Volume 2010, Article ID 134897,11pages doi:10.1155/2010/134897
Research Article
Coupled Coincidence Point and Coupled Common Fixed Point Theorems in Partially Ordered Metric Spaces with w-Distance
Mujahid Abbas,
1Dejan Ili ´c,
2and Muhammad Ali Khan
11Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan
2Department of Mathematics, Faculty of Sciences and Mathematics, University of Niˆs, Viˆsegradska 33, 18000 Niˆs, Serbia
Correspondence should be addressed to Dejan Ili´c,[email protected] Received 7 April 2010; Accepted 18 October 2010
Academic Editor: Hichem Ben-El-Mechaiekh
Copyrightq2010 Mujahid Abbas 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.
We introduce the concept of aw-compatible mapping to obtain a coupled coincidence point and a coupled point of coincidence for nonlinear contractive mappings in partially ordered metric spaces equipped withw-distances. Related coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend, and unify several well-known comparable results in the literature.
1. Introduction and Preliminaries
In 1996, Kada et al.1introduced the notion ofw-distance. They elaborated, with the help of examples, that the concept ofw-distance is general than that of metric on a nonempty set.
They also proved a generalization of Caristi fixed point theorem employing the definition of w-distance on a complete metric space. Recently, Ili´c and Rakoˇcevi´c2obtained fixed point and common fixed point theorems in terms ofw-distance on complete metric spacessee also 3–9.
Definition 1.1. LetX, dbe a metric space. A mappingp : X×X → 0,∞is called aw- distance onXif the following are satisfied:
w1px, z≤px, y py, zfor allx, y, z∈X,
w2for anyx∈X,px,·:X → 0,∞is lower semicontinuous,
w3for any ε > 0 there existsδε > 0 such thatpz, x ≤ δ and pz, y ≤ δ imply px, y≤ε, for anyx, y, z∈X.
The metricdis aw-distance onX. For more examples ofw-distances, we refer to10.
Definition 1.2. LetX be a nonempty set with aw-distance onX. Ones denotes thew-closure of a subsetBofXby clωBwhich is defined as
clωB
x∈X:pxn, x−→0 for some sequence{xn}inB
∪B. 1.1
The next Lemma is crucial in the proof of our results.
Lemma 1.3see1. LetX, dbe a metric space, and letpbe aw-distance onX. Let{xn}and {yn}be sequences inX, letαn andβnbe sequences in 0,∞converging to 0, and letx, y, z ∈ X.
Then the following hold.
1Ifpxn, y≤αn andpxn, z≤βnfor anyn∈N, thenyz. In particular, ifpx, y 0,px, z 0 thenyz.
2Ifpxn, yn≤αnandpxn, z≤βnfor anyn∈N, thenynconverges toz.
3Ifpxn, xm≤αnfor anym, n∈Nwithn≺m, thenxnis a Cauchy sequence.
4Ifpy, xn≤αnfor anyn∈N, thenxnis a Cauchy sequence.
Bhaskar and Lakshmikantham in11introduced the concept of coupled fixed point of a mappingF:X×X → Xand investigated some coupled fixed point theorems in partially ordered sets. They also discussed an application of their result by investigating the existence and uniqueness of solution for a periodic boundary value problem. Sabetghadam et al. in12 introduced this concept in cone metric spaces. They investigated some coupled fixed point theorems in cone metric spaces. Recently, Lakshmikantham and ´Ciri´c13proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces which extend the coupled fixed point theorem given in11. The following are some other definitions needed in the sequel.
Definition 1.4see12. LetXbe any nonempty set. LetF :X×X → Xandg :X → Xbe two mappings. An ordered pairx, y∈X×Xis called
1a coupled fixed point of a mappingF:X×X → XifxFx, yandyFy, x, 2a coupled coincidence point of hybrid pair {F, g} if gx Fx, y and gy
Fy, xandgx, gyis called coupled point of coincidence,
3a common coupled fixed point of hybrid pair {F, g} if x gx Fx, yand ygy Fy, x.
Note that ifx, yis a coupled fixed point ofF, theny, xis also a coupled fixed point of the mappingF.
Definition 1.5. LetXbe any nonempty set. Mappings F :X×X → X andg :X → Xare calledw-compatible ifgFx, y Fgx, gywhenevergx Fx, yandgy Fy, x.
Definition 1.6. LetX, dbe a metric space withw-distancep. A mappingF : X ×X → X is said to bew-continuous at a pointx, y ∈ X×X with respect to mappingg : X → X if for everyε > 0 there exists aδε > 0 such thatpgu, gx pgv, gy < δ implies that pFx, y, Fu, v< εfor allu, v∈X.
Definition 1.7. Let X be a partially ordered set. Mapping g : X → X is called strictly monotone increasing mapping if
xy⇐⇒gxgy or equivalentlyxy⇐⇒gxgy. 1.2 Definition 1.8. LetX be a partially ordered set. A mappingF : X ×X → X is said to be a mixed monotone ifFx, yis monotone nondecreasing inxand monotone nonincreasing in y, that is, for anyx, y∈X,
x1, x2∈X, x1x2⇒F x1, y
F
x2, y , y1, y2∈X, y1y2 ⇒F
x, y1
F x, y2
. 1.3
Kada et al.1gave an example to show thatp is not symmetric in general. We denote by MX and M1X, respectively, the class of all w-distances on X and the class of all w- distances onX which are symmetric for comparable elements inX. Also in the sequel, we will consider thatx, yandu, vare comparable with respect to ordering inX×Xifxu andyv.
2. Coupled Coincidence Point
In this section, we prove coincidence point results in the frame work of partially ordered metric spaces in terms of aw-distance.
Theorem 2.1. LetX, dbe a partially ordered metric space with aw-distancepandg :X → Xa strictly monotone increasing mapping. Suppose that a mixed monotone mappingF :X×X → Xis w-continuous with respect togsuch that
p F
x, y
, Fu, v
≤a1p gu, gx
a2p gv, gy
, 2.1
for allx, y, u, v∈Xwithxu, yvorxu, yvanda1a2<1. LetFX×X⊆gXand py, x 0 wheneverpx, y 0, for somex, y∈clωFX×X. IfgXis complete and there exist x0, y0∈Xsuch thatgx0Fx0, y0andFy0, x0gy0, thenFandghave a coupled coincidence point.
Proof. Letgx1 Fx0, y0and gy1 Fy0, x0for somex1, y1 ∈ X; this can be done since FX ×X ⊆ gX. Following the same arguments, we obtain gx2 Fx1, y1 and gy2 Fy1, x1. Put
F1 x0, y0
gx1, F2 x0, y0
F x1, y1
gx2,
F2 y0, x0
F y1, x1
gy2.
2.2
Similarly for alln∈N,
gxn1Fn1 x0, y0
, gyn1 Fn1 y0, x0
. 2.3
Sincegis strictly monotone increasing andFhas the mixed monotone property, we have gx2F2
x0, y0 F
x1, y1
F
x0, y0
gx1, gy2gy1. 2.4
Similarly
gx0 F x0, y0
gx1F2 x0, y0
gx2· · ·
Fn1
x0, y0
gxn1· · ·,
gy0 F y0, x0
gy1F2 y0, x0
gy2· · ·
Fn1
y0, x0
· · ·.
2.5
Now for alln≥2, using2.1, we get
p Fn
x0, y0 , Fn1
x0, y0 p
F
xn−1, yn−1 , F
xn, yn
≤a1p
gxn, gxn−1 a2p
gyn, gyn−1 a1
p Fn
x0, y0 , Fn−1
x0, y0 a2 p
Fn y0, x0
, Fn−1
y0, x0 , p
Fn y0, x0
, Fn1
y0, x0
≤a1 p
Fn y0, x0
, Fn−1
y0, x0 a2 p
Fn x0, y0
, Fn−1
x0, y0 .
2.6
From2.6,
p Fn
x0, y0
, Fn1 x0, y0
p Fn
y0, x0
, Fn1 y0, x0
≤h p
Fn x0, y0
, Fn−1 x0, y0
p Fn
y0, x0
, Fn−1
y0, x0 ,
2.7
whereha1a2. Continuing, we conclude that
p Fn
x0, y0 , Fn1
x0, y0 p
Fn y0, x0
, Fn1
y0, x0
≤hn p
gx1, gx0
p
gy1, gy0
hnδ1
2.8
ifnis odd, whereδ1pgx1, gx0 pgy1, gy0. Also, p
Fn x0, y0
, Fn1 x0, y0
p Fn
y0, x0
, Fn1 y0, x0
≤hn p
gx0, gx1 p
gy0, gy1 hnδ2
2.9
ifnis even, where
δ2p
gx0, gx1 p
gy0, gy1
. 2.10
LetδnpFnx0, y0, Fn1x0, y0 pFny0, x0, Fn1y0, x0; then for everyninNwe have
δn≤hnδ0, 2.11
where
δ0max{δ1, δ2}. 2.12
Hence,
p Fn
x0, y0 , Fn1
x0, y0
−→0, p Fn
y0, x0 , Fn1
y0, x0
−→0 as n−→ ∞. 2.13
Form > n, we get p
Fn x0, y0
, Fm
x0, y0 p
Fn y0, x0
, Fm
y0, x0
≤p Fn
x0, y0 , Fn1
x0, y0 p
Fn1 x0, y0
, Fn2
x0, y0 · · · p
Fm−1 x0, y0
, Fm
x0, y0 p
Fn y0, x0
, Fn1
y0, x0 p
Fn1 y0, x0
, Fn2
y0, x0 · · · p
Fm−1 y0, x0
, Fm
y0, x0
δnδn1· · ·δm−1≤hnδ0hn1δ0· · ·hm−1δ0≤ hn 1−hδ0
2.14
which further implies that
p Fn
x0, y0
, Fm x0, y0
≤ hn 1−hδ0
p Fn
y0, x0
, Fm y0, x0
≤ hn 1−hδ0.
2.15
Lemma 1.33 implies that {Fnx0, y0} {gxn} and {Fny0, x0} {gyn} are Cauchy sequences in gX. SincegXis complete, there existx, y ∈ X such thatgxn → gxand gyn → gy. Sincepgxn,·is lower semicontinuous, we have
p Fn
x0, y0 , gx
≤lim inf
m→ ∞ p
gxn, gxm
≤ hn
1−hδ0 2.16
which implies that
p Fn
x0, y0
, gx
−→0 asn−→ ∞. 2.17
Similarly
p Fn
y0, x0
, gy
−→0 asn−→ ∞. 2.18
Letε >0 be given. SinceFisw-continuous atx, ywith respect tog, there existsδ >0 such that for eachn
p
gxn, gx p
gyn, gy
< δ implies thatp F
x, y , F
xn, yn
< ε
2. 2.19
Sincepgxn, gx → 0 andpgyn, gy → 0, forγ minε/2, δ/2, there existsn0such that, for alln≥n0,
p
gxn, gx
< γ, p
gyn, gy
< γ. 2.20
Now,
p F
x, y , gx
≤p F
x, y , Fn01
x0, y0
p Fn01
x0, y0
, gx
p F
x, y , F
xn0, yn0
p
gxn01, gx
< ε 2γε
2.21
implies thatpFx, y, gx 0. Since p
Fn x0, y0
, F x, y
≤p Fn
x0, y0
, gx p
gx, F x, y
≤ hn 1−hδ0,
2.22
usingLemma 1.31, we obtainFx, y gx. Similarly, we can prove thatFy, x gy. Hence x, yis coupled coincidence point ofFandg.
Theorem 2.2. LetX, dbe a partially ordered metric space with aw-distancephaving the following properties.
1If{xn}is inXwithxn xn1for allnandxn → xfor somex∈X, thenxn xfor all n.
2If{yn}is inXwithyn1 ynfor allnandyn → yfor somey∈X, thenyynfor all n.
LetF:X×X → Xbe a mixed monotone andg:X → Xa strict monotone increasing mapping such that
p F
x, y
, Fu, v
≤a1p gu, gx
a2p gv, gy
, 2.23
for allx, y, u, v ∈Xwithxu, yvorxu, yvanda1a2<1. LetFX×X⊆gX andpy, x 0 wheneverpx, y 0, for somex, y∈clωFX×X. IfgXis complete and there existx0, y0 ∈ X such thatgx0 Fx0, y0and Fy0, x0 gy0, thenF andg have a coupled coincidence point.
Proof. Construct two sequences {gxn} {Fnx0, y0} and {gyn} {Fny0, x0} such that gxn gxn1 andgyn gyn1for allnandgxn → gxandgyn → gyfor somex ∈X, as given in the proof ofTheorem 2.1. Now, we need to show thatFx, y gxandFy, x gy.
Letε >0. SincepFnx0, y0, gx → 0 andpFny0, x0, gy → 0, there existsn1 ∈ Nsuch that, for alln≥n1, we have
p Fn
x0, y0 , gx
< ε
3, p Fn
y0, x0 , gy
< ε
3. 2.24
Consider
p F
x, y , gx
≤p F
x, y , Fn1
x0, y0
p Fn1
x0, y0
, gx
p F
x, y , F
xn, yn p
Fn1 x0, y0
, gx
≤a1p
gxn, gx a2p
gyn, gy p
Fn1 x0, y0
, gx a1p
Fn x0, y0
, gx a2p
Fn y0, x0
, gy p
Fn1 x0, y0
, gx
< a1ε 3 a2ε
3 ε 3
< ε,
2.25
which implies thatpFx, y, gx 0. Also, fromTheorem 2.1, we have
p Fn
x0, y0
, gx
≤ hn
1−hδ0. 2.26
Therefore,
p Fn
x0, y0
, F x, y
≤p Fn
x0, y0 , gx
p gx, F
x, y
≤ hn 1−hδ0
2.27
implies thatgxFx, y. Similarly, we can prove thatFy, x gy. Hencex, yis coupled coincidence point ofFandg.
3. Coupled Common Fixed Point
In this section, using the concept ofw-compatible maps, we obtain a unique coupled common fixed point of two mappings.
Theorem 3.1. Let all the hypotheses ofTheorem 2.1(resp.,Theorem 2.2) hold witha1a2<1/2. If for everyx, y,x∗, y∗∈X×Xthere existsu, v∈X×Xthat is comparable tox, yandx∗, y∗ with respect to ordering inX×X, then there exists a unique coupled point of coincidence ofFandg.
Moreover ifFandgarew-compatible, thenFandghave a unique coupled common fixed point.
Proof. Letgx∗, gy∗be another coupled coincidence point of F andg. We will discuss the following two cases.
Case 1. Ifx, yis comparable tox∗, y∗with respect to ordering inX×X, then
p
gx, gx∗ p
gy, gy∗ p
F x, y
, F
x∗, y∗ p
F y, x
, F
y∗, x∗
≤a1p
gx∗, gx a2p
gy∗, gy a1p
gy∗, gy a2p
gx∗, gx
≤a1a2 p
gx, gx∗ p
gy, gy∗
3.1
implies thatpgx, gx∗ pgy, gy∗ 0. Hencepgx, gx∗ 0pgy, gy∗. Also,
p gx, gx
p gy, gy
pFx, x, Fx, x p F
y, y , F
y, y
≤2a1p gx, gx
2a2p
gy, gy 3.2
gives thatpgx, gx 0pgy, gy. The result follows usingLemma 1.31.
Case 2. If x, y is not comparable to x∗, y∗, then there exists an upper bound or lower bound u, vofx, y,x∗, y∗. Again sinceg is strictly monotone increasing mapping and F satisfies mixed monotone property, therefore, for all n 0,1, . . .,Fnu, v, Fnv, u is
comparable toFnx, y, Fny, x gx, gyandFny, x, Fnx, y gy, gx. Following similar arguments to those given in the proof ofTheorem 2.1, we obtain
p
gx, gx∗ p
gy, gy∗ p
Fn x, y
, Fn
x∗, y∗ p
Fn y, x
, Fn
y∗, x∗
≤ p
Fn x, y
, Fnu, v p
Fnu, v, Fn
x∗, y∗
p Fn
y, x
, Fnv, u p
Fnv, u, Fn
y∗, x∗
p Fn
x, y
, Fnu, v p
Fn y, x
, Fnv, u
p
Fnu, v, Fn
x∗, y∗ p
Fnv, u, Fn
y∗, x∗
≤hnβ0hnγ0,
3.3
whereβ0 max{pgu, gx pgv, gy, pgx, gu pgy, gv} andγ0 max{pgx∗, gu pgy∗, gv, pgu, gx∗ pgv, gy∗}. On taking limit as n → ∞on both sides of3.3, we have
p
gx, gx∗ p
gy, gy∗
0 3.4
andpgx, gx∗ 0 pgy, gy∗. By the same lines as in Case1, we prove thatpgx, gx 0 pgy, gy. AgainLemma 1.31implies thatgx gx∗andgy gy∗. Hencegx, gyis unique coupled point of coincidence ofFandg. Note that ifgx,gyis a coupled point of coincidence ofF andg, then gy, gxare also a coupled points of coincidence ofF andg.
Thengxgyand thereforegx, gxis unique coupled point of coincidence ofFandg. Let ugx. SinceFandgare w-compatible, we obtain
gug gx
gFx, x F gx, gx
Fu, u. 3.5
Consequentlygugx. ThereforeuguFu, u. Henceu, uis a coupled common fixed point ofFandg.
Remark 3.2. If in addition to the hypothesis ofTheorem 2.1resp.,Theorem 2.2we suppose thatp∈M1X,x0andy0are comparable, thengxgy.
Proof. Recall thatgx0 Fx0, y0. Now, ifx0 y0, thengx0 gy0. We claim that, for all n ∈N,gxn gyn. Sinceg is strictly monotone increasing andF satisfies mixed monotone property, we have
gx1F x0, y0
F
y0, x0
gy1. 3.6
Assuming thatgxn gyn, sincegis strictly monotone increasing, soxn yn. By the mixed monotone property ofF, we have
gxn1Fn1 x0, y0
F xn, yn
F yn, xn
gyn1. 3.7
Therefore,
gxngyn ∀n. 3.8
Lettingε >0, there exists ann0∈Nsuch thatpgx, Fnx0, y0< ε/4 andpFny0, x0, gy<
ε/4 for alln≥n0. Now,
p gx, gy
≤p
gx, Fn01 x0, y0
Fn01
x0, y0
, gy
≤p
gx, Fn01 x0, y0
p Fn01
x0, y0
, Fn01 y0, x0
Fn01
y0, x0
, gy
< ε 4hp
Fn0 x0, y0
, Fn0
y0, x0 ε
4
≤ ε 2h
p Fn0
x0, y0 , gx
p gx, gy
gy, Fn0
y0, x0
< ε 2hε
4 hp gx, gy
hε 4
< εhp gx, gy
3.9 implies that1−hpgx, gy< ε. Sinceh <1, thereforepgx, gy 0. Similarly we can prove thatpgx, gx 0. Hence byLemma 1.31, we havegxgy. Similarly, ifgx0 gy0, we can show thatgxngynfor eachnandgxgy.
Acknowledgment
The present version of the paper owes much to the precise and kind remarks of the learned referees.
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