Some Coupled Fixed Point Theorems in Cone Metric Spaces

(1)

Volume 2009, Article ID 125426,8pages doi:10.1155/2009/125426

Research Article

Some Coupled Fixed Point Theorems in Cone Metric Spaces

F. Sabetghadam,

¹

H. P. Masiha,

¹

and A. H. Sanatpour

²

1Department of Mathematics, K. N. Toosi University of Technology, Tehran, 16315-1618, Iran

2Department of Mathematics, Tarbiat Moallem University, Tehran, 15618-36314, Iran

Correspondence should be addressed to H. P. Masiha,[email protected] Received 17 July 2009; Revised 23 September 2009; Accepted 28 September 2009 Recommended by Jerzy Jezierski

We prove some coupled fixed point theorems for mappings satisfying diﬀerent contractive conditions on complete cone metric spaces.

Copyrightq2009 F. Sabetghadam 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.

1. Introduction

Recently, Huang and Zhang in1generalized the concept of metric spaces by considering vector-valued metrics cone metrics with values in an ordered real Banach space. They proved some fixed point theorems in cone metric spaces showing that metric spaces really doesnot provide enough space for the fixed point theory. Indeed, they gave an example of a cone metric spaceX, dand proved existence of a unique fixed point for a selfmapT ofX which is contractive in the category of cone metric spaces but is not contractive in the category of metric spaces. After that, cone metric spaces have been studied by many other authorssee 1–9and the references therein.

Regarding the concept of coupled fixed point, introduced by Bhaskar and Laksh- mikantham 10, we consider the corresponding definition for the mappings on complete cone metric spaces and prove some coupled fixed point theorems in the next section. First, we recall some standard notations and definitions in cone metric spaces.

A conePis a subset of a real Banach spaceEsuch that iPis closed, nonempty andP /{0};

iiifa, bare nonnegative real numbers andx, y∈P, thenaxby∈P;

iiiP∩−P {0}.

(2)

For a given coneP ⊆ E, the partial ordering≤with respect toP is defined byx≤ yif and only ify−x∈ P. The notation x ywill stand fory−x ∈intP, where intP denotes the interior ofP. Also, we will usex < yto indicate thatx≤yandx /y.

The conePis called normal if there exists a constantM >0 such that for everyx, y∈E if 0≤x≤ythen||x|| ≤M||y||. The least positive number satisfying this inequality is called the normal constant ofP see1. The coneP is called regular if every increasingdecreasing and bounded abovebelowsequence is convergent inE. It is known that every regular cone is normalsee1, or7, Lemma 1.1.

Huang and Zhang defined the concept of a cone metric space in1as follows.

Definition 1.1see1. LetX be a nonempty set and letEbe a real Banach space equipped with the partial ordering ≤ with respect to the cone P ⊆ E. Suppose that the mapping d:X×X → Esatisfies the following conditions:

d10≤dx, yfor allx, y∈Xanddx, y 0 if and only ifxy;

d2dx, y dy, xfor allx, y∈X;

d3dx, y≤dx, z dz, yfor allx, y, z∈X.

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

Definition 1.2see1. LetX, dbe a cone metric space,x∈Xand{xn}_n≥1be a sequence in X. Then

i{xn}_n≥1 converges tox, denoted by limn→ ∞xn x, if for every c ∈ Ewith 0 c there exists a natural numberNsuch thatdxn, xcfor alln≥N;

ii{xn}_n≥1 is a Cauchy sequence if for everyc ∈ Ewith 0 cthere exists a natural numberNsuch thatdxn, xmcfor alln, m≥N.

A cone metric spaceX, dis said to be complete if every Cauchy sequence in X is convergent inX. If for any sequence{xn}inXthere exists a subsequence{xni}of{xn}such that{xni}is convergent inX, then the cone metric spaceX, dis called sequentially compact.

Clearly, every sequentially compact cone metric space is complete. Huang and Zhang in 1 investigated the existence and uniqueness of the fixed point for a selfmapT on a cone metric space X, d. They considered diﬀerent types of contractive conditions on T. They also assumedX, dto be complete whenP is a normal cone, andX, dto be sequentially compact whenPis a regular cone. Later, in7, Rezapour and Hamlbarani improved some of the results in1by omitting the normality assumption of the coneP, whenX, dis complete.

See4,6,7,9for more related results aboutcompletecone metric spaces and fixed point theorems for diﬀerent types of mappings on these spaces.

In the rest of this paper, we always suppose thatEis a real Banach space,P ⊆ Eis a cone with intP /∅and≤is partial ordering with respect toP. We also note that the relations PintP⊆intPandλintP ⊆intP λ >0always hold true.

2. Main Results

For a given partially ordered set X, Bhaskar and Lakshmikantham in10 introduced the concept of coupled fixed point of a mappingF :X×X → X. Later in11Lakshmikantham and ´Ciri´c investigated some more coupled fixed point theorems in partially ordered sets. The following is the corresponding definition of coupled fixed point in cone metric spaces.

(3)

Definition 2.1. LetX, dbe a cone metric space. An elementx, y ∈ X×X is said to be a coupled fixed point of the mappingF:X×X → XifFx, y xandFy, x y.

In the next theorems of this section, we investigate some coupled fixed point theorems in cone metric spaces.

Theorem 2.2. LetX, dbe a complete cone metric space. Suppose that the mappingF:X×X → X satisfies the following contractive condition for allx, y, u, v∈X:

d F

x, y

, Fu, v

≤kdx, u ld y, v

, 2.1

wherek, lare nonnegative constants withkl <1. ThenFhas a unique coupled fixed point.

Proof. Choosex0, y0 ∈ Xand set x1 Fx0, y0,y1 Fy0, x0, . . . , x_n1 Fxn, yn,yn1 Fyn, xn. Then by2.1we have

dxn, xn1 d F

xn−1, yn−1 , F

xn, yn

≤kdx_n−1, xn ld y_n−1, yn

, 2.2

and similarly,

d

yn, yn1 d

F

yn−1, xn−1 , F

yn, xn

≤kd y_n−1, yn

ldx_n−1, xn. 2.3

Therefore, by letting

dndxn, xn1 d

yn, yn1

, 2.4

we have

dn dxn, xn1 d

yn, yn1

≤kdx_n−1, xn ld y_n−1, yn

kd

y_n−1, yn

ldx_n−1, xn

≤kl

dxn−1, xn d yn−1, yn

kld_n−1.

2.5

Consequently, if we setδklthen for eachn∈Nwe have

0≤dn≤δdn−1≤δ²dn−2≤ · · · ≤δⁿd0. 2.6

(4)

Ifd00 thenx0, y0is a coupled fixed point ofF. Now, letd0 >0. For eachn≥mwe have dxn, xm≤dxn, xn−1 dxn−1, xn−2 · · ·dxm1, xm,

d yn, ym

≤d

yn, yn−1 d

yn−1, yn−2

· · ·d

ym1, ym

. 2.7

Therefore,

dxn, xm d yn, ym

≤dn−1dn−2· · ·dm

≤

δⁿ⁻¹δⁿ⁻²· · ·δ^m d0

≤ δ^m 1−δ d0,

2.8

which implies that{xn}and{yn}are Cauchy sequences inX, and there existx^∗, y^∗∈Xsuch that lim_n_{→ ∞}xn x^∗and lim_n_{→ ∞}yn y^∗. Letc∈Ewith 0c. For everym∈Nthere exists N∈Nsuch thatdxn, x^∗c/2manddyn, y^∗c/2mfor alln≥N. Thus

d F

x^∗, y^∗ , x^∗

≤d F

x^∗, y^∗ , xN1

dxN1, x^∗ d

F x^∗, y^∗

, F xN, yN

dxN1, x^∗

≤kdxN, x^∗ ld yN, y^∗

dx_N1, x^∗ kl c

2m c 2m ≤ c

m.

2.9

Consequently,dFx^∗, y^∗, x^∗ c/mfor allm ≥ 1. Thus,dFx^∗, y^∗, x^∗ 0 and hence Fx^∗, y^∗ x^∗. Similarly, we haveFy^∗, x^∗ y^∗meaning thatx^∗, y^∗is a coupled fixed point ofF.

Now, ifx, yis another coupled fixed point ofF,then

d x, x^∗

d F

x, y , F

x^∗, y^∗

≤kd x, x^∗

ld y, y^∗

, d

y, y^∗ d

F y, x

, F

y^∗, x^∗

≤kd y, y^∗

ld x, x^∗

, 2.10

and therefore,

d x, x^∗

d y, y^∗

≤kl d

x, x^∗ d

y, y^∗

. 2.11

Sincekl <1,2.11implies thatdx, x^∗ dy, y^∗ 0. Hence, we havex, y x^∗, y^∗ and the proof of the theorem is complete.

It is worth noting that when the constants in Theorem 2.2 are equal we have the following corollary.

(5)

Corollary 2.3. LetX, dbe a complete cone metric space. Suppose that the mappingF:X×X → X satisfies the following contractive condition for allx, y, u, v∈X:

d F

x, y

, Fu, v

≤ k 2

dx, u d y, v

, 2.12

wherek∈0,1is a constant. ThenFhas a unique coupled fixed point.

Example 2.4. LetER²,P {x, y∈R² :x, y≥0} ⊆R²,andX 0,1. Defined:X×X → E with dx, y |x−y|,|x−y|. ThenX, d is a complete cone metric space. Consider the mappingF:X×X → XwithFx, y xy/6. ThenFsatisfies the contractive condition 2.12fork1/3, that is,

d F

x, y

, Fu, v

≤ 1 6

dx, u d y, v

. 2.13

Therefore, byCorollary 2.3,F has a unique coupled fixed point, which in this case is0,0.

Note that if the mappingF :X×X → Xis given byFx, y xy/2, thenFsatisfies the contractive condition2.12fork1, that is,

d F

x, y

, Fu, v

≤ 1 2

dx, u d y, v

. 2.14

In this case,0,0and1,1are both coupled fixed points ofF and hence the coupled fixed point ofF is not unique. This shows that the conditionk < 1 in corollary2.12and hence kl <1 inTheorem 2.2are optimal conditions for the uniqueness of the coupled fixed point.

Theorem 2.5. LetX, dbe a complete cone metric space. Suppose that the mappingF:X×X → X satisfies the following contractive condition for allx, y, u, v∈X:

d F

x, y

, Fu, v

≤kd F

x, y , x

ldFu, v, u, 2.15

Proof. Choosex0, y0 ∈ Xand set x1 Fx0, y0,y1 Fy0, x0, . . . , xn1 Fxn, yn,yn1 Fyn, xn. Then by applying2.15we get

dxn, x_n1≤δdxn, x_n−1, d

yn, yn1

≤δd

yn, yn−1

, 2.16

whereδ k/1−l < 1. This implies that{xn} and{yn} are Cauchy sequences inX, d and therefore by the completeness ofX, there existx^∗,y^∗ ∈Xsuch that lim_n_{→ ∞}xn x^∗and

(6)

limn→ ∞yny^∗. Letm∈Nand choose a natural numberNsuch thatdxn, x^∗ 1−l/4mc for alln≥N. Thus,

d F

x^∗, y^∗ , x^∗

≤d

x_N1, F

x^∗, y^∗

dx_N1, x^∗ d

F xN, yN

, F

x^∗, y^∗

dxN1, x^∗

≤kd F

xN, yN

, xN

ld

F x^∗, y^∗

, x^∗

dxN1, x^∗,

2.17

which implies that

d F

x^∗, y^∗ , x^∗

≤ k

1−l dxN1, xN 1

1−l dxN1, x^∗ c

m. 2.18

Sincem∈Nwas arbitrary,dFx^∗, y^∗, x^∗ 0 or equivalentlyFx^∗, y^∗ x^∗. Similarly, one can getFy^∗, x^∗ y^∗showing thatx^∗, y^∗is a coupled fixed point ofF.

Now, ifx, yis another coupled fixed point ofF,then by applying2.15we have d

x, x^∗ d

F x, y

, F

x^∗, y^∗

≤kd F

x, y , x

ld F

x^∗, y^∗ , x^∗

0, 2.19

and thereforexx^∗. Similarly, we can getyy^∗and hencex, y x^∗, y^∗.

Theorem 2.6. LetX, dbe a complete cone metric space. Suppose that the mappingF:X×X → X satisfies the following contractive condition for allx, y, u, v∈X,

d F

x, y

, Fu, v

≤kd F

x, y , u

ldFu, v, x, 2.20

Proof. First, note that the uniqueness of the coupled fixed point is an obvious result ofkl <1 in2.20. To prove the existence of the fixed point, letx0, y0 ∈ X and choose the sequence {xn}and{yn}like in the proof ofTheorem 2.5, that isx1Fx0, y0,y1Fy0, x0, . . . , xn1 Fxn, yn,yn1 Fyn, xn. Then by applying2.20we have

dxn, x_n1 d F

x_n−1, y_n−1 , F

xn, yn

≤kd F

xn−1, yn−1 , xn

ld

F xn, yn

, xn−1

≤l d

F xn, yn

, xn

dxn, xn−1 ,

2.21

which implies

dxn, xn1≤ l

1−ldxn, xn−1. 2.22

(7)

Similarly, one can get

d

yn, yn1

≤ l 1−ld

yn, yn−1

. 2.23

Therefore,{xn}and{yn}are Cauchy sequences inX, dand hence by the completeness of X, there existx^∗, y^∗ ∈Xsuch that lim_{n→ ∞}xn x^∗and lim_n_{→ ∞}yn y^∗. Letc∈Ewith 0c and for eachm ∈Nchoose a natural numberNsuch thatdxn, x^∗ 1−l/4mcfor all n≥N. Thus,

d F

x^∗, y^∗ , x^∗

≤d

xN1, F

x^∗, y^∗

dxN1, x^∗ d

F xN, yN

, F

x^∗, y^∗

dx_N1, x^∗

≤kd F

xN, yN

, x^∗ ld

F x^∗, y^∗

, xN

dxN1, x^∗,

2.24

which implies

d F

x^∗, y^∗ , x^∗

≤ 1k

1−ldxN1, x^∗ l

1−ldxN, x^∗ c

m. 2.25

Sincem∈Nwas arbitrary,dFx^∗, y^∗, x^∗ 0 or equivalentlyFx^∗, y^∗ x^∗. Similarly, one can getFy^∗, x^∗ y^∗and hencex^∗, y^∗is a coupled fixed point ofF.

When the constants in Theorems2.5and2.6are equal, we get the following corollaries.

d F

x, y

, Fu, v

≤ k 2

d F

x, y , x

dFu, v, u

, 2.26

d F

x, y

, Fu, v

≤ k 2

d F

x, y , u

dFu, v, x

, 2.27

Remark 2.9. Note that inTheorem 2.5, if the mappingF:X×X → Xsatisfies the contractive condition2.15for allx, y, u, v∈X, thenFalso satisfies the following contractive condition:

d F

x, y

, Fu, v d

Fu, v, F x, y

≤kdFu, v, u ld F

x, y , x

. 2.28

(8)

Consequently, by adding2.15and2.28,Falso satisfies the following:

d F

x, y

, Fu, v

≤ kl 2 d

F x, y

, x kl

2 dFu, v, u, 2.29

which is a contractive condition of the type 2.26in Corollary 2.7with equal constants.

Therefore, one can also reduce the proof of general case2.15inTheorem 2.5to the special case of equal constants. A similar argument is valid for the contractive conditions2.20in Theorem 2.6and2.27inCorollary 2.8.

Acknowledgment

The authors would like to thank the referees for their valuable and useful comments.

References

1 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.

2 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.

3 R. H. Haghi and Sh. Rezapour, “Fixed points of multifunctions on regular cone metric spaces,”

Expositiones Mathematicae. In press.

4 D. Ili´c and V. Rakoˇcevi´c, “Common fixed points for maps on cone metric space,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 876–882, 2008.

5 D. Klim and D. Wardowski, “Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5170–

5175, 2009.

6 S. Radenovi´c, “Common fixed points under contractive conditions in cone metric spaces,” Computers and Mathematics with Applications, vol. 58, no. 6, pp. 1273–1278, 2009.

7 Sh. Rezapour and R. Hamlbarani, “Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 719–724, 2008.

8 Sh. Rezapour and R. H. Haghi, “Fixed point of multifunctions on cone metric spaces,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 825–832, 2009.

9 Sh. Rezapour and M. Derafshpour, “Some common fixed point results in cone metric spaces,” to appear in Journal of Nonlinear and Convex Analysis.

10 T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006.

11 V. Lakshmikantham and L. ´Ciri´c, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009.