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

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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

1_{Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} 2_{Department of Mathematical Sciences, LUMS, DHA Lahore, Lahore 54792, Pakistan}

Correspondence should be addressed to N. Hussain,[email protected]

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 mixedg-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 aQ-functionq. 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.

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Recall that if X,≤ is a partially ordered set and F : X → X such that for x, y ∈ X, x ≤ y implies Fx ≤ Fy, 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 ×X → X. 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, y∈X,

x1, x2∈X, x1≤x2⇒F

x1, y

≤Fx2, y

,

y1, y2∈X, y1≤y2⇒F

x, y1

≥Fx, 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×X → Xif

Fx, yx, Fy, xy. 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 < tandlimr→tϕr < tfor eacht >0, and also

suppose thatF:X×X → Xandg:X → Xsuch thatFhas the mixedg-monotone property and

dFx, y, Fu, v≤ϕ

dgx, gudgy, gv 2

1.3

for allx, y, u, v∈Xfor whichgx≤guandgy≥gv.Suppose thatFX×X⊆gX,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 xn≤xfor alln,

iiif a nonincreasing sequence{yn} → y,theny≤ynfor alln.

If there existsx0, y0∈Xsuch that

gx0≤F

x0, y0

, gy0

≥Fy0, x0

, 1.4

then there existx, y∈Xsuch that

gx Fx, y, gyFy, x, 1.5

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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, y∈X,

M2dx, y 0 if and only ifxy,

M3dx, y≤dx, z dz, yfor allx, y, z∈X.

The pairX, dis called a quasi-metric space.

Definition 1.5. Let X, 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, z∈X,

Q2if x ∈ X andynn≥1 is a sequence inX such that it converges to a pointywith respect to the quasi-metricandqx, yn≤ Mfor someMMx,thenqx, y ≤ M;

Q3for any >0, there existsδ >0 such thatqz, x≤ δ, andqz, y≤ δimplies 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 yn_n_≥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 anyx∈X, 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

dx, y

⎧ ⎨ ⎩

0, if xy,

_y _, _otherwise_, 1.6

andq:X×X → Rby

qx, y y , ∀x, y∈X. 1.7

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bLetX 0,1.Defined:X×X → Rby

dx, y

⎧ ⎨ ⎩

y−x, ifx≤y,

2x−y, otherwise, 1.8

andq:X×X → Rby

qx, y x−y , ∀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 let{αn}n∈Nand{βn}n∈Nbe such that they converge to0andx, y, z∈X.Then,

the following hold:

1ifqxn, y≤αnandqxn, z≤βnfor alln∈N, thenyz. In particular, ifqx, y 0

andqx, z 0, thenyz;

2ifqxn, yn≤αnandqxn, z≤βnfor alln∈N, then{yn}n∈Nconverges toz;

3ifqxn, xm≤αnfor alln, m∈Nwithm > n, then{xn}_n_∈Nis a Cauchy sequence;

4ifqy, xn≤αnfor alln∈N, 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 ×X → X and g : X → X. 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, y∈X,

x1, x2∈X, gx1≤gx2impliesF

x1, y

≤Fx2, y

,

y1, y2∈X, g

y1

≤gy2

impliesFx, y1

≥Fx, y2

.

2.1

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

Fx, ygx, Fy, xgy. 2.2

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

gFx, yFgx, gy 2.3

for allx, y∈X.

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×X → X;g :X → Xare such thatF has the mixed

g-monotone property and

qFx, y, Fu, v≤kϕ

qgx, guqgy, gv 2

2.5

aFis continuous or

bXhas the following property:

iif a nondecreasing sequence{xn} → x, thenxn ≤xfor alln,

iiif a nonincreasing sequence{yn} → y, theny≤ynfor alln.

gx0≤F

x0, y0

, gy0

≥Fy0, x0

, 2.6

gx Fx, y, gyFy, x, 2.7

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Proof. Choosex0, y0∈Xto be such thatgx0≤Fx0, y0andgy0≥Fy0, x0.SinceFX× X ⊆ gX, we can choosex1, y1 ∈ X such that gx1 Fx0, y0and gy1 Fy0, x0. Again fromFX ×X ⊆ gX, we can choose x2, y2 ∈ X 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, gyn1

Fyn, xn, ∀n≥0. 2.8

We will show that

gxn≤gxn1, ∀n≥0, 2.9

gyn≥gyn1

, ∀n≥0. 2.10

We will use the mathematical induction. Let n 0.Since gx0 ≤ Fx0, y0and gy0 ≥ Fy0, x0,and asgx1 Fx0, y0andgy1 Fy0, x0,we havegx0≤gx1andgy0≥ gy1.Thus,2.9and2.10hold forn0.Suppose now that2.9and2.10hold for some fixedn ≥ 0.Then, sincegxn ≤ gxn1 andgyn1 ≤ gyn,and as F has the mixed g -monotone property, from2.8and2.9,

gxn1 F

xn, yn≤Fxn1, yn

, Fyn1, xn

≤Fyn, xngyn1

, 2.11

and from2.8and2.10,

gx_n2 F

x_n1, yn1

≥Fx_n1, yn

, Fy_n1, xn

≥Fy_n1, xn1

gy_n2

. 2.12

Now from2.11and2.12, we get

gxn1≤gxn2,

gyn1

≥gyn2

.

2.13

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

gx0≤gx1≤gx2≤gx3≤ · · · ≤gxn≤gxn1≤ · · ·,

gy0

≥gy1

≥gy2

≥gy3

≥ · · · ≥gyn≥gyn1

≥ · · · .

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Denote

δnqgxn, gxn1

qgyn, gyn1

. 2.15

We show that

δn≤2kϕ

δn−1 2

. 2.16

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

qgxn, gxn1

qFxn−1, yn−1

, Fxn, yn

≤kϕ

qgxn−1, gxn

qgyn−1

, gyn 2

kϕ

δn−1 2

.

2.17

Similarly, from2.11and2.5, asgyn≤gyn−1andgxn≥gxn−1,

qgyn1

, gyn

qFyn, xn

, Fyn−1, xn−1

≤kϕ

qgyn−1

, gynqgxn−1, gxn

2

kϕ

δn−1 2

.

2.18

Adding2.17and2.18, we obtain2.16. Sinceϕt< tfort >0,it follows, from2.16, that

0≤δn≤kδn−1≤k2δn−2 ≤ · · · ≤knδ0, 2.19

and so, by squeezing, we get

lim

n→ ∞δn0. 2.20

Thus,

lim n→ ∞

qgxn, gxn1

qgyn, gyn1

lim

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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

qgyn

, gym

≤qgxn, gxn1

qgyn

, gyn1

qgxn1, gxn2

qgyn1

, gyn2

· · ·qgxm−1, gxm

qgym−1

, gym

δnδn1δn2· · ·δm−1

≤δn2kϕ

δn 2 2kϕ δn1 2

· · ·2kϕ

δm−2 2

≤δn2k

δn 2

δn1 2 · · ·

δm−2 2

≤δnkδnδn1δn2· · ·

≤δnk

δn2kϕ

_δ n 2 2kϕ _δ n1 2 · · ·

≤δnkδnkδnkδn1· · ·

≤δnk

δnkδnk2δnk3δn· · ·

δn

1kk2k3· · ·

1 1−k

δnλδn → 0, asn−→ ∞

λ 1 1−k

.

2.22

This means that form > n > n0,

qgxn, gxm≤λδn, qgyn, gym≤λδn. 2.23

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

lim

n→ ∞gxn x, nlim→ ∞g

yn

y, _2.24

and2.24combined with the continuity ofgyields

lim n→ ∞g

gxngx, lim n→ ∞g

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From2.11and commutativity ofFandg,

ggxn1

gFxn, yn

Fgxn, g

yn

,

ggyn1

gFyn, xnFgyn, 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, gynF

lim

n→ ∞gxn,nlim→ ∞g

ynFx, y,

gy lim n→ ∞g

gy_n1

lim n→ ∞F

gyn

, gxn

F

lim n→ ∞g

yn

, lim n→ ∞gxn

Fy, x.

2.27

Thus,

gx Fx, y, gyFy, x. 2.28

Case 2. Suppose that the assumption (b) holds. Let hx ggx. Now, sinceg is continuous, {gxn} is nondecreasing with gxn → x, gxn ≤ x for all n ∈ N, and {gyn} is nonincreasing with gyn → y, gyn ≥ y for alln ∈ N, so hxn_n_≥₁ is nondecreasing, that is,

hx0≤hx1≤hx2≤hx3≤ · · · ≤hxn≤hxn1≤ · · · 2.29

withhxn ggxn → gx,hxn ≤ gxfor alln ∈ N, andhynn≥1 is nonincreasing, that is,

hy0

≥hy1

≥hy2

≥hy3

≥ · · · ≥hyn≥hyn1

≥ · · · 2.30

withhyn ggyn → gy,hyn≥gyfor alln∈N. Let

γnqhxn, hxn1 q

hyn

, hyn1

. 2.31

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

We show that

lim n→ ∞q

hxn, gxqhyn, gy0,

lim n→ ∞q

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Inδnm, replacinggbyhandδbyγ, we get

qhxn, hxm q

hyn

, hym

≤λγn−→0, asn−→ ∞, 2.33

that is, form > n > n0,

qhxn, hxm≤λγn, qhyn, hym≤ λγn

2 , 2.34

or form > nn01,

qhx_n01, hxm≤λγn01,

qhyn01

, hym

≤ λγn01 2 .

2.35

LetMgx λγn01, andMgy λ/2γn01.Then, sincehxm → gx, hym → gy, and hxn01, hyn01∈X,by axiomQ2of theQ-function, we get

qhxn01, gx

≤Mgx, qhyn01

, gy≤Mgy. ∗

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

Case 3.

qhxn, gx

qhyn

, gy≤qhxn, hxn1 q

hyn

, hy_n1

qhxn1, gx

qhyn1

, gy

≤γnMgxMgy.

∗∗

This implies that

qhxn, gx

≤γnMgxMgy,

qhyn, gy≤γnMgxMgy.

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Case 4. Also, we have

qhxn, Fx, yphyn, Fy, x

≤qhxn, hxn1 q

hyn, hyn1

qhxn1, F

x, yqhyn1

, Fy, x

γnqFgxn, gyn, Fx, y

qFgyn

, gxn

, Fy, x

≤γnkϕ

qggxn, gx

qggyn

, gy 2

kϕ

qggyn

, gyqggxn, gx

2

2.37

or

qhxn, F

x, yqhyn

, Fy, x

γnkϕ

qhxn, gx

qhyn

, gy 2

kϕ

qhyn

, gyqhxn, gx

2

γn2kϕ

qhxn, gx

qhyn

, gy 2

≤γnkqhxn, gxqhyn, gy ≤γnk

γnMgxMgy by∗∗

μγn, whereμ1k

1λλ 2

.

2.38

That is, forn > n0,

qhxn, Fx, y≤μγn, qhyn, Fy, x≤μγn. 2.39

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

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

dx, y

⎧ ⎨ ⎩

y−x, ifx≤y,

2x−y, otherwise, 2.40

andq:X×X → Rby

qx, y x−y , ∀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×X → Xby

Fx, y

⎧ ⎪ ⎨ ⎪ ⎩

x−y

5 , ifx≥y, 0, ifx < y,

2.42

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

gFx, y

⎧ ⎪ ⎨ ⎪ ⎩

x−y

k , ifx≥y 0, ifx < y,

⎫ ⎪ ⎬ ⎪

⎭F

gx, gy, 2.43

andF,gare both continuous on their domains andFX×X ⊆ gX. Letx, y, u, v ∈X be such thatgx≤guandgy≥gv.There are four possibilities for2.5to hold. We first compute expression on the left of2.5for these cases:

ix≥andu≥v,

qFx, y, Fu, v Fx, y−Fu, v

x−y 5 −

u−v 5

1

5 x−u−

y−v

≤ 1 5

|x−u| y−v .

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iix≥yandu < v,

qFx, y, Fu, v Fx, y−0

x−y 5

1

5 x−u−

y−u

≤ 1

5 x−u−

y−v u < v

≤ 1 5

|x−u| y−v .

2.45

iiix < yandu≥v,

qFx, y, Fu, v|0−Fu, v|

u−v

5

1

5|u−x x−v| ≤ 1

5 u−x

y−v x < y

≤ 1 5

|x−u| y−v .

2.46

ivx < yandu < v,

qFx, y, Fu, v|0−0|0. 2.47

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

kϕ

qgx, guqgy, gv 2

k

qgx, guqgy, gv/2 2

k

4

5 k

|x−u| y−v

5

4

|x−u| y−v .

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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 thatxn → xandyn → y, then byTheorem 2.4,xn≤xandyn≥yfor alln≥1.

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

gx0 0F

x0, y0

, asx0 < y0 andg

y0

25> kFy0, x0

. 2.49

Therefore, byTheorem 2.4, there existsx, y∈Xsuch thatgx Fx, yandgy Fy, x.

Corollary 2.6. LetX,≤, dbe a partially ordered complete quasi-metric space with aQ-functionq

onX. SupposeF:X×X → Xandg:X → Xare such thatFhas the mixedg-monotone property and assume that there existsk∈0,1such that

qFx, y, Fu, v≤ k 2

qgx, guqgy, gv 2.50

aFis continuous or

bXhas the following properties:

gx0≤F

x0, y0

, gy0

≥Fy0, x0

, 2.51

gx Fx, y, gyFy, x, 2.52

that is,Fandghave a coupled coincidence.

Proof. Takingϕt tinTheorem 2.4, we obtainCorollary 2.6.

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following partial order:

forx, y,u, v∈S×S, x, y≤u, v⇐⇒x≤u, y≥v. 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∗, x∗∈X×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 Fx, y, ygyFy, x. 2.54

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

gx gx∗, gygy∗. 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, v1 ∈ X so thatgu1 Fu0, v0andgv1 Fv0, u0.Then, as in the proof ofTheorem 2.4, we can inductively define sequences{gun}and{gvn}such that

gu_n1 Fun, vn, gvn1 Fvn, un. 2.56

Further, set x0 x, y0 y, x∗₀ x∗, y∗₀ y∗, and, as above, define the sequences {gxn},{gyn}and{gx∗n},{gy∗n}.Then it is easy to show that

gxn Fx, y, gynFy, x, gx_n∗ Fx∗, y∗, gy∗_nFy∗, x∗

2.57

for alln ≥ 1.SinceFx, y, Fy, x gx1, gy1 gx, gyandFu, v, Fv, u

gu1, gv1are comparable; thereforegx≤gu1andgy≥gv1.It is easy to show that

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

qgun1, gx

qgvn1, g

y

qFun, yn

, Fx, yqFvn, un, F

y, x

≤kϕ

qgun, gxqgyn, gy 2

kϕ

qgvn, gyqgun, gx 2

by2.6

2kϕ

qgun, gxqgvn, gy 2

≤kqgun, gx

qgvn, g

y k

≤k2ϕ

qgun−1, gx

qgvn−1, g

y 2

k2ϕ

qgvn−1, g

yqgun−1, gx

2

by2.6

2k2ϕ

qgvn−1, g

yqgun−1, gx

2

≤k2qgun−1, gx

qgvn−1, g

y k2

≤k3ϕ

qgun−2, gx

qgvn−2, g

y 2

by2.6

k3ϕ

qgvn−2, g

yqgun−2, gx

2

2k3ϕ

qgun−2, gx

qgvn−2, g

y 2

≤k3_q_g_v n−2, g

yqgun−2, gx

k3

≤ · · · ≤knqgu0, gx

qgv0, g

y kn

knt0−→0, asn−→ ∞,

2.58

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

qgun1, gx

≤knt0, q

gvn1, g

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Similarly, one can prove that

qgun1, gx∗

≤knt₀, qgvn1, g

y∗≤knt₀, n∈N, 2.60

wheret₀ qgu0, gx∗ qgv0, gy∗.Thus byLemma 1.8,gx gx∗andgy gy∗. Sincegx Fx, yandgy Fy, x, by commutativity ofFandg, we have

ggxgFx, yFgx, gy, ggygFy, xFgy, 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.

onX. Assume that the functionϕ : 0,∞ → 0,∞is such that ϕt < tfor eacht > 0.Let

k∈0,1,and letF:X×X → Xbe a mapping having the mixed monotone property onXand

qFx, y, Fu, v≤kϕ

qx, u qy, v 2

, for eachx≤u, y≥v. 2.65

Also suppose that either

aFis continuous or

iiif a non-increasing sequence{yn} → y, theny≤ynfor alln.

x0≤F

x0, y0

, y0≥F

y0, x0

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then, there existx, y∈Xsuch that

xFx, y, yFy, 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,

xFx, y, yFy, x.

2.68

We show thatxy.Let us suppose thatx0≤y0.We will show thatxn, ynare comparable for alln≥0,that is,

xn≤yn, ∀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≤Fyn, xnyn1 2.70

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

qx_n1, x q

Fxn, yn

, Fx, y

≤kϕ

qxn, x qyn, y 2

≤kqxn, x q

yn, y 2

≤ k 2

kϕ

qxn−1, x q

yn−1, y

2

kϕ

qyn−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−→ ∞,

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

qx_n1, y

qFxn, yn

, Fy, x≤kn1w0−→0, asn−→ ∞, 2.72

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

onX. LetF : X×X → Xbe a mapping having the mixed monotone property onX. Assume that there exists ak∈0,1such that

qFx, y, Fu, v≤ k 2

qx, u qy, v, for eachx≤u, y≥v. 2.73

Also, suppose that either

aFis continuous or

iif a nondecreasing sequence{xn} → x, thenxn≤xfor alln,

x0≤F

x0, y0

, y0≥F

y0, x0

, 2.74

then, there existx, y∈Xsuch that

xFx, y, yFy, 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

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