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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).

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(ii) (X,d)is complete if and only if(X,ρ)is complete. (iii) Forx,x, . . . ,xnX,y,y, . . . ,ynXandλ, . . . ,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: XXis 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,yX.

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:XX 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 pointxXthe Picard iteration sequence(Tnx)converges toξ.

(iii) A priori error estimate.For every pointxXthe following a priori error estimate holds:

dTnx,ξλ

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,xYandAY. We say that

xAif there exists at least one vectoryAsuch 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

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for allx,yX. 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 allxXthe 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:XXis 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,yX.

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:XX 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 pointxXthe 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:

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(C) Ifxnxandλ∈R, thenλxnλx.

(C) IfλnλinRandxY, thenλnxλx.

A real vector spaceYendowed with convergence is said to be avector space with conver-gence. Ifxnx, 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) Ifxnx,yny,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) Ifxy, thenxy.

(S) Ifxyandyz, thenxz. (S) Ifxy, thenx+zy+z. (S) Ifλ> andxy, thenλxλy.

(S) Ifxnx,ynyandxy, thenxnynfor 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 andKYbe 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 yxK,

xy if and only if yxK◦.

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).

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(i) d(x,y)for allx,yXandd(x,y) = if and only ifx=y; (ii) d(x,y) =d(y,x)for allx,yX;

(iii) d(x,y)d(x,z) +d(z,y)for allx,y,zX.

The pair (X,d) is called acone metric spaceoverY.

Let (X,d) be a cone metric space over a solid vector space (Y,,≺),x∈XandrY

withr. Then the setU(x,r) ={xX:d(x,x)≺r}is called anopen ballwith center

xand 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 toxXif and only if for every vectorcYwithc,d(xn,x)≺cfor all but finitely manyn.

Recall also that a sequence (xn) inXis called aCauchy sequenceif for everycY 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 mn,

where(bn)is a sequence in Y which converges to.Then(xn)converges to a pointξX 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:XX.Suppose that for some xX,the Picard iteration(Tnx)converges to a point

ξX.Suppose also that there exist nonnegative numbersαandβsuch that

d(ξ,)αd(x,ξ) +βd(Tx,ξ) for each xX. ()

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, . . . ,xnY. 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 allixco{y, . . . ,yn};

(P) xco{x, . . . ,xn,y}andyco{y, . . . ,ym} ⇒xco{x, . . . ,xn,y, . . . ,ym};

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(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:XX be a quasi-contraction with contraction constantλ∈[, ),and let xX.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 numbermn. 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 numberinthe following inequality holds:

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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 numberinthe 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 somein, 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

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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 pointxX, 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:XX be a quasi-contraction with contraction constantλ∈[, ),and let xX.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 numbermn. 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+cod(x,Tx), . . . ,dx,Tn+x. ()

By the induction hypothesis, we have that () holds for allmn. 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+ 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

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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:XX be a quasi-contraction with contraction constantλ∈[, ).Then for all x,yX,

we have

d(x,Tx)αd(x,y) +βd(x,Ty), ()

whereα=λ/( –λ)andβ= ( +λ)/( –λ).

Proof Letx,yXbe 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:XX has at most one fixed point in X.

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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

rYwithr, the setU(x,r) ={xX:d(x,x)r}is called aclosed ballwith center

xand radiusr.

Theorem . Let(X,d)be a complete cone metric space over a solid vector space(Y,),

and let T: XX 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 pointxXthe Picard sequence(Tnx)remains in the closed ballU(x,r)and converges toξ.

(iii) A priori error estimate.For every pointxXthe following a priori estimate holds:

dTnx,ξ λ

n

 –λd(x,Tx) for alln≥. ()

(iv) A posteriori error estimates.For every pointxXthe 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∈Nwithmn, 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 pointξXsuch that (Tnx) converges toξandd(Tnx,ξ)bnfor everyn≥. The last inequality coincides with the estimate (). Settingn=  in (), we get

d(x,ξ) 

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which means thatξU(x,r). The inequality () holds for every pointxX. 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,yX, 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:XX|d(Tx,Ty)λd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx),

C=T:XX|d(Tx,Ty)λcod(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

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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:

co{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). Thenmax{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,γβ} ≤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

γαδ<γ and δα+γβ

 .

Then the functionsa(t) =α,b(t) =β+δtandc(t) =γδtsatisfy 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×XYbe defined by

d(x,y) =a, d(x,z) =b, d(y,z) =c,

d(u,v) =d(u,v) andd(u,u) =  foru,vX. Then (X,d) is a complete cone metric space overYsince the triple (a,b,c) satisfies property (C). Consider now the mappingT:XX

defined by

Tx=x, Ty=x, Tz=y.

Using Lemma ., it is easy to prove thatTCif and only ifco{b,c}for someλ

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and only ifak{b,c}for somek∈[, ). Now taking into account that the triple (a,b,c) satisfies property (C), we conclude thatTCandT∈/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

References

1. Collatz, L: Functional Analysis and Numerical Mathematics. Academic Press, New York (1966) 2. Zabrejko, PP:K-Metric andK-normed linear spaces: survey. Collect. Math.48, 825-859 (1997)

3. Jankovi´c, S, Kadelburg, Z, Radenovi´c, S: On cone metric spaces: a survey. Nonlinear Anal.74, 2591-2601 (2011) 4. Proinov, PD: A unified theory of cone metric spaces and its applications to the fixed point theory. Fixed Point Theory

Appl.2013, Article ID 103 (2013)

5. Du, W-S: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal.72, 2259-2261 (2010) 6. Amini-Harandi, A, Fakhar, M: Fixed point theory in cone metric spaces obtained via the scalarization method.

Comput. Math. Appl.59, 3529-3534 (2010)

7. Feng, Y, Mao, W: The equivalence of cone metric spaces and metric spaces. Fixed Point Theory11, 259-263 (2010) 8. Kadelburg, Z, Radenovi´c, S, Rakoˇcevi´c, V: A note on the equivalence of some metric and cone metric fixed point

results. Appl. Math. Lett.24, 370-374 (2011)

9. Asadi, M, Rhoades, BE, Soleimani, H: Some notes on the paper ‘The equivalence of cone metric spaces and metric spaces’. Fixed Point Theory Appl.2012, Article ID 87 (2012)

10. Ercan, Z: On the end of the cone metric spaces. Topol. Appl.166, 10-14 (2014)

11. Banach, S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations integrals. Fundam. Math.3, 133-181 (2009)

12. Berinde, V: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics, vol. 1912. Springer, Berlin (2007) 13. Radenovi´c, S, Kadelburg, Z: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal.5,

38-50 (2011)

14. ´Ciri´c, LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc.45, 267-273 (1974)

15. Proinov, PD: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal.67, 2361-2369 (2007)

16. Proinov, PD: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complex.26, 3-42 (2010)

17. Zhang, X: Fixed point theorem of generalized operator quasi-contractive mapping in cone metric spaces. Afr. Math. 25, 135-146 (2014)

18. Ali, MU, Kamran, T, Karapinar, E: A new approach to (α,ψ)-contractive nonself multivalued mappings. J. Inequal. Appl.2014, Article ID 71 (2014)

19. Zhang, X: Fixed point theorem of generalized quasi-contractive mapping in cone metric space. Comput. Math. Appl. 62, 1627-1633 (2011)

20. Ili´c, D, Rakoˇcevi´c, V: Quasi-contraction on a cone metric space. Appl. Math. Lett.22, 728-731 (2009)

21. Kadelburg, Z, Radenovi´c, S, Rakoˇcevi´c, V: Remarks on ‘Quasi-contraction on a cone metric space’. Appl. Math. Lett.22, 1674-1679 (2009)

22. Gaji´c, L, Rakoˇcevi´c, V: Quasi-contractions on a nonnormal cone metric spaces. Funct. Anal. Appl.46, 62-65 (2012) 23. Rezapour, S, Haghi, RH, Shahzad, N: Some notes on fixed points of quasi-contraction maps. Appl. Math. Lett.23,

498-502 (2010)

24. Ding, H-S, Jovanovi´c, M, Kadelburg, Z, Radenovi´c, S: Common fixed point results for generalized quasicontractions in tvs-cone metric spaces. J. Comput. Anal. Appl.15, 463-470 (2013)

25. Karapinar, E: Fixed point theorems in cone Banach spaces. Fixed Point Theory Appl.2009, Article ID 609281 (2009) 26. Karapinar, E: Some nonunique fixed point theorems of ´Ciri´c type on cone metric spaces. Abstr. Appl. Anal.2010,

Article ID 123094 (2010)

27. Karapinar, E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl.59, 3656-3668 (2010)

28. Khamsi, MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl.2010, Article ID 315398 (2010)

29. Duki´c, D, Paunovi´c, L, Radenovi´c, S: Convergence of iterates with errors of uniformly quasi-Lipschitzian mappings in cone metric spaces. Kragujev. J. Math.35, 399-410 (2011)

30. Karapinar, E, Kumam, P, Sintunavarat, W: Coupled fixed point theorems in cone metric spaces with ac-distance and applications. Fixed Point Theory Appl.2012, Article ID 194 (2012)

31. Radenovi´c, S: A pair of non-self mappings in cone metric spaces. Kragujev. J. Math.36, 189-198 (2012)

(14)

33. Du, W-S, Karapinar, E: A note on coneb-metric and its related results: generalizations or equivalence? Fixed Point Theory Appl.2013, Article ID 210 (2013)

34. Kadelburg, Z, Radenovi´c, S: A note on various types of cones and fixed point results in cone metric spaces. Asian J. Math. Appl.2013, 0104 (2013)

35. Popovi´c, B, Radenovi´c, S, Shukla, S: Fixed point results to tvs-coneb-metric spaces. Gulf J. Math.1, 51-64 (2013) 36. Xu, S, Radenovi´c, S: Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach

algebra without assumption of normality. Fixed Point Theory Appl.2014, Article ID 101 (2014)

37. Proinov, PD, Cholakov, SI: Semilocal convergence of Chebyshev-like root-finding method for simultaneous approximation of polynomial zeros. Appl. Math. Comput.236, 669-682 (2014)

38. Proinov, PD, Cholakov, SI: Convergence of Chebyshev-like method for simultaneous computation of multiple polynomial zeros. C. R. Acad. Bulg. Sci.67(2014, in press)

39. Abbas, M, Rhoades, BE: Fixed and periodic point results in cone metric spaces. Appl. Math. Lett.22, 511-515 (2009) 40. Olaleru, JO: Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory Appl.2009,

Article ID 657914 (2009)

41. Azam, A, Beg, I, Arshad, M: Fixed point in topological vector space-valued cone metric spaces. Fixed Point Theory Appl.2010, Article ID 604084 (2010)

42. Song, G, Sun, X, Zhao, Y, Wang, G: New common fixed point theorems for maps on cone metric spaces. Appl. Math. Lett.23, 1033-1037 (2010)

10.1186/1029-242X-2014-226

References

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