Coupled Fixed Point Theorems under Weak Contractions

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Volume 2012, Article ID 184534,9pages doi:10.1155/2012/184534

Research Article

Coupled Fixed Point Theorems under Weak Contractions

Y. J. Cho,

¹

Z. Kadelburg,

²

R. Saadati,

³

and W. Shatanawi

⁴

1Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

2Faculty of Mathematics, University of Belgrade, 11000 Beograd, Serbia

3Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran

4Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

Correspondence should be addressed to R. Saadati,[email protected] Received 6 October 2011; Accepted 28 December 2011

Academic Editor: Binggen Zhang

Copyrightq2012 Y. J. Cho 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.

Cho et al.Comput. Math. Appl. 612011, 1254–1260studied common fixed point theorems on cone metric spaces by using the concept of c-distance. In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of c-distance in cone metric spaces.

1. Introduction

Many fixed point theorems have been proved for mappings on cone metric spaces in the sense of Huang and Zhang1. For some more results on fixed point theory and applications in cone metric spaces, we refer the readers to2–15. Recently, Bhaskar and Lakshmikantham 16introduced the concept of a coupled coincidence point of a mappingFfromX×XintoX and a mappinggfromXintoXand studied fixed point theorems in partially ordered metric spaces. For some more results on couple fixed point theorems, refer to17–23.

Recently, Cho et al.7introduced a new concept ofc-distance in cone metric spaces, which is a cone version ofw-distance of Kada et al.24 see also25and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using thec-distance.

In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept ofc-distance.

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2. Preliminaries

In this paper, assume thatEis a real Banach space. LetP be a subset ofEwith intP/∅.

ThenP is called a cone if the following conditions are satisfied:

1Pis closed andP /{θ};

2a, b∈R,x, y∈Pimpliesaxby∈P;

3x∈P∩ −P impliesxθ.

For a coneP, define the partial ordering with respect toP byx yif and only if y−x∈P. We writex≺yto indicate thatxybutx /y, whilexystand fory−x∈intP.

It can be easily shown thatλ intP⊆intPfor all positive scalarsλ.

Definition 2.1see1. LetXbe a nonempty set. Suppose that the mappingd:X×X → E satisfies the following conditions:

1θdx, yfor allx, y∈Xanddx, y θif and only ifxy;

2dx, y dy, xfor allx, y∈X;

3dx, ydx, z dy, zfor allx, y, z∈X.

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

Definition 2.2see1. LetX, dbe a cone metric space. Letxnbe a sequence inX and x∈X.

1If, for anyc ∈ X with θ c, there existsN ∈ N such thatdxn, x cfor all n≥N, thenxnis said to be convergent to a pointx∈X andxis the limit ofxn. We denote this by limn→ ∞xnxorxn → xasn → ∞.

2If, for any c ∈ Ewithθ c, there existsN ∈ N such thatdxn, xm cfor all n, m≥N, thenxnis called a Cauchy sequence inX.

3The spaceX, dis called a complete cone metric space if every Cauchy sequence is convergent.

Definition 2.3 see 7. Let X, be a partially ordered set, and let F : X ×X → X be a function. Then the mapping F is said to have the mixed monotone property if Fx, y is monotone nondecreasing inxand is monotone nonincreasing iny; that is,

x1 x2 impliesF x1, y

F x2, y

2.1

for ally∈Xand

y1y2 impliesF x, y2

F x, y1

2.2

for allx∈X.

Definition 2.4see7. An elementx, y∈X×Xis called a coupled fixed point of a mapping F:X×X → XifFx, y xandFy, x y.

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Recently, Cho et al.7introduced the concept ofc-distance on cone metric spaceX, d which is a generalization ofw-distance of Kada et al.24.

Definition 2.5see7. LetX, dbe a cone metric space. Then a functionq:X×X → Eis called ac-distance onXif the following are satisfied:

q1θqx, yfor allx, y∈X;

q2qx, zqx, y qy, zfor allx, y, z∈X;

q3for anyx∈X, if there existsuu_x ∈Psuch thatqx, ynufor eachn≥1, then qx, yuwheneverynis a sequence inXconverging to a pointy∈X;

q4for anyc ∈Ewithθ c, there existse ∈Ewith 0 ≤esuch thatqz, x eand qz, ycimplydx, yc.

Cho et al.7noticed the following important remark in the concept ofc-distance on cone metric spaces.

Remark 2.6see7. Letqbe ac-distance on a cone metric spaceX, d. Then 1qx, y qy, xdoes not necessarily hold for allx, y∈X,

2qx, y θis not necessarily equivalent toxyfor allx, y∈X.

The following lemma is crucial in proving our results.

Lemma 2.7see7. LetX, dbe a cone metric space, and letqbe ac-distance onX. Letxnand ynbe sequences inXandx, y, z∈X. Suppose thatunis a sequence inPconverging toθ. Then the following hold:

1ifqxn, yu_nandqxn, zu_n, thenyz;

2ifqxn, y_nu_nandqxn, zu_n, thenynconverges to a pointz∈X;

3ifqxn, xmunfor eachm > n, thenxnis a Cauchy sequence inX;

4Ifqy, xnu_n, thenxnis a Cauchy sequence inX.

3. Main Results

In this section, we prove some coupled fixed point theorems by usingc-distance in partially ordered cone metric spaces.

Theorem 3.1. LetX,be a partially ordered set, and suppose thatX, dis a complete cone metric space. Letqbe ac-distance onX, and letF :X×X → Xbe a continuous function having the mixed monotone property such that

q F

x, y , F

x^∗, y^∗ k

2

qx, x^∗ q

y, y^∗ 3.1

for somek∈0,1and allx, y, x^∗, y^∗∈Xwithxx^∗∧yy^∗orxx^∗∧yy^∗. If there existx0, y0∈Xsuch thatx0Fx0, y0andFy0, x0y0, thenFhas a coupled fixed pointu, v.

Moreover, one hasqv, v θandqu, u θ.

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Proof. Letx0, y0 ∈ X be such thatx0 Fx0, y0andFy0, x0 y0. Letx1 Fx0, y0and y₁ Fy0, x₀. Since F has the mixed monotone property, we have x₀ x₁ andy₁ y₀. Continuing this process, we can construct two sequencesxnandyninXsuch that

xnF

x_n−1, y_n−1

x_n1F xn, yn

, y_n1F

yn, xn

ynF

y_n−1, x_n−1

. 3.2

Letn∈N. Now, by3.1, we have qxn, x_n1 q

F

x_n−1, y_n−1 , F

x_n, y_n k

2

qxn−1, xn q y_n−1, yn

, qxn1, x_n q

F x_n, y_n

, F

x_n−1, y_n−1 k

2

qxn, x_n−1 q

yn, y_n−1 .

3.3

From3.3, it follows that

qxn, x_n1 qxn1, xn k 2

qxn, x_n−1 q

yn, y_n−1

. 3.4

Similarly, we have

q

yn, y_n1 q

y_n1, yn

k 2

qxn, x_n−1 q

yn, y_n−1

. 3.5

Thus it follows from3.4and3.5that qxn, x_n1 qxn1, xn q

yn, y_n1 q

y_n1, yn

k

qxn−1, x_n q

y_n−1, y_n

qxn, x_n−1 q

y_n, y_n−1

. 3.6

Repeating3.6n-times, we get

qxn, x_n1 qx_n1, xn q

yn, y_n1 d

y_n1, yn

kⁿ

qx1, x0 q y1, y0

qx0, x1 q y0, y1

. 3.7

Thus we have

qxn, x_n1kⁿ

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ , q

yn, y_n1 kⁿ

qx1, x0 q y1, y0

qx0, x1 q y0, y1

. 3.8

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Letm, n∈N withm > n. Since

qxn, xm^m−1

in

qxi, x_i1,

q y_n, y_m

^m−1

in

q y_i, y_i1

,

3.9

andk <1, we have

qxn, x_m kⁿ 1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ , q

y_n, y_m kⁿ

1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ .

3.10

FromLemma 2.73, it follows thatxnandynare Cauchy sequences inX, d. SinceXis complete, there existu, v∈Xsuch thatx_n → uandy_n → v. SinceFis continuous, we have

x_n1F x_n, y_n

−→Fu, v, y_n1F

y_n, x_n

−→Fv, u. 3.11

By the uniqueness of the limits, we getu fu, vandvFv, u. Thusu, vis a coupled fixed point ofF.

Moreover, by3.1, we have

qu, u qFu, v, Fu, v k 2

qu, u qv, v , qv, v qFv, u, Fv, u k

2

qv, v qu, u .

3.12

Therefore, we get

qu, u qv, vk

qv, v qu, u

. 3.13

Sincek <1, we conclude thatqu, u qv, v θ, and hencequ, u θandqv, v θ. This completes the proof.

Theorem 3.2. In addition to the hypotheses ofTheorem 3.1, suppose that any two elementsxandy inXare comparable. Then the coupled fixed point has the formu, u, whereu∈X.

Proof. As in the proof ofTheorem 3.1, there exists a coupled fixed pointu, v∈X×X. Here uFu, vandvFv, u. By the additional assumption and3.1, we have

qu, v qFu, v, Fv, u k 2

qu, v qv, u , qv, u qFv, u, Fu, v k

2

qv, u qu, v .

3.14

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Thus we have

qu, v qv, uk

qv, u qu, v

. 3.15

Sincek < 1, we getqu, v qv, u θ. Hencequ, v θandqv, u θ. Letun θand x_nu. Then

qxn, uun,

qxn, vun. 3.16

FromLemma 2.71, we haveuv. Hence the coupled fixed point ofF has the formu, u.

This completes the proof.

Theorem 3.3. LetX,be a partially ordered set, and suppose thatX, dis a complete cone metric space. Letqbe ac-distance onX, and letF :X×X → Xbe a function having the mixed monotone property such that

q F

x, y , F

x^∗, y^∗ k

4

qx, x^∗ q

y, y^∗ 3.17

for somek∈0,1and allx, y, x^∗, y^∗ ∈Xwithxx^∗∧yy^∗orxx^∗∧y y^∗. Also, suppose thatXhas the following properties:

aifxnis a nondecreasing sequence inXwithx_n → x, thenx_nxfor alln≥1;

bifxnis a nonincreasing sequence inXwithx_n → x, thenxx_nfor alln≥1.

Assume there existx0, y0∈Xsuch thatx0Fx0, y0andFy0, x0y0. Ify0 x0, thenFhas a coupled fixed point.

Proof. As in the proof ofTheorem 3.1, we can construct two Cauchy sequencesxnandyn inXsuch that

x₀x₁ · · · x_n · · ·,

y₀ y₁ · · ·y_n · · ·. 3.18

Moreover, we have thatxnconverges to a pointu∈Xandynconverges tov∈X,

qxn, x_m kⁿ 1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ , q

yn, ym

kⁿ 1−k

qx1, x0 q y1, y0

qx0, x1 q y0, y1

3.19

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for eachn > m≥1. Byq3, we have qxn, u kⁿ

1−k

qx1, x0 q y1, y0

qx0, x1 q y0, y1

,

q y_n, v

kⁿ 1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ ,

3.20

and so

qxn, u q y_n, v

2kⁿ 1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁

. 3.21

By the propertiesaandb, we have

vy_ny₀x₀x_nu. 3.22

By3.17, we have

qxn, Fu, v q F

x_n−1, y_n−1

, Fu, v k

4

qxn−1, u q

y_n−1, v , q

yn, Fv, u q

F

y_n−1, x_n−1

, Fv, u k

4 q

y_n−1, v

qxn−1, u .

3.23

Thus we have

qxn, Fu, v q

yn, Fv, u k

2

qxn−1, u q

y_n−1, v

. 3.24

By3.21, we get

qxn, Fu, v q

y_n, Fv, u k

2 ·2kⁿ⁻¹ 1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ kⁿ

1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ .

3.25

Therefore, we have

qxn, Fu, v kⁿ 1−k

qx1, x₀ q y₁, y₀

qx0, x₁ q

y₀, y₁ , q

yn, Fv, u kⁿ

1−k

qx1, x0 q y1, y0

qx0, x1 q y0, y1

.

3.26

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By using3.20and3.26,Lemma 2.71shows thatuFu, vandvFv, u. Therefore, u, vis a coupled fixed point ofF. This completes the proof.

Example 3.4. LetEC¹_R0,1withx x∞x∞andP {x∈E:xt≥0, t∈0,1}.

LetX 0,∞ with usual order, and letd:X×X → Ebe defined bydx, yt |x−y|e^t. ThenX, dis an ordered cone metric spacesee7, Example 2.9. Further, letq:X×X → E be defined byqx, yt ye^t. It is easy to check thatqis a c-distance. Consider now the functionF:X×X → Xdefined by

F x, y

⎧⎨

⎩ 1 8

x−y

, x≥y, 0, x < y.

3.27

Then it is easy to see that

q F

x, y

, Fu, v 1

6

qx, u q

y, v 3.28

for allx, y, u, v ∈ Xwithx ≤ u∧y ≥ vorx ≥ u∧y ≤ v. Note that 0 ≤ F0,1and 1≥F1,0. Thus, byTheorem 3.1, it follows thatFhas a coupled fixed point inE. Here0,0 is a coupled fixed point ofF.

Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and TechnologyGrant No.: 2011–0021821.

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