R E S E A R C H
Open Access
Iterative approximation of fixed points of
quasi-contraction mappings in cone metric
spaces
Petko D Proinov
*and Ivanka A Nikolova
*Correspondence:
proinov@uni-plovdiv.bg Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv, 4000, Bulgaria
Abstract
In this paper, we establish a full statement of ´Ciri´c’s fixed point theorem in the setting of cone metric spaces. More exactly, we obtaina priorianda posteriorierror estimates for approximating fixed points of quasi-contractions in a cone metric space. Our result complements recent results of Zhang (Comput. Math. Appl. 62:1627-1633, 2011), Dinget al.(J. Comput. Anal. Appl. 15:463-470, 2013) and others.
MSC: Primary 54H25; secondary 47H10; 46A19
Keywords: Picard iteration; cone metric space; solid vector space; fixed points; quasi-contractions; error estimates
1 Introduction
In this paper, we study fixed points of quasi-contraction mappings in a cone metric space (X,d) over a solid vector space (Y,). Cone metric spaces have a long history (see Col-latz [], Zabrejko [], Jankovićet al.[], Proinov [] and references therein). A unified theory of cone metric spaces over a solid vector space was developed in a recent paper of Proinov []. Recall that an ordered vector space with convergence structure (Y,) is called:
• a solid vector space if it can be endowed with a strict vector ordering(≺); • a normal vector space if the convergence ofYhas the sandwich property.
Every metric space (X,d) is a cone metric space overR(with usual ordering and usual convergence). On the other hand, every cone metric space over a solid vector space is a metrizable topological space (see Proinov [] and references therein). It is well known that a lot of fixed point results in cone metric setting can be directly obtained from their metric versions (see Du [], Amini-Harandi and Fakhar [], Feng and Mao [], Kadelburget al.
[], Asadiet al.[], Proinov [], and Ercan []).
For instance, for this purpose we can use the following theorem. This theorem shows that every cone metric is equivalent to a metric which preserves the completeness as well as some inequalities.
Theorem .([, Theorem .]) Let(X,d)be a cone metric space over a solid vector space
(Y,).Then there exists a metricρon X such that the following statements hold true.
(i) The topology of(X,d)coincides with the topology of(X,ρ).
(ii) (X,d)is complete if and only if(X,ρ)is complete. (iii) Forx,x, . . . ,xn∈X,y,y, . . . ,yn∈Xandλ, . . . ,λn∈R,
d(x,y) n
i=
λid(xi,yi) implies ρ(x,y)≤ n
i=
λiρ(xi,yi).
In , Banach [] proved his famous fixed point theorem for contraction mappings. Banach’s contraction principle is one of the most useful theorems in the fixed point the-ory. It has two versions: a short version and a full version. In a metric space setting its full statement can be seen, for example, in the monograph of Berinde [, Theorem .]. Recently, full statements of Banach’s fixed point theorem in a cone metric spaces over a solid vector space were given by Radenović and Kadelburg [, Theorem .] and Proinov [, Theorem .].
Definition .([]) Let (X,d) be a metric space. A mappingT: X→Xis called a quasi-contraction(with contraction constantλ) if there existsλ∈[, ) such that
d(Tx,Ty)≤λmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx) ()
for allx,y∈X.
There are a large number of generalizations of Banach’s contraction principle (see, for example, [–] and references therein). In , Ćirić [] introduced contraction map-pings and proved the following well known generalization of Banach’s fixed point theorem.
Theorem . Let(X,d)be a complete metric space and T:X→X be a quasi-contraction with contraction constantλ.Then the following statements hold true:
(i) Existence and uniqueness.Thas a unique fixed pointξ inX.
(ii) Convergence of Picard iteration.For every starting pointx∈Xthe Picard iteration sequence(Tnx)converges toξ.
(iii) A priori error estimate.For every pointx∈Xthe following a priori error estimate holds:
dTnx,ξ≤ λ
n
–λd(x,Tx) for alln≥. ()
Following Zhang [], in the next definition, we define a useful binary relation between an ordered vector spaceYand the set of all subsets ofY. It plays a very important role in this paper as it is used to prove our main result.
Definition .([]) Let (Y,) be an ordered vector space,x∈YandA⊂Y. We say that
xAif there exists at least one vectory∈Asuch thatxy.
In , Ilić and Rakočević [] generalized the concept of quasi-contraction to cone metric space as follows: A selfmappingT of a cone metric space (X,d) over an ordered vector space (Y,) is called a quasi-contraction onXif there existsλ∈[, ) such that
for allx,y∈X. They proved the following result [, Theorem .]: Let (X,d) be a cone metric space over a normal solid Banach space (Y,); then every quasi-contractionT of the type () has a unique fixed point inX, and for allx∈Xthe Picard iterative sequence (Tnx) converges to this fixed point. Kadelburget al.[, Theorem .] improved this result by omitting the assumption of normality provided thatλ∈[, /). Gajić and Rakočević [, Theorem ] proved this result for any contraction constantλ∈[, ). Rezapouret al.
[, Theorem .] proved this result in the case whenYis a solid topological vector space andλ∈[, ). Furthermore, Kadelburget al.[, Theorem .(b)] proved that this result is equivalent to the short version of Ćirić’s fixed point theorem.
In , Zhang [] presented the following new definition for quasi-contractions in cone metric spaces.
Definition .([]) Let (X,d) be a cone metric space over an ordered vector space (Y,). A mappingT:X→Xis called aquasi-contraction(with contraction constantλ) if there existsλ∈[, ) such that
d(Tx,Ty)λcod(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx) ()
for allx,y∈X.
By applying Theorem . to the first two conclusions of Theorem ., we obtain the following fixed point theorem in a cone metric setting.
Theorem . Let(X,d)be a complete cone metric space over a solid vector space(Y,)
and T:X→X be a quasi-contraction.Then the following statements hold true:
(i) Existence and uniqueness.T has a unique fixed pointξinX.
(ii) Convergence of Picard iteration.For every starting pointx∈Xthe Picard iteration sequence(Tnx)converges toξ.
In , Zhang [] proved Theorem . in the case whenY is a normal solid Banach space. In , Dinget al.[] proved this theorem in the case whenYis a solid topological vector space.
In this paper, we establish a full statement of Ćirić’s fixed point theorem in the setting of cone metric spaces. Our result complements Theorem .. Thus it extends and comple-ments the corresponding results of Zhang [], Dinget al.[], and others.
For some recent results on the topic, we refer the reader to [–]. In the papers [, ] one can find some applications of cone metric spaces to iterative methods for finding all zeros of polynomial simultaneously.
2 Preliminaries
In this section, we introduce some basic definitions and theorems of cone metric spaces over a solid vector space.
Definition .([]) LetYbe a real vector space andSbe the set of all infinite sequences inY. A binary relation→betweenSandY is called aconvergenceonY if it satisfies the following axioms:
(C) Ifxn→xandλ∈R, thenλxn→λx.
(C) Ifλn→λinRandx∈Y, thenλnx→λx.
A real vector spaceYendowed with convergence is said to be avector space with conver-gence. Ifxn→x, then (xn) is said to be aconvergent sequenceinY, and the vectorxis said to be alimitof (xn).
Definition .([]) Let (Y,→) be a vector space with convergence. An orderingon
Y is said to be avector ordering if it is compatible with the algebraic and convergence structures onYin the sense that the following are true:
(V) Ifxy, thenx+zy+z. (V) Ifλ≥andxy, thenλxλy.
(V) Ifxn→x,yn→y,xnynfor alln, thenxy.
A vector space with convergenceY endowed with vector ordering is called anordered vector space with convergence.
Definition .([]) Let (Y,,→) be an ordered vector space with convergence. A strict ordering≺onY is said to be astrict vector orderingif it is compatible with the vector ordering, the algebraic structure and the convergence structure onYin the sense that the following are true:
(S) Ifx≺y, thenxy.
(S) Ifxyandy≺z, thenx≺z. (S) Ifx≺y, thenx+z≺y+z. (S) Ifλ> andx≺y, thenλx≺λy.
(S) Ifxn→x,yn→yandx≺y, thenxn≺ynfor all but finitely manyn.
It turns out that an ordered vector space can be endowed with at most one strict vector ordering (see Proinov [, Theorem .]).
Definition .(Solid vector space) An ordered vector space with convergence endowed with a strict vector ordering is said to be asolid vector space.
Let us consider an important example of a solid vector space.
Example . Let (Y,τ) be a topological vector space andK⊂Ybe a cone with nonempty interior K◦. Define the vector orderingonY and the strict vector ordering≺onY, respectively, by means of
xy if and only if y–x∈K,
x≺y if and only if y–x∈K◦.
ThenY is a solid vector space called asolid topological vector space.
Now let us recall the definition of a cone metric space known also as ‘K-metric spaces’ (see Zabrejko [], Proinov [] and references therein).
(i) d(x,y)for allx,y∈Xandd(x,y) = if and only ifx=y; (ii) d(x,y) =d(y,x)for allx,y∈X;
(iii) d(x,y)d(x,z) +d(z,y)for allx,y,z∈X.
The pair (X,d) is called acone metric spaceoverY.
Let (X,d) be a cone metric space over a solid vector space (Y,,≺),x∈Xandr∈Y
withr. Then the setU(x,r) ={x∈X:d(x,x)≺r}is called anopen ballwith center
xand radiusr.
Every cone metric spaceXover a solid vector spaceY is a Hausdorff topological space with topology generated by the basis of all open balls. Then a sequence (xn) of points inX converges tox∈Xif and only if for every vectorc∈Ywithc,d(xn,x)≺cfor all but finitely manyn.
Recall also that a sequence (xn) inXis called aCauchy sequenceif for everyc∈Y with
c there isN∈Nsuch thatd(xn,xm)≺cfor alln,m>N. A cone metric spaceXis called
completeif each Cauchy sequence inXis convergent.
In order to prove our main result we need the following two theorems.
Theorem .([]) Let(X,d)be a complete cone metric space over a solid vector space
(Y,).Suppose(xn)is a sequence in X satisfying
d(xn,xm)bn for all n,m≥with m≥n,
where(bn)is a sequence in Y which converges to.Then(xn)converges to a pointξ∈X with error estimate
d(xn,ξ)bn for all n≥.
Theorem .([]) Let(X,d)be a cone metric space over a solid vector space(Y,)and T:X→X.Suppose that for some x∈X,the Picard iteration(Tnx)converges to a point
ξ∈X.Suppose also that there exist nonnegative numbersαandβsuch that
d(ξ,Tξ)αd(x,ξ) +βd(Tx,ξ) for each x∈X. ()
Thenξ is a fixed point of T.
3 Auxiliary results
LetAbe a subset of a real vector spaceY. Recall that theconvex hullofA, denotedcoA, is the smallest convex set includingA. Supposex,x, . . . ,xn∈Y. It is well known thatx∈ co{x, . . . ,xn}if and only if there exist nonnegative numbersα, . . . ,αnsuch that
n
i=αi= andx=ni=αixi.
Lemma . Let(Y,)be an ordered vector space.Suppose that x,y,x, . . . ,xn,y, . . . ,ym
are vectors in Y andλis a real number.Then:
(P) xco{x, . . . ,xn} ⇒xco{x, . . . ,xn,y};
(P) xco{x, . . . ,xn}andxiyifor alli⇒xco{y, . . . ,yn};
(P) xco{x, . . . ,xn,y}andyco{y, . . . ,ym} ⇒xco{x, . . . ,xn,y, . . . ,ym};
(P) xco{λx,x, . . . ,xn} ⇔xco{x, . . . ,xn}ifλ< andxifor somei;
(P) xco{x, . . . ,xn,y} ⇔xco{x, . . . ,xn}ify=xifor somei.
Proof We only prove the necessity of (P) since the proofs of the other properties are similar. The inequalityxco{λx,x, . . . ,xn,} implies that there exist nonnegative
num-bersα,α, . . . ,αn such thatα+
n
i=αi= andxαλx+
n
i=αixi. From this inequality andαλ< , we deduce
x
n
i=
βixi, ()
whereβi=αi/( –αλ). We have
n
i=βi= ( –α)/( –αγ) < . By the assumptions, we have
xi for somei. Without loss of generality we may assume thatx. Define the
non-negative numbersγ, . . . ,γnbyγ= –nj=βjandγi=βifori≥. From () andβ≤γ,
we obtain
x
n
i=
γixi.
This impliesxco{x, . . . ,xn}since
n
i=γi= .
Remark . Note that Lemma . remains true if we omit the expression ‘co’ from its
formulation.
The following lemma was given by Zhang [, Lemma ] in a slightly different form. We give a simple proof of this lemma.
Lemma . Let (X,d) be a cone metric space over an ordered vector space (Y,),
T:X→X be a quasi-contraction with contraction constantλ∈[, ),and let x∈X.Then for every m∈N,we have
dTix,Tmxλicod(x,Tx), . . . ,dx,Tmx for i= , . . . ,m. ()
Proof We prove the statement by induction onm. It is obviously true form= . Assume thatn∈Nand assume that () is satisfied for any natural numberm≤n. We have to prove that
dTix,Tn+xλicod(x,Tx), . . . ,dx,Tn+x fori= , . . . ,n+ . ()
We divide the proof of () into three steps.
Step . We claim that for every natural numberi≤nthe following inequality holds:
By the definition of the quasi-contraction mapping, we obtain
dTix,Tn+x
=dTTi–x,TTnx
λcodTi–x,Tnx,dTi–x,Tix,dTnx,Tn+x,dTi–x,Tn+x,dTix,Tnx.
From the induction hypothesis and properties (P) and (P), we get the following three inequalities:
dTi–x,Tnxλi–cod(x,Tx), . . . ,dx,Tnx,
dTi–x,Tixλi–cod(x,Tx), . . . ,dx,Tnx,
dTix,Tnxλi–cod(x,Tx), . . . ,dx,Tnx.
From the last four inequalities and properties (P) and (P), we obtain the desired inequal-ity.
Step . We claim that for every natural numberi≤nthe following inequality holds:
dTix,Tn+xcoλid(x,Tx), . . . ,λidx,Tn+x,λdTnx,Tn+x.
We prove this by finite induction oni. Settingi= in the claim of Step , we immediately arrive at the following inequality:
dTx,Tn+xcoλd(x,Tx), . . . ,λdx,Tnx,λdTnx,Tn+x,λdx,Tn+x,
which proves the claim of Step fori= . Assume that for somei≤n, the claim of Step holds. Now we shall show that
dTi+x,Tn+xcoλi+d(x,Tx), . . . ,λi+dx,Tn+x,λdTnx,Tn+x. ()
It follows from Step that
dTi+x,Tn+x
coλi+d(x,Tx), . . . ,λi+dx,Tn+x,λdTnx,Tn+x,λdTix,Tn+x. ()
By the finite induction hypothesis and property (P), we have
dTix,Tn+xcoλid(x,Tx), . . . ,λidx,Tn+x,dTnx,Tn+x. ()
From (), (), and properties (P) and (P), we obtain ().
Step . Now we shall prove (). From the claim of Step withi=n, we get
dTnx,Tn+xcoλnd(x,Tx), . . . ,λndx,Tn+x,λdTnx,Tn+x.
According to the property (P), this inequality is equivalent to
which by (P) implies
dTnx,Tn+xλi–cod(x,Tx), . . . ,dx,Tn+x. ()
Finally, by the claim of Step and the inequality (), taking into account the properties (P) and (P), we obtain (). This completes the proof of the lemma.
In the following lemma, we show that ifTis a quasi-contraction of a cone metric spaceX, then for every starting pointx∈X, the Picard iteration sequence (Tnx) is bounded in the spaceX.
Lemma . Let (X,d) be a cone metric space over an ordered vector space (Y,),
T:X→X be a quasi-contraction with contraction constantλ∈[, ),and let x∈X.Then for every m∈N,we have
dx,Tmx
–λd(x,Tx). ()
Proof We prove the statement by induction onm. Ifm= , then inequality () holds since ≤λ< . Assume thatn∈Nand assume that () is satisfied for any natural numberm≤n. Then we have to prove that
dx,Tn+x
–λd(x,Tx). ()
From the triangle inequality, we obtain
dx,Tn+xd(x,Tx) +dTx,Tn+x. ()
By Lemma ., we get
dTx,Tn+xλcod(x,Tx), . . . ,dx,Tn+x. ()
By the induction hypothesis, we have that () holds for allm≤n. Then it follows from (), (P), and (P) that
dTx,Tn+xco
λ
–λd(x,Tx),λd
x,Tn+x .
This inequality implies that there existsα∈[, ] such that
dTx,Tn+xα λ
–λd(x,Tx) + ( –α)λd
x,Tn+x. ()
Combining () and (), we get
dx,Tn+xd(x,Tx) +α λ
–λd(x,Tx) + ( –α)λd
x,Tn+x,
which is equivalent to the following inequality:
( –λ+αλ)dx,Tn+x –λ+αλ
Multiplying both sides of this inequality by /( –λ+αλ), we obtain (). This completes the proof of the lemma.
Lemma . Let(X,d)be a cone metric space over an ordered vector space(Y,),and let T:X→X be a quasi-contraction with contraction constantλ∈[, ).Then for all x,y∈X,
we have
d(x,Tx)αd(x,y) +βd(x,Ty), ()
whereα=λ/( –λ)andβ= ( +λ)/( –λ).
Proof Letx,y∈Xbe fixed. First we shall prove that
d(Tx,Ty)λd(x,y) +d(x,Tx) +d(x,Ty). ()
It follows from Definition . that there exist five nonnegative numbersα,β,γ,μ,νsuch thatα+β+γ +μ+ν= and
d(Tx,Ty)λαd(x,y) +βd(x,Tx) +γd(y,Ty) +μd(x,Ty) +νd(y,Tx).
From this and the inequalities
d(y,Ty)d(x,y) +d(x,Ty) and d(y,Tx)d(x,y) +d(x,Tx),
we obtain
d(Tx,Ty)λ(α+γ +ν)d(x,y) + (β+ν)d(x,Tx) + (γ+μ)d(x,Ty).
This inequality yields () sinceα+γ+ν≤,β+ν≤ andγ+μ≤. Now we are ready to prove (). From the triangle inequality, we get
d(x,Tx)d(x,Ty) +d(Tx,Ty).
From this and (), we obtain
d(x,Tx)d(x,Ty) +λd(x,y) +d(x,Tx) +d(x,Ty),
which can be presented in the following equivalent form:
( –λ)d(x,Tx)λd(x,y) + ( +λ)d(x,Ty).
Multiplying both sides of this inequality by /( –λ), we get ().
Lemma . Let(X,d)be a cone metric space over an ordered vector space(Y,).Then every quasi-contraction T:X→X has at most one fixed point in X.
4 Main result
Now we are ready to state the main result of this paper. Let (X,d) be a complete cone metric space over an ordered vector spaceY. Recall that for a pointx∈Xand a vector
r∈Ywithr, the setU(x,r) ={x∈X:d(x,x)r}is called aclosed ballwith center
xand radiusr.
Theorem . Let(X,d)be a complete cone metric space over a solid vector space(Y,),
and let T: X→X be a quasi-contraction with contraction constantλ∈[, ).Then the following statements hold true:
(i) Existence, uniqueness and localization.Thas a unique fixed pointξ which belongs to the closed ballU(x,r)with radius
r=
–λd(x,Tx),
wherexis any point inX.
(ii) Convergence of Picard iteration.Starting from any pointx∈Xthe Picard sequence(Tnx)remains in the closed ballU(x,r)and converges toξ.
(iii) A priori error estimate.For every pointx∈Xthe following a priori estimate holds:
dTnx,ξ λ
n
–λd(x,Tx) for alln≥. ()
(iv) A posteriori error estimates.For every pointx∈Xthe following a posteriori estimate holds:
dTnx,ξ
–λd
Tnx,Tn+x for alln≥, ()
dTnx,ξ λ
–λd
Tnx,Tn–x for alln≥. ()
Proof Letxbe an arbitrary point inX. By Lemma ., for allm,n∈Nwithm≥n, we have
dTnx,Tmxλncod(x,Tx), . . . ,dx,Tmx.
From this, Lemma ., and properties (P) and (P), we deduce
dTnx,Tmxbn, wherebn=
λn
–λd(x,Tx). ()
Note that (bn) is a sequence inYwhich converges to sinceλn→ inR. Now applying Theorem . to the Picard sequence (Tnx), we conclude that there exists a pointξ∈Xsuch that (Tnx) converges toξandd(Tnx,ξ)bnfor everyn≥. The last inequality coincides with the estimate (). Settingn= in (), we get
d(x,ξ)
which means thatξ∈U(x,r). The inequality () holds for every pointx∈X. Applying () to the pointTnx, we obtain (). Settingn= in (), we get
d(Tx,ξ) λ
–λd(x,Tx). ()
Applying () to the pointTn–x, we get (). Settingn= in (), we obtaind(x,Tmx)b
for everym≥. Hence, the sequence (Tnx) lies in the ballU(x,r) sincer=b
.
It follows from Lemma . that ξ satisfies condition (). Hence, by Theorem ., we conclude that ξ is a fixed point of T. The uniqueness of the fixed point follows from
Lemma ..
Theorem . extends and complements the recent results of Dinget al.[, Theorem .] and Zhang [, Theorem ] as well as previous results due to Ilić and Rakočević [, Theorem .], Kadelburget al.[, Theorem .], Rezapouret al.[, Theorem .] and Kadelburget al.[, Theorem .(b)] who have studied quasi-contraction mappings of the type ().
Theorem . also extends and complements the results of Abbas and Rhoades [, Corollary .], Olaleru [, Theorem .], Azamet al.[, Theorem .], Songet al.[, Corollary .]. These authors have studied the class of mappings satisfying a contractive condition of the type
d(Tx,Ty)αd(x,y) +βd(x,Tx) +γd(y,Ty) +μd(x,Ty) +νd(y,Tx) ()
for allx,y∈X, whereα,β,γ,μandνare five nonnegative constants such thatα+β+γ+
μ+ν< . In this case Theorem . holds withλ=α+β+γ +μ+νsince condition () implies condition () with thisλ. Let us note that in this special case Theorem . holds even withλ= (α+δ)/( –δ), whereδ= (β+γ +μ+ν)/.
5 Examples
Zhang [, Example ] gives an example showing that the set of all quasi-contractions of the type () is a proper subset of the set of all quasi-contractions defined by (). In order to prove this, he considers a selfmapping of a cone metric spaceXover a normal solid vector spaceY. Dinget al.[, Example .] provide a similar example, but for the case of a non-normal solid vector spaceY.
The aim of this section is to unify these two examples. LetBdenote the set of all quasi-contractions of the type (), and letCdenote the set of all quasi-contractions defined by (), that is,
B=T:X→X|d(Tx,Ty)λd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx),
C=T:X→X|d(Tx,Ty)λcod(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).
Definition . Let (Y,) be an ordered vector space, and leta,b,c be three vectors inY. We say that the triple (a,b,c) satisfies property (C) if the following two statements hold:
• aλco{b,c}for someλ∈[, ),ba+candca+b. • ak{b,c}is wrong for everyk∈[, ).
Proposition . Let Y=Rbe endowed with the usual ordering≤.Then there are no triples
(a,b,c)in Y satisfying property(C).
Proof Assume that there is a triple (a,b,c) inY with property (C). Thenaλmax{b,c}
for someλ∈[, ). On the other hand,akmax{b,c}is wrong for everyk∈[, ). This is a contradiction which proves the proposition.
Proposition . Let Y=Rn(n≥)be endowed with coordinate-wise ordering.Then in Y there exist infinitely many triples(a,b,c)satisfying property(C).
Proof Choose three real numbersα,βandγ such that
β<γ< β and max{β,γ–β} ≤α<β+γ .
Then the vectorsa= (α, . . . ,α),b= (β, . . . ,β,γ) andc= (γ, . . . ,γ,β) satisfy property (C) withλ= α
β+γ.
Proposition . Let Y=Cn[, ] (n≥)be endowed with point-wise ordering.Then in
Y there exist infinitely many triples(a,b,c)satisfying property(C).
Proof Choose three real numbersα,βandγas in the proof of Proposition ., then choose a real numberδsuch that
γ –α≤δ<γ and δ≤α+γ –β
.
Then the functionsa(t) =α,b(t) =β+δtandc(t) =γ –δtsatisfy property (C) withλ=
α
β+γ.
Example . Let (Y,) be an arbitrary solid vector space, and let (a,b,c) be any triple in
Y with property (C). Furthermore, letX={x,y,z}, and letd: X×X→Ybe defined by
d(x,y) =a, d(x,z) =b, d(y,z) =c,
d(u,v) =d(u,v) andd(u,u) = foru,v∈X. Then (X,d) is a complete cone metric space overYsince the triple (a,b,c) satisfies property (C). Consider now the mappingT:X→X
defined by
Tx=x, Ty=x, Tz=y.
Using Lemma ., it is easy to prove thatT∈Cif and only ifaλco{b,c}for someλ∈
and only ifak{b,c}for somek∈[, ). Now taking into account that the triple (a,b,c) satisfies property (C), we conclude thatT∈CandT∈/B. Hence,Bis a proper subset ofC.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
Acknowledgements
The research is supported by Project NI13 FMI-002 of Plovdiv University.
Received: 19 February 2014 Accepted: 23 May 2014 Published:03 Jun 2014
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