Normed Ordered and E-Metric Spaces

(1)

Volume 2012, Article ID 272137,11pages doi:10.1155/2012/272137

Research Article

Normed Ordered and E-Metric Spaces

Ahmed Al-Rawashdeh,

¹

Wasfi Shatanawi,

²

and Muna Khandaqji

²

1Department of Mathematical Sciences, UAEU, Al-Ain 17551, UAE

2Department of Mathematics, Hashemite University, Zarqa 13115, Jordan

Correspondence should be addressed to Ahmed Al-Rawashdeh,[email protected] Received 31 December 2011; Revised 23 February 2012; Accepted 26 February 2012 Academic Editor: Naseer Shahzad

Copyrightq2012 Ahmed Al-Rawashdeh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space E, and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties ofR, and we discuss their extension to E. We introduce the notion of E-metric spaces and characterize cone metric space. Afterwards, we get generalizations of notions of convergence and Cauchy theory. In particular, we get a fixed point theorem of a contractive mapping in E-metric spaces. Finally, by extending the notion of a contractive sequence in a real-valued metric space, we show that in E-metric spaces, a contractive sequence is Cauchy.

1. Introduction

Recall that the space of real numbersRis a normed space which having the usual ordering

≤, such that it is translation invariant, that is, for all x, y, and z inR, x ≤ yimplies that xz ≤ yz. Also, for any spaceX, a metricdreal-valued metricdefines a metric space X, d.

Recently, the concept of a cone metric space has been studied in1–15, and others.

Indeed, they proved some fixed-point theorems of generalized contractive mappings. In particular, Huang and Zhang in7introduced the following.

Definition 1.1see7. LetPbe a subset of a normed spaceEwith IntP/∅. ThenPis called a cone if

1Pis closed andP /{0}.

2a, b∈R, x, y∈P ⇒axby∈P.

3x∈Pand−x∈P ⇒x0.

(2)

A conePinduces a partial ordering≤with respect toPbyx≤yif and only ify−x∈P. The notionx < ymeans thatx≤yandx /y, whilexystands fory−x∈IntP. A conePis called normal if there exists a numberk >0 such that for allx, y∈E, we have

0≤x≤y⇒ x ≤ky. 1.1 The least positive numberk, satisfying1.1, is called the normal constant ofP. Vandergraft in16 or see Example 2.1 of10presented an example of a nonnormal cone, that is, a cone of a normed spaceEwhich does not satisfy1.1.

Definition 1.2See7. LetXbe a nonempty set, and letEbe a normed space having a normal coneP with IntP/∅. Suppose, the mappingd:X×X → Esatisfies the following:

10< dx, y, for allx, y∈Xanddx, y 0 if and only ifxy, 2dx, y dy, x, for allx, y∈X,

3dx, y≤dx, z dz, y, for allx, y, z∈X.

Thendis called a cone metric onX, andX, dis called a cone metric space.

Remark 1.3. In7and in order to define a cone metric space, the authors started by a Banach spaceEsee7, page 1469. But in fact, and up to the results of the current paper, they just needEto be a normed space only.

More generally, Karapınar in17and Shatanawi in18,19studied the couple fixed-point theorems in cone metric spaces. Also, Shatanawi in20studied common coincidence point in cone metric spaces.

In this paper, we start with different approach, and without starting by cones, we introduce the notion of normed ordered spaces in general see Definition 2.1 which generalizes most of the properties ofRalso the notion of ordered field is already defined, see21, page 7. We show that many results can be extended to a normed ordered space E; however, some properties cannot, as depending on crucial properties ofR. In the second part, and considering a nonempty setX, we replace the real-valued metric with anE-valued metricdenoted bydÊ, andX, dÊis calledE-metric space, we discuss some examples. We generalize results of real-valued metric spacessee, e.g., 22concerning sequences to the case ofE-metric spaces. Afterwards, we introduce the notions of convergence and Cauchy sequences inE-metric space. ConsideringEto be a normed ordered space and thatXhas an E-valued matric, we give a characterization of cone metric spaces in the sense of Huang and Zhang in7. Then we get results concerning convergence and Cauchy theory in the case of E-metric space. In particular, we get a fixed-point theorem of a contractive mapping. Finally, and in a similar way of a real-valued metric space, we introduce the notion of a contractive sequence in anE-metric space, and we prove that every contractive sequence is Cauchy.

Notice that ordered spaces need not be totally ordered. In fact, many properties ofR are deduced asRbeing totally ordered. We say thatEhas the completeness propertyCPif every subset of an upper bound has a supremum inEand every subset of a lower bound has an infemum inE. IfEis an ordered space, then all types of intervals can be defined. Indeed the unbounded intervalsleft and right raysare

a,∞ {x∈E; a < x}, −∞, a {x∈E; x < a}. 1.2

(3)

2. Normed Ordered Spaces

In this section, we define normed ordered spaces, and we give diﬀerent examples. Then we discuss generalizations of main results in the real space to normed ordered spaces. In particular, the Bolzano-Weierstrass theorem and intermediate value theorem are not true in general normed ordered spaces. The properties CP and totally ordered of R, are in fact seems to be crucial.

Definition 2.1. An ordered spaceEis a vector space over the real numbers, with a partial order relation≤such that

O1for allx, y, andzinE,x≤yimpliesxz≤yztranslation invariant, O2for allα∈Randx∈Ewithx≥0,αx≥0.

Moreover, ifEis equipped with a norm · such that

O3there exists a real numberk >0 and for allx, y∈E, 0≤x≤yimpliesx ≤ky, thenEis called a normed ordered space.

By the translation invariantO1,x ≤ymeansy−x≥ 0. The strict inequalityx < ystands forx≤yandx /y.

Proposition 2.2. LetEbe an ordered space. Then for allxandyinE, we have ix≥0 implies that−x≤0,

iiIfx≥0 and−x≥0, thenx0, iiiIfx≥0 andy≥0, thenxy≥0, ivfor allα∈R⁻andx≥0,αx≤0,

vx > yandα∈Rimplies thatαx > αy, vix > yandα∈R⁻implies thatαx < αy.

Proof. iAsx≥0, by translation invariant, we have−xx≥ −x, hence−x≤0.iiAssuming that−x≥ 0 implies byithatx −−x≤ 0 so we getx ≤ 0 ≤x, and as the relation is an antisymmetric, we havex0.iiiAsx≥0, we getxy≥y≥0, hence the result holds by transitivity.ivAsα∈R⁻, we have−α∈Rand byO2this implies that−αx≥0. Usingi

−−αx≤0, henceαx≤0.vAsx > y, thenx−y >0, and byO2we getαx−y>0, hence αx > αy.viis similar.

Now let us introduce the following examples of normed ordered spaces.

Example 2.3. aThe set of real numbersRwith usual ordering and the absolute value.

bThe setRⁿwith the ordering defined by x1, x2, . . . , xn≤

y1, y2, . . . , yn

⇐⇒xi≤yi, 1≤i≤n. 2.1

This ordering is called the simplicial ordering of Rⁿ. Then Rⁿ together with simplicial ordering and the Euclidean norm is a normed ordered space.

(4)

cThe set of rational numbersQas a vector space over itself together with absolute value and usual ordering.

dThe complex spaceCtogether with the modulus, and the order defined by

z≤w⇐⇒w−z∈R, w−z≥0 2.2

is a normed ordered space.

eThe spaceC0,1of all continuous real-valued functions on0,1together with the supremum norm,

fsup

fx; x∈0,1

2.3

and the pointwise ordering,

f≤g⇐⇒fx≤gx; ∀x∈0,1 2.4

is a normed ordered space.

fLetX, μbe any measure space. Then for all 1≤p ≤ ∞, the spaceL^pXwith the norm,

f

p

X

f^p dμ 1/p

2.5

and the pointwise ordering is a normed order space.

Definition 2.4. LetEbe an ordered space. Then an elementv ∈Eis called positive ifv ≥0, and it is called strict positive ifv >0. The set of all positive elements inEis denoted byE. Remark 2.5. The following result is valid for any ordered space having a norm, and not necessarily satisfying the propertyO3. So the space needs not to be a normed ordered space.

Then according toDefinition 2.1with the characterization inTheorem 3.8, this is equivalent to say that a conePis nonnormalindeed this exists, see Example 2.1 of10.

Proposition 2.6. LetEbe an ordered space having a norm. ThenEis a closed set.

Proof. We will prove thatE\E is an open subset ofE. Given thata ∈ E\E. Define that dinf{x−a; x∈E}, then

Ba, d {x∈E;x−a< d} 2.6

is an open ball containing a. Now we claim thatBa, d ⊆ E\E: if not, then there exists x₀ ∈ Ba, d, which means thatx0 −a < d, andx₀ ∈ E, x0 −a ≥ d, hence we get a contradiction. Then the claim is proved.

The interior of E is denoted by IntE. In the following example, we show that IntC0,1consists of functions that never touch thex-axis.

(5)

Example 2.7. IfEC0,1, then IntE {f ∈E; fx/0; ∀x}.

Proof. Suppose that f ∈ E and fx/0, for allx ∈ 0,1. Choose > 0 such that <

inf{fx; ∀x∈0,1}. Now given anyg in the open ballBf, , and assume thatgx0 0 for somex0∈0,1. Then

<inf fx < fx0 <sup fx−gx ; x∈0,1

f−g<

. 2.7

Therefore,{f ∈E; fx/0; ∀x}is an open subset, so we get that{f ∈E; fx/0; ∀x} ⊆ IntE. Conversely, ifg∈IntE, then there exists0>0 such thatBg, 0⊆E. If for some x0∈0,1,gx0 0, then define thathx gx−0/2. Therefore,h∈Bg, 0andh /∈ E ashx0 −0/2 which is a contradiction, sogx/0, for allx∈0,1.

The following example shows that IntEcan be empty.

Example 2.8. InR², consider the subspaceE{x,−x; ∀x∈R}, with the induced norm and simplicial ordering as inExample 2.3b. ThenE{0,0}, hence IntE ∅.

For allxandyinE, byxywe meany−x∈IntE, sox0 meansx∈IntE. Then we have the following properties.

Proposition 2.9. LetEbe an ordered space having a norm. Then iThe zero element is not in IntE,

iiIfx0, then−x0,

iiiIfx0 andα∈R, thenαx0, ivIfx0 andα∈R⁻, thenαx0.

Proof. iAssume that 0∈IntE, then there is an-neighborhoodB0, of 0 such that

B0, {x∈E; x< } ⊆E, 2.8

so this open neighborhood contains a nonzeroxif not thenB0, {0}; hence,{0}is an open subset which is a contradiction, as metric spaces are Hausdorﬀ. Also, we have−x ∈ B0, , hence byProposition 2.2ii, we get a contradiction. Ifx 0, then 0−−x x ∈ IntE; hence, this provesii. To proveiiiassume thatx ∈ IntEand letα > 0. There exists an-neighborhoodBx, ofxsuch that

Bx, {e∈E;e−x< } ⊆E. 2.9

Consider that ballBαx, αofαx, we claim that it is contained inE. Ife ∈Bαx, α, then e/α−x < which means thate/α ∈ Bx, and thereforee/α ∈E so byO2, we get e ≥ 0, which proves the claim.ivfollows byiiandiii, hence the proposition has been checked.

Proposition 2.10. LetEbe an ordered space having a norm with IntE/∅. If > 0, then there existsc0, such thatc< .

(6)

Proof. Given that > 0 and let x ∈ IntE, by Proposition 2.9i, we have x /0 and by Proposition 2.9iii, we have x/x 0. Choose a real number t ∈ 0, and put c:tx/x. Thenc0 andct < , hence the lemma is checked.

Now let us consider a sequence{xn}^∞_n1in a normed ordered spaceE. We recall some properties that hold forRand try to extend it toE. AsRis totally ordered, we know thatxn

have a monotonic subsequence. This is not true in general even thoughx_nis assumed to be bounded sequence, the following example explains this.

Example 2.11. Consider thatER², with the simplicial ordering. Letγ be a bijection fromN ontoQ∩1,2and define the sequence{xn}^∞_n1byx_n γn,1/γn. Thenx_nis a bounded sequence in E; for example, 2,1is an upper bound, with no monotonic subsequence, as every two terms are not comparable. Also, it is a divergent sequence.

Remark 2.12. Recall that by the completeness propertyCP, we know thatRhas the bounded monotone convergence theorem. Also, the spacesRⁿandC0,1 Example 2.3b,e, resp.

have the bounded monotone convergence theorem.

The following example shows that the bounded monotone convergence theorem is not true in a normed ordered spaceE, in general. TheCPdoes not hold in the normed ordered spaceQ. Moreover, the Bolzano-Weierstrass theoremB-Wis not true in general.

Example 2.13. Consider the normed ordered space inExample 2.3cand the sequencex_n 11/nⁿ. It is bounded and increasing sequence but has no limit inQ. As every subsequence ofxnis convergent toe, so no subsequence is convergent inQ, that is,B-Wfails.

Now, let us consider a continuous functionf defined on a normed ordered space E into a normed ordered spaceF. The following examples show that the maximum-minimum theorem Max-Min and the intermediate value theorem IVT in the case of real-valued functions do not hold forf.

Example 2.14. Consider that the continuous functionf :R → R², which is defined byfx x,−x, and consider thatM 0,1. ThenMis a compact subset ofRandfMis the line segment from0,0to1,−1. Indeed, supfM 1,0and inffM 0,−1and both do not belong tofM.

More precisely, using known theorems about normed spaces and compactnesssee, e.g.,22, one may easily deduce the following.

Proposition 2.15. LetEbe a normed ordered space having (CP), letXbe a metric space andf:X → Ebe any continuous function. IfMis any compact subset ofX, thenfMattains its sup and inf in E.

Example 2.16IVT is invalid. Consider the functiong:R → R², which is defined by

gx

⎧⎨

⎩

x,−√ 1−x²

, −1≤x≤1,

x,0, x≥1 or x≤ −1. 2.10

(7)

Leth : R² → R² be the rotation mapping by the angleπ/4, that is, for any z x, y ∈ R², hz e^iπ/4z. Setf :h◦g, sofis a continuous function defined on real numbers with values inR². Now

f−1 −1,−1≤0,0≤1,1 f1, 2.11 but noc∈−1,1withfc 0,0.

3. E-Metric Spaces

Recall that in order to define a metric on a setX, it is necessary to have an ordered spaceE only. In this section, we defineE-metric spaces, and we give main examples. We characterize cone metric spaces as in 7, by using the notion ofE-metric space of a normed ordered spaceE. Then we generalize main theorems in real-valued metric spaces such as Cauchy, convergence theories, and contractive sequencessee, e.g.,21,22to the case ofE-metric spaces. Moreover, we get a fixed point theorem of a contractive mapping.

Definition 3.1. LetXbe any nonempty set and letEbe any ordered space, over the real scalars.

An orderedE-metric onXis anE-valued functiond^E:X×X → Esuch that for allx, y, and zinX, we have

idÊx, y>0 anddÊx, y 0 if and only ifxy, iidÊx, y dÊy, x,

iiidÊx, y≤dÊx, z dÊz, y.

Then the pairX, d^Eis calledE-metric space.

Now, consider an ordered spaceEwith a norm and consider anE-metric spaceX, d^E, letpbe a point inXandc∈IntE. Then the open ball inXcentered atpof radiuscis

B p, c

x∈X;d^E x, p

< c

. 3.1

Example 3.2. aFixingX E R, with the usual distance, is reduced to the usual metric space of real numbers, having the open intervals as open balls.

bConsider thatXR²andER²with the simplicial ordering. Define the function d^R²fromR²×R²intoR²by

d^R² x₁, y₁

,

x₂, y₂

|x2−x₁|, y₂−y₁ . 3.2 Thend^R² is a metric, and henceR², d^R²is anR²-metric space. Moreover, the open balls are realized as the open rectangles inR²space.

cRecallExample 2.3e, letX C0,1, and consider the normed spaceEC0,1.

Define the functiondÊ:X×X → EbydÊf, g |f−g|. ThenX, dÊis anE-metric space.

Moreover, fixg∈C0,1and letxbe a function with all its values are strictly positive; that

(8)

is,xnever touch thex-axis, so byExample 2.7we havex∈IntE. Then the open ball Bg, is a fiber that consists of all functionsfwith values

gx−x< fx< gx x, ∀x∈0,1. 3.3 Definition 3.3. LetEbe a normed ordered space, and letX, d^Ebe anE-metric space. Then a sequence{xn}^∞_n1inXis called convergent to a pointx₀ ∈X, if for allc∈IntE, there exists a positive integerNsuch thatd^Exn, x0< c, for alln > N.

If a sequence{xn}^∞_n1converges to a pointx₀∈X, then we write

nlim→ ∞xnx0, or simplyxn −→x0. 3.4

Definition 3.4. LetEbe a normed ordered space, and letX, d^Ebe anE-metric space. Then a sequence{xn}^∞_n1inXis called Cauchy, if for allc∈IntE, there exists a positive integerN such thatd^Exn, x_m< c, for alln, m > N.

Proposition 3.5. Let{xn}^∞_n1be a sequence in aE-metric spaceX, d^E, whereEis a normed ordered space. Ifx_nis convergent, then it is a Cauchy sequence.

Proof. Assume that for somex₀ ∈ X,x_n → x₀. Letc ∈IntE. Then there exists a positive integerNsuch thatdÊxn, x0< c/2, for alln > N. Therefore for alln, m > N,dÊxn, xm≤ dÊxn, x₀ dÊxm, x₀< c.

Proposition 3.6. LetEbe a normed ordered space with IntE/∅and letX, d^Ebe anE-metric space. Then any convergent sequence{xn}ⁿ_n1inXhas a unique limit.

Proof. Assume thatx_n → x₁andx_n → x₂and letc 0. Then there exist positive integers N₁andN₂such that

d^Exn, x₁< c

2, ∀n > N1, d^Exn, x₂< c

2, ∀n > N2.

3.5

Choose thatNmax{N1, N₂}1, then we get the following:

0≤dÊx1, x2≤dÊx1, xN dÊxN, x2≤ c 2 c

2 c. 3.6

AsEis a normed ordered space, there existsk > 0 such that dÊx1, x₂ ≤ kc. Asc an arbitrary, thenc → 0, which givesdÊx1, x20, thendÊx1, x2 0, hencex1x2. Definition 3.7. LetEbe a normed ordered space, and letX, dÊbe anE-metric space. Then X, dÊis called completeE-metric space if every Cauchy sequence inXis convergent.

Now, using the concepts ofE-metric spaces, let us have the following characterization of cone metric spacesDefinition 1.2in the sense of Huang and Zhang as in7.

(9)

Theorem 3.8. LetXbe any nonempty set andEbe a space over the real scalars. Then the following are equivalent.

aThe pairX, d^Eis anE-metric space, whereEis a normed ordered space, with IntE/∅, bThe pairX, dis a cone metric space.

Proof. Firstaimpliesb. Consider thatP E, then byProposition 2.6,P is closed, and as IntE/ ∅, we have P /{0}. If a, b ≥ 0 real numbers and x, y ∈ P, then, by O2 of Definition 2.1, we getax, by ≥ 0, then, byProposition 2.2iii, we have axby ∈ P. Now if x ∈ P and −x ∈ P, then by Proposition 2.2ii, we get x 0, so P is a cone in the sense ofDefinition 1.1. AsEis a normed space, then by definition,Pis a normal cone. Then considering thatdd^E, we get thatX, dis a cone metric space.

Conversely, assume thatX, dis a cone metric space. Then we have a conePinE, and the order with respect toP x ≤ y ⇔ y−x ∈ Pdefines a partial ordering onE. To check the translation invariantO1: given anyx, yandz∈ E, withx≤y. Then by the definition of the order,y−x∈P, therefore,yz−xz∈P, hencexz≤yz. To proveO2: by iiofDefinition 1.1, we know that 0∈P, and takingy0, we getax−0∈P, which means ax ≥ 0.O3is direct, henceEis a normed ordered space. By the definition of cone metric space, IntP/∅. Finally, the cone metricdonXis chosen to be the orderedE-metric as in Definition 3.1; hence, the theorem has been checked.

Now, using the above characterization, we have the following results, including a fixed-point theorem of a contractive mapping. Indeed,Proposition 2.10is used in the proof of Lemmas 1 and 4 of Haung and Zhang in7.

Corollary 3.9. Let Ebe a normed ordered space with IntE/∅, and letX, d^Ebe anE-metric space. Then a sequence{xn}^∞_n1inXconverges tox∈Xif and only ifd^Exn, x → 0 inE.

Proof. Direct by usingTheorem 3.8together with Lemma 1 in7.

Corollary 3.10. LetEbe a normed ordered space with IntE/∅, and letX, d^Ebe anE-metric space. Then a sequence{xn}^∞_n1 is Cauchy if and only if d^Exn, xm → 0 inE as n → ∞and m → ∞.

Proof. The proof is directed by usingTheorem 3.8together with Lemma 4 in7.

Finally, we have the following fixed-point theorem in normed ordered spaces.

Theorem 3.11. Let Ebe a normed ordered space with IntE/∅, and let X, d^Ebe a complete E-metric space. If a functionf :X → Xsatisfies the following contractive condition:

d^E

fx, f y

≤kd^E x, y

, for some k∈0,1, 3.7

thenfhas a unique fixed point inX.

Proof. It follows directly by usingTheorem 3.8and Theorem 1 in7.

Finally, and extending the notion in real-valued metric spaces, let us introduce the notion of a contractive sequence inE-metric space, then proving that it is indeed a suﬃcient to be a Cauchy sequence.

(10)

Definition 3.12. Let E be an ordered space, and let X, dÊ be an E-metric space. Then a sequence{xn}^∞_n1inX is called contractive if there exists a real numberl ∈ 0,1, such that dÊxn2, x_n1≤ldÊxn1, x_n.

Theorem 3.13. LetEbe a normed ordered space with IntE/∅, and letX, d^Ebe anE-metric space. Then every contractive sequence inXis Cauchy.

Proof. Suppose thatx_nis a contractive sequence inX. Then for some real numberl ∈ 0,1, we have

dÊx_n2, x_n1≤ldÊx_n1, xn≤ · · · ≤lⁿdÊx2, x1. 3.8 Therefore assuming thatm > n, we get the following:

dÊxm, x_n≤dÊxm, x_m−1 dÊx_m−1, x_m−2 · · ·dÊx_n1, x_n

≤

l^m−2l^m−3· · ·lⁿ⁻¹

d^Ex2, x₁

≤ lⁿ⁻¹

1−ld^Ex2, x₁.

3.9

Therefore, for somek∈0,1,

d^Exm, x_n≤klⁿ⁻¹ 1−l

dÊx2, x₁, 3.10 which implies thatdÊxm, x_n → 0 and thendÊxm, x_n → 0 inE, hence byCorollary 3.10, x_nis a Cauchy sequence.

Acknowledgment

The authors would like to thank the editor and the referee for their valuable comments and suggestions.

References

1 M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416–420, 2008.

2 M. Abbas and B. E. Rhoades, “Fixed and periodic point results in cone metric spaces,” Applied Mathematics Letters, vol. 22, no. 4, pp. 511–515, 2009.

3 M. A. Alghamadi, S. Radenovi´c, and N. Shahzad, “On some generalizations of commuting mappings,” Abstract and Applied Analysis, vol. 2012, Article ID 952052, 6 pages, 2012.

4 T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, vol. 65, no. 7, pp. 1379–1393, 2006.

5 D. Duki´c, Z. Kadelburg, and S. Radenovi´c, “Fixed point of geraghty type mappings in various generalized metric spaces,” Abstract and Applied Analysis, vol. 2011, Article ID 561245, 13 pages, 2011.

6 Z. Golubovi´c, Z. Kadelburg, and S. Radenovi´c, “Coupled coincidence points od mappings in ordered partial metric spaces,” Abstract and Applied Analysis. In press.

(11)

7 L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,”

Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.

8 X. Huang, C. Zhu, and X. Wen, “A common fixed point theorem in cone metric spaces,” International Journal of Mathematical Analysis, vol. 4, no. 13–16, pp. 721–726, 2010.

9 S. Jankovi´c, Z. Kadelburg, and S. Radenovi´c, “On cone metric spaces: a survey,” Nonlinear Analysis, vol. 74, no. 7, pp. 2591–2601, 2011.

10 Z. Kadelburg, M. Pavlovi´c, and S. Radenovi´c, “Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3148–3159, 2010.

11 Z. Kadelburg, P. P. Murthy, and S. Radenovi´c, “Common fixed points for expansive mappings in cone metric spaces,” International Journal of Mathematical Analysis, vol. 5, no. 25–28, pp. 1309–1319, 2011.

12 J. R. Morales and E. Rojas, “Cone metric spaces and fixed point theorems of T-Kannan contractive mappings,” International Journal of Mathematical Analysis, vol. 4, no. 1–4, pp. 175–184, 2010.

13 H. K. Nashine, Z. Kadelburg, and S. Radenovi´c, “Couple common fixed point theorems for W^∗- compatible mappings in ordered cone metric spaces,” Applied Mathematics and Computation, vol. 218, pp. 5422–5432, 2012.

14 S. Radenovi´c and Z. Kadelburg, “Generalized weak contractions in partially ordered metric spaces,”

Computers & Mathematics with Applications, vol. 60, no. 6, pp. 1776–1783, 2010.

15 W. Shatanawi, “Some coincidence point results in cone metric spaces,” Mathematical and Computer Modelling, vol. 55, no. 7-8, pp. 2023–2028, 2011.

16 J. S. Vandergraft, “Newton’s method for convex operators in partially ordered spaces,” SIAM Journal on Numerical Analysis, vol. 4, pp. 406–432, 1967.

17 E. Karapınar, “Couple fixed point theorems for nonlinear contractions in cone metric spaces,”

Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3656–3668, 2010.

18 W. Shatanawi, “Partially ordered cone metric spaces and coupled fixed point results,” Computers &

Mathematics with Applications, vol. 60, no. 8, pp. 2508–2515, 2010.

19 W. Shatanawi, “Some common coupled fixed point results in cone metric spaces,” International Journal of Mathematical Analysis, vol. 4, no. 45–48, pp. 2381–2388, 2010.

20 W. Shatanawi, “Onω-Compatible mappings and common coincidence point in cone metric spaces,”

Applied Mathematics Letters, vol. 25, no. 6, pp. 925–931, 2012.

21 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, NY, USA, Third edition, 1976.

22 E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, NY, USA, 1978.

(12)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Probability and Statistics

Journal of

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Operations Research

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Algebra

Discrete Dynamics in Nature and Society

Decision Sciences

Discrete Mathematics

^{Journal of}

Normed Ordered and E-Metric Spaces

Research Article