Coupled Coincidence Points in Partially Ordered Cone Metric Spaces with a c-Distance

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Volume 2012, Article ID 312078,15pages doi:10.1155/2012/312078

Research Article

Coupled Coincidence Points in Partially Ordered Cone Metric Spaces with a c-Distance

Wasfi Shatanawi,

¹

Erdal Karapınar,

²

and Hassen Aydi

³

1Department of Mathematics, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

2Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey

3Institut Sup´erieur d’Informatique et des Technologies de Communication De Hammam Sousse, Universit´e de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia

Correspondence should be addressed to Erdal Karapınar,[email protected] Received 22 May 2012; Revised 25 June 2012; Accepted 25 June 2012

Academic Editor: Alexander Timokha

Copyrightq2012 Wasfi Shatanawi 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.2012proved some coupled fixed point theorems in partially ordered cone metric spaces by using the concept of a c-distance in cone metric spaces. In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of a c-distance. Our results generalize several well-known comparable results in the literature. Also, we introduce an example to support the usability of our results.

1. Introduction and Preliminaries

Fixed point theory is an essential tool in functional nonlinear analysis. Consequently, fixed point theory has wide applications areas not only in the various branches of mathematics see, e.g., 1, 2but also in many fields, such as, chemistry, biology, statistics, economics, computer science, and engineeringsee, e.g., 3–11. For example, fixed point results are incredibly useful when it comes to proving the existence of various types of Nash equilibria see, e.g.,7in economics. On the other hand, fixed point theorems are vital for the existence and uniqueness of diﬀerential equations, matrix equations, integral equations see, e.g., 1, 2. Banach contraction mapping principle Banach fixed point theorem is one of the most powerful theorems of mathematics and hence fixed point theory. Huang and Zhang12 generalized the Banach contraction principle by replacing the notion of usual metric spaces by the notion of cone metric spaces. Then many authors obtained many fixed and common fixed point theorems in cone metric spaces. For some works in cone metric spaces, we may refer the readeras examplesto13–24. The concept of a coupled fixed point of a mapping F:X×X → Xwas initiated by Bhaskar and Lakshmikantham25, while Lakshmikantham

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and ´Ciri´c26initiated the notion of coupled coincidence point of mappingsF:X×X → X andg :X → X and studied some coupled coincidence point theorems in partially ordered metric spaces. For some coupled fixed point and coupled coincidence point theorems, we refer the reader to27–34.

In the present paper,N^∗is the set of positive integers andEstands for a real Banach space. LetPbe a subset ofE. We will always assume that the conePhas a nonempty interior IntP such cones are called solid. ThenP is called a cone if the following conditions are satisfied:

1P is closed andP /{θ},

2a, b∈R,x, y∈Pimpliesaxby∈P, 3x∈P∩ −P impliesxθ.

For a coneP, define a partial orderingwith respect toP byxyif and only ify−x∈P.

We will writex≺yto indicate thatxybutx /y, whilexywill stand fory−x∈IntP.

It can be easily shown thatλIntP⊆IntPfor all positive scalarλ.

Definition 1.1see12. LetX be a nonempty set. Suppose the mapping d : X ×X → E satisfies

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.

Bhaskar and Lakshmikantham 25 introduced the notion of mixed monotone property of the mappingF:X×X → X.

Definition 1.2 see 25. Let X,≤ be a partially ordered set and F : X × X → X be a mapping. Then the mapping F is said to have mixed monotone property if Fx, y is monotone nondecreasing inxand is monotone nonincreasing iny; that is, for anyx, y∈X,

x1≤x2 impliesF x1, y

≤F x2, y

, ∀y∈X, y₁≤y₂ impliesF

x, y₂

≤F x, y₁

, ∀x∈X. 1.1

Inspired byDefinition 1.2, Lakshmikantham and ´Ciri´c in26introduced the concept of ag-mixed monotone mapping.

Definition 1.3 see26. Let X,≤ be a partially ordered set and F : X ×X → X. Then the mapping F is said to have mixed gg-monotone property if Fx, y is monotone g- nondecreasing inxand is monotoneg-nonincreasing iny; that is:

gx₁≤gx₂ impliesF x₁, y

≤F x₂, y

, ∀y∈X, gy1≤gy2 impliesF

x, y2

≤F x, y1

, ∀x∈X. 1.2

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Definition 1.4see25. An elementx, y∈X×Xis called a coupled fixed point of a mapping F:X×X → Xif

F x, y

x, F y, x

y. 1.3

Definition 1.5see26. An elementx, y∈X×X is called a coupled coincidence point of the mappingsF :X×X → Xandg:X → Xif

F x, y

gx, F y, x

gy. 1.4

Recently, Cho et al. 35 introduced the concept ofc-distance on cone metric space X, dwhich is a generalization ofw-distance of Kada et al.36 see also37,38.

Definition 1.6see35. LetX, dbe a cone metric space. Then a functionq:X×X → Eis called ac-distance onXif the following is satisfied:

q1θqx, yfor allx, y∈X,

q2qx, zqx, y qy, zfor allx, y, z∈X,

q3for eachx ∈ X andn ≥ 1, ifqx, yn ufor someu ux ∈ P, thenqx, y u wheneverynis a sequence inXconverging to a pointy∈X,

q4for allc ∈Ewithθ c, there existse ∈Ewith 0 ≤e such thatqz, x e, and qz, yeimpliesdx, yc.

Cho et al.35noticed the following important remark in the concept ofc-distance on cone metric spaces.

Remark 1.7see35. 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.

Very recently, Cho et al.39proved the following existence theorems.

Theorem 1.8see39. LetX,be a partially ordered set and suppose thatX, dis a complete cone metric space. Letqbe ac-distance onXand 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∗

, 1.5

for somek∈0,1and allx, y, x∗, y∗ ∈Xwithxx∗∧yy∗orxx∗∧yy∗. If there existx₀, y₀∈Xsuch thatx₀Fx0, y₀andy₀Fy0, x₀, thenFhas a coupled fixed pointu, v.

Moreover, one hasqv, v qu, u θ.

Theorem 1.9see39. 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∗

, 1.6

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

1ifxnis a nondecreasing sequence inXwithx_n → x, thenx_nxfor alln≥1, 2ifxnis a nonincreasing sequence inXwithx_n → x, thenxx_nfor alln≥1.

If there existx₀, y₀ ∈ Xsuch thatx₀ Fx0, y₀andy₀ Fy0, x₀, thenFhas a coupled fixed pointu, v. Moreover, one has qv, v qu, u θ.

For other fixed point results using ac-distance, see40.

In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion ofc-distance. Our results generalize Theorems1.8and1.9.

We consider an application to illustrate our result is usefulseeSection 3.

2. Main Results

The following lemma is essential in proving our results.

Lemma 2.1see35. LetX, dbe a cone metric space, and letqbe a cone distance onX. Letxn andynbe sequences inXandx, y, z ∈X. Suppose thatunis a sequence inP converging toθ.

Then the following holds.

1Ifqxn, yu_nandqxn, zu_n, thenyz.

2Ifqxn, y_nu_nandqxn, zu_n, thenynconverges toz.

3Ifqxn, x_mu_nform > n, thenxnis a Cauchy sequence inX.

4Ifqy, xnu_n, thenxnis a Cauchy sequence inX.

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

Theorem 2.2. LetX,≤be a partially ordered set and suppose thatX, dis a cone metric space. Let qbe ac-distance onX. LetF:X×X → Xandg:X → Xbe two mappings such that

q F

x, y , F

x∗, y∗

q F

y, x , F

y∗, x∗

k q

gx, gx∗

q

gy, gy∗

, 2.1

for somek∈0,1and for allx, y, x∗, y∗ ∈Xwithgx≤gx∗∧gy≥gy∗orgx≥gx∗∧gy≤ gy∗. Assume thatFandgsatisfy the following conditions:

1Fis continuous,

2gis continuous and commutes withF, 3FX×X⊆gX,

4 X, dis complete,

5Fhas the mixedg-monotone property.

If there existx0, y0∈Xsuch thatgx0≤Fx0, y0andFy0, x0≤gy0, thenFandghave a coupled coincidence pointu, v. Moreover, one hasqgu, gu θandqgv, gv θ.

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Proof. Letx0, y0 ∈ X be such thatgx0 ≤ Fx0, y0and Fy0, x0 ≤ gy0. SinceFX×X ⊆ gX, we can choosex₁, y₁ ∈ X such thatgx₁ Fx0, y₀andgy₁ Fy0, x₀. Again since FX×X ⊆ gX, we can choosex₂, y₂ ∈ Xsuch thatgx₂ Fx1, y₁andgy₂ Fy1, x₁. SinceFhas the mixedg-monotone property, we havegx0 ≤gx1 ≤gx2andgy₂≤gy1≤gy0. Continuing this process, we can construct two sequencesxnandyninXsuch that

gx_nF

x_n−1, y_n−1

≤gx_n1F x_n, y_n

, gy_n1F

y_n, x_n

≤gy_nF

y_n−1, x_n−1

. 2.2

Letn∈N^∗. Then by2.1, we have q

gx_n, gx_n1 q

gy_n, gy_n1 q

F

x_n−1, y_n−1 , F

x_n, y_n q

F

y_n−1, x_n−1 , F

y_n, x_n k

q

gx_n−1, gxn

q

gy_n−1, gyn

.

2.3

Repeating2.3n-times, we get q

gx_n, gx_n1 q

gy_n, gy_n1 kⁿ

q

gx₀,gx₁ q

gy₀, gy₁

. 2.4

Thus, we have

q

gxn, gx_n1 kⁿ

q

gx0, gx1

q

gy0, gy1

, 2.5

q

gy_n, gy_n1 kⁿ

q

gx₀, gx₁ q

gy₀, gy₁

. 2.6

Letm, n∈N^∗withm > n. Then byq2and2.5, we have

q

gx_n, gx_m ^m−1

in

q

gx_i, gx_i1

^m−1

in

kⁱ q

gx₀, gx₁ q

gy₀, gy₁

kⁿ 1−k

q

gx₀, gx₁ q

gy₀, gy₁ .

2.7

Similarly, we have

q

gy_n, gy_m kⁿ

1−k q

gx₀, gx₁ q

gy₀, gy₁

. 2.8

From part3ofLemma 2.1, we conclude thatgxnandgynare Cauchy sequences in in X, d. SinceX is complete, there areu, v ∈ X such thatgx_n → uand gy_n → v. Using

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the continuity ofg, we getggxn → guandggyn → gv. Also, by continuity ofFand commutativity ofFandg, we have

gu lim

n→ ∞g gx_n1

lim

n→ ∞g F

xn, yn

lim

n→ ∞F

gxn, gyn

Fu, v,

gv lim

n→ ∞g gy_n1

lim

n→ ∞g F

y_n, x_n lim

n→ ∞F

gy_n, gx_n

Fv, u. 2.9

Hence,u, vis a coupled coincidence point ofFandg. Moreover, by2.1we have q

gu, gu q

gv, gv

qFu, v, Fu, v qFv, u, Fv, u k

q gu, gu

q

gv, gv

. 2.10

Since k < 1, we conclude that qgu, gu qgv, gv θ, and hence qgu, gu θ and qgv, gv θ.

The continuity ofFinTheorem 2.2can be dropped. For this, we present the following useful lemma which is a variant ofLemma 2.1,1.

Lemma 2.3. LetX, dbe a cone metric space, and letqbe ac-distance onX. Letxnbe a sequence in X. Suppose thatαnandβnare sequences inPconverging toθ. Ifqxn, yα_nandqxn, zβ_n, thenyz.

Proof. Letc θbe arbitrary. Sinceα_n → θ, so there existsN₁ ∈Nsuch thatα_n c/2 for alln ≥ N₁. Similarly, there existsN₂ ∈ Nsuch thatβ_n c/2 for alln ≥ N₂. Thus, for all N≥max{N1, N2}, we have

q x_n, y

c

2, qxn, z c

2. 2.11

Takeec/2, so byq4, we get thatdy, zcfor eachcθ; henceyz.

Theorem 2.4. LetX,≤be a partially ordered set and suppose thatX, dis a cone metric space. Let qbe ac-distance onX. LetF:X×X → Xand letg :X → Xbe two mappings such that

q F

x, y , F

x∗, y∗

q F

y, x , F

y∗, x∗

k q

gx, gx∗

q

gy, gy∗

, 2.12

for somek∈0,1and for allx, y, x∗, y∗ ∈Xwithgx≤gx∗∧gy≥gy∗orgx≥gx∗∧gy≤ gy∗. Assume thatFandgsatisfy the following conditions:

1FX×X⊆gX,

2gXis a complete subspace ofX, 3Fhas the mixedg-monotone property.

Suppose thatXhas the following properties:

iif a nondecreasing sequencex_n → x, thenx_n≤xfor alln, iiif a nonincreasing sequencex_n → x, thenx≤x_nfor alln.

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Assume there existx0, y0 ∈ Xsuch thatgx0 ≤ Fx0, y0andFy0, x0≤ gy0. ThenF andg have a coupled coincidence point, sayu, v∈X×X. Also,qgu, gu qgv, gv θ.

Proof. As in the proof ofTheorem 2.2, we can construct two Cauchy sequencesgxnand gynin the complete cone metric spacegX, d. Then, there existu, v∈Xsuch thatgxn → guandgy_n → gv. Similarly we have for allm > n≥1

q

gxn, gxm

kⁿ 1−k

q

gx0, gx1

q

gy0, gy1

,

q

gyn, gym

kⁿ 1−k

q

gx0, gx1

q

gy0, gy1

.

2.13

Byq3, we get that

q

gxn, gu kⁿ

1−k q

gx0, gx1

q

gy0, gy1

, 2.14

q

gyn, gv kⁿ

1−k q

gx0, gx1

q

gy0, gy1

. 2.15

By summation, we get that

q

gxn, gu q

gyn, gv 2 kⁿ

1−k q

gx0, gx1

q

gy0, gy1

. 2.16

Sincegxnis nondecreasing andgynis nonincreasing, using the propertiesi,iiofX, we have

gxn≤gu, gv≤gyn, ∀n≥0. 2.17

From this and2.14, we have q

gx_n, Fu, v q

gy_n, Fv, u q

F

x_n−1, y_n−1

, Fu, v q

F

y_n−1, x_n−1

, Fv, u k

q

gx_n−1, gu q

gy_n−1, gv .

2.18

Therefore q

gx_n, Fu, v q

gy_n, Fv, u k

q

gx_n−1, gu q

gy_n−1, gv

. 2.19

By2.16, we have q

gx_n, Fu, v q

gy_n, Fv, u k

q

gx_n−1, gu q

gy_n−1, gv k2kⁿ⁻¹

1−k q

gx₀, gx₁ q

gy₀, gy₁ 2kⁿ

1−k q

gx0, gx1

q

gy0, gy1

.

2.20

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This implies that

q

gx_n, Fu, v 2kⁿ

1−k q

gx₀, gx₁ q

gy₀, gy₁

, 2.21

q

gy_n, Fv, u 2kⁿ

1−k q

gx₀, gx₁ q

gy₀, gy₁

. 2.22

By 2.14, 2.21 and Lemma 2.3, we obtain gu Fu, v. Similarly, by 2.15, 2.22, and Lemma 2.3, we obtaingvFv, u. Also, adjusting as the proof ofTheorem 2.2, we get that

q gu, gu

q gv, gv

θ. 2.23

Corollary 2.5. LetX,≤be a partially ordered set and suppose thatX, dis a cone metric space. Let qbe ac-distance onX. LetF:X×X → X, and letg:X → Xbe two mappings such that

q F

x, y , F

x∗, y∗

aq

gx, gx∗

bq

gy, gy∗

, 2.24

for somea, b∈0,1withab <1 and for allx, y, x∗, y∗ ∈Xwithgx≤gx∗∧gy≥gy∗or gx≥gx∗∧gy≤gy∗. Assume thatFandgsatisfy the following conditions:

1Fis continuous,

2gis continuous and commutes withF, 3FX×X⊆gX,

4 X, dis complete,

5Fhas the mixedg-monotone property.

If there existx₀, y₀∈Xsuch thatgx₀≤Fx0, y₀andFy0, x₀≤gy₀, thenFandghave a coupled coincidence pointu, v. Moreover, one hasqgu, gu θandqgv, gv θ.

Proof. Givenx, x∗, y, y∗ ∈Xsuch thatgx≤gx∗∧gy≥gy∗. By2.24, we have q

F x, y

, F

x∗, y∗

aq

gx, gx∗

bq

gy, gy∗

, q

F y, x

, F

y∗, x∗

aq

gy, gy∗

bq

gx, gx∗

. 2.25

Thus q

F x, y

, F

x∗, y∗

q F

y, x , F

y∗, x∗

ab q

gx, gx∗

q

gy, gy∗

. 2.26 Sinceab <1, the result follows fromTheorem 2.2.

Corollary 2.6. LetX,≤be a partially ordered set and suppose thatX, dis a complete cone metric space. Letqbe ac-distance onX. LetF : X×X → Xbe a continuous mapping having the mixed monotone property such that

q F

x, y , F

x∗, y∗

aqx, x∗ bq y, y∗

, 2.27

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for somea, b ∈ 0,1with ab < 1 and for allx, y, x∗, y∗ ∈ X with x ≤ x∗∧y ≥ y∗or x≥x∗∧y ≤y∗. If there existx₀, y₀ ∈Xsuch thatx₀ ≤Fx0, y₀andFy0, x₀≤y₀, thenF has a coupled fixed pointx, y. Moreover, one hasqx, x θand qy, y θ.

Proof. It follows fromCorollary 2.5by takinggIXthe identity map.

Corollary 2.7. LetX,≤be a partially ordered set and suppose thatX, dis a cone metric space. Let qbe ac-distance onX. LetF:X×X → Xandg:X → Xbe two mappings such that

q F

x, y , F

x∗, y∗

aq

gx, gx∗

bq

gy, gy∗

, 2.28

for somea, b∈0,1withab <1 and for allx, y, x∗, y∗ ∈Xwithgx≤gx∗∧gy≥gy∗or gx≥gx∗∧gy≤gy∗. Assume thatFandgsatisfy the following conditions:

1FX×X⊆gX,

2gXis a complete subspace ofX, 3Fhas the mixedg-monotone property.

Suppose thatXhas the following properties:

iif a nondecreasing sequencex_n → x, thenx_n≤xfor alln, iiif a nonincreasing sequencex_n → x, thenx≤x_nfor alln.

Assume there existx0, y0 ∈ Xsuch thatgx0 ≤ Fx0, y0andFy0, x0≤ gy0. ThenF andg have a coupled coincidence point.

Proof. It follows from Theorem 2.4 by similar arguments to those given in proof of Corollary 2.5.

Corollary 2.8. LetX,≤be a partially ordered set and suppose thatX, dis a complete cone metric space. Letqbe ac-distance onX. LetF : X×X → X be a mapping having the mixed monotone property such that

q F

x, y , F

x∗, y∗

aq

x, x∗ bq y, y∗

, 2.29

for somea, b ∈ 0,1with ab < 1 and for allx, y, x∗, y∗ ∈ X with x ≤ x∗∧y ≥ y∗or x≥x∗∧y≤y∗. Suppose thatXhas the following properties:

iif a nondecreasing sequencex_n → x, thenx_n≤xfor alln, iiif a nonincreasing sequencex_n → x, thenx≤x_nfor alln.

Assume there existx0, y0 ∈X such thatx0 ≤ Fx0, y0andFy0, x0≤ y0. ThenFhas a coupled fixed point.

Proof. It follows fromCorollary 2.7by takinggI_Xthe identity map.

Corollary 2.9. 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 mapping having the mixed monotone property such that

q F

x, y , F

x∗, y∗

q F

y, x , F

y∗, x∗

k

qx, x∗ q y, y∗

, 2.30

for somek∈0,1and for allx, y, x∗, y∗ ∈Xwithx≤x∗∧y≥y∗orx≥x∗∧y≤y∗.

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If there existx0, y0 ∈ Xsuch thatx0 ≤ Fx0, y0andFy0, x0≤ y0, thenFhas a coupled fixed pointx, y. Moreover, we haveqx, x θandqy, y θ.

Proof. It follows fromTheorem 2.2by takinggI_X.

Corollary 2.10. LetX,≤be a partially ordered set and suppose thatX, dis a complete cone metric space. Letqbe ac-distance onX. LetF : X×X → X be a mapping having the mixed monotone property such that

q F

x, y , F

x∗, y∗

q F

y, x , F

y∗, x∗

k

qx, x∗ q y, y∗

, 2.31

for somek ∈0,1and for allx, y, x∗, y∗ ∈Xwithx≤ x∗∧y ≥y∗orx≥x∗∧y≤ y∗.

Suppose thatXhas the following properties:

iif a nondecreasing sequencex_n → x, thenx_n≤xfor alln, iiif a nonincreasing sequencexn → x, thenx≤xnfor alln.

Assume there existx₀, y₀ ∈X such thatx₀ ≤ Fx0, y₀andFy0, x₀≤ y₀. ThenFhas a coupled fixed point.

Proof. It follows fromTheorem 2.4by takinggI_X.

Example 2.11. LetEC¹_R0,1with||x||||x||∞||x||∞andP{x∈E:xt≥0, t∈0,1}.

LetX 0,1with usual order≤. Defined : X ×X → X bydx, yt |x−y|e^tfor all x, y ∈ X. ThenX, dis a partially ordered cone metric space. Define q : X ×X → Eby qx, yt ye^tfor allx, y∈X. Thenqis ac-distance. DefineF :X×X → Xby

F x, y

⎧⎨

⎩ x−y

2 , x≥y,

0, x < y. 2.32

Then,

1qFx, y, Fx∗, y∗ qFy, x, Fy∗, x∗≤1/2qx, x∗ qy, y∗, for allx≤x∗

andy≥y∗,

2there is nok∈0,1such thatqFx, y, Fx∗, y∗≤k/2qx, x∗ qy, y∗for allx≤x∗andy≥y∗,

3there is nok∈0,1such thatqFx, y, Fx∗, y∗≤k/4qx, x∗ qy, y∗for allx≤x∗andy≥y∗.

Note that 0≤F0,0and 0≥F0,0. Thus byCorollary 2.10, we haveFwhich has a coupled fixed point. Here0,0is a coupled fixed point ofF.

Proof. The proof of2.1is easy. To prove2.3, suppose the contrary; that is, there isk∈0,1 such that qFx, y, Fx∗, y∗ ≤ k/2qx, x∗ qy, y∗ for allx ≤ x∗and y ≥ y∗. Take x0, y1, x∗1 andy∗0. Then

qF0,1, F1,0t≤ k 2

q0,1 q1,0

t. 2.33

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Thus

q

0,1 2

t 1

2e^t≤ k

2e^t. 2.34

Hencek≥1 is a contradiction. The proof of2.5is similar to proof of2.3.

Remark 2.12. Note that Theorems 3.1 and 3.2 of39are not applicable toExample 2.11.

Remark 2.13. Theorem 3.1 of39is a special case ofCorollary 2.6andCorollary 2.9.

Remark 2.14. Theorem 3.3 of39is a special case ofCorollary 2.8andCorollary 2.10.

3. Application

Consider the integral equations xt

_T

0

f

t, xs, ys

ds, t∈0, T,

yt _T

0

f

t, ys, xs

ds, t∈0, T,

3.1

where T > 0 andf : 0, T×R×R → R. Let X C0, T,R denote the space of R- valued continuous functions onI 0, T. The purpose of this section is to give an existence theorem for a solutionx, yto3.1that belongs toX, by using the obtained result given by Corollary 2.10. LetER², and letP ⊂Ebe the cone defined by

P

x, y

∈R²| x≥0, y≥0

. 3.2

We endowXwith the cone metricd:X×X → Edefined by du, v

sup

t∈I |ut−vt|,sup

t∈I|ut−vt|

, ∀u, v∈X. 3.3

It is clear thatX, dis a complete cone metric space. Letqx, y dx, yfor allx, y ∈ X.

Then,qis ac-distance.

Now, we endowXwith the partial order≤given by

u, v∈X, u≤v⇐⇒ux≤vx, ∀x∈I. 3.4

Also, the product spaceX×Xcan be equipped with the partial orderstill denoted≤given as follows:

x, y

≤u, v⇐⇒x≤u, y≥v. 3.5

It is easy thatiandiigiven inCorollary 2.10are satisfied.

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Now, we consider the following assumptions:

af :0, T×R×R → Ris continuous,

bfor allt∈0, T, the functionft,·,·:R → Rhas the mixed monotone property, cfor allt∈0, T, for allp, q, p, q∈Rwithp≤qandp≥q, we have

f t, q, q

−f t, p, p

≤ 1 Tϕ

q−pp−q 2

, 3.6

whereϕ :0,∞ → 0,∞is continuous nondecreasing an satisfies the following condition: There exists 0< k <1 such that

ϕr≤kr ∀r≥0, 3.7

dthere existsx0, y0∈C0, T,Rsuch that

x0t≤ _T

0

f

t, x0s, y0s ds,

_T

0

f

t, y0s, x0s

ds≤y0t, ∀t∈0, T. 3.8

We have the following result.

Theorem 3.1. Suppose that a–d hold. Then, 3.1 has at least one solution x^∗, y^∗ ∈ C0, T,R×C0, T,R.

Proof. Define the mappingA:C0, T,R×C0, T,R → C0, T,Rby

A x, y

t _T

0

f

t, xs, ys

ds, x, y∈C0, T,R, t∈0, T. 3.9

We have to prove that A has at least one coupled fixed point x^∗, y^∗ ∈ C0, T,R × C0, T,R.

Fromb, it is clear thatAhas the mixed monotone property.

Now, letx, y, u, v ∈ C0, T,Rsuch thatx ≤ uand y ≥ vorx ≥ uandy ≤ v.

Usingc, for allt∈0, T, we have Au, vt−A

x, y t≤

_T

0

ft, us, vs−f

t, xs, ys ds

≤ 1 T

_T

0

ϕ

us−xs ys−vs 2

ds

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≤ 1 T

_T

0

ϕ

sup_z∈0,T|uz−xz|sup_z∈0,Tyz−vz 2

ds

ϕ

≤k

,

3.10 which implies that

sup_z∈0,TAu, vt−A x, y

t≤k

. 3.11 Similarly, one can get

sup_z∈0,TAv, ut−A y, x

t≤k

. 3.12

We deduce

sup_z∈0,TAu, vt−A x, y

tsup_z∈0,TAv, ut−A y, x

t

≤k

sup_z∈0,T|uz−xz|sup_z∈0,Tyz−vz.

3.13

Thus

d

Au, v, A x, y

d

Av, u, A y, x

k

dx, u d y, v

. 3.14

Thus, we proved that condition2.31ofCorollary 2.10is satisfied. Moreover, fromd, we have x0 ≤ Ax0, y0 and Ay0, x0 ≤ y0. Finally, applying our Corollary 2.10, we get the desired result.

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