This document summarizes a research paper that proves some new fixed point theorems for generalized weakly contractive mappings in ordered partial metric spaces. The paper extends previous theorems proved by Nashine and Altun in 2017. It presents definitions of partial metric spaces and properties. It proves a new fixed point theorem (Theorem 2.1) for nondecreasing mappings on ordered partial metric spaces that satisfy a generalized contractive condition. The theorem shows the mapping has a fixed point and the partial metric of the fixed point to itself is 0. It uses properties of partial metrics, contractive conditions and continuity to prove the sequence generated by iterating the mapping is Cauchy and converges.
Unique fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces
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Unique Fixed Point Theorems for Generalized Weakly
Contractive Condition in Ordered Partial Metric Spaces
RajeshShrivastava1,Sudeep Kumar Pandey2,Ramakant Bhardwaj3,R.N.Yadav4
1.Govt. Science and Commerce College, Benazir, Bhopal (M.P.)India
2.Millennium Group of Institution, Bhopal (M.P.)India
3.Truba Institute of Technology and Science, Bhopal (M.P.)India
4.Patel Group of Institutions, Bhopal (M.P.)India
Abstract
The aim of this paper to prove some fixed point theorems for generalized weakly contractive condition in
ordered partial metric spaces. The result extend the main theorems of Nashine and altun[17] on the class of
ordered partial metric ones.
Keywords: - Partial metric, ordered set, fixed point, common fixed point.
AMS subject classification: - 54H25, 47H10, 54E50
Introduction and preliminaries
The concept of partial metric space was introduced by Methuews [16] in 1994.In such spaces the distance of a
point to itself may not be zero. Specially from the point of sequences, a convergent sequence need not have
unique limit. Methuews [16] extended the well known Banach contraction principle to complete partial metric
spaces. After that many interesting fixed point results were established in such spaces. In this direction we refer
the reader to Velero[21]. Oltra and Velero[23]. Altun et at[4]. Ramaguera[24]. Altun and Erduran[2] and
Aydi[6,7,8].
First we recall some definitions and properties of
partial metric spaces [see 2, 4, 16, 22,23,24,25 for more details]
Definition 1.1:- A partial metric on non empty set X is a function
p: X Ă X â
+
such that for all
x, y , z â X .
( p1 ) x = y â p ( x, y) = p(x, y) = p ( y, y)
( p2 ) p(x, x) †p(x, y)
? p3 ) p(x, y) = p( y, x)
(
( p4 ) p(x, y) †p(x, x) + p( z, y) â p( z, z)
A partial metric space is a pair (X, p ) such that X is a non-empty set and p is a partial metric on X.
Remarks 1.2:- It is clear that if p ( x , y ) = 0 , then from (p1) and (p2).
? = y But if x = y , p ( x , y ) may not be 0. A basic example of a partial metric space is the pair
x
(
+
, p ) where p ( x, y ) = max{ x, y} for all x, y â
+
Each partial metric p on X generates a T0 topology Ïp on X which has a base the family of open p-balls
{Bp ( x, Δ ), x â X , Δ > 0}
Where
? p ( x Δ ) = { y â X : p( x, y) < p( x, x) + Δ } for all x â X and Δ > 0.
B
If p is a partial metric on X then function P
s
: X Ă X â R+
P ( x , y ) = 2 p ( x, y ) â p ( x, x ) â p ( y , y )
given by
s
(1.1)
Is metric on X.
Definition 1.3:- let
( X , p)
be a partial metric space and
ççç{
( i )ççççxn } converges to a point x â X
ççç{
( ii )ççççxn } is called Cauchy sequence if
{ xn }
be a sequence in X. then
if and only if p ( x, x) = lim p ( x, xn ).
there exists
120
p â+â
( and is finite )
lim ? ( xn xm ) .
p
n , m â+â
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Definition 1.4:- A partial metric space
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( X , p ) is said to be complete if every Cauchy sequence (x ) in x
n
converges, with respect to Ïp to a point xâX, such that p(x,x)=lim p(xn,xm).
Lemma 1.5:- Let (X,p)be a partial metric space then.
(a)
{ x n} is a Cauchy sequence in (X,p)if and only if it is a Cauchy sequence in the metric space (X,ps).
(b)
(X,p) is complete if and only if the metric space (X,ps) is complete. Further more
lim p s ( xn , x) = 0 if and only if
n â+â
p(x, x)= lim p(xn , x)= lim p(xn , xm )
nâ+â
n ,mâ+â
Definition 1.6:- (([2]) suppose that (X, p) is a metric space. A mapping
F : ( X , p ) â ( X , p ) is said to be continuous at x â X , if for every Δ>0 there exists ÎŽ>0 such that
F ( B p ( x, ÎŽ ) ) â B p ( Fx, Δ ).
The following results are easy to check.
Lemma 1.7 :- let (x,p) be a partial metric space F:XâX is continuous if and only if given a sequence { xn }ââ
and xâX such that p(x,x)=
hence p(Fx,Fx)=
lim
nâ+â
p(x, xn),
lim p(Fx,Fxn).
nâ+â
Remarks 1.8: ([22]) let (x, p) be a partial metric space and F :( x, p) â(x, p) if F is continuous on (X, p) then F:
(X, ps) â(X, ps) is continuous.
On the other hand. Fixed point problems of contractive mapping in metric spaces endowed with a partially order
have been studied by many authors (see [1,3,5,9,10,11,12,14,15,17,18,19,20,21]. In particular Nashine and altun
[17] proved the following.
Theorem 1.9:- Let (X, â€) be a partially ordered set and (X, d) be a complete metric space. Suppose that T:XâX
is a nondecreasing mapping such that for every to comparable elements. x,yâX
Κ (d(Tx,Ty)) †Κ (m(x,y)) â
(m(x,y))
.(1.2)
Where
M(x,y)=a1d(x,y)+ a2d(x,Tx)+ a3d(y,Ty)+ a4 [d(y,Tx)+d(x,Ty)]+ a5[d(y,Ty)+d(x,Tx)]
With a1>0 , a2,a3,a4,a5 >0,a1+a2+a3+2(a4+a5 )†1 and Κ
:[(0,+â] â [0,+â]
is a continuous non decreasing
,
Ï is a lower semi continues function and Κ(t)=0=
(t) If and only if t=0. Also suppose there exists x0âX with
x0†Tx0. Assume that
i.
T is continuous or
ii.
If a nondcreasing sequence { x n} converges to x, then x n â x for all n.
Then T has a fixed point
The purpose of this paper is to extend theorem (1.9) on the class of ordered partial metric space. Also a common
fixed point result is given.
Ï
Ï
Ï
Ï
2. Main results
Theorem 2.1:- (X, â€) be a partially ordered set and (X, p) be a complete partial metric space. Suppose that T:
XâX is a nondecreasing such that for every two comparable element x, yâX.
Κ(p(Tx,Ty))<=Κ (Ξ(x,y)) â
(Ξ(x,y))
(2.1)
Where
Ξ(x,y)=a1p(x,y) + a2p(x,Tx) + a3p(y,Ty) + a4 [p(y,Tx) + p(x,Ty)]
(2.2)
+ a5[p(y,Ty) + p(x,Tx)]
With . (a1, a4,a5)>0 , (a2,a3) >=0,(a1+a2+a3+2(a4+a5 ))†1 and Κ
:[(0,+â] â [0,+â] Κ is a continuous
,
nondecreasing,
is a lower semi continuous function and
Ï
Ï
Ï
Ï
Κ(t)=0=
(t). if and only if t=0 . Also suppose there exists there exists x0âX with x0†Tx0.
Assume that:
i.
T is continuous or
ii.
If a non decreasing sequence { xn } converges to x in (X, p) then xnâ€x for all n.
Then T has a fixed point, say z moreover p(z,z)=0.
121
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Proof: - If Tx0=x0 then the proof is completed. Suppose that Tx0â x0. Now since x0<Tx0 and T is non decreasing
we have
x0<Tx0â€T2x0†âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ..â€Tnx0â€Tn+1x0â€âŠâŠâŠ
Put xn=Tnx0, hence xn+1=Txn. If there exists n0â{1,2,âŠ.} such that Ξ
(
(
n0
, xn0 â1
) =0 then by definition (2.2), it
)
p xn 0 â1 , xn 0 = p xn0 , Txn0 â1 = 0 ,
is clear that
So
)
(x
xn0 â1 = xn0 = Txn0 â1 and so we are finished. Now we can suppose
Ξ(xn,xn-1)>0
For all nâ„1, let us check that
lim
(2.3)
p(xn+1, xn)=0
nâ+â
(2.4)
By (2.2), we have using condition (p4)
Ξ(xn,xn-1)=a1p(xnxn-1) + a2p(xn,Txn) + a3p(xn-1,Txn-1 ) + a4[p(xn-1,Txn)+p(xn,Txn-1)
+ a5[p(xn-1,Txn-1)+p(xn,Txn)]
= a1p(xn, xn-1) + a2p(xn, xn+1) +a3p(xn-1, xn)+a4[p(xn-1, xn+1)+p(xn,xn)]
+a5[p(xn-1,xn)+p(xn,xn+1)]
â€(a1+a3+a4+a5)p(xn,xn-1)+(a2+a4+a5)p(xn,xn+1)[by (p4)]
Now we claim that
P(xn+1,xn)â€p(xn,xn-1 )
For all nâ„1. Suppose that is not true, that is there exists n0â„1 such that
since
xn0 †xn0 +1
( (
)) = Κ ( p (Tx , Tx ))
†Κ (Ξ ( x , x ) â Ï (Ξ ( x , x ))
†Κ (a + a + a + a ) p ( x , x )
+ ( a + a + a ) p ( x , x ) â Ï (Ξ ( x , x
))
†Κ ( a + a + a + 2a + 2a ) p ( x , x ) â Ï (Ξ ( x
†Κ ( p ( x x ) â Ï (Ξ ( x , x ) )
1
2
5
5
1
2
n0
) now
n0 â1
n0
4
4
(
n0 â1
n0
n0 â1
3
)
p xn0 +1 , xn0 > p xn0 , xn0 â1
we can use the inequality( 2.1) then we have
Κ p xn0 +1 , xn0
n0
(
( 2.5)
n0 +1
n0
3
n0 +1
Which implies that
n â1
n0
4
n0
5
n0
n0
n0 ? 1
n0 +1
n0
, xno â1
))
no â1
Ï ( Ξ ( xn , xn â1 ) ) †0
0
0
and by property of
Ï
given that Ξ
(x
n0
)
, xn0 â1 = 0 , this
contradict (2.3) hence( 2.5 )holds and so the sequence p(xn+1,xn) is non increasing and bounded below. Thus there
exists Ï>0 such that limit
lim p(xn+1,xn)=Ï. Assume that Ï >0, by (2.2), we have
nâ+â
a1Ï=
lim a1p( xn,xn-1)†lim supΞ( xn,xn-1)
nâ+â
nâ+â
=
lim sup [(a1+a3) p(xn,xn-1)+a2p( xn,xn+1)
nâ+â
+a4[p(xn-1,xn+1)+p(xn,xn)]+a5[p(xn-1,xn)+p(xn,xn+1)]
â€
lim sup[( a1+a3+a4+a5 )p( xn,xn-1)+ (a2+a4+a5 )p(xn,xn-1)]
nâ+â
This implies that
0<a1Ïâ€
lim supΞ (xn, xn-1)†(a1+a2+a3+2a4+2a5 )Ï â€ Ï
nâ+â
And so there exists Ï1>0 and subsequence {xn(k)} of {xn} such that
lim Ξ (xn (k) ,xn (k) â1 )= Ï1 †Ï
k â+â
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By the lower semi-continuity of
Ï
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we have
Ï ( Ï1 ) †lim inf Ï (Ξ (xn (k) ,xn (k) +1 ))
k â+â
From( 2.1) we have
Κ (p(xn(k)+1,xn(k)))= Κ (p(Txn(k),Txn(k)-1))
†Κ (Ξ(xn(k),xn(k)-1))-
Ï
(Ξ(xn(k),xn(k)-1))
And taking upper limit as Kâ+â we have using the properties of Κ and
Κ(Ï)â€ Ï ( Ï1 ) â
Ï
lim inf Ï (Ξ (xn (k) ,xn (k) +1 ))
k â+â
Ï (Ï )
â€Îš(Ï)- Ï (Ï )
That is Ï (Ï ) =0 thus by the property of Ï
â€Îš(Ï1)-
1
1
we have Ï1=0 which is a contradiction. Therefore we have Ï=0
1
that is (2.4) holds.
Now we show that{xn} is a cauchy sequence in the partial metric space (x,p).
From lemma 1.5 it is sufficient to prove that {xn} is a Cauchy sequence in the metric space (X,ps)suppose to the
contrary. Then there is a â>0 such that for and integer K there exist integer m(k)>n(k)>k such that
ps(xn(k),xm(k))>Δ
(2.6)
For ever integer K let m(k) be the least positive integer exceeding n(k) satisfying
(2.6 )and such that
ps(xn(k),xm(k)-1)â€Î”
( 2.7)
Now using (2.4)
Δ< ps(xn(k),xm(k))†ps(xn(k),xm(k)-1)+ ps(xm(k)-1,xm(k))
â€Î”+ ps(xm(k)-1,xm(k))
Then by (2.4) it follows that
lim p s ( xn ( k ) , xm ( k ) ) = Δ
(2.8)
k â+â
Also by the triangle inequality. We have
p s ( x n ( k ) , x m ( k ) â1 ) â p s ( x n ( k ) , x m ( k ) ) †p s ( x m( k ) â1 , x m ( k ) )
By using (2.4), (2.8) we get
lim p s ( xn ( k ) , xm ( k ) â1 ) = Δ
(2.9)
k â+â
On the other hand by definition of ps .
p s ( xn ( k ) , xm ( k ) â1 ) = 2 p
p s ( xn ( k ) , xm ( k ) â1 ) = 2 p
(x
(x
n(k )
, xm ( k ) ) â p
n ( k ) , xm ( k ) â1
(x
)â p (x
n(k )
, xn( k ) ) â p
n ( k ) , xn( k )
(x
)â p (x
m( k )
, xm ( k ) )
m( k ) â1 , xm ( k ) â1 )
letting kâ+â, we find thanks to (2.8), (2.9) and the condition p3 in (2.4)
lim
p
(x
n(k )
, xm (k )
lim
p
(x
n(k )
, x m ( k )â1
k â +â
k â +â
)=
Δ
( 2 .1 0 )
2
)=
Δ
( 2 .1 1)
2
In view of (2.2) we get
123
Hence
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a1 p ( xn ( k ) , xm ( k )â1 ) †Ξ ( xn ( k ) , xm ( k )â1 )
=a1 p ( xn ( k ) , xm ( k ) â1 ) + a2 p ( xn ( k ) , T xn( k ) )
+a3 p ( xm( k ) â1 , T xm ( k ) â1 )
+ a4 [ p ( xm( k ) â1 , T xn( k ) â1 ) + p ( xn ( k ) , T xm ( k )â1 )]
+a5 ïŁź p ( xm( k )â1 , T xm ( k ) â1 ) + p ( xn( k ) , T xn( k ) ) ïŁč
ïŁ°
ïŁ»
=a1 p ( xn ( k ) , xm ( k ) â1 ) + a2 p ( xn ( k ) , xn( k )+1 )
+a3 p ( xm( k ) â1 , xm ( k ) )
+ a4 [ p ( xm( k ) â1 , xn( k ) ) + p ( xn ( k ) , xn( k )+1 )]
+a5 ïŁź p ( xm( k )â1 , xm ( k ) ) + p ( xn( k ) , xn( k ) +1 ) ïŁč
ïŁ°
ïŁ»
†a1 p ( xn ( k ) , xm ( k ) â1 ) + a2 p ( xn ( k ) , xn( k )+1 )
+a3 p ( xm( k ) â1 , xm ( k ) )
+ a4 ïŁź p ( xm( k )â1 , xn( k ) ) + p ( xn ( k ) , xn( k ) +1 ) + p ( xn ( k ) , xm ( k ) ) ïŁč
ïŁ°
ïŁ»
+a5 ïŁź p ( xm( k )â1 , xm ( k ) ) + p ( xn( k ) , xn( k )+1 ) ïŁč
ïŁ°
ïŁ»
Taking upper limit as Kâ +â and using (2.4),(2.10) and (2.11) we have
0 < a1
Δ
2
(x
†lim sup Ξ
k â+â
n ( k ) , xm ( k ) â1 ) †(a 1 + 2 a4 + 2 a 5 )
Δ
2
this implies that there exists Δ1>0 And subsequence {xn(k(p))} of { xn(k) } Such that
Ξ
lim
pâ +â
(x
n ( k (p ))
By the lower semi continuity of
, xm
Ï
( k (p )) â 1
Δ1 â€
Δ
2
we have
(
Ï (Δ 1 ) †lim inf Ï Îž ( xn ( k ) , xm ( k ) â1 )
k â+â
)=
)
Now by (2.1) we getâŠ
ïŁ«Î” ïŁ¶
Ï ïŁŹ ïŁ· = lim supÏ p ( xn ( k (p)) , xm ( k (p)) )
p â+â
2
ïŁ ïŁž
(
†lim supÏ ( p ( x
p â+â
=
n ( k (p))
)
) (
, xm ( k (p)) +1 ) + p (Txn ( k (p)) , T xm ( k (p)) â1 )
lim sup Ï (p(Tx n(k(p)) ),Tx m(k(p)â1) ))
p â+â
124
)
â€
Δ
2
.
âŠ
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(
)
(
)
†lim su p ïŁźÏ Îž ( x n ( k (p ) ) , x m ( k (p ) ) â 1 ) â Ï Îž ( x n ( k (p ) ) , x m ( k (p ) ) â 1 ) ïŁč
ïŁ°
ïŁ»
p â +â
(
= Ï ( Δ 1 ) â lim in f Ï Îž ( x n ( k (p ) ) , x m ( k (p ) ) â 1 )
p â +â
)
â€ Ï (Δ 1 ) â Ï (Δ 1 )
ïŁ«Î” ïŁ¶
â€Ï ïŁŹ
ïŁ· â Ï (Δ 1 )
ïŁ 2 ïŁž
Which is a contradiction? Therefore {xn} is a Cauchy sequence in the metric space
(X, ps) from lemma (1.5) (X, ps) is a complete metric space. Then there is zâX such that
lim p s ( xn , z) = 0
n â +â
Again from lemma (1.5), we have thanks to( 2.4) and the condition (p2).
p(z, z) = lim p ( xn , z) = lim p ( xn , xn ) = 0
n â+â
(2.12)
n â+â
We will prove that Tz=z
1. Assume that (i) hold, that is T is continuous. By( 2.12) the sequence converges in(X,p)to z, and since T
is continuous hence the sequence.. converges to Tz that is
p(Tz,Tz) = lim p (T xn ,Tz)
(2.13)
nâ+â
Again thanks to (2.12)
P(z,Tz)= lim p ( xn , z) =
n â+â
lim p (T x n â1 , Tz) = p (Tz, Tz)
n â +â
(2.14)
On the other hand by(2.1),(2.14)
( Ξ(z,z))
Κ( (p(z,Tz))=Κ (p(Tz,Tz))â€Îš (Ξ(z,z)Where from (2.12) and the condition p2
Ξ(z,z) = a1p(z,z) +( a2+a3+2a4+2a5 )p( z,Tz)
= (a2+a3+2a4+2a5 )p (z,Tz )â€p(z,Tz)
Thus, Κ (p(z,Tz), â€Îš(Ξ(z,z))( (z,z))
(
â€Îš(p(z,Tz))(Ξ( z,z))
Ï
Ï
Ï
Ï
In follows that
(Ξ(z, Z))=0 so Ξ (z,z )=( a2+a3+2a4+2a5 )p(z,Tz) =0 that is p(z,Tz)=0, because Δ> 0.
Hence z=Tz that is z is a fixed point of T
Assume that ii holds than we have xnâ€z for all n, Therefore all n, we can use the inequality (2.1) for xn and z
since
Ξ(z,xn)=a1p(z,xn)+a2p(z,Tz)+a3p(xn,Txn)+a4[p(xn,Tz)+p(z,Txn)+a5[p(xn,Txn)+p(z,Tz)]
= a1p(z,xn )+a2p(z,Tz)+a3p(xn,xn+1)+a4[p(xn,Tz)+p(z,xn+1)+a5[p(xn,xn+1)+p(z,Tz)]
Hence from (2.4), (2.12)
lim Ξ (z, xn ) = (a 1 + a 4 + a 5 ) p( z , Tz)
n â +â
we have,
Ï ( p(z, Tz) ) = lim supÏ ( p (Tz, xn+1 ) )
n â+â
= lim supÏ ( p (Tz, T xn ) )
n â+â
†lim supÏ [ (Ï ( z , xn ) ) -Ï (Ξ (z,x n ))
n â+â
â€Îš ((a1 +a4+a5)p(Tz,z))â€Îš(p(Tz,z))-
Ï
Ï
Ï
]
(a2+a4+a5)p(Tz,z).
((a2+a4+a5)p(Tz,z)).
Ï
Then
((a2+a4+a5)p(Tz,z))=0 And since (a4,a5) >0 hence by the property of
we have p(Tz,z)=0 so Tz=z,
This complete the proof of theorem (2.1)
Remarks 2.2 Theorem 2.1 holds for ordered partial metric spaces, so it is an extension of the result of Noshine
and altun (17) given in theorem (1.9) which is verified just for ordered metric ones.
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Corollary 2.3 :- Let (X,â€) be a partially ordered set and (x,p) be a complete partial metric space suppose that
T:Xâ X be a non decreasing mapping such that for every two comparable elements x,yâX
P(Tx,Ty) ,â€Îž(x,y)(Ξ(x,y))
(2.15)
Where
Ξ(x,y)=a1p(x,y)+a2p(x,Tx)+a3p(y,Ty)
+a4[p(y,Tx)+p(x,Ty)+a5[p(y,Ty)+p(x,Tx)]
(2.16)
: (0,+â) â (0,+â).
is a lower semi continuous
With (a1,a4,a5)>0.(a2,a3)â„0,(a1+a2+a3+2a4+2a5)â€1 and
Ï
function and
Ï (t)=0 if and
Ï
Ï
only if t=0 also suppose that there exists x0âX with xo†Tx0, Assume that
Ï
i.
T is continuous or
ii.
If a nondcreasing sequence {xn} converges to x, in (X,p),then xnâ€x for all n.
Then T has a fixed point, say z moreover p(z,z)=0
Proof: - It is sufficient to take Κ (t)=t in theorem.
Corollary 2.4:- Let (X.â€) be a partially ordered set and (X,p) be a complete partial metric space suppose
that T:Xâ X be a non decreasing mapping such that for every two comparable elements x,yâX
P (Tx, Ty)†kΞ(x,y)
(2.17)
Where
Ξ(x,y)=a1p(x,y)+a2p(x,Tx)+a3p(y,Ty)
+a4[p(y,Tx)+p(x,Ty)+a5[p(y,Ty)+p(x,Tx)]
(2.18)
With k â [0,1],(a1,a4,a5)>0,(a2,a3)â„0,(a1+a2+a3+2a4+2a5)â€1 also suppose , there exists x0 âX with xo†Tx0,
Assume that
i.
T is continuous or
ii.
If a nondcreasing sequence {xn} converges to x in(X,p) then xnâ€x for all n.
Then T has a fixed point, say z moreover p(z,z)=0
Proof: - It sufficient to take Κ(t)=(1-k)t in corollary (2.3)
We give in the following a sufficient condition for the uniqueness of the fixed point of the mapping T.
Theorem 2.5 :- Let all the conditions of the theorem (2.1) be fulfilled and let the following condition hold for
arbitrary two points x,yâX there exists zâX which is comparable with both x and y. If (a1+2a2+2a4+2a5 )†1
or (a1+2a3+2a4+2a5) â€1. Then the fixed point of T is unique.
Proof :- Let u and v be two fixed point of T, i.e Tu=u and Tv=v. we have in mind, p(u,u)=p(v,v)=0. Consider the
following two cases.
1. U and v are comparable. Then we can apply condition 2.1 and obtain that
Κ(p(u,v))=Κ(p(Tu,Tv))<=Κ(Ξ(u,v))(Ξ(u,v))
Where
Ξ(u,v)=a1p(u,v)+a2p(u,Tu)+a3p(v,Tv)+a4[p(u,Tv)+p(v,Tu)]+a5[p(v,Tv)+p(u,Tu)]
= a1p(u,v )+a2p(u,u)+a3p(v,v)+a4[p(u,v)+p(v,u)]+a5[p(v,v)+p(u,u)]
=((a1 +2a4+2a5)p(u,v))+ a2p(u,u)+a3p(v,v)
â€(a1 +a2+a3+2a4+2a5)p(u,v)â€p(u,v).
We deduce
Κ(p(u,v)) â€Îš(p(u,v)(Ξ (u,v)) i.e Ξ(u,v)=0
So p(u,v)=0 meaning that u=v, that is the uniqueness of the fixed point of T.
2. Suppose that u and v are not comparable. Choose and element wâX comparable with both of them.
Then also u=Tnu is comparable is Tnw for each n (Since T is nondecreasing) Appling (2.1) one obtain
that
Κ(p(u,Tnw))=Κ(p(TTn-1u, TTn-1w))
â€Îš(Ξ(Tn-1u, Tn-1w))(Ξ(Tn-1u, Tn-1w))
Ï
Ï
Ï
=Κ(Ξ(u, Tn-1w))-
Ï (Ξ(u, T
n-1
w))
Where
Ξ(u,Tn-1w)=a1p (u, Tn-1w)+ a2p(u,T Tn-1u)+a3p(Tn-1w,T Tn-1w)
+a4[p(u,T Tn-1w)+p(Tn-1w, Tu)]+a5[p((Tn-1w,T Tn-1w)+p(u, TTn-1u)]
=a1p (u, Tn-1w)+ a2p(u,u)+a3p(Tn-1w, Tnw)
+a4[p(u, Tnw)+p(Tn-1w, u)]+a5[p((Tn-1w, Tnw)+p(u,u)]
=(a1+a4) p (u, Tn-1w)+a3p(Tn-1w, Tnw)+a4p(u, Tnw)+a5p(Tn-1w, Tnw)
=(a1 +a3+a4+a5)p(u,Tn-1w)+ (a3+a4+a5)p(u,Tnw)
Similarly as in the proof of theorem (2.1). It can be shown that under the condition (a1+2a3+2a4+2a5) â€1
126
8. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper)
www.iiste.org
ISSN 2225-0522 (Online)
Vol.3, No.12, 2013
P(u,Tnw)â€p(u,Tn-1w)
Note that when we consider
Κ(p(Tn w,u))†Κ(Ξ(Tn-1w,u))(Ξ(Tn-1w,u))
Where
Ξ(Tn-1w,u))= (a1 +a2)p(u, Tn-1w)+a2p (Tn-1w, Tnw)+ a4 p(u, Tnw)+ a5 p(Tn-1w, Tnw)
â€(a1 +a2+a4+a5)p(u,Tn-1w)+( a2+a4+a5) p(u, Tnw))
Hence one finds under (a1 +2a2+2a4+2a5) †that p(Tnw,u)â€p(Tn-1w,u)
In each case, it follows that the sequence {p(u,fnw)} is non increasing and it has a limit lâ„0 adjusting again
in the proof of theorem (2.1). one can finds that l=0 in the same way it can be deduced that p(v,Tnw)â0 as
nâ+âNow passing to the limit in p(u,v)†p(u,Tnw)+ p(Tnw,v) it follow that P(u,v)=0 so u=v, and the
uniqueness of the fixed point is proved.
Acknowledgement: One of the authors (R.B.) is thank full to MPCST Bhopal for project no. 2556
Ï
References
1. R. P. Agarwal, M. A. ElGebeily and D. OâRegan, âGeneralized contractions in
partially ordered
metric spacesâ, Applicable Analysis. 87(1)(2008) 109116.
2. I. Altun and A. Erduran, âFixed point theorems for monotone mappings on partial metric spacesâ, Fixed Point
Theory and Applications, Volume 2011, Article ID 508730, 10 pages, doi:10.1155/2011/508730.
3. I. Altun and H. Simsek, âSome fixed point theorems on ordered metric spaces and application, Fixed Point
Theory and Applicationsâ, Volume 2010, Article ID 621469, 17 pages, 2010.
4. I. Altun, F. Sola and H. Simsek, âGeneralized contractions on partial metric spacesâ, Topology and Its
Applications. 157 (18) (2010) 27782785.
5. H. Aydi,â Coincidence and common fixed point results for contraction type maps in partially ordered metric
spacesâ, Int. Journal of Math. Analysis, Volume 5, 2011, Number 13, 631642.
6. H. Aydi, âSome fixed point results in ordered partial metric spacesâ, Accepted in J. Nonlinear Sciences. Appl,
(2011).
7. . H. Aydi, âfixed point theorems for generalized weakly contractive condition in ordered partial metric
spacesâ, in J. Nonlinear Analysis and Optimization Vol 2 No. 2 . (2011),269-284.
8. H. Aydi, âSome coupled fixed point results on partial metric spacesâ, International Journal of Mathematics
and Mathematical Sciences, Volume 2011, Article ID 647091, 11 pages doi:10.1155/2011/647091.
9. H. Aydi, âFixed point results for weakly contractive mappings in ordered partial metric spacesâ, Journal of
Advanced Mathematical and Studies, Volume 4, Number 2, (2011).
10. I. Beg and A. R. Butt, âFixed point for setvalued mappings satisfying an implicit relation in partially ordered
metric spacesâ, Nonlinear Analysis. 71 (9) (2009) 36993704.
11. T.G. Bhaskar and V. Lakshmikantham, âFixed point theorems in partially ordered metric spaces and
applicationsâ, Nonlinear Analysis. 65 (2006) 13791393.
12. B.S. Choudhury and A. Kundu, âA coupled coincidence point result in partially ordered metric spaces for
compatible mappingsâ, Nonlinear Analysis (2010), doi:10.1016/j.na.2010.06.025.
13. Lj. CÂŽ iricÂŽ , N. CakicÂŽ , M. RajovicÂŽ and J. S. Ume,â Monotone generalized nonlinear contractions in
partially ordered metric spacesâ, Fixed Point Theory and Applications, Volume 2008, Article ID 131294, 11
pages, 2008.
14. B. C. Dhage, âCondensing mappings and applications to existence theorems for common solution of
di€erential equationsâ, Bull. Korean Math. Soc, 36 (3) (1999), 565578.
15. J. Harjani and K. Sadarangani, âGeneralized contractions in partially ordered metric spaces and applications
to ordinary differential equationsâ, Nonlinear Analysis. 72 (34) (2010) 11881197.
16. V. Lakshmikantham and Lj. CÂŽ iricÂŽ , âCoupled fixed point theorems for nonlinear contractions in partially
ordered metric spacesâ, Nonlinear Analysis. 70 (2009) 43414349.
17. S.G. Matthews, âPartial metric topology, in Proceedings of the 8th Summer Conference on General
Topology and Applicationsâ, vol. 728, pp. 183197,
Annals of the New York Academy of Sciences, 1994.
18. H.K. Nashine and I. Altun, âFixed point theorems for generalized weakly contractive condition in ordered
metric spacesâ, Fixed Point Theory and Applications, Volume 2011, Article ID 132367, 20 pages
doi:10.1155/2011/132367.
19. H.K. Nashine and B. Samet, âFixed point results for mappings satisfying
(Κ ,
)weakly contractive condition in partially ordered metric spacesâ, Nonlinear Analysis. 74 (2011)
22012209.
Ï
127
9. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper)
www.iiste.org
ISSN 2225-0522 (Online)
Vol.3, No.12, 2013
20. J.J. Nieto and R.R. LÂŽopez, âContractive mapping theorems in partially ordered sets and applications to
ordinary differential equationsâ, Order. 22 (3) (2005) 223239.
21. A.C.M. Ran and M.C.B. Reurings, âA fixed point theorem in partially ordered sets and some applications to
matrix equationsâ, Proceedings of the American Mathematical Society. 132 (5) (2004) 14351443.
22. B. Samet, âCoupled fixed point theorems for a generalized MeirKeeler
contraction in partially ordered metric spacesâ, Nonlinear Analysis. 72 (2010) 45084517. 284 H. AYDI/JNAO :
VOL. 2, NO. 2, (2011), 269284
23. S.J. Oâ Neill, âTwo topologies are better than oneâ, Tech. report, University of Warwick, Coventry, UK,
http://www.dcs.warwick.ac.uk/reports/283.html, 1995.
24. S. Oltra and O. Valero, âBanachâs fixed point theorem for partial metric spacesâ, Rendiconti dellâIstituto di
Matematica dellâUniversit `a di Trieste. 36 (12) (2004) 1726.
25. S. Romaguera, âA Kirk type characterization of completeness for partial metric
spacesâ, Fixed Point
Theory and Applications, Volume 2010, Article ID 493298, 6 pages, 2010.
26. O. Valero, âOn Banach fixed point theorems for partial metric spacesâ, Applied General Topology. 6
(2)(2005) 229240.
128
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