Volume 2012, Article ID 327273,20pages doi:10.1155/2012/327273
Research Article
Coupled Fixed Points for Meir-Keeler Contractions in Ordered Partial Metric Spaces
Thabet Abdeljawad,
1Hassen Aydi,
2and Erdal Karapınar
31Department of Mathematics and Computer Sciences, C¸ankaya University, 06530 Ankara, Turkey
2Institut Sup´erieur d’Informatique et des Technologies de Communication de Hammam Sousse, Universit´e de Sousse, Route GP1, 4011 H. Sousse, Tunisia
3Department of Mathematics, Atılım University, ˙Incek, 06836 Ankara, Turkey
Correspondence should be addressed to Erdal Karapınar,erdalkarapinar@yahoo.com Received 18 February 2012; Revised 19 April 2012; Accepted 2 May 2012
Academic Editor: Rafael Martinez-Guerra
Copyrightq2012 Thabet Abdeljawad 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.
In this paper, we prove the existence and uniqueness of a new Meir-Keeler type coupled fixed point theorem for two mappingsF:X×X → Xandg:X → Xon a partially ordered partial metric space. We present an application to illustrate our obtained results. Further, we remark that the metric case of our results proved recently in Gordji et al.2012have gaps. Therefore, our results revise and generalize some of those presented in Gordji et al.2012.
1. Introduction and Preliminaries
Fixed point theory is an important tool in the study of nonlinear analysis as it is considered to be the key connection between pure and applied mathematics with wide applications in economics, physical sciences, such as biology, chemistry, physics, differential equations, and almost all engineering fields see, e.g., 1–13. From the engineering point of view there are numerous problems in adaptive systems where convergence, optimal performance, and stability are key issues. In this direction many case studies with engineering applications can be described by contraction mappings and their fixed point iterations, such as linear and nonlinear filters, image restoration and image retrieval, and in many other areas where this theory helps to describe and/or understand the phenomenon. Indeed, the relaxation in linear systems, and relaxation and stability in neural networks can be analyzed in this light, where examples for a posteriori and normalized learning algorithms for adaptive filters for monophonic and stereophonic echo cancelation can be presented14,15.
Also it is worth mentioning that Matthews introduced the notion of partial metric space, which provides an area with great potential for the development of fixed point theory, as well as tools of conducting studies on denotational semantics of data-flow networks16.
As a result it is evident that the importance of fixed point theory cannot be ruled out. Banach fixed point theorem17is the cornerstone of this topic. The result of Banach has drawn considerable interest of many authors. There are very different approaches in the study of generalization of a Banach fixed point theorem.
One of the interesting generalizations was announced by Matthews16. The author introduced the notion of partial metric spaces and proved the analog of Banach fixed point theorem. Roughly speaking, a partial metric space is a generalization of a metric spaces in which self distance of some points may not be zero. Matthews 16 discovered this phenomena when he tried to overcome problems of applying metric space techniques in the subfield of computer science: semantics and domain theory see, e.g., 18, 19. After the pioneer result of Mathews, remarkably good results have been reported on partial metric spacessee, e.g.,20–40.
On the other hand, considering the existence and uniqueness of a fixed point in partially ordered sets initiated a new trend in fixed point theory. The first result in this direction was given by Turinici 41, where he extended Banach contraction principle in partially ordered sets. Ran and Reurings 42 presented some applications of Turinici’s theorem to matrix equations. After this intriguing paper, so many exceptionally good results have been revealed in this directionsee, e.g.,43–50. Worth mentioning, Gnana Bhaskar and Lakshmikantham 44 introduced the notion of a coupled fixed point in the class of partially ordered metric spaces. Motivated by the above history, we devote this paper to prove the existence and uniqueness of coupled fixed points for a new Meir-Keeler type mappings in ordered partial metric spaces.
First, we recall basic definitions and crucial results. Hereafter, we assume thatX /∅ and we use the notation
XkX ×X× · · · ×X.
k-many 1.1
Definition 1.1 see44. Let X,≤ be a partially ordered set and F : X ×X → X. F is said to have the mixed monotone property ifFx, yis monotone nondecreasing inxand is monotone nonincreasing iny, that is, for anyx, y∈X,
x1≤x2⇒F x1, y
≤F x2, y
, forx1, x2∈X, y1≤y2⇒F
x, y2
≤F x, y1
, fory1, y2∈X. 1.2
Definition 1.2see44. An elementx, y∈X×Xis said to be a coupled fixed point of the mappingF:X×X → Xif
F x, y
x, F y, x
y. 1.3
The following two results of Bhaskar and Lakshmikantham in44were proved in the context of cone metric spaces in51.
Theorem 1.3see44. LetX,≤be a partially ordered set and suppose that there is a metricdon Xsuch thatX, dis a complete metric space. LetF:X×X → Xbe a continuous mapping having the mixed monotone property onX. Assume that there existsk∈0,1with
d F
x, y
, Fu, v
≤ k 2
dx, u d
y, v , ∀u≤x, y≤v. 1.4
If there existsx0, y0∈Xsuch thatx0≤Fx0, y0andFy0, x0≤y0, then there existx, y∈Xsuch thatxFx, yandyFy, x.
Theorem 1.4see44. LetX,≤be a partially ordered set and suppose that there is a metricdon Xsuch thatX, dis a complete metric space. LetF :X×X → Xbe a mapping having the mixed monotone property onX. Suppose thatXhas the following properties:
iif a nondecreasing sequence{xn} → x, thenxn≤x, for alln, iiif a nonincreasing sequence{yn} → y, theny≤yn, for alln.
Assume that there exists ak∈0,1with
d F
x, y
, Fu, v
≤ k 2
dx, u d
y, v , ∀u≤x, y≤v. 1.5
If there existsx0, y0∈Xsuch thatx0≤Fx0, y0andFy0, x0≤y0, then there existx, y∈Xsuch thatxFx, yandyFy, x.
Inspired byDefinition 1.1, the following concept of ag-mixed monotone mapping was introduced by Lakshmikantham and ´Ciri´c47.
Definition 1.5see47. LetX,≤be partially ordered set andF:X×X → Xandg :X → X.
Fis said to have mixedg-monotone property ifFx, yis monotoneg-nondecreasing inxand is monotoneg-nonincreasing iny, that is, for anyx, y∈X,
gx1≤gx2 ⇒F x1, y
≤F x2, y
, forx1, x2∈X, g
y1
≤g y2
⇒F x, y2
≤F x, y1
, for y1, y2 ∈X. 1.6
It is clear thatDefinition 1.5reduces toDefinition 1.1whengis the identity.
Definition 1.6see47. An elementx, y∈X×X is called a coupled coincidence point of mappingsF:X×X → Xandg:X → Xif
F x, y
gx, F y, x
g y
, 1.7
and is called a coupled common fixed ofFandg, if F
x, y
gx x, F y, x
g y
y. 1.8
The mappingsFandgare said to commute if g
F x, y
F
gx, g y
, 1.9
for allx, y∈X.
Very recently, Gordji et al.31replaced mixed g-monotone property with a mixed strict g- monotone property and improved the results in47.
Definition 1.7see31. LetX,≤be a partially ordered set andF : X ×X → X andg : X → X.F is said to have the mixed strictg-monotone property ifFx, yis monotoneg- nondecreasing inxand is monotoneg-nonincreasing iny, that is, for anyx, y∈X,
gx1< gx2 ⇒F x1, y
< F x2, y
, forx1, x2∈X, g
y1
< g y2
⇒F x, y1
> F x, y2
, for y1, y2 ∈X. 1.10 If we replace g with identity map in 1.10, we get the definition of mixed strict monotone property ofF.
A partial metric is a functionp:X×X → 0,∞satisfying the following conditions:
P1Ifpx, x px, y py, y, thenxy, P2px, y py, x,
P3px, x≤px, y,
P4px, z py, y≤px, y py, z,
for allx, y, z∈X. ThenX, pis called a partial metric space. Ifpis a partial metricponX, then the functiondp:X×X → 0,∞given by
dp x, y
2p x, y
−px, x−p y, y
1.11
is a metric onX. Each partial metricponXgenerates aT0topologyτponXwith a base of the family of openp-balls{Bpx, ε:x∈X, ε >0}, whereBpx, ε {y∈X:px, y< px, x ε}
for allx ∈ X andε > 0. Similarly, closedp-ball is defined asBpx, ε {y ∈ X : px, y ≤ px, x ε}. For more details see for example16,21.
Definition 1.8see16,21,33. LetX, pbe a partial metric space.
iA sequence{xn}inXconverges tox∈Xwhenever limn→ ∞px, xn px, x.
iiA sequence{xn} inX is called Cauchy whenever limn,m→ ∞pxn, xmexistsand finite.
iii X, pis said to be complete if every Cauchy sequence{xn}inX converges, with respect toτp, to a pointx∈X, that is, limn,m→ ∞pxn, xm px, x.
ivA mappingf :X → X is said to be continuous atx0 ∈ Xif for eachε > 0 there existsδ >0 such thatfBx0, δ⊂Bfx0, ε.
Lemma 1.9see16,21,33. LetX, pbe a partial metric space.
aA sequence{xn}is Cauchy if and only if{xn}is a Cauchy sequence in the metric space X, dp,
b X, pis complete if and only if the metric spaceX, dpis complete. Moreover,
nlim→ ∞dpx, xn 0⇐⇒ lim
n→ ∞px, xn lim
n,m→ ∞pxn, xm px, x. 1.12 Lemma 1.10see20. LetX, pbe a partial metric space. Then
AIfpx, y 0 thenxy.
BIfx /y, thenpx, y>0.
Remark 1.11. Ifxy,px, ymay not be 0.
The following two lemmas can be derived from the triangle inequalityP4.
Lemma 1.12see20. Letxn → zasn → ∞in a partial metric spaceX, p, wherepz, z 0.
Then limn→ ∞pxn, y pz, yfor everyy∈X.
Lemma 1.13 see 36. Let limn→ ∞pxn, y py, y and limn→ ∞pxn, z pz, z. If py, y pz, zthenyz.
Remark 1.14. Limit of a sequence{xn}in a partial metric spaceX, pis not unique.
Example 1.15. ConsiderX 0,∞withpx, y max{x, y}. ThenX, pis a partial metric space. Clearly,p is not a metric. Observe that the sequence {11/n2}converges both for example tox2 andy3, so no uniqueness of the limit.
We give the partial case of a definition given in31.
Definition 1.16 see 31. Let X, p,≤ be a partially ordered partial metric space. Let F : X×X → Xandg :X → X. The mappingFis said to be ag-Meir-Keeler type contraction if for anyε >0 there exists aδε>0 such that
ε≤ 1 2
p
gx, gu
p g
y
, gv < εδε ⇒p F
x, y
, Fu, v
< ε, 1.13 for allx, y, u, v∈Xwithgx≤gu,gy≥gv.
If we replacegwith the identity in1.13andpa metric onX, theFis called a Meir- Keeler type contraction.
Definition 1.17. LetX, p,≤be a partially ordered partial metric space. LetF :X×X → X andg :X → X. The mappingFis said to be a strictg-Meir-Keeler type contraction if there exists 0< k <1 such that for anyε >0 there exists aδε>0 such that
ε≤ k 2
p
gx, gu p
g y
, gv < εδε ⇒p F
x, y
, Fu, v
< ε, 1.14 for allx, y, u, v∈Xwithgx≤gu,gy≥gv.
If we replace g with the identity in1.14and if p a metric onX, the F is called a strict Meir-Keeler type contraction. Further, it can be shown easily that every strict Meir- Keelerresp., strictg-Meir-Keelertype contraction is a Meir-Keelerresp.,g-Meir-Keeler type contraction.
LetX, pbe a partial metric space. Note that the mappingsρk :Xk×Xk → 0,∞ defined by
ρkx,y:p x1, y1
p x2, y2
· · ·p xk, yk
, 1.15
forms a partial metric onXkwherex x1, x2, . . . , xkandy y1, y2, . . . , yk∈Xk. The following fact can be derived easily fromDefinition 1.16.
Lemma 1.18. LetX, p,≤be a partially ordered partial metric space. LetF : X ×X → X and g:X → X. IfFis ag-Meir-Keeler type contraction, then one has
p F
x, y
, Fu, v
< 1 2
p
gx, gu
p g
y
, gv , 1.16
for allx, y, u, v∈Xwithgx< gu,gy≥gvorgx≤gu,gy> gv.
Proof. Without loss of generality, suppose that gx < gu and gy ≥ gv where x, y, u, v ∈ X. It is clear thatpgx, gu pgy, gv> 0. Setε 1/2pgx, gu pgy, gv > 0. SinceF is a g-Meir-Keeler type contraction, then, for thisε, there exits δδε>0 such that
ε≤ 1 2
p
gx0, gu0 p
g y0
, gv0 < εδ⇒p F
x0, y0
, Fu0, v0
< ε, 1.17 for allx0, y0, u0, v0 ∈Xwithgx0< gu0andgy0≥gv0. The result follows by choosing xx0,yy0,uu0,zz0, that is,
p F
x, y
, Fu, v
< 1 2
p
gx, gu
p g
y
, gv . 1.18
Remark 1.19. LetX, p,≤be a partially ordered partial metric space. LetF:X×X → Xand g:X → X. IfFis a strictg-Meir-Keeler type contraction, then we have
p F
x, y
, Fu, v
< k 2
p
gx, gu p
g y
, gv , 1.19
for allx, y, u, v∈Xwithgx< gu,gy≥gvorgx≤gu,gy> gv.
Proof. The proof is similar toLemma 1.18above.
2. Existence of Coupled Fixed Points
The following theorem is our first main result.
Theorem 2.1. Let X, p,≤ be a partially ordered partial metric space. Suppose that X has the following properties:
aif{xn}is a sequence such thatxn1 > xnfor eachn1,2, . . .andxn → x, thenxn < x for eachn1,2, . . .,
bif{yn}is a sequence such thatyn1 < yn for eachn1,2, . . .andyn → y, thenyn > y for eachn1,2, . . ..
Letg :X → XandF:X2 → Xbe mappings such thatFX×X⊂gXandgXis a complete subspace ofX, p. Suppose thatFsatisfies the following conditions:
iFhas the mixed strictg-monotone property, iiFis ag-Meir-Keeler type contraction, iiithere existx0, y0∈Xsuch that
gx0< F x0, y0
, g
y0
≥F y0, x0
. 2.1
ThenFandghave a coupled coincidence point, that is, there existx, y∈Xsuch that F
x, y
gx, F y, x
g y
. 2.2
Proof. Letx, y x0, y0 ∈ X2 be such thatgx0 < Fx0, y0and gy0 ≥ Fy0, x0. We construct the sequence{xn}and{yn}in the following way. Due to the assumptionFX×X⊂ gX, we are able to choosex1, y1∈X2 such thatgx1 Fx0, y0andgy1 Fy0, x0. By repeating the same argument, we can choosex2, y2∈X2such thatgx2 Fx1, y1and gy2 Fy1, x1. Inductively, we observe that
gxn1 F
xn, yn , g
yn1 F
yn, xn
∀n0,1,2, . . . . 2.3
We claim that, for alln≥2
· · ·> gxn> gxn−1>· · ·> gx1> gx0,
· · ·< g yn
< g yn−1
<· · ·< g y1
≤g y0
. 2.4
We will use the mathematical induction to show2.4. By assumptioniii, we have gx0< F
x0, y0
gx1, g y0
≥F y0, x0
g y1
. 2.5
Assume that the inequalities in 2.4 hold for some n ≥ 2. Regarding the mixed g-strict monotone property ofF, we have
gxn−1< gxn ⇒ F
xn−1, yn−1
< F
xn, yn−1 , F
yn−1, xn−1
> F yn−1, xn
. 2.6
By repeating the same arguments, we observe that
g yn−1
> g yn
⇒ F
xn, yn−1
< F xn, yn
, F
yn−1, xn
> F yn, xn
. 2.7
Combining the above inequalities, together with2.3, we get
gxn F
xn−1, yn−1
< F xn, yn
gxn1, g
yn F
yn−1, xn−1
> F yn, xn
g yn1
. 2.8
So,2.4holds for alln≥2. Set Δnp
gxn, gxn1 p
g yn
, g yn1
. 2.9
TakingLemma 1.18and2.4into account, we get p
gxn, gxn1 p
F
xn−1, yn−1 , F
xn, yn
< 1 2p
gxn−1, gxn p
g yn−1
, g yn
,
p g
yn , g
yn1 p
F
yn−1, xn−1 , F
yn, xn
< 1 2p
gxn−1, gxn p
g yn−1
, g yn
.
2.10
If we add the previous two inequalities side by side, we obtain thatΔn<Δn−1. Hence,{Δn}is monotone decreasing sequence inR. Since the sequence{Δn}is bounded below, there exists L≥0 such that limn→ ∞ΔnL.
We proveL0. Suppose on the contrary thatL /0. Thus, there is a positive integerk such that for anyn≥k, we have
ε≤ Δn
2 1 2
p
gxn, gxn1 p
g yn
, g
yn1 < εδε, 2.11
whereεL/2 andδεis chosen byii. In particular, fornk, we have ε≤ Δk
2 1 2
p
gxk, gxk1 p
g yk
, g
yk1 < εδε. 2.12
Regarding the assumptioniiitogether with2.12and2.4, we have p
F xk, yk
, F
xk1, yk1
< ε, 2.13
which is equivalent to
p
gxk1, gxk2
< ε. 2.14
Similarly, we have
p g
yk1 , g
yk2
< ε. 2.15
Summing the two above inequalities Δk1
2 1 2
p
gxk1, gxk2 p
g yk1
, g
yk2 < ε, 2.16
which contradicts2.11fornk1. Thus,εL/20. That is,
nlim→ ∞Δn lim
n→ ∞
p
gxn, gxn1 p
g yn
, g
yn1 0. 2.17
Consequently, we have
nlim→ ∞p
gxn, gxn1
0 lim
n→ ∞p g
yn
, g yn1
. 2.18
By conditionP3, we have p
gxn, gxn
≤p
gxn, gxn1
, 2.19
so lettingn → ∞, we get
nlim→ ∞p
gxn, gxn
0. 2.20
Analogously, we have
nlim→ ∞p g
yn , g
yn
0. 2.21
We claim that the sequences{gxn}and{gyn}are Cauchy ingX, p.
Take an arbitraryε >0. It follows from2.17that there existsk∈Nsuch that 1
2 p
gxk, gxk1 p
g yk
, g
yk1 < δε. 2.22
Without loss of the generality, assume thatδε≤εand define the following set Π:
x, y
∈X2:p
x, gxk p
y, g yk
<2εδεandx > gxk, y≤g yk
. 2.23 Take∧ gX, gX∩Π. We claim that
F p, q
, F q, p
∈ ∧ ∀ x, y
g
p , g
q
∈ ∧wherep, q∈X. 2.24
Takex, y gp, gq ∈Π. Then, by2.22and the triangle inequalitywhich still holds for partial metricswe have
1 2
p
gxk, F p, q
p gyk, F
q, p ≤ 1 2
p
gxk, gxk1 p
gxk1, F p, q 1
2 p
g yk
, gyk1 p
g yk1
, F q, p 1
2 p
gxk, gxk1 p
g yk
, g yk1 1
2p F
p, q , F
xk, yk 1
2p F
yk, xk , F
q, p
< δε 1 2p
F p, q
, F xk, yk
1
2p F
yk, xk , F
q, p .
2.25
We distinguish two cases.
First Case.1/2px, gxk py, gyk 1/2pgp, gxk pgq, gyk≤ε.
ByLemma 1.18and the definition ofΠ, the inequality2.25turns into 1
2 p
gxk, F p, q
p g
yk
, F
q, p < δε 1 2d
F p, q
, F xk, yk
1
2d F
yk, xk , F
q, p
< δε 1 2
1 2
p g
p , gxk
p g
q , g
yk
1 2
p g
yk , g
q p
gxk, g p
δε 1 2
p g
p , gxk
p g
q , g
yk
≤δε ε.
2.26
Second Case. ε < 1/2px, gxk py, gyk 1/2pgp, gxk pgq, gyk< εδε.
In this case, we have
ε < 1 2
p g
p , gxk
p g
q , g
yk < εδε. 2.27
Sincexgp > gxkandygq≤gyk, byii, we get p
F p, q
, F
xk, yk
< ε. 2.28
Also, we have
ε < 1 2
p
gyk, gq p
gxk, gp < εδε. 2.29
Sincexgp> gxkandygq≤gyk, byii, we get p
F yk, xk
, F q, p
< ε. 2.30
Thus, combining2.25,2.28and2.30, we obtain 1
2 p
gxk, F p, q
p g
yk
, F
q, p < εδε. 2.31
On the other hand, usingi, it is obvious that F
p, q
> gxk, F q, p
≤g yk
. 2.32
We conclude thatFp, q, Fq, p∈Π. SinceFX×X⊂gX, so F
p, q , F
q, p
∈ ∧, 2.33
that is,2.24holds. By2.22, we havegxk1, gyk1∈ ∧. This implies with2.24that gxk1, g
yk1
∈ ∧⇒ F
xk1, yk1 , F
yk1, xk1
gxk2, g yk2
∈ ∧ ⇒
F
xk2, yk2 , F
yk2, xk2
gxk3, g yk3
∈ ∧ ⇒ · · ·⇒
gxn, g yn
∈ ∧⇒ · · ·.
2.34
Then, for alln > k, we havegxn, gyn∈ ∧. This implies that for alln, m > k, we have p
gxn, gxm p
g yn
, g ym
≤p
gxn, gxk p
g yn
, g yk
p
gxk, gxm p
g yk
, g ym
<4εδε≤8ε.
2.35
Thus, the sequences{gxn}and{gyn}are Cauchy ingX, p. ByLemma 1.9,{gxn}and {gyn}are also Cauchy ingX, dp. Again byLemma 1.9,gX, dpis complete. Thus, there existx, y∈Xsuch that by using2.20and2.21, we arrive at
nlim→ ∞dp
gx, gxn
0⇐⇒p
gx, gx lim
n→ ∞p
gx, gxn lim
n→ ∞p
gxn, gxn 0,
nlim→ ∞dp
g y
, g yn
0⇐⇒p g
y , g
y lim
n→ ∞p g
y , g
yn
lim
n→ ∞p g
yn
, g yn
0.
2.36 Since the sequences{gxn}and{gyn}are monotone increasing and monotone decreasing, respectively, by propertiesaandb, we conclude that
gxn< gx, g yn
> g y
, 2.37
for eachn≥0. Therefore, having in mind thatFis ag-Meir-Keeler type contraction, by2.37 andLemma 1.18, we get
p
gxn1, F x, y
p F
xn, yn
, F x, y
< 1 2
p
gxn, gx p
g yn
, g
y .
2.38 From2.36, byLemma 1.12, we obtain
p
gx, F x, y
lim
n→ ∞p
gxn1, F x, y
≤ 1 2 lim
n→ ∞
p
gxn, gx p
g yn
, g
y 0,
2.39
soFx, y gx. Analogously we getFy, x gy.
Remark 2.2. We remak that Theorem 2.1 has been proved recently in31in the category of partially ordered metric spaces. However, they proceed the proof without using the assump- tionsaandbstated in our Theorem. They claimed thatgxn< xandgyn> yby using the fact that the sequences{gxn}and{gyn}are increasing and decreasing, respectively.
In our belief, this step is not true and cannot be achieved without using the assumptionsa andb. Actually, this may not be true if the partial ordering, for example, is obtained via nonstrongly minihedral cones.
Corollary 2.3. Let X, p,≤ be a complete ordered partial metric space. Suppose that X has the following properties:
aif{xn}is a sequence such thatxn1 > xnfor eachn1,2, . . .andxn → x, thenxn < x for eachn1,2, . . .,
bif{yn}is a sequence such thatyn1 < yn for eachn1,2, . . .andyn → y, thenyn > y for eachn1,2, . . ..
LetF:X2 → Xbe a given mapping. Suppose thatFsatisfies the following conditions:
iFhas the mixed strict monotone property, iiFis a Meir-Keeler type contraction, iiithere existx0, y0∈Xsuch that
x0 < F x0, y0
, y0≥F y0, x0
. 2.40
Then,Fhas a coupled fixed point, that is, there existx, y∈Xsuch that F
x, y
x, F y, x
y. 2.41 Proof. It follows by takinggIX, the identity mapping onX, inTheorem 2.1.
3. Uniqueness of Coupled Fixed Points
LetX,≤be a partially ordered set. We endowX×Xby the following orderdenoted≤g u, v≤g
x, y
⇐⇒gu< gx, g y
≤gv, ∀ x, y
,u, v∈X×X. 3.1 Moreover,u, vandx, yare calledg-comparable if eitheru, v≤gx, yoru, v≤gx, y.
In caseg IX, we shortly say thatu, vandx, yare comparable and denote byu, v ≤ x, y. In this section, we will prove the uniqueness of the coupled fixed point.
Theorem 3.1. In addition to the hypotheses of Theorem 2.1, assume that for all nong-comparable points x, y,x∗, y∗ ∈ X2, there exists a, b ∈ X2 such that Fa, b, Fb, a is comparable to bothgx, gyandgx∗, gy∗. Further, assume that F and g commute andF is a strict g-Meir-Keeler type contraction. Then,F andg have a unique coupled common fixed point, that is, there existsu, v∈X2such that
ugu Fu, v, vgv Fv, u. 3.2
Proof. The set of coupled coincidence points ofFandg is not empty due toTheorem 2.1. If x, yis the only coupled coincidence point ofFandg, then commutativity ofFandgimplies that
g
gx
g F
x, y F
gx, g y
, g
g y
g F
y, x F
g y
, gx .
3.3 Hence,u, v gx, gyis a coupled coincidence point ofFandgand by uniqueness we conclude that
F x, y
gx x, F y, x
g y
y. 3.4
Now suppose thatx, y,x∗, y∗∈X2are two coupled coincidence points ofFandg.
We show thatgx gx∗andgy gy∗. To this end we distinguish the following two cases.
First Case.x, yisg-comparable tox∗, y∗with respect to the ordering inX2, where F
x, y
gx, F y, x
g y
, F
x∗, y∗
gx∗, F y∗, x∗
g y∗
. 3.5
Without loss of the generality, we may assume that
gx F
x, y
< F x∗, y∗
gx∗, g y
F y, x
≥F y∗, x∗
g y∗
. 3.6
By definition ofρ2andLemma 1.18we have 0< ρ2
gx, g y
,
gx∗, g y∗
p
gx, gx∗ p
g y∗
, g y p
F x, y
, F
x∗, y∗ p
F y∗, x∗
, F y, x
,
< p
gx, gx∗ p
g y∗
, g y
ρ2
gx, g y
,
gx∗, g y∗
,
3.7
which is a contradiction. Therefore, we havegx, gy gx∗, gy∗. Hence gx gx∗, g
y g
y∗
. 3.8
Second Case.x, yis notg-comparable tox∗, y∗.
By assumption, there exists a, b ∈ X2 such that Fa, b, Fb, ais comparable to bothgx, gyandgx∗, gy∗. Then, we have
gx F
x, y
< Fa, b, F x∗, y∗
gx∗< Fa, b, g
y F
y, x
≥Fb, a, F
y∗, x∗ g
y∗
≥Fb, a, 3.9
Settingxx0,yy0,aa0,bb0, andx∗x∗0,y∗y∗0as in the proof ofTheorem 2.1, we get
gxn1 F xn, yn
, g yn1
F yn, xn
∀n0,1,2, . . . , gan1 Fan, bn, gbn1 Fbn, an ∀n0,1,2, . . . , g
x∗n1 F
x∗n, y∗n , g
yn1∗ F
y∗n, x∗n
∀n0,1,2, . . . .
3.10
SinceFx, y, Fy, x gx, gy gx1, gy1is comparable withFa, b, Fb, a ga1, gb1, we havegx< ga1andgb1≤gy. By using thatFhas the mixedg-strict
monotone property, we observe thatgx < ganandgbn ≤ gyfor alln ≥ 1. Thus, by Remark 1.19, we get that
p
gx, gan1 p
g y
, gbn1 p
F x, y
, Fan, bn p
Fbn, an, F y, x
< k 2
p
gx, gan p
g y
, gbn k
2 p
g y
, gbn p
gx, gan kp
gx, gan p
g y
, gbn .
3.11 Inductively, we derive that
p
gx, gan1 p
g y
, gbn1 < kn p
gx, ga1 p
g y
, gb1 . 3.12 The right hand side of above inequality tends to zero asn → ∞. Hence,
nlim→ ∞
p
gx, gan1 p
g y
, gbn1 0. 3.13
Analogously, we get that
nlim→ ∞
p
gx∗, gan1 p
g y∗
, gbn1 0. 3.14
By the triangle inequality, we have p
gx, gx∗
≤p
gx, gan1 p
gx∗, gan1
−p
gan1, gan1
≤p
gx, gan1 p
gx∗, gan1
−→0 asn−→ ∞, p
g y
, g y∗
≤p g
y
, gbn1 p
g y∗
, gbn1
−p
gbn1, gbn1
≤p g
y
, gbn1 p
g y∗
, gbn1
−→0 asn−→ ∞.
3.15 Combining all observation above, we get that pgx∗, gx 0 andpgy∗, gy 0.
Therefore,
gx gx∗, g y
g y∗
. 3.16
In both cases above, we have shown that3.16holds. Now, letgx uandgy v. By the commutativity ofFandgwith the fact thatgx Fx, yandFy, x gy, we have
gu g
gx
g F
x, y F
gx, g y
Fu, v,
gv g
g y
g F
y, x F
g y
, gx
Fv, u. 3.17
Thus,u, vis a coupled coincidence point ofF andg. Settingu x∗andv y∗in3.17.
Then, by3.16we have
ugx gx∗ gu, vg
y g
y∗
gv. 3.18
From3.17we get that
ugu Fu, v, vgv Fv, u. 3.19
Hence, the pairu, vis the coupled common fixed point ofFandg.
Finally, we prove the uniqueness of the coupled common fixed point of F and g.
Actually, ifz, wis another coupled common fixed point ofFandg, then
ugu gz z, vgv gw w 3.20
follows from3.16.
Remark 3.2. We remark that Theorem 3.1 above has been recently proved in31 without assuming that the mapping F is a strictg-Meir-Keeler contraction. This leads to a gap in the proof of Theorem 2.6 there.
Corollary 3.3. Suppose that all the hypotheses of Corollary 2.3 hold, and further, for all x, y, x∗, y∗ ∈ X2, there existsa, b ∈ X2 that is comparable tox, yand x∗, y∗. Further, assume thatFis a strict Meir-Keeler type contraction. Then,Fhas a unique coupled fixed point.
4. Applications
Motivated by Suzuki52and on the same lines of Theorem 3.1 of53, one can prove the following result.
Theorem 4.1. LetX, p,≤be a partially ordered partial metric space. LetF:X2 → Xandg:X → Xbe given mappings such thatFX2⊂ gX. Assume that there exists a functionθfrom0,∞ into itself satisfying the following:
Iθ0 0 andθt>0 for everyt >0, IIθis nondecreasing and right continuous, IIIfor everyε >0, there existsδε>0 such that
ε≤θ 1
2 p
gx, gu p
gy, gv < εδε ⇒θp F
x, y
, Fu, v
< ε, 4.1
for allgx≤guandgy≥gv.
Then,Fis ag-Meir-Keeler type function.
The following result is an immediate consequence of Theorems2.1and4.1.
Corollary 4.2. LetX, p,≤be a partially ordered complete partial metric space. GivenF:X2 → X andg :X → Xsuch thatFX2⊂gX,gXis a complete subspace and the following hypotheses hold:
iFhas the mixedg-strict monotone property, iifor everyε >0, there existsδε>0 such that
ε≤
1/2pgx,gupgy,gv
0
φtdt < εδε ⇒
pFx,y,Fu,v
0
φtdt < ε, 4.2
for allgx≤guandgy≥gv, whereφ:0,∞ → 0,∞is a locally integrable function satisfying s
0φtdt >0 for alls >0,
iiithere existx0, y0∈Xsuch that gx0 < F
x0, y0
, gy0≥F y0, x0
. 4.3
Assume that the hypothesesaand bgiven inTheorem 2.1hold. Then, F andg have a coupled coincidence point.
To end this paper, we give the following corollary.
Corollary 4.3. Let X, p,≤be a partially ordered partial metric space. Given F : X2 → X and g:X → Xsuch thatFX2⊂gX,gXis a complete subspace and the following hypotheses hold:
iF has the mixedg-strict monotone property, iifor allgx≤guandgy≥gv,
pFx,y,Fu,v
0
φtdt≤k
1/2pgx,gupgy,gv
0
φtdt, 4.4
wherek∈0,1andφis a locally integrable function from0,∞into itself satisfyings
0φtdt >0 for alls >0,
iiithere existx0, y0∈Xsuch that gx0 < F
x0, y0
, gy0≥F y0, x0
. 4.5
Assume that the hypothesesaandbofTheorem 2.1hold. Then,Fandghave a coupled coincidence point.
Proof. For allε >0, we takeδε 1/k−1εand we applyCorollary 4.2.
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