Coupled Fixed-Point Theorems for Contractions in Partially Ordered Metric Spaces and Applications

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Volume 2012, Article ID 150363,20pages doi:10.1155/2012/150363

Research Article

Coupled Fixed-Point Theorems for Contractions in Partially Ordered Metric Spaces and Applications

M. Eshaghi Gordji,

¹

Y. J. Cho,

²

S. Ghods,

³

M. Ghods,

⁴

and M. Hadian Dehkordi

⁴

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea

3Department of Mathematics, Islamic Azad University, Semnan Branch, Semnan, Iran

4Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

Correspondence should be addressed to M. Eshaghi Gordji,madjid.eshaghi@gmail.comand Y. J.

Cho,yjcho@gnu.ac.kr

Received 30 September 2011; Revised 20 December 2011; Accepted 24 December 2011 Academic Editor: Stefano Lenci

Copyrightq2012 M. Eshaghi Gordji 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.

Bhaskar and Lakshmikantham2006showed the existence of coupled coincidence points of a mappingFfromX×XintoXand a mappinggfromXintoXwith some applications. The aim of this paper is to extend the results of Bhaskar and Lakshmikantham and improve the recent fixed- point theorems due to Bessem Samet2010. Indeed, we introduce the definition of generalizedg- Meir-Keeler type contractions and prove some coupled fixed point theorems under a generalized g-Meir-Keeler-contractive condition. Also, some applications of the main results in this paper are given.

1. Introduction

The Banach contraction principle1is a classical and powerful tool in nonlinear analysis and has been generalized by many authorssee2–15and others.

Recently, Bhaskar and Lakshmikantham16introduced the notion of a coupled fixed- point of the given two variables mapping. More precisely, letX be a nonempty set andF : X×X → Xbe a given mapping. An elementx, y∈X×Xis called a coupled fixed-point of the mappingFif

F x, y

x, F y, x

y. 1.1

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They also showed the uniqueness of a coupled fixed-point of the mapping F and applied their theorems to the problems of the existence and uniqueness of a solution for a periodic boundary value problem.

Theorem 1.1see Zeidler15. LetX,≤be a partially ordered set and suppose that there is a metric d onX such thatX, dis a complete metric space. LetF : X × X → X be a continuous mapping having the mixed monotone property onX. Assume that there existsk∈0,1such that

d F

x, y

, Fu, v

≤ k 2

dx, u d

y, v 1.2

for allx≥uandy≤v. Moreover, if there existx₀, y₀∈Xsuch that x0≤F

x0, y0

, y0≥F y0, x0

, 1.3

then there existx, y∈Xsuch thatxFx, yandyFy, x.

Later, in17, Lakshmikantham and ´Ciri´c investigated some more coupled fixed-point theorems in partially ordered sets, and some others obtained many results on coupled fixed- point theorems in cone metric spaces, intuitionistic fuzzy normed spaces, ordered cone metric spaces and topological spacessee, e.g.,18–25.

In9, Meir and Keeler generalized the well-known Banach fixed-point theorem1as follows.

Theorem 1.2Meir and Keeler9. LetX, dbe a complete metric space andT :X → X be a given mapping. Suppose that, for any >0, there existsδ>0 such that

≤d x, y

< δ ⇒d

Tx, T y

< 1.4

for allx, y∈X. Then T admits a unique fixed-pointx₀∈Xand, for allx∈X, the sequence{Tⁿx}

converges tox₀.

Proposition 1.3see17. LetX, dbe a partially ordered metric space andF : X ×X → X be a given mapping. If the contraction 1.2 is satisfied, then F is a generalized Meir-Keeler type contraction.

Motivated by the results of Bhaskar and Lakshmikantham 16, Lakshmikantham and ´Ciri´c17, and Samet26, in this paper, we introduce the definition ofg-Meir-Keeler- contractive mappings and prove some coupled fixed-point theorems under a generalized g-Meir-Keeler contractive condition.

2. Main Results

LetXbe a nonempty set. We note that an elementx, y∈X×Xis called a coupled coincidence point of a mappingF : X×X → X andg : X → X ifFx, y gxandFy, x gy for all x, y ∈ X. Also, we say thatF andg are commutativeor commuting ifgFx, y Fgx, gyfor allx, y∈X.

We introduce the following two definitions.

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Definition 2.1. LetX,≤be a partially ordered set andF :X×X → X andg :X → X. We say thatFhas the mixed strictg-monotone property if, for anyx, y∈X,

x1, x2∈X, gx1< gx2 ⇒F x1, y

< F x2, y

, y1, y2∈X, g

y1

< g y2

⇒F x, y1

> F x, y2

. 2.1

Definition 2.2. LetX,≤be a partially ordered set anddbe a metric onX. LetF:X×X → X and g : X → X be two given mappings. We say thatF is a generalizedg-Meir-Keeler type contraction if, for all >0, there existsδ>0 such that, for allx, y, u, v∈Xwithgx≤gu andgy≥gv,

≤ 1 2

d

gx, gu d

g y

, gv

< δ ⇒d F

x, y

, Fu, v

< . 2.2

Lemma 2.3. LetX,≤be a partially ordered set anddbe a metric onX. LetF :X×X → Xand g : X → X be two given mappings. IfF is a generalizedg-Meir-Keeler type contraction, then we have

d F

x, y

, Fu, v

< 1 2

d

gx, gu

d g

y

, gv 2.3

for allx, y, u, vwithgx< gu,gy≥gvorgx≤gu,gy> gv.

Proof. Letx, y, u, v ∈X such that gx < guandgy ≥ gvorgx ≤ guandgy >

gv. Then dgx, gu dgy, gv > 0. SinceF is a generalized g-Meir-Keeler type contraction, for 1/2dgx, gu dgy, gv, there existsδ >0 such that, for allx₀, y₀, u₀, v₀ ∈Xwithgx0≤gu0andgy0≥gv0,

≤ 1 2

d

gx0, gu0 d

g y₀

, gv0

< δ ⇒d F

x₀, y₀

, Fu0, v₀

< . 2.4

Therefore, puttingx₀x,y₀y,u₀uandv₀v, we have

d F

x, y

, Fu, v

< 1 2

d

gx, gu

d g

y , gv

. 2.5

This completes the proof.

From now on, we suppose thatX,≤is a partially ordered set, and there exists a metric donXsuch thatX, dis a complete metric space.

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Theorem 2.4. LetF:X×X → Xandg :X → Xbe such thatFX×X⊆gX,gis continuous and commutative withF. Also, suppose that

aFhas the mixed strictg-monotone property;

bFis a generalizedg-Meir-keeler type contraction;

cthere existx₀, y₀∈Xsuch thatgx0< Fx0, y₀andgy0> Fy0, x₀.

Then there existx, y ∈ X such thatgx Fx, yandgy Fy, x; that is,F and g have a coupled coincidence inX×X.

Proof. Letx0, y0 ∈Xbe such thatgx0< Fx0, y0andgy0> Fy0, x0. SinceFX×X ⊆ gX, we can choosex₁, y₁∈Xsuch thatgx1 Fx0, y₀andgy1 Fy0, x₀. Again, from FX×X⊆gX, we can choosex₂, y₂∈Xsuch thatgx2 Fx1, y₁andgy2 Fy1, x₁. Continuing this process, we can construct the sequences{xn}and{yn}inXsuch that

gxn1 F xn, yn

, g y_n1

F yn, xn

2.6

for alln≥0.

Now, we show that

gxn< gx_n1, g y_n

> g y_n1

2.7

for alln≥0. Forn0, we have gx0< F

x₀, y₀

gx1, g y₀

> F y₀, x₀

g y₁

. 2.8

SinceFhas the mixed strictg-monotone property, then we have gx0< gx1 ⇒F

x0, y1

< F x1, y1

,

g y₀

> g y₁

⇒F x₀, y₀

< F x₀, y₁

. 2.9

It follows thatFx0, y0< Fx1, y1, that is,gx1< gx2. Similarly, we have

g y₁

< g y₀

⇒F y₁, x₀

< F y₀, x₀

, gx1> gx0 ⇒F

y₁, x₁

< F y₁, x₀

. 2.10

Thus it follows thatFy1, x1< Fy0, x0, that is,gy2< gy1. Again, we have

gx1< gx2 ⇒F x₁, y₂

< F x₂, y₂

, g

y1

> g y2

⇒F x1, y1

< F x1, y2

. 2.11

Thus it follows thatFx1, y₁< Fx2, y₂, that is,gx2< gx3.

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Similarly, we have g

y₂

< g y₁

⇒F y₂, x₁

< F y₁, x₁

, gx2> gx1 ⇒F

y₂, x₂

< F y₂, x₁

. 2.12

Thus it follows thatFy2, x2< Fy1, x1, that is,gy3< gy2. Continuing this process for eachn≥1, we get the following:

gx0< gx1< gx2<· · ·< gxn< gxn1<· · ·, g

y0

> g y1

> g y2

>· · ·g yn

> g y_n1

>· · ·. 2.13

Denote that

δ_n:d

gxn, gx_n1 d

g y_n

, g y_n1

. 2.14

Sincegxn−1< gxnandgyn−1> gyn, it follows from2.6andLemma 2.3that d

gxn, gx_n1 d

F

x_n−1, y_n−1 , F

x_n, y_n

< 1 2

d

gxn−1, gxn d

g y_n−1

, g y_n

.

2.15

Sincegyn< gyn−1andgxn> gxn−1, it follows from2.6andLemma 2.3that d

g y_n1

, g y_n

d F

y_n, x_n , F

y_n−1, x_n−1

< 1 2

d g

y_n , g

y_n−1 d

gxn, gxn−1 .

2.16

Thus it follows from2.14–2.16 thatδn < δ_n−1. This means that the sequence{δn/2} is monotone decreasing. Therefore, there existsδ^∗≥0 such that limn→ ∞ δn/2 δ^∗, that is,

nlim→ ∞

1 2

d

gxn, gx_n1 d

g y_n

, g y_n1

δ^∗. 2.17

Now, we show thatδ^∗ 0. Suppose thatδ^∗ > 0 hold. Letδ^∗ . Then there exists a positive integermsuch that

≤ 1 2

d

gxm, gx_m1 d

g y_m

, g

y_m1

< δ. 2.18

Then, by using2.7and the conditionb, we have d

F

x_m, y_m , F

x_m1, y_m1

< , 2.19

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and so, by2.6, we have

d

gxm1, gxm2

< . 2.20

On the other hand, by2.15, we have 1

2 d

gxm, gxm1 d

g y_m

, g

y_m1

< , 2.21

which is a contradiction with2.18. Thus we haveδ^∗0, that is,

nlim→ ∞

1 2

d

gxn, gx_n1 d

g y_n

, g y_n1

0, 2.22

that is,

nlim→ ∞δ_n0. 2.23

Now, we prove that{gxn}and{gyn}are Cauchy sequences inX. Suppose that at least one of{gxn} or{gyn}is not a Cauchy sequence. Then there exist > 0 and two subsequences{lk},{mk}of integers such thatm_k> l_k≥kand

d

gxlk, gxmk

≥

2, d g

ylk

, g ymk

≥

2 2.24

for allk≥1. Thus we have r_kd

gxlk, gxmk d

g y_l_k

, g y_m_k

≥ 2.25

for allk≥1. Letm_kbe the smallest number exceedingl_ksuch that2.25holds. Then we have d

gxlk, gxmk−1 d

g y_l_k

, g y_m_k₋₁

< . 2.26

Thus, from2.14,2.25,2.26and the triangle inequality, it follows that ≤rk

≤d

gxlk, gxmk−1 d

gxmk−1, gxmk d

g y_l_k

, g y_m_k₋₁

d g

y_m_k₋₁ , g

y_m_k

< δmk−1

2.27

and so

≤ lim

k→ ∞rk≤ lim

k→ ∞δmk−1. 2.28

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Hence, by2.23, we have

klim→ ∞r_k. 2.29

It follows from2.6,2.14, and the triangle inequality that rkd

gxlk, gxmk d

g ylk

, g ymk

≤d

gxlk, gxlk1 d

gxlk1, gxmk1 d

gxmk1, gxmk d

g y_l_k

, g y_l_k₁

d g

y_l_k₁ , g

y_m_k₁ d

g y_m_k₁

, g y_m_k δlkδmkd

gxlk1, gxmk1 d

g ylk1

, g ymk1 δ_l_kδ_m_kd

F x_l_k, y_l_k

, F

x_m_k, y_m_k d

F y_l_k, x_l_k

, F

y_m_k, x_m_k .

2.30

Form2.13we havegxlk < gxmkandgylk > gymk. Now, it follows fromLemma 2.3 and2.30that

r_k< δ_l_kδ_m_kd

gxlk, gxmk d

g y_l_k

, g y_m_k

, 2.31

that is,

r_k< δ_l_kδ_m_kr_k. 2.32

This is a contradiction. Therefore, {gxn} and {gyn} are Cauchy sequences. Since X is complete, there existx, y∈Xsuch that

nlim→ ∞gxn x, lim

n→ ∞g y_n

y. 2.33

Since{gxn}is monotone increasing and{gyn}is monotone decreasing, we have gxn< x, g

yn

> y 2.34

for alln≥1. Thus it follows from2.33and the continuity ofgthat

nlim→ ∞g gxn

gx, lim

n→ ∞g g

yn

g y

. 2.35

Thus, for allm≥1, there exists a positive integern₀such that, for alln≥n₀,

d g

gxn , gx

< 1

4m, d

g g

y_n , g

y

< 1

4m. 2.36

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Hence, from2.6, the commutativity ofFandgand the triangle inequality, we have d

F x, y

, gx

≤d F

x, y , g

gxn d

g gxn

, gx d

F x, y

, g F

x_n−1, y_n−1 d

g gxn

, gx d

F x, y

, F

gx_n−1, g y_n−1

d g

gxn , gx

.

2.37

Thus, it follows from2.34,2.36, andLemma 2.3that d

F x, y

, gx

< 1 2

d g

gxn−1 , gx

d g

g y_n−1

, g

y

d g

gxn , gx

< 1 8m 1

8m 1 4m 1

2m −→0

2.38

asm → ∞. Therefore, we haveFx, y gx. Similarly, we can show thatFy, x gy.

This means that F and g have a coupled coincidence point in X ×X. This completes the proof.

Corollary 2.5. LetF :X×X → Xbe a mapping satisfying the following conditions:

aFhas the mixed strict monotone property;

bFis a generalized Meir-Keeler type contraction;

cthere existsx₀, y₀∈Xsuch thatx₀< Fx0, y₀andy₀> Fy0, x₀. Then there existx, y∈Xsuch thatxFx, yandyFy, x.

Proof. The conclusion follows fromTheorem 2.4by puttingg I: the identity mappingon X.

Now, we introduce the product spaceX×X with the following partial order: for all x, y,u, v∈X×X,

u, v≤ x, y

⇐⇒u < x, v≥y. 2.39

Theorem 2.6. Suppose that all the hypotheses of Theorem 2.4 hold and, further, for all x, y,x^∗, y^∗ ∈ X ×X, there exists u, v ∈ X ×X such that Fu, v, Fv, uis comparable with Fx, y, Fy, xand Fx^∗, y^∗, Fy^∗, x^∗. Then F and g have a unique coupled common fixed-point, that is, there exists a uniquex, y∈X×Xsuch that

xgx F

x, y

, yg

y F

y, x

. 2.40

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Proof. ByTheorem 2.4, the set of coupled coincidences of the mappingFandgis nonempty.

First, we show that, ifx, yand x^∗, y^∗are coupled coincidence points of F andg, that is, if

gx F x, y

, g

y F

y, x

, gx^∗ F

x^∗, y^∗

, g

y^∗ F

y^∗, x^∗ ,

2.41

then we have

gx gx^∗, g

y g

y^∗

. 2.42

Putu₀ u,v₀ vand chooseu₁, v₁ ∈ Xsuch thatgu1 Fu0, v₀andgv1 Fv0, u₀. Then, similarly as in the proof of Theorem 2.4, we can inductively define the sequences {gun}and{gvn}such that

gun1 Fun, v_n, gvn1 Fvn, u_n 2.43

for alln ≥ 0. Also, if we setx₀ x,y₀ y,x^∗₀ x^∗, andy^∗₀ y^∗, then we can define the sequences{gxn},{gyn},{gx^∗_n}, and{gy_n^∗}as follows:

gx_n1 F x_n, y_n

, g

y_n1 F

y_n, x_n , g

x^∗_n1 F

x^∗_n, y^∗_n

, g

y_n1^∗ F

y^∗_n, x_n^∗ 2.44

for alln≥0. Since F

x, y , F

y, x

gx1, g y₁

gx, g y

, Fu, v, Fv, u

gu1, gv1 2.45

are comparable each other, then gx < gu1 and gy ≥ gv1. It is easy to show that gx, gy, andgun, gvnare comparable each other, that is,gx < gunandgy ≥ gvnfor alln≥1. Thus it follows fromLemma 2.3that

d

gx, gu_n1 d

g y

, gv_n1 d

F x, y

, Fun, v_n d

F y, x

, Fvn, u_n

< 1 2

d

gx, gun d

g y

, gvn 1

2 d

g y

, gvn d

gx, gun d

gx, gun d

g y

, gvn

2.46

and so 1 2

d

gx, gun1 d

g y

, gvn1

< 1 2ⁿ

d

gx, gu1 d

g y

, gv1

−→0 2.47

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asn → ∞. Therefore, we have

nlim→ ∞d

gx, gu_n1

0, lim

n→ ∞d g

y

, gv_n1

0. 2.48

Similarly, we can prove that

n→ ∞limd

gx^∗, gun1

0, lim

n→ ∞d g

y^∗

, gvn1

0. 2.49

Thus, by the triangle inequality,2.48and2.49, we have

d

gx, gx^∗

≤d

gx, gun1 d

gx^∗, gun1

−→0, d

g y

, g y^∗

≤d g

y

, gv_n1 d

g y^∗

, gv_n1

−→0 2.50

asn → ∞, which imply thatgx gx^∗andgy gy^∗.

Now, we prove thatgx xandgy y. Denote thatgx zandgx w. Since gx Fx, yandgy Fy, x, by the commutativity ofFandg, we have

gz g

gx

g F

x, y F

gx, g y

Fz, w, 2.51

gw g

g y

g F

y, x F

g y

, gx

Fw, z. 2.52

Thus,z,wis a coupled coincidence point ofFandg.

Puttingx^∗zandy^∗win2.52, it follows from2.42that

zgx gx^∗ gz, wg

y g

y^∗

gw 2.53

and so, from2.51and2.52,

zgz Fz, w, wgw Fw, z. 2.54

Therefore,z, wis a coupled common fixed-point ofFandg.

Finally, to prove the uniqueness of the coupled common fixed-point ofFandg, assume thatp, qis another coupled common fixed-point ofF andg. Then, by2.42, we havep gp gz zandqgq gw w. This completes the proof.

Corollary 2.7. Suppose that all the hypotheses ofCorollary 2.5hold and, further, for allx, yand x^∗, y^∗∈X×X, there existsu, v∈X×Xthat is comparable withx, yandx^∗, y^∗. Then there exists a uniquex∈Xsuch thatxFx, x.

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3. Applications

Now, we give some applications of the main results inSection 2.

Theorem 3.1. LetF :X × X → X andg :X → X be two given mappings. Assume that there exists a functionϕ:0,∞ → 0,∞satisfying the following conditions:

aϕ0 0 andϕt>0 for anyt >0;

bϕis nondecreasing and right continuous;

cfor any >0, there existsδ>0 such that, for allx, y, u, v∈Xwithgx≤guand gy≥gv,

≤ϕ 1

2 d

gx, gu

d g

y

, gv

< δ ⇒ϕ d

F x, y

, Fu, v

< . 3.1

ThenFis a generalizedg-Meir-Keeler type contraction.

Proof. For any >0, it follows fromathatϕ>0 and so there existsα >0 such that, for allu, v, u^∗, v∈Xwithgu≤gu^∗andgv≥gv^∗,

ϕ≤ϕ

1 2

d

gu, gu^∗ d

gv, gv^∗

< ϕ α ⇒ϕdFu, v, Fu^∗, v^∗< ϕ.

3.2

From the right continuity ofϕ, there exists δ > 0 such that ϕδ < ϕ α. For any x, y, u, v∈Xsuch thatgx≤gu,gy≥gvand

≤ 1 2

d

gx, gu

d g

y , gv

< δ, 3.3

sinceϕis nondecreasing function, we get the following:

ϕ≤ϕ

1 2

d

gx, gu

d g

y

, gv

< ϕα< ϕ α. 3.4

By3.2, we haveϕdFx, y, Fu, v < ϕand sodFx, y, Fu, v < . Therefore, it follows thatFis a generalizedg-Meir-Keeler type contraction. This completes the proof.

Corollary 3.2see26, Theorem 3.1. LetF : X × X → X be a given mapping. Assume that there exists a functionϕ:0,∞ → 0,∞satisfying the following conditions:

aϕ0 0 andϕt>0 for anyt >0;

bϕis nondecreasing and right continuous;

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cfor any >0, there existsδ>0 such thatx≤u,y≥vand

≤ϕ 1

2

dx, u d y, v

< δ ⇒ϕ d

F x, y

, Fu, v

< . 3.5

ThenFis a generalized Meir-Keeler type contraction.

The following result is an immediate consequence of Theorems2.4and3.1.

Corollary 3.3. LetF:X×X → Xandg:X → Xbe two given mappings such thatFX×X⊆ gX,gis continuous and commutative withF. Also, suppose that

bfor any >0, there existsδ>0 such that, for allx, y, u, v∈Xwithgx≤guand gy≥gv,

≤

1/2dgx,gudgy,gv

0

ϕtdt < δ ⇒

_dFx,y,Fu,v

0

ϕtdt < , 3.6

where ϕ is a locally integrable function from0,∞into itself satisfying the following condition:

_s

0

ϕtdt >0 3.7

for alls >0;

cthere existx₀, y₀∈Xsuch thatgx0< Fx0, y₀andgy0> Fy0, x₀.

Then there existsx, y∈X×X such thatgx Fx, yandgy Fy, x. Moreover, ifgx0 andgy0are comparable to each other, thenF andghave a unique coupled common fixed-point in X×X.

Corollary 3.4. LetF :X×X → Xbe a mapping satisfying the following conditions:

aFhas the mixed strict monotone property;

bfor any >0, there existsδ>0 such thatx≤u,y≥vand

≤

1/2dx,udy,v

0

ϕtdt < δ ⇒

_dFx,y,Fu,v

0

ϕtdt < , 3.8

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whereϕis a locally integrable function from0,∞into itself satisfying _s

0

ϕtdt >0 3.9

for alls >0;

cthere existx₀, y₀∈Xsuch thatx₀< Fx0, y₀andy₀> Fy0, x₀.

Then there existsx, y∈X×Xsuch thatxFx, yandyFy, x. Moreover, ifx₀andy₀are comparable to each other, thenFhas a unique coupled common fixed-point inX×X.

Corollary 3.5. LetF:X×X → Xandg:X → Xbe two given mappings such thatFX×X⊆ gX,gis continuous and commutes withF. Also, suppose that

bfor anyx, y, u, v∈Xwithgx≤guandgy≥gv,

_dFx,y,Fu,v

0

ϕtdt≤k

1/2dgx,gudgy,gv

0

ϕtdt, 3.10

wherek∈0,1andϕis a locally integrable function from0,∞into itself satisfying _s

0

ϕtdt >0 3.11

for alls >0;

cthere existx0, y0∈Xsuch thatgx0< Fx0, y0andgy0> Fy0, x0.

Then there existsx, y∈X×X such thatgx Fx, yandgy Fy, x. Moreover, ifgx0 andgy0are comparable to each other, thenF andghave a unique coupled common fixed-point in X×X.

Proof. For any >0, if we takeδ 1/k−1and applyCorollary 3.3, then we can get the conclusion.

Corollary 3.6. Let F :X×X → Xbe a mapping satisfying the following conditions:

aFhas the mixed strict monotone property, bfor anyx, y, u, v∈Xwithx≤uandy≥v,

_dFx,y,Fu,v

0

ϕtdt≤k

1/2dx,udy,v

0

ϕtdt, 3.12

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wherek∈0,1andϕis a locally integrable function from0,∞into itself satisfying _s

0

ϕtdt >0 3.13

for alls >0;

cthere existx₀, y₀∈Xsuch thatx₀< Fx0, y₀andy₀> Fy0, x₀.

Then there exist x, y ∈ X such that x Fx, y and y Fy, x. Moreover, if x0 and y0 are comparable to each other, thenFhas a unique coupled common fixed-point inX×X.

Finally, by using the above results, we show the existence of solutions for the following integral equation:

xt, yt ^T

0

Gt, s

fs, xs λxs

− f

s, ys

λys ds, _T

0

Gt, s f

s, ys

λys

−

fs, xs λxs

ds

,

3.14

wherex, y∈CI,R : the set of continuous functions fromIintoR,T >0,f :I×R → Ris a continuous function and

Gt, s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

e^λTs−t

e^λT−1 , if 0≤s < t≤T; e^λs−t

e^λT−1, if 0≤t < s≤T.

3.15

Definition 3.7. A lower solution for the integral equation3.14is an elementα, β∈C¹I,R× C¹I,Rsuch that

αt λβt≤ft, αt−f t, βt

, α0< αT, βt λαt≥f

t, βt

−ft, αt, β0≥βT, 3.16

whereC¹I,Rdenotes the set of diﬀerentiable functions fromIintoR.

Now, we prove the existence of solutions for the integral equation3.14by using the existence of a lower solution for the integral equation3.14.

Theorem 3.8. Let A be the class of the functionsϕ : 0,∞ → 0,∞satisfying the following conditions:

aϕis increasing;

bfor anyx≥0, there existsk∈0,1such thatϕx<k/2x.

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In the integral equation3.14, suppose that there existsλ >0 such that, for allx, y∈Rwithy > x, 0< f

t, y

λy−

ft, x λx

≤λϕ y−x

, 3.17

whereϕ∈ A. If a lower solution of the integral equation3.14exists, then a solution of the integral equation3.14exists.

Proof. Define a mappingF:CI,R×CI,R → CI,Rby

F

xt, yt

_T

0

Gt, s

fs, xs λxs

− f

s, ys

λys

ds. 3.18

Note that, if xt, yt ∈ CI,R × CI,R is a coupled fixed-point of F, then xt, ytis a solution of the integral equation3.14.

Now, we check the hypotheses inCorollary 2.5as follows:

1X×XCI,R×CI,Ris a partially ordered set if we define the order relation in X×Xas follows:

ut, vt≤

xt, yt

iﬀut< xt, vt≥yt 3.19

for allxt, yt,ut, vt∈X×Xandt∈I.

2 X, dis a complete metric space if we define a metricdas follows:

d

xt, yt sup

t∈I

xt−yt:xt, yt∈X

. 3.20

3The mappingF has the mixed strict monotone property. In fact, by hypothesis, if x2> x1, then we have

ft, x2 λx2> ft, x1 λx1, 3.21

which implies that, for anyt∈I, _T

0

fs, x2s λx₂s−f s, ys

−λys

Gt, sds

>

_T

0

fs, x1s λx1s−f s, ys

−λys

Gt, sds,

3.22

that is,

F

x₂t, yt

> F

x₁t, yt

. 3.23

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Similarly, ify1< y2, then we have

f t, y2

λy2> f t, y1

λy1, 3.24

which implies that, for anyt∈I,

_T

0

fs, xs λxs−f

s, y₂s

−λy₂s

Gt, sds

<

_T

0

fs, xs λxs−f

s, y1s

−λy1s

Gt, sds,

3.25

that is,

F

xt, y2t

< F

xt, y1t

. 3.26

Now, we show thatFsatisfies1.2. In fact, letx, y≤u, vandt∈I. Then we have

d F

xt, yt

, Fut, vt supF

xt, yt

−Fut, vt:t∈I sup

t∈I

_T

0

Gt, s

fs, xs λxs−f s, ys

−λys ds

− _T

0

Gt, s

fs, us λus−fs, vs−λvs ds

≤sup

t∈I

_T

0

Gt, sfs, xs λxs−fs, us−λus fs, vs λvs−f

s, ys

−λysds.

3.27

Since the functionϕxis increasing andx, y≤u, v, we have

ϕxs−us≤ϕdxs, us, ϕ

vs−ys

≤ϕ d

vs, ys

, 3.28

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we obtain the following:

d F

xt, yt

, Fut, vt

≤sup

t∈I

_T

0

Gt, sλϕxs−us λϕ

vs−ysds

≤λsup

t∈I

_T

0

Gt, sϕdxs, us ϕ d

vs, ysds λ

ϕdxs, us ϕ

d

vs, ys

·sup

t∈I

_T

0

Gt, sds

λ

ϕdxs, us ϕ

d

vs, ys

·sup

t∈I

1 e^λT−1

1

λe^λTs−t _t

0

1

λe^λs−t _T

t

λ

ϕdxs, us ϕ

d

vs, ys

· 1 λe^λT−1

e^λT−1

ϕdxs, us ϕ

d

vs, ys

< k 2

dxs, us d

vs, ys

≤ k

2sup{|xt−ut|:t∈I} k

2supvt−yt:t∈I k

2

dxt, ut d

yt, vt

.

3.29 Then, byProposition 1.3,Fis a generalized Meir-Keeler type contraction.

Finally, letαt, βt∈C¹I,R×C¹I,Rbe a lower solution for the integral equation 3.14. Then we show that

α < F α, β

, β≥F β, α

. 3.30

Indeed, we haveαt λβt≤ft, αt−ft, βtfor anyt∈Iand so αt λαt≤ft, αt−f

t, βt

λαt−λβt 3.31

for anyt∈I. Multiplying bye^λtin3.31, we get the following:

αte^λt

≤

ft, αt λαt

− f

t, βt

λβt

e^λt 3.32

for anyt∈I, which implies that

αte^λt ≤α0 _t

0

fs, αs λαs

−f s, βs

−λβs

e^λsds 3.33

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for anyt∈I. This implies that

α0e^λt< αTe^λT ≤α0 _T

0

fs, αs λαs−f s, βs

−λβs

e^λsds 3.34

and so

α0<

_T

0

e^λs e^λT−1

−λβs

ds. 3.35

Thus it follows from3.35and3.33that

αte^λt<

_T

t

e^λs e^λT−1

fs, αs λαs−f s, βs

−λβs ds

_t

0

e^λTs e^λT−1

−λβs ds,

3.36

and so

αt<

_t

0

e^λTs−t e^λT−1

fs, αs λαs−f s, βs

−λβs ds

_T

t

e^λs−t e^λT−1

−λβs ds.

3.37

Hence we have

αt<

_T

0

Gt, s

−λβs

dsF

αt, βt 3.38

for anyt∈I.

Similarly, we haveβt ≥ Fβt, αt. Therefore, by Corollary 2.5,F has a coupled fixed-point.

Example 3.9. In the integral equation3.14, we putλ1.5,fu, v u−vfor allu, v∈I×R andT 0.5. Thenfis a continuous function, and we have

xt, yt

^0.5

0

Gt, s

0.5xs−0.5ys ds,

_0.5

0

Gt, s

0.5ys−0.5xs ds

,

3.39

(19)

wherex, y∈CI,R, and

Gt, s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

e^1.50.5s−t

e^0.75−1 , if 0≤s < t≤0.5, e^1.5s−t

e^0.75−1, if 0≤t < s≤0.5.

3.40

Also,αt, βt −2e^−0.5t,3e^−0.5tis a lower solution of3.39. Moreover, if we defineϕx x/3 for allx∈0,∞, thenϕis increasing and, for anyx > 0, there existsk 1/1.1∈ 0,1 such thatϕx x/3<k/2xx/2.2. For allx, y∈Rwithy > x, we have

0< f t, y

λy−

ft, x λx 0.5

y−x

≤λϕ y−x

1.5y−x 3 0.5

y−x .

3.41

Therefore, all the conditions ofTheorem 3.8hold, and a solution of3.39exists.

Acknowledgment

This work 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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