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Volume 2011, Article ID 703938,21pages doi:10.1155/2011/703938

Research Article

Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a Q-Function

N. Hussain,

1

M. H. Shah,

2

and M. A. Kutbi

1

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2Department of Mathematical Sciences, LUMS, DHA Lahore, Lahore 54792, Pakistan

Correspondence should be addressed to N. Hussain,nhusain@kau.edu.sa Received 20 August 2010; Accepted 16 September 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 N. Hussain 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.

Using the concept of a mixed g-monotone mapping, we prove some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a Q-function q. The presented theorems are generalizations of the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham 2006, Lakshmikantham and ´Ciri´c2009and many others.

1. Introduction

The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directionscf.1–31. Recently, Bhaskar and Lakshmikantham 8, Nieto and Rodr´ıguez-L ´opez 28, 29, Ran and Reurings 30, and Agarwal et al. 1 presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham 8 noted that their theorem can be used to investigate a large class of problems and discussed the existence and uniqueness of solution for a periodic boundary value problem. For more on metric fixed point theory, the reader may consult the book22.

Recently, Al-Homidan et al. 2introduced the concept of a Q-function defined on a quasi-metric space which generalizes the notions of a τ-function and a ω-distance and establishes the existence of the solution of equilibrium problemsee also3–7. The aim of this paper is to extend the results of Lakshmikantham and ´Ciri´c24for a mixed monotone nonlinear contractive mapping in the setting of partially ordered quasi-metric spaces with a Q-functionq. We prove some coupled coincidence and coupled common fixed point theorems for a pair of mappings. Our results extend the recent coupled fixed point theorems due to Lakshmikantham and ´Ciri´c24and many others.

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Recall that if X,≤ is a partially ordered set and F : XX such that for x, yX, xy implies FxFy, then a mapping F is said to be nondecreasing.

Similarly, a nonincreasing mapping is defined. Bhaskar and Lakshmikantham8introduced the following notions of a mixed monotone mapping and a coupled fixed point.

Definition 1.1Bhaskar and Lakshmikantham 8. LetX,≤be a partially ordered set and F : X ×XX. The mapping F is said to have the mixed monotone property if F is nondecreasing monotone in its first argument and is nonincreasing monotone in its second argument, that is, for anyx, yX,

x1, x2X, x1x2F x1, y

F x2, y

, y1, y2X, y1y2F

x, y1

F x, y2

. 1.1

Definition 1.2 Bhaskar and Lakshmikantham 8. An element x, y ∈ X ×X is called a coupled fixed point of the mappingF:X×XXif

F x, y

x, F y, x

y. 1.2 The main theoretical result of Lakshmikantham and ´Ciri´c in 24is the following coupled fixed point theorem.

Theorem 1.3Lakshmikantham and ´Ciri´c24, Theorem 2.1. LetX,≤be a partially ordered set, and suppose, there is a metricdonX such thatX, dis a complete metric space. Assume there is a functionϕ :0,∞ → 0,∞withϕt < tand limrtϕr < tfor eacht >0, and also suppose thatF:X×XXandg:XXsuch thatFhas the mixedg-monotone property and

d F

x, y

, Fu, v

ϕ d

gx, gu d

g y

, gv 2

1.3

for allx, y, u, vXfor whichgxguandgygv.Suppose thatFX×XgX,and gis continuous and commutes withF, and also suppose that either

aFis continuous or

bXhas the following property:

iif a nondecreasing sequence{xn} → x,then xnxfor alln, iiif a nonincreasing sequence{yn} → y,thenyynfor alln.

If there existsx0, y0Xsuch that gx0F

x0, y0

, g y0

F y0, x0

, 1.4

then there existx, yXsuch that gx F

x, y

, g

y F

y, x

, 1.5

that is,Fandghave a coupled coincidence.

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Definition 1.4. LetXbe a nonempty set. A real-valued functiond:X×X → Ris said to be quasi-metric onXif

M1dx, y≥0 for allx, yX, M2dx, y 0 if and only ifxy,

M3dx, ydx, z dz, yfor allx, y, zX.

The pairX, dis called a quasi-metric space.

Definition 1.5. LetX, d be a quasi-metric space. A mapping q : X×X → R is called a Q-function onXif the following conditions are satisfied:

Q1for allx, y, zX,

Q2if xX andynn≥1 is a sequence inX such that it converges to a pointywith respect to the quasi-metricandqx, ynMfor someMMx,thenqx, yM;

Q3for any >0, there existsδ >0 such thatqz, xδ, andqz, yδimplies that dx, y.

Remark 1.6see2. IfX, dis a metric space, and in addition toQ1–Q3,the following condition is also satisfied:

Q4for any sequence xnn≥1 in X with limn→ ∞sup{qxn, xm : m > n} 0 and if there exists a sequence ynn≥1 in X such that limn→ ∞qxn, yn 0, then limn→ ∞dxn, yn 0,

then aQ-function is called aτ-function, introduced by Lin and Du27. It has been shown in27that everyw-distance orw-function, introduced and studied by Kada et al.21, is a τ-function. In fact, if we considerX, das a metric space and replaceQ2by the following condition:

Q5for anyxX, the functionpx,· → Ris lower semicontinuous,

then aQ-function is called aw-distance onX. Several examples ofw-distance are given in 21. It is easy to see that if qx,·is lower semicontinuous, then Q2 holds. Hence, it is obvious that everyw-function is aτ-function and everyτ-function is aQ-function, but the converse assertions do not hold.

Example 1.7see2. aLetXR. Defined:X×X → Rby

d x, y

⎧⎨

0, if xy,

y , otherwise, 1.6

andq:X×X → Rby

q x, y

y , ∀x, y∈X. 1.7

Then one can easily see thatdis a quasi-metric andqis aQ-function onX, butqis neither a τ-function nor aw-function.

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bLetX 0,1.Defined:X×X → Rby

d x, y

⎧⎨

yx, ifxy, 2

xy

, otherwise, 1.8

andq:X×X → Rby

q x, y

xy , ∀x, y∈X. 1.9 Thenqis aQ-function onX.However, qis neither aτ-function nor aw-function, because X, dis not a metric space.

The following lemma lists some properties of aQ-function onX which are similar to that of aw-functionsee21.

Lemma 1.8see2. Letq : X ×X → R be aQ-function onX.Let{xn}n∈Nand{yn}n∈Nbe sequences inX, and letn}n∈Nandn}n∈Nbe such that they converge to 0 andx, y, zX.Then, the following hold:

1ifqxn, yαnandqxn, zβnfor allnN, thenyz. In particular, ifqx, y 0 andqx, z 0, thenyz;

2ifqxn, ynαnandqxn, zβnfor allnN, then{yn}n∈Nconverges toz;

3ifqxn, xmαnfor alln, mNwithm > n, then{xn}n∈Nis a Cauchy sequence;

4ifqy, xnαnfor allnN, then{xn}n∈Nis a Cauchy sequence;

5if q1, q2, q3, . . . , qn are Q-functions on X, then qx, y max{q1x, y, q2x, y, . . . , qnx, y}is also aQ-function onX.

2. Main Results

Analogous with Definition 1.1, Lakshmikantham and ´Ciri´c 24 introduced the following concept of a mixedg-monotone mapping.

Definition 2.1 Lakshmikantham and ´Ciri´c24. Let X,≤ be a partially ordered set, and F : X ×XX and g : XX. We say F has the mixed g-monotone property if F is nondecreasingg-monotone in its first argument and is nondecreasingg-monotone in its second argument, that is, for anyx, yX,

x1, x2X, gx1gx2impliesF x1, y

F x2, y

, y1, y2X, g

y1

g y2

impliesF x, y1

F x, y2

. 2.1

Note that ifgis the identity mapping, thenDefinition 2.1reduces toDefinition 1.1.

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Definition 2.2see24. An elementx, y∈X×Xis called a coupled coincidence point of a mappingF:X×XXandg:XXif

F x, y

gx, F y, x

g y

. 2.2

Definition 2.3see24. LetXbe a nonempty set andF :X×XXandg :XX.one saysFandgare commutative if

g F

x, y F

gx, g y

2.3

for allx, yX.

Following theorem is the main result of this paper.

Theorem 2.4. LetX,≤, dbe a partially ordered complete quasi-metric space with aQ-functionq onX. Assume that the functionϕ:0,∞ → 0,∞is such that

ϕt< t, for eacht >0. 2.4

Further, suppose thatk ∈0,1andF : X×XX;g :XXare such thatF has the mixed g-monotone property and

q F

x, y

, Fu, v

q

gx, gu q

g y

, gv 2

2.5

for allx, y, u, vXfor whichgxguandgygv.Suppose thatFX×XgX,and gis continuous and commutes withF, and also suppose that either

aFis continuous or

bXhas the following property:

iif a nondecreasing sequence{xn} → x, thenxnxfor alln, iiif a nonincreasing sequence{yn} → y, thenyynfor alln.

If there existsx0, y0Xsuch that gx0F

x0, y0

, g y0

F y0, x0

, 2.6

then there existx, yXsuch that gx F

x, y

, g

y F

y, x

, 2.7

that is,Fandghave a coupled coincidence.

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Proof. Choosex0, y0Xto be such thatgx0Fx0, y0andgy0Fy0, x0.SinceFX× XgX, we can choosex1, y1X such that gx1 Fx0, y0and gy1 Fy0, x0. Again fromFX ×XgX, we can choose x2, y2X such that gx2 Fx1, y1and gy2 Fy1, x1.Continuing this process, we can construct sequences{xn}and{yn}inX such that

gxn1 F xn, yn

, g yn1

F yn, xn

, ∀n≥0. 2.8

We will show that

gxngxn1, ∀n≥0, 2.9

g yn

g yn1

, ∀n≥0. 2.10

We will use the mathematical induction. Let n 0.Since gx0Fx0, y0and gy0Fy0, x0,and asgx1 Fx0, y0andgy1 Fy0, x0,we havegx0gx1andgy0gy1.Thus,2.9and2.10hold forn0.Suppose now that2.9and2.10hold for some fixedn ≥ 0.Then, sincegxngxn1 andgyn1gyn,and as F has the mixed g- monotone property, from2.8and2.9,

gxn1 F xn, yn

F xn1, yn

, F yn1, xn

F yn, xn

g yn1

, 2.11

and from2.8and2.10,

gxn2 F

xn1, yn1

F

xn1, yn

, F

yn1, xn

F

yn1, xn1 g

yn2

. 2.12

Now from2.11and2.12, we get

gxn1gxn2, g

yn1

g yn2

.

2.13

Thus, by the mathematical induction, we conclude that2.9and 2.10hold for alln ≥ 0.

Therefore,

gx0gx1gx2gx3≤ · · · ≤gxngxn1≤ · · ·, g

y0

g y1

g y2

g y3

≥ · · · ≥g yn

g yn1

≥ · · · .

2.14

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Denote

δnq

gxn, gxn1 q

g yn

, g yn1

. 2.15

We show that

δn≤2kϕ δn−1

2

. 2.16

Sincegxn−1gxnandgyn−1gyn,from2.11and2.5, we have q

gxn, gxn1 q

F

xn−1, yn−1 , F

xn, yn

q

gxn−1, gxn q

g yn−1

, g yn

2

δn−1

2

.

2.17

Similarly, from2.11and2.5, asgyngyn−1andgxngxn−1, q

g yn1

, g yn

q F

yn, xn , F

yn−1, xn−1

q

g yn−1

, g yn

q

gxn−1, gxn 2

δn−1

2

.

2.18

Adding2.17and2.18, we obtain2.16. Sinceϕt< tfort >0,it follows, from2.16, that 0≤δnn−1k2δn−2 ≤ · · · ≤knδ0, 2.19

and so, by squeezing, we get

nlim→ ∞δn0. 2.20

Thus,

n→ ∞lim q

gxn, gxn1 q

g yn

, g yn1

lim

n→ ∞δn0. 2.21

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Now, we prove that{gxn}and{gyn}are Cauchy sequences. Form > n,and sinceϕt< t for eacht >0,we have

δnmq

gxn, gxm q

g yn

, g ym

q

gxn, gxn1 q

g yn

, g yn1

q

gxn1, gxn2 q

g yn1

, g yn2 · · ·

q

gxm−1, gxm q

g ym−1

, g ym δnδn1δn2· · ·δm−1

δn2kϕ δn

2

2kϕ δn1

2

· · ·2kϕ δm−2

2

δn2k δn

2 δn1

2 · · ·δm−2 2

δnnδn1δn2· · ·

δnk

δn2kϕ δn

2

2kϕ δn1

2

· · ·

δnnnn1· · ·

δnk

δnnk2δnk3δn· · · δn

1kk2k3· · ·

1 1−k

δnλδn → 0, asn−→ ∞

λ 1 1−k

.

2.22

This means that form > n > n0,

q

gxn, gxm

λδn, q g

yn

, g ym

λδn. 2.23

Therefore, byLemma 1.8,{gxn}and{gyn}are Cauchy sequences. SinceX is complete, there existsx, yXsuch that

nlim→ ∞gxn x, lim

n→ ∞g yn

y, 2.24

and2.24combined with the continuity ofgyields

nlim→ ∞g gxn

gx, lim

n→ ∞g g

yn

g y

. 2.25

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From2.11and commutativity ofFandg, g

gxn1 g

F

xn, yn F

gxn, g yn

, g

g yn1

g F

yn, xn

F g

yn

, gxn

. 2.26

We now show thatgx Fx, yandgy Fy, x.

Case 1. Suppose that the assumption (a) holds. Taking the limit asn → ∞in2.26, and using the continuity ofF, we get

gx lim

n→ ∞g

gxn1 lim

n→ ∞F

gxn, g yn

F

nlim→ ∞gxn, lim

n→ ∞g yn

F x, y

, g

y lim

n→ ∞g g

yn1 lim

n→ ∞F g

yn , gxn

F

nlim→ ∞g yn

, lim

n→ ∞gxn

F y, x

. 2.27

Thus,

gx F x, y

, g

y F

y, x

. 2.28

Case 2. Suppose that the assumption (b) holds. Lethx ggx. Now, sinceg is continuous, {gxn} is nondecreasing with gxnx, gxnx for all n ∈ N, and {gyn} is nonincreasing with gyny, gyny for alln ∈ N, so hxnn≥1 is nondecreasing, that is,

hx0hx1hx2hx3≤ · · · ≤hxnhxn1≤ · · · 2.29 withhxn ggxngx,hxngxfor alln ∈ N, andhynn≥1 is nonincreasing, that is,

h y0

h y1

h y2

h y3

≥ · · · ≥h yn

h yn1

≥ · · · 2.30 withhyn ggyngy,hyngyfor alln∈N.

Let

γnqhxn, hxn1 q h

yn , h

yn1

. 2.31

Then replacinggbyhandδbyγin2.16, we getγn≤2kϕγn−1/2such that limn→ ∞γn 0.

We show that

nlim→ ∞q

hxn, gx q

h yn

, g y

0,

nlim→ ∞q

hxn, F x, y

q h

yn

, F y, x

0. 2.32

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Inδnm, replacinggbyhandδbyγ, we get

qhxn, hxm q h

yn , h

ym

λγn−→0, asn−→ ∞, 2.33

that is, form > n > n0,

qhxn, hxmλγn, q h

yn

, h ym

λγn

2 , 2.34

or form > nn01,

qhxn01, hxmλγn01,

q h

yn01 , h

ym

λγn01 2 .

2.35

LetMgx λγn01, andMgy λ/2γn01.Then, sincehxmgx, hymgy, and hxn01, hyn01X,by axiomQ2of theQ-function, we get

q

hxn01, gx

Mgx, q h

yn01 , g

y

Mgy.

Therefore, by the triangle inequality and∗, we haveforn > n0 Case 3.

q

hxn, gx q

h yn

, g y

qhxn, hxn1 q h

yn , h

yn1

q

hxn1, gx q

h yn1

, g

y

γnMgxMgy.

∗∗

This implies that

q

hxn, gx

γnMgxMgy, q

h yn

, g y

γnMgxMgy.

2.36

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Case 4. Also, we have

q

hxn, F x, y

p h

yn

, F y, x

qhxn, hxn1 q h

yn

, h yn1

q

hxn1, F x, y

q h

yn1 , F

y, x γn

q F

gxn, g yn

, F x, y q

F g

yn , gxn

, F y, x

γn q

ggxn, gx q

gg yn

, g y 2

q

gg yn

, g y

q

ggxn, gx 2

2.37

or

q

hxn, F x, y

q h

yn , F

y, x γn

q

hxn, gx q

h yn

, g y 2

q

h yn

, g y

q

hxn, gx 2

γn2kϕ q

hxn, gx q

h yn

, g y 2

γnk q

hxn, gx q

h yn

, g

y

γnk

γnMgxMgy by∗∗

μγn, whereμ1k

1λλ 2

.

2.38

That is, forn > n0,

q

hxn, F x, y

μγn, q h

yn

, F y, x

μγn. 2.39

Hence, byLemma 1.8,gx Fx, yand gy Fy, x.Thus, F and g have a coupled coincidence point.

The following example illustratesTheorem 2.4.

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Example 2.5. LetX 0,∞with the usual partial order≤.Defined:X×X → Rby

d x, y

⎧⎨

yx, ifxy, 2

xy

, otherwise, 2.40

andq:X×X → Rby

q x, y

xy , ∀x, y∈X. 2.41 Thend is a quasi-metric andqis aQ-function onX.Thus, X,≤, dis a partially ordered complete quasi-metric space with a Q-function q onX. Let ϕt t/2,for t > 0. Define F:X×XXby

F x, y

⎧⎪

⎪⎩ xy

5 , ifxy, 0, ifx < y,

2.42

and g : XX bygx 5x/k, where 0 < k < 1. Then, F has the mixed g-monotone property with

g F

x, y

⎧⎪

⎪⎩ xy

k , ifxy 0, ifx < y,

⎫⎪

⎪⎭F

gx, g y

, 2.43

andF,gare both continuous on their domains andFX×XgX. Letx, y, u, vX be such thatgxguandgygv.There are four possibilities for2.5to hold. We first compute expression on the left of2.5for these cases:

ix≥anduv,

q F

x, y

, Fu, v F

x, y

Fu, v

xy

5 −u−v 5

1

5 x−uyv

≤ 1 5

|x−u| yv .

2.44

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iixyandu < v, q

F x, y

, Fu, v F

x, y

−0

xy

5

1

5 x−uyu

≤ 1

5 x−u

yv u < v

≤ 1 5

|x−u| yv .

2.45

iiix < yanduv, q

F x, y

, Fu, v

|0−Fu, v|

u−v

5 1

5|u−x xv|

≤ 1

5 u−x

yv x < y

≤ 1 5

|x−u| yv .

2.46

ivx < yandu < v, q

F x, y

, Fu, v

|0−0|0. 2.47

On the other hand,in all the above four cases, we have

q

gx, gu q

g y

, gv 2

k q

gx, gu q

g y

, gv /2 2

k 4

5 k

|x−u| yv 5

4

|x−u| yv .

2.48

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Thus, F satisfies the contraction condition 2.5 of Theorem 2.4. Now, suppose that xnn≥1;ynn≥1be, respectively, nondecreasing and nonincreasing sequences such thatxnxandyny, then byTheorem 2.4,xnxandynyfor alln≥1.

Letx0 0, y05k.Then, this point satisfies the relations gx0 0F

x0, y0

, asx0 < y0 andg y0

25> kF y0, x0

. 2.49

Therefore, byTheorem 2.4, there existsx, yXsuch thatgx Fx, yandgy Fy, x.

Corollary 2.6. LetX,≤, dbe a partially ordered complete quasi-metric space with aQ-functionq onX. SupposeF:X×XXandg:XXare such thatFhas the mixedg-monotone property and assume that there existsk∈0,1such that

q F

x, y

, Fu, v

k 2

q

gx, gu q

g y

, gv 2.50

for allx, y, u, vXfor whichgxguandgygv.Suppose thatFX×XgX,and gis continuous and commutes withF, and also suppose that either

aFis continuous or

bXhas the following properties:

iif a nondecreasing sequence{xn} → x, thenxnxfor alln, iiif a nonincreasing sequence{yn} → y, thenyynfor alln.

If there existsx0, y0Xsuch that gx0F

x0, y0

, g

y0

F y0, x0

, 2.51

then there existx, yXsuch that gx F

x, y

, g

y F

y, x

, 2.52

that is,Fandghave a coupled coincidence.

Proof. Takingϕt tinTheorem 2.4, we obtainCorollary 2.6.

Now, we will prove the existence and uniqueness theorem of a coupled common fixed point. Note that ifS,≤is a partially ordered set, then we endow the productS×Swith the

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following partial order:

for x, y

,u, v∈S×S, x, y

≤u, v⇐⇒xu, yv. 2.53

FromTheorem 2.4, it follows that the setCF, gof coupled coincidences is nonempty.

Theorem 2.7. The hypothesis ofTheorem 2.4holds. Suppose that for everyx, y,y, xX×X there exists au, v ∈ X×X such thatFu, v, Fv, uis comparable toFx, y, Fy, xand Fx, y, Fy, x.Then,Fandghave a unique coupled common fixed point; that is, there exist a uniquex, y∈X×Xsuch that

xgx F x, y

, yg

y F

y, x

. 2.54

Proof. By Theorem, 2.1CF, g/φ. Let x, y,x, yCF, g. We show that if gx Fx, y, gy Fy, xandgx Fx, y,gy Fy, x,then

gx gx, g y

g y

. 2.55

By assumption there is u, v ∈ X × X such that Fu, v, Fv, u is comparable with Fx, y, Fy, x and Fx, y, Fy, x. Put u0 u,v0 v and choose u1, v1X so thatgu1 Fu0, v0andgv1 Fv0, u0.Then, as in the proof ofTheorem 2.4, we can inductively define sequences{gun}and{gvn}such that

gun1 Fun, vn, gvn1 Fvn, un. 2.56

Further, set x0 x, y0 y, x0 x, y0 y, and, as above, define the sequences {gxn},{gyn}and{gxn},{gyn}.Then it is easy to show that

gxn F x, y

, g

yn

F y, x

, gxn F x, y

, g

yn F

y, x 2.57

for alln ≥ 1.SinceFx, y, Fy, x gx1, gy1 gx, gyandFu, v, Fv, u gu1, gv1are comparable; thereforegxgu1andgygv1.It is easy to show that gx, gyandgun, gvnare comparable, that is,gxgunandgygvnfor all

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n≥1.From2.5and properties ofϕ, we have

q

gun1, gx q

gvn1, g y q

F un, yn

, F x, y

q

Fvn, un, F y, x

q

gun, gx q

g yn

, g y 2

q

gvn, g y

q

gun, gx 2

by2.6

2kϕ q

gun, gx q

gvn, g y 2

k q

gun, gx q

gvn, g

y

k

k2ϕ q

gun−1, gx q

gvn−1, g y 2

k2ϕ q

gvn−1, g y

q

gun−1, gx 2

by2.6

2k2ϕ q

gvn−1, g y

q

gun−1, gx 2

k2 q

gun−1, gx q

gvn−1, g

y

k2

k3ϕ q

gun−2, gx q

gvn−2, g y 2

by2.6

k3ϕ q

gvn−2, g y

q

gun−2, gx 2

2k3ϕ q

gun−2, gx q

gvn−2, g y 2

k3 q

gvn−2, g y

q

gun−2, gx

k3

≤ · · · ≤kn q

gu0, gx q

gv0, g

y

kn knt0−→0, asn−→ ∞,

2.58

wheret0qgu0, gx qgv0, gy.From this, it follows that, for eachn∈N,

q

gun1, gx

knt0, q

gvn1, g y

knt0. 2.59

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Similarly, one can prove that q

gun1, gx

knt0, q

gvn1, g y

knt0, n∈N, 2.60 wheret0 qgu0, gx qgv0, gy.Thus byLemma 1.8,gx gxandgy gy. Sincegx Fx, yandgy Fy, x, by commutativity ofFandg, we have

g

gx

g F

x, y F

gx, g y

, g

g y

g F

y, x F

g y

, gx .

2.61

Denotegx z, gy w.Then from2.61,

gz Fz, w, gw Fw, z. 2.62

Thus,z, wis a coupled coincidence point. Then, from2.55, withx zand y w, it follows thatgz gxandgw gy; that is,

gz z, gw w. 2.63

From2.62and2.63,

zgz Fz, w, wgw Fw, z. 2.64

Therefore, z, w is a coupled common fixed point of F and g. To prove the uniqueness, assume that p, q is another coupled common fixed point. Then, by 2.55, we have p gp gz zandqgq gw w.

Corollary 2.8. LetX,≤, dbe a partially ordered complete quasi-metric space with aQ-functionq onX. Assume that the functionϕ : 0,∞ → 0,∞is such that ϕt < tfor eacht > 0.Let k∈0,1,and letF:X×XXbe a mapping having the mixed monotone property onXand

q F

x, y

, Fu, v

qx, u q y, v 2

, for eachxu, yv. 2.65

Also suppose that either aFis continuous or

bXhas the following properties:

iif a nondecreasing sequence{xn} → x, thenxnxfor alln, iiif a non-increasing sequence{yn} → y, thenyynfor alln.

If there existsx0, y0Xsuch that x0F

x0, y0

, y0F y0, x0

, 2.66

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then, there existx, yXsuch that xF

x, y

, yF

y, x

. 2.67

Furthermore, ifx0, y0are comparable, thenxy,that is,xFx, x.

Proof. Following the proof ofTheorem 2.4withgIthe identity mapping onX, we get xngxn−→x, yng

yn

−→y, xF

x, y

, yF

y, x

. 2.68

We show thatxy.Let us suppose thatx0y0.We will show thatxn, ynare comparable for alln≥0,that is,

xnyn, ∀n≥0, 2.69

wherexnFxn−1, yn−1, yn Fyn−1, yn−1,n∈ {1,2, . . .}.Suppose that2.69holds for some fixedn≥0.Then, by mixed monotone property ofF,

xn1F xn, yn

F yn, xn

yn1 2.70

and2.69follows. Now from2.69,2.65, and properties ofϕ,we have qxn1, x q

F xn, yn

, F x, y

qxn, x q yn, y 2

kqxn, x q yn, y 2

k 2

qxn−1, x q yn−1, y 2

q yn−1, y

qxn−1, x 2

k2ϕ

qxn−1, x q yn−1, y 2

k3ϕ

qxn−2, x q yn−2, y 2

≤ · · · ≤kn1ϕ

qx0, x q y0, y 2

kn1s0−→0, asn−→ ∞,

2.71

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wheres0ϕqx0, x qy0, y/2.Similarly, we get q

xn1, y q

F xn, yn

, F y, x

kn1w0−→0, asn−→ ∞, 2.72

wherew0ϕqx0, y qy0, x/2. Hence, byLemma 1.8,xy,that is,xFx, x.

Corollary 2.9. LetX,≤, dbe a partially ordered complete quasi-metric space with aQ-functionq onX. LetF : X×XXbe a mapping having the mixed monotone property onX. Assume that there exists ak∈0,1such that

q F

x, y

, Fu, v

k 2

qx, u q y, v

, for eachxu, yv. 2.73

Also, suppose that either aFis continuous or

bXhas the following properties:

iif a nondecreasing sequence{xn} → x, thenxnxfor alln, iiif a nonincreasing sequence{yn} → y, thenyynfor alln.

If there existsx0, y0Xsuch that x0F

x0, y0

, y0F y0, x0

, 2.74

then, there existx, yXsuch that xF

x, y

, yF

y, x

. 2.75

Furthermore, ifx0, y0are comparable, thenxy,that is,xFx, x.

Proof. Takingϕt tinCorollary 2.8, we obtainCorollary 2.9.

Remark 2.10. As an application of fixed point results, the existence of a solution to the equilibrium problem was considered in2–7. It would be interesting to solve Ekeland-type variational principle, Ky Fan type best approximation problem and equilibrium problem utilizing recent results on coupled fixed points and coupled coincidence points.

Acknowledgment

The first and third author are grateful to DSR, King Abdulaziz University for supporting research project no.3-74/430.

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