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,
1M. H. Shah,
2and M. A. Kutbi
11Department 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.
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
≤F x2, y
, y1, y2∈X, y1≤y2⇒F
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×X → Xif
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 limr→tϕr < tfor eacht >0, and also suppose thatF:X×X → Xandg:X → Xsuch 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, 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
, g y0
≥F y0, x0
, 1.4
then there existx, y∈Xsuch that gx F
x, y
, g
y F
y, x
, 1.5
that is,Fandghave a coupled coincidence.
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. 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, 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 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 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
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.
bLetX 0,1.Defined:X×X → Rby
d x, y
⎧⎨
⎩
y−x, ifx≤y, 2
x−y
, otherwise, 1.8
andq:X×X → Rby
q x, 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 to 0 andx, 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
≤F x2, y
, y1, y2∈X, g
y1
≤g y2
impliesF x, y1
≥F x, y2
. 2.1
Note that ifgis the identity mapping, thenDefinition 2.1reduces toDefinition 1.1.
Definition 2.2see24. An elementx, y∈X×Xis called a coupled coincidence point of a mappingF:X×X → Xandg:X → Xif
F x, y
gx, F y, x
g y
. 2.2
Definition 2.3see24. LetXbe a nonempty set andF :X×X → Xandg :X → X.one saysFandgare commutative if
g F
x, y F
gx, g y
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
q F
x, y
, Fu, v
≤kϕ q
gx, gu q
g y
, gv 2
2.5
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, thenxn ≤xfor alln, iiif a nonincreasing sequence{yn} → y, theny≤ynfor alln.
If there existsx0, y0∈Xsuch that gx0≤F
x0, y0
, g y0
≥F y0, x0
, 2.6
then there existx, y∈Xsuch that gx F
x, y
, g
y F
y, x
, 2.7
that is,Fandghave a coupled coincidence.
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
, g yn1
F yn, xn
, ∀n≥0. 2.8
We will show that
gxn≤gxn1, ∀n≥0, 2.9
g yn
≥g yn1
, ∀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
≤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
gxn1≤gxn2, g
yn1
≥g yn2
.
2.13
Thus, by the mathematical induction, we conclude that2.9and 2.10hold for alln ≥ 0.
Therefore,
gx0≤gx1≤gx2≤gx3≤ · · · ≤gxn≤gxn1≤ · · ·, g
y0
≥g y1
≥g y2
≥g y3
≥ · · · ≥g yn
≥g yn1
≥ · · · .
2.14
Denote
δnq
gxn, gxn1 q
g yn
, g yn1
. 2.15
We show that
δn≤2kϕ δn−1
2
. 2.16
Sincegxn−1≤gxnandgyn−1≥gyn,from2.11and2.5, we have q
gxn, gxn1 q
F
xn−1, yn−1 , F
xn, yn
≤kϕ q
gxn−1, gxn q
g yn−1
, g yn
2
kϕ δn−1
2
.
2.17
Similarly, from2.11and2.5, asgyn≤gyn−1andgxn≥gxn−1, q
g yn1
, g yn
q F
yn, xn , F
yn−1, xn−1
≤kϕ q
g yn−1
, g yn
q
gxn−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
nlim→ ∞δn0. 2.20
Thus,
n→ ∞lim q
gxn, gxn1 q
g yn
, g yn1
lim
n→ ∞δn0. 2.21
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
≤δnkδnδn1δn2· · ·
≤δnk
δn2kϕ δn
2
2kϕ δn1
2
· · ·
≤δnkδnkδnkδn1· · ·
≤δnk
δnkδnk2δnk3δn· · · 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, y∈Xsuch 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
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 gxn → x, gxn ≤ x for all n ∈ N, and {gyn} is nonincreasing with gyn → y, gyn ≥ y for alln ∈ N, so hxnn≥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,
h y0
≥h y1
≥h y2
≥h y3
≥ · · · ≥h yn
≥h yn1
≥ · · · 2.30 withhyn ggyn → gy,hyn≥gyfor 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
Inδnm, replacinggbyhandδbyγ, we get
qhxn, hxm q h
yn , h
ym
≤λγn−→0, asn−→ ∞, 2.33
that is, form > n > n0,
qhxn, hxm≤λγn, q h
yn
, h ym
≤ λγn
2 , 2.34
or form > nn01,
qhxn01, hxm≤λγn01,
q h
yn01 , h
ym
≤ λγn01 2 .
2.35
LetMgx λγn01, andMgy λ/2γn01.Then, sincehxm → gx, hym → gy, and hxn01, hyn01∈X,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
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
≤γnkϕ q
ggxn, gx q
gg yn
, g y 2
kϕ q
gg yn
, g y
q
ggxn, gx 2
2.37
or
q
hxn, F x, y
q h
yn , F
y, x γnkϕ
q
hxn, gx q
h yn
, g y 2
kϕ 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.
Example 2.5. LetX 0,∞with the usual partial order≤.Defined:X×X → Rby
d x, y
⎧⎨
⎩
y−x, ifx≤y, 2
x−y
, otherwise, 2.40
andq:X×X → Rby
q x, 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
F x, 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
g F
x, y
⎧⎪
⎨
⎪⎩ x−y
k , ifx≥y 0, ifx < y,
⎫⎪
⎬
⎪⎭F
gx, g y
, 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,
q F
x, y
, Fu, v F
x, y
−Fu, v
x−y
5 −u−v 5
1
5 x−u− y−v
≤ 1 5
|x−u| y−v .
2.44
iix≥yandu < v, q
F x, y
, Fu, v F
x, 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, q
F x, 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, q
F x, y
, Fu, v
|0−0|0. 2.47
On the other hand,in all the above four cases, we have
kϕ q
gx, gu q
g y
, gv 2
k q
gx, gu q
g y
, gv /2 2
k 4
5 k
|x−u| y−v 5
4
|x−u| y−v .
2.48
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> kF y0, 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
q F
x, y
, Fu, v
≤ k 2
q
gx, gu q
g y
, gv 2.50
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 properties:
iif a nondecreasing sequence{xn} → x, thenxn ≤xfor alln, iiif a nonincreasing sequence{yn} → y, theny≤ynfor alln.
If there existsx0, y0∈Xsuch that gx0≤F
x0, y0
, g
y0
≥F y0, x0
, 2.51
then there existx, y∈Xsuch 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
following partial order:
for x, 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 F x, y
, yg
y F
y, 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∗, 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, 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
gun1 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 F x, y
, g
yn
F y, x
, gx∗n F x∗, y∗
, g
y∗n F
y∗, 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 gx, gyandgun, gvnare comparable, that is,gx≤gunandgy≥gvnfor all
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
≤kϕ q
gun, gx q
g yn
, g y 2
kϕ 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
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 gx∗andgy 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×X → Xbe a mapping having the mixed monotone property onXand
q F
x, y
, Fu, v
≤kϕ
qx, u q y, v 2
, for eachx≤u, y≥v. 2.65
Also suppose that either aFis continuous or
bXhas the following properties:
iif a nondecreasing sequence{xn} → x, thenxn ≤xfor alln, iiif a non-increasing sequence{yn} → y, theny≤ynfor alln.
If there existsx0, y0∈Xsuch that x0≤F
x0, y0
, y0≥F y0, x0
, 2.66
then, there existx, y∈Xsuch 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 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
≤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
≤kϕ
qxn, x q yn, y 2
≤kqxn, x q yn, y 2
≤ k 2
kϕ
qxn−1, x q yn−1, y 2
kϕ
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
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×X → Xbe 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 eachx≤u, y≥v. 2.73
Also, suppose that either aFis continuous or
bXhas the following properties:
iif a nondecreasing sequence{xn} → x, thenxn≤xfor alln, iiif a nonincreasing sequence{yn} → y, theny≤ynfor alln.
If there existsx0, y0∈Xsuch that x0≤F
x0, y0
, y0≥F y0, x0
, 2.74
then, there existx, y∈Xsuch 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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