnx–n–,xn–x–n
kβ
∨xn+x–n,xnx–n+ kβ
By the definition of partial ordering inEwith respect toP, one has
∨xnx–n–,xn–x–n
kβ
∨xn+x–n,xnx–n+ –(–k)β
∈P.
That is
∨xn+x–n ,xnx–n+
–k∨
xnx–n–,xn–x–n
k
.
From Lemma .(), one can obtain
∨xn+x–n ,xnx–n+
∨xnx–n–,xn–x–n
k
–k.
As in the proof of Theorem .,{xn}is a Cauchy sequence and there existsx∗∈E such thatlimn→∞xn=x∗. Also,x∗is a fixed point ofA.
It remains to be shown thatx∗is a unique fixed point ofA.
Suppose that there existsy∗∈E,y∗x∗, such thatAy∗=y∗. Due to the condition (), we have
∨Ay∗Ax∗–,Ax∗Ay∗–∨Ay∗y∗–,y∗Ay∗–k∨Ax∗x∗–,x∗Ax∗–k.
That is
∨y∗x∗–,x∗y∗–∨y∗y∗–,y∗y∗–k∨x∗x∗–,x∗x∗–k=e.
Since∨(y∗x∗–,x∗y∗–) =x∗y∗–. Thenx∗y∗–e, which implies (x∗y∗–)–∈P. Now, ap-plying Definition ., we obtain the desired result.
Similar to the proof of Theorem ., one can verify
dx∗,x
≤ –k
– kNM(Ax)x– .
Letu,v∈E, anduv. By [u,v] from now on we denote the order interval{u∈E:
uuv}. Finally, we consider the fixed-point theorem in the order interval.
Theorem . Let[u,v]be the order interval in E.If A: [u,v]→[u,v]satisfies the
conditions in Theorem.,then A has a unique fixed point in the comparable sense.
Proof Consider sequences{un}and{vn}defined byun=Anuandvn=Anv, respectively. Then{un},{vn} ⊆[u,v]. SinceAsatisfies Theorem .(), then for anyn∈N,unandvn are comparable, and
e∨unvn–,vnun–
∨uv–,vu– (–kk)n
.
Notice thatPis a normal cone,
d(un,vn)≤NMuv–
k
–k
n
Againuu, hence for alln∈N,unandun+are comparable. Then
e∨unun+–,un+un–
∨uu–,uu– (–kk)n
,
from which one deduces
d(un,un+)≤NMuu–
k
–k
n
.
Hence, we conclude that{un}is a Cauchy sequence with a limit pointu∗∈[u,v] for any <k<
.
Similarly,{vn}is a Cauchy sequence with a limit pointv∗∈[u,v] for any <k<. Now, as
du∗,v∗= lim
n→∞d(un,vn)≤nlim→∞NMuv–
k
–k
n
= ,
thenu∗=v∗.
The rest of the proof is analogue to that in Theorem ..
If one checks the proof of Theorem ., then one can easily obtain the following result.
Corollary . Let [u,v] be the order interval in E. If the continuous mapping A : [u,v]→[u,v]satisfies the k-ordered contractive condition, then A admits a unique
fixed point in the comparable sense.
3 Examples
We give some examples to illustrate the main result of this paper in the following.
Example . Consider the space (Rn,d) given in Example .. LetP={x= (x
,x, . . . ,xn)∈ Rn:xi≥, ≤i≤n}be a normal cone. The partial ordering inEwith respect toPis given
by
xy ⇐⇒ xi≤yi, ≤i≤n,
thenEis a lattice. It is known that the operators onRnare in one-to-one correspondence
with then×nmatrices. Consider a matrix
A= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
a · · · a · · ·
· · · ·
· · · ·
· · · an
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,
Example . Consider the integral equation of Fredholm type
x(t) =
Kt,s,x(s)ds+g(t), t∈[, ].
Assume that
() K: [, ]×[, ]×R→Randg: [, ]→Rare continuous; () K(t,s,·) :R→Ris monotonous fort,s∈R;
() there exist a continuous functionϕ: [, ]×[, ]→Randk∈(, )such that
K(t,s,u) –K(t,s,v)≤kϕ(t,s)(u–v),
fort,s∈[, ],u,v∈R. () supt∈[,]|ϕ(t,s)|ds≤;
() there existsx∈C[, ]such that for anyt∈[, ],x(t)≤
K(t,s,x(s))ds+g(t)or
x(t)≥
K(t,s,x(s))ds+g(t).
Then the integral equation has a unique solutionx∗inC[, ].
In fact, letE=C[, ] with the metric induced by the supremum norm,i.e.,
d(x,y) =x–y=maxx(t) –y(t):t∈[, ]
for x,y∈C[, ], and P={x∈C[, ] :x(t)≥ for anyt∈[, ]} be a normal cone in
C[, ]. The partial ordering inEinduced byPis given as follows:
xy ⇐⇒ x(t)≤y(t) for anyt∈[, ].
DefineA:C[, ]→C[, ] by
Ax(t)=
Kt,s,x(s)ds+g(t), t∈[, ].
From the condition (),Ais monotonous. The monotonicity condition implies thatAx
andAyare comparable ifxandyare comparable. Observe that forx,y∈E, ifxandyare comparable, then
Ay(t)–Ax(t)≤
Kt,s,y(s)–Kt,s,x(s)ds
≤k
ϕ(t,s)y(s) –x(s)ds
≤k
ϕ(t,s)dsy–x
≤ky–x,
fort∈[, ], which implies thatAy–Ax ≤ky–x. Thus,Ais continuous. Again,Eis a lattice, by Theorem . and Remark ., the integral equation has a unique solutionx∗in
Observe that the functions satisfying the conditions in Example . do exist. For exam-ple,
K(t,s,u) =ktsu, wherek∈(, ),
g(t)≥ or g(t)≤ is continuous,
ϕ(t,s) =ts,
x(t) = ,
fort,s∈[, ] andu∈R.
Notice that the examples given above are in linear spaces. As to the noncommutative case, it is under consideration now.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their many valuable suggestions that have greatly contributed to improve the quality of this paper. This work is supported financially by the NSFC (10971011, 11371222).
Received: 10 September 2013 Accepted: 21 January 2014 Published:07 Feb 2014
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