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1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.8, 2014
78
Common Random Fixed Point Theorems of Contractions in
Partial Cone Metric Spaces over Non normal Cones
PIYUSH M. PATEL(1)
Research Scholar of Sainath University,Ranchi(Jarkhand).
pmpatel551986@gmail.com
RAMAKANT BHARDWAJ(2)
Associate Professor in Truba Institute of Engineering & Information Technology, Bhopal (M.P).
drrkbhardwaj100@gmail.com
SABHAKANT DWIVEDI(3)
Department of Mathematics, Institute for Excellence in higher education Bhopal
1. ABSTRACT: The purpose of this paper is to prove existence of common random fixed point in the setting of
partial cone metric space over the non-normal cones.
Key wards: common fixed point,cone metric space, random variable
2. INTRODUCTION AND PRELIMINARIES
Random nonlinear analysis is an important mathematical discipline which is mainly concerned with the study of
random nonlinear operators and their properties and is needed for the study of various classes of random
equations. The study of random fixed point theory was initiated by the Prague school of Probabilities in the
1950s [4, 13, and 14]. Common random fixed point theorems are stochastic generalization of classical common
fixed point theorems. The machinery of random fixed point theory provides a convenient way of modeling many
problems arising from economic theory and references mentioned therein. Random methods have revolutionized
the financial markets. The survey article by Bharucha-Reid [1] attracted the attention of several mathematicians
and gave wings to the theory. Itoh [18] extended Spacek's and Hans's theorem to multivalued contraction
mappings. Now this theory has become the full edged research area and various ideas associated with random
fixed point theory are used to obtain the solution of nonlinear random system (see [2,3,7,8,9 ]). Papageorgiou
[11, 12], Beg [5,6] studied common random fixed points and random coincidence points of a pair of compatible
random and proved fixed point theorems for contractive random operators in Polish spaces.
In 2007, Huang and Zhang [9] introduced the concept of cone metric space and establish some fixed point
theorems for contractive mappings in normal cone metric spaces. Subsequently, several other authors [10, 17, ]
studied the existence of fixed points and common fixed points of pings satisfying contractive type condition on a
normal cone metric space. In 2008, Rezapour and Hamlbarani [17] omitted the assumption of normality in cone
metric space, which is a milestone in developing fixed point theory in cone metric space. In this paper we prove
existence of common random fixed point in the setting of cone random metric spaces under weak contractive
condition. Recently, Dhagat et al. [19] introduced the concept of cone random metric space and proved an
existence of random fixed point under weak contraction condition in the setting of cone random metric spaces.
The purpose of this paper to find common random fixed point theorems of contractions in partial cone metric
spaces over non normal cones.
Definition 2.1. Let X be a nonempty set and let 𝑃 be a cone of a topological vector space E . A partial cone
metric on X is a mapping PXX:p  such that, for each Xthtgtf )(),(),( , t ,

2. Mathematical Theory and Modeling www.iiste.org
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Vol.4, No.8, 2014
79





t
ththptgthpthtfptgtfpp
tgtfptftfpp
tftgptgtfpp
tgtftgtgptftfptgtfpp
))(),(())(),(())(),(())(),(()3(
))(),(())(),(()3(
))(),(())(),(()2(
),()())(),(())(),(())(),(()1(
The pair ),( pX is called a partial cone metric space over 𝑃.
Definition 2.2. A function f: Ω → C is said to be measurable if
1
( )f B C
  for every Borel subset B of
H.
Definition 2.3. A function :F C C  is said to be a random operator if (., ):F x C  is measurable
for every x C
Definition 2.4. A measurable :g C  is said to be a random fixed point of the random operator
:F C C  if ( , ( )) ( )F t g t g t for all t 
Definition 2.5. A random operator F: Ω×C → C is said to be continuous if for fixed t , ( ,.):F t C C
is continuous.
Lemma:2.6 Let P be solid cone of a topological vector space E and     .)(,)(,)( Ethtgtf nnn  if
ntgthtf nnn  )()()( , and there exists some Et )( Such that
)()()()()()( tththenttgandttf nnn  

Lemma:2.7 Let P be solid cone of a normed vector space  .,E then for each sequence   .)( Etfn 
)()()()(
.
ttfimpliesttf nn  
 moreover if P is normal, then )()( ttfn 

implies )()(
.
ttfn 
Lemma:2.8 Let P be solid cone of a normed vector space    andKE n ,.,   .)( Ptfn 
 
 )(,sup)( tfKthenKandtf nnnnn .
Theorem 2.9 Let ),( pX  be partial cone metric space. The mapping :,ST XX  are called
contractions restricted with variable positive linear bounded mappings if there exist
)4,3,2,1(:  iXXLi  such that
))),((),(()),((),(())(),((
)),((),(())(),((
))),((),(())(),(())(),(())(),(())),(((
4
3
21
ttfTtgpttgStfptgtfL
ttgStgptgtfL
ttfTtfptgtfLtgtfptgtfLttfTp



(*))(),(  Xtgtf
In particular if (*) is holds with
then T and S are called contractions restricted with positive linear bounded mappings
3. Main Result:
)4,3,2,1())(),((  iAandAtgtfL iii 

3. Mathematical Theory and Modeling www.iiste.org
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Vol.4, No.8, 2014
80
Theorem 3.1. Let ( , )X p be a 𝜃-complete partial cone metric space over a solid cone 𝑃 of a normed vector
space ( , . )E and let , :T S X X  be contractions restricted with variable positive linear bounded
random mappings. If
3 4( ( ( ), ( )) ( ( ), ( ))) 1p L f t g t L f t g t  and 2 4( ( ( ), ( )) ( ( ), ( ))) 1p L f t g t L f t g t 
( ), ( ) (1)f t g t X  
1 2 1l l  and 3l   , where (.)p denotes the spectral radius of linear bounded mappings,
1 1
(t),g(t) X
2 2
(t),g(t) X
3 3
(t),g(t) X
l sup ( ( ), ( ))
l sup ( ( ), ( ))
l sup ( ( ), ( )) , (2)
f
f
f
K f t g t
K f t g t
K f t g t





                               
1 1 1 2 4
2 2 1 3 4
3 2 3 4
( ( ), ( )) ( ( ), ( ))[L ( ( ), ( )) ( ( ), ( )) ( ( ), ( ))]
( ( ), ( )) ( ( ), ( ))[L ( ( ), ( )) ( ( ), ( )) ( ( ), ( ))]
( ( ), ( )) ( ( ), ( ))[I ( ( ), ( )) ( ( ), ( ))]
K f t g t L f t g t f t g t L f t g t L f t g t
K f t g t L f t g t f t g t L f t g t L f t g t
K f t g t L f t g t L f t g t L f t g t
f
  
  
  
 ( ), ( ) (3)t g t X                                      
where 1( ( ), ( ))L f t g t and 2 ( ( ), ( ))L f t g t denote the inverse of
3 4I ( ( ), ( )) ( ( ), ( ))L f t g t L f t g t  and 2 4I ( ( ), ( )) ( ( ), ( ))L f t g t L f t g t  respectively, then andT S
have common random fixed point in X Moreover if
1 2 3 4( ( ( ), ( )) ( ( ), ( )) ( ( ), ( )) 2 ( ( ), ( ))) 1 ( ), ( )
(4)
p L f t g t L f t g t L f t g t L f t g t f t g t X     
         
then andT S have unique common random fixed point (t)h X such that, for each
(t)nh X , (t) (t)p
nh h

 where (t)nh is defined by
1
( (t),t) ;n is even number
(t) (5)
S( (t),t) ;n is odd number
n
n
n
T h
h
h

                        

Proof. For each Xyx , by (1), the inverse of ))(),(())(),((1 43 tgtfLtgtfL  and
))(),(())(),((1 42 tgtfLtgtfL  exist. then, it is clear that 1L and 2L are meaningful
and 321 ,, KKK are well defined.
 



0
431 ))(),(())(),(())(),((
i
i
tgtfLtgtfLtgtfL
Xyx , , which is together with )4,3,2(:  iXXLi  implies that )2,1(:  iXXLi 
and hence ).3,2,1(:  iXXKi  by(*),(5) , (p4)  XXL :4
  )6())(),(())(),(())(),((
0
422  

i
i
tgtfLtgtfLtgtfL

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Vol.4, No.8, 2014
81
)7(
))](),(())(),(())[(),((
))(),(())(),((
))(),(())(),((
))(),(())(),((
))](),(())(),(())[(),((
))(),(())(),((
))(),(())(),((
))(),(())(),((
)),((),),((())(),((
22121221224
22121223
1221222
1221221
12122221224
22121223
1221222
1221221
1222212



















NktftfptftfptftfL
tftfptftfL
tftfptftfL
tftfptftfL
tftfptftfptftfL
tftfptftfL
tftfptftfL
tftfptftfL
ttfSttfTptftfp
kkkkkk
kkkk
kkkk
kkkk
kkkkkk
kkkk
kkkk
kkkk
kkkk
And so
 
)8(
))](),((
))(),((
))(),(())(),((
))(),(())(),(())(),((
122
1224
12221221
221212241222















Nktftfp
tftfL
tftfLtftfL
tftfptftfLtftfLI
kk
kk
kkkk
kkkkkk
Act the above inequality with ))(),(( 1221 tftfL kk  ; then,  XXL :1 ,
)9()),(),(())(),(())(),(( 12212212212   NktftfptftfKtftfp kkkkkk
Similarly ,by (*),(p3), (p4) and  XXL :4
)10(
))](),(())(),(())[(),((
))(),(())(),((
))(),(())(),((
))(),(())(),((
))](),(())(),(())[(),((
))(),(())(),((
))(),(())(),((
))(),(())(),((
)),((),),(((
))(),(())(),((
3222122212224
221212223
322212222
122212221
3212222212224
221212223
322212222
122212221
1222
22323222





















NktftfptftfptftfL
tftfptftfL
tftfptftfL
tftfptftfL
tftfptftfptftfL
tftfptftfL
tftfptftfL
tftfptftfL
ttfSttfTp
tftfptftfp
kkkkkk
kkkk
kkkk
kkkk
kkkkkk
kkkk
kkkk
kkkk
kk
kkkk
And so

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Vol.4, No.8, 2014
82
 
)11(
))](),((
))(),((
))(),(())(),((
))(),(())(),(())(),((
1222
12224
1222312221
32221222412222















Nktftfp
tftfL
tftfLtftfL
tftfptftfLtftfLI
kk
kk
kkkk
kkkkkk
Act the above inequality with ))(),(( 12222 tftfL kk  ; then, by: ,:2  XXL
)12(.,))(),(())(),(()(),(( 2212122221222   NktftfptftfKtftfp kkkkkk
Moreover ,:, 21  XXKK
)13()),(),(())(),((........
))........(),(())(),(())(),((
10101
122212212212

 
NktftfptftfK
tftfKtftfKtftfp kkkkkk
In the following, we will prove that
)14())(),((  
tftfp mn
For m>n, we have four cases
2qn2p,m(4)12qn2p,(3)m2qn1,+2p=m(2)1;+2q=n1,+2p=m(1) 
Where p and q are two non negative integers such that qp  . We only show that (14) holds for
case (1) the proof of three cases is similar. It follows for (p4) and (13) that
)15(.
))(),(())(),((.....))(),(())(),((
))(),(())(),(()).....(),(())(),((
.....))(),(())(),(()).....(),(())(),(())(),((
))(),(())(),(())........(),(())(),((
))(),((
))(),(())(),((.........))(),(())(),((
))(),((
))(),((
10101122211222
101013222212221
101011222122112222
1010112221221
1021
12221232222212
1212















qptftfptftfKtftfKtftfK
tftfptftfKtftfKtftfK
tftfptftfKtftfKtftfKtftfK
tftfptftfKtftfKtftfK
tftfKpK
tftfptftfptftfptftfp
tftfp
tftfp
ppqp
ppqq
qqqqqq
qqqq
ppppqqqq
pq
mn
By
,121 ll
   
  
)16(,
1
))(),((
))(),((
))(),((.....
))(),((
21
1021211
10
1
211211
10212
1
1
1
2
1
12
1
1
1021














  

qp
ll
tftfplllll
tftfpllllll
tftfpllllllll
tftfKpK
q
p
qi
p
qi
ii
ppppqqqq
Which implies that .
))(),(( tftfp mn ,and hence 
))(),(( tftfp mn by
lemma(2.7) Thus by (15) and lemma (2.6) 
))(),(( tftfp mn ;that is (14) holds. It is prove that
 )(tfn is a Cauchy sequence in  pX, ,and so by the  completeness of  pX, , there exits Xth )(
such that )()( thtf p
n 

and ))(),(( ththp ; that is,

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Vol.4, No.8, 2014
83
)17())(),((  
thtfp n .
For all ,Nk  by (*) and (p4),
   
    
 
  
))(),((
)),((),())(),(())(),((
))(),(())(),((
)),((),())(),((
))(),(())(),((
))(),((),(,),(
))(),(()(),(),),(()(),),((
2
122124
212123
122
12121
212
22
thtfp
tthTtfptfthptfthL
tftfptfthL
tthTthptfthL
tfthptfthL
thtfpttfStthTp
thtfptfthtthTpthtthTp
k
kkk
kkk
k
kk
kk
kk












 
 
 
 
  )18())(),(()),((),(
))(),(()),((),(
)),((),())(),((
))(),((
))(),(()))(),(())(),((
)),((),())(),((
))(),(())(),((
2
2
122
124
212123
122
12121


















thtfptthTthp
thtfptthTthp
tthtfptfthp
tfthL
tfthpthtfptfthL
tthTthptfthL
tfthptfthL
k
k
kk
k
kkk
k
kk
And so
 
 
 
)19(
))(),(())(),(())(),((
))(),(())(),(())(),(())(),((
))(),),((())(),(())(),((
2124123
12124123121
124122







NkthtfptfthLtfthLI
tfthptfthLtfthLtfthL
thtthTptfthLtfthLI
kkk
kkkk
kk
Act the above inequality with ))(),(( 122 tfthL k ;then,  XXL :1 ,
)20())(),((,)(),((,))(,),((( 212312122   NkthtfpKtfthpKthtthTp kkkk
Where ))(),(( 12212,2 tfthpKK kk   and ))(),(( 12212,3 tfthpKK kk   .
It is clear that { 12,2 kK },{ 12,3 kK } subsets of  and   12,312,2 sup,sup kkkk KK by 21ll
<1and 3l .then its follows from the lemma (3) and (17)that
)21(,)(),(()(),(( 1212,31212,2   
tfthpKtfthpK kkkk
Which together with lemma (2.6)and (20) implies that   )),((),( tthTthp
Therefore )()),(( thtthT  by (p1) and( p3).similarly we can show that )()),(( thtthS  .
Hence )(th is common random fixed point of T and

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84
Now we show the uniqueness of fixed point. Let )(tf and )(th be two common random fixed point of T
and S then by (*) and (p3) )3,2(:  iXXLi 
 
 
  )22()),(),(())(),((2))(),(())(),(())(),((
))(),(())(),(())(),(())(),((
))(),(())(),((2))(),((
))),((),(())),((),(())(),((
))),((),(())(),(()),((),(())(),(()),(),(())(),((
))),(()),),((())(),((
4321
32
41
4
321






tfthptfthLtfthLtfthLtfthL
tftfptfthLththptfthL
tfthptfthLtfthL
tthTtfpttfSthptfthL
ttfStfptfthLtthTthptfthLtfthptfthL
ttfStthTptfthp
And so
  )23()),(),(())(),((2))(),(())(),(())(),(( 4321  tfthptfthLtfthLtfthLtfthLI
It follow from (3) that the inverse of
 ))(),((2))(),(())(),(())(),(( 4321 tfthLtfthLtfthLtfthLI  exists(denoted by)
  1
4321 ))(),((2))(),(())(),(())(),((

 tfthLtfthLtfthLtfthLI and
  
1
4321 ))(),((2))(),(())(),(())(),(( tfthLtfthLtfthLtfthLI by Neumann’s formula
with  1
4321 ))(),((2))(),(())(),(())(),((

 tfthLtfthLtfthLtfthLI ;then ))(),(( tfthp
and hence )()( tfth  by (p1) and (p3)the proof is completed.
REFERENCES:
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American Mathematical Society, 82(5) (1976), 641-657.
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ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.8, 2014
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Banja, Serbia, May-June (2008).
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