A05920109

IOSR Journal of Engineering (IOSRJEN) ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 05, Issue 09 (September. 2015), ||V2|| PP 01-09

www.iosrjen.org

Random Fixed Point TheoremsIn Metric Space Balaji R Wadkar1, Ramakant Bhardwaj2, Basantkumar Singh3 1

(Dept. of Mathematics, “S R C O E” Lonikand, Pune & Research scholar of “AISECT University”, Bhopal, India) wbrlatur@gmail.com 2 (Dept. of Mathematics, “TIT Group of Institutes (TIT &E)” Bhopal (M.P), India) rkbhardwaj100@gmail.com 3 (Principal, “AISECT University”, Bhopal-Chiklod Road, Near BangrasiaChouraha, Bhopal, (M.P), India) dr.basantsingh73@gmail.com

Abstract: - The present paper deals with some fixed point theorem for Random operator in metric spaces. We find unique Random fixed point operator in closed subsets of metric spaces by considering a sequence of measurable functions.

AMS Subject Classification: 47H10, 54H25.

Keywords: Fixed point,Common fixed point,Metric space, Borelsubset, Random Operator.

I.

INTRODUCTION

Fixed point theory is one of the most dynamic research subjects in nonlinear sciences. Regarding the feasibility of application of it to the various disciplines, a number of authors have contributed to this theory with a number of publications. The most impressing result in this direction was given by Banach, called the Banach contraction mapping principle: Every contraction in a complete metric space has a unique fixed point. In fact, Banach demonstrated how to find the desired fixed point by offering a smart and plain technique. This elementary technique leads to increasing of the possibility of solving various problems in different research fields. This celebrated result has been generalized in many abstract spaces for distinct operators.

Random fixed point theory is playing an important role in mathematics and applied sciences. At present it received considerable attentation due to enormous applications in many important areas such as nonlinear analysis, probability theory and for thestudy of Random equations arising in various applied areas.

In recent years, the study of random fixed point has attracted much attention some of the recent literature in random fixed point may be noted in [1, 5, 6, 8]. The aimof this paper is to prove some random fixed point theorem.Before presenting our results we need some preliminaries that include relevant definition.

II.

PRELIMINARIES

Throughout this paper, (Ω,Σ)denotes a measurable space, X be a metric space andC is non-empty subset of X. Definition 2.1: A function f:Ω→C is said to be measurable if f '(B∩C)∈ Σfor every Borelsubset B of X. Definition 2.2: A function f:Ω×C →C is said to be random operator, iff (.,X):Ω→C is measurable for every X ∈ C. Definition 2.3: A random operator f:Ω×C →C is said to be continuous if for fixedt ∈ Ω,f(t,.):C×C is continuous.

Definition 2.4: A measurable function g:Ω→C is said to be random fixed pointof the random operator f:Ω×C →C, if f (t, g(t) )=g(t), ∀ t ∈ Ω.

III. Main Results Theorem (3.1): Let (X, d) be a complete metric space and E be a continuous self-mappings such that  d  g (  ), E {  , g (  )} d  h (  ), E {  , h (  )} d  g (  ), E {  , h (  )}       d  g (  ), h (  ) d  h (  ), E {  , g (  )} d  h (  ), E {  , h (  )}   d ( E {  , g (  )}, E {  , h (  )})     2  d  g (  ), E {  , h (  )} d  h (  ), E {  , h (  )}   [ d  g (  ), h (  ) ]   

  d  g (  ), E { , g (  )}   d  h (  ), E { , h (  )}    [ d  g (  ), E { , h (  )}   d  h (  ), E { , g (  )} ]

International organization of Scientific Research

1|Page

Random Fixed Point TheoremInMetric Spaces 

d  g (  ), E { , g (  )} d  h (  ), E { , h (  )}  d  g (  ), h (  ) 

(3.1.1)

  [ d  g (  ), h (  ) ]

Forall g (  ) ) and

with g (  )  h (  ) , where  ,  ,  ,  ,  1 , then E has a unique fixed point in X.

h ( )  X

  2   2     

: R



 [ 0 ,1 )

are such that

Proof:Let {gn}be a sequence, define d as follows g n {  }  E {  , g n  1 (  )} , n = 1,2,3,4…

If

g n { }  g n  1 { }

So let

for some n then the result follows immediately.

g n ( )  g n 1 ( )

for all n then

d  g n (  ), g n  1 (  )   d  E { , g n  1 (  )}, E { , g n (  )} 

 d  g n  1 (  ), E {  , g n  1 (  )} d  g n (  ), E {  , g n (  )} d  g n  1 (  ), E {  , g n (  )}       d  g n  1 (  ), g n (  ) d  g n (  ), E {  , g n  1 (  )} d  g n (  ), E {  , g n (  )}    2    d  g n  1 (  ), g n (  )   d  g n  1 (  ), E {  , g n (  )} d  g n (  ), E {  , g n (  )}     

  d  g n  1 (  ), E { , g n  1 (  )}   d  g n (  ), E { , g n (  )}    d  g n  1 (  ), E { , g n (  )}   d  g n (  ), E { , g n  1 (  )} 



d  g n  1 (  ), E {  , g n  1 (  )} d  g n (  ), E {  , g n (  )}  d  g n  1 (  ), g n (  ) 

  . d  g n  1 (  ), g n (  ) 

d  g n  1 (  ), g n (  ) d  g n (  ), g n  1 (  ) d  g n  1 (  ), g n  1 (  )       d  g n  1 (  ), g n (  ) d  g n (  ), g n (  ) d  g n (  ), g n  1 (  )      2 d  g n  1 (  ), g n (  )   d  g n  1 (  ), g n  1 (  ) d  g n (  ), g n  1 (  )          d  g n  1 (  ), g n (  )   d  g n (  ), g n  1 (  )    d  g n  1 (  ), g n  1 (  )   d  g n (  ), g n (  )  

d  g n  1 (  ), g n (  ) d  g n (  ), g n  1 (  )  d  g n  1 (  ), g n (  ) 

  . d  g n  1 (  ), g n (  ) 

 d  g n  1 (  ), g n (  ) d  g n (  ), g n  1 (  ) d  g n  1 (  ), g n  1 (  )     d  g n  1 (  ), g n  1 (  ) d  g n (  ), g n  1 (  )   

  d  g n  1 (  ), g n (  )   d  g n (  ), g n  1 (  )    d  g n  1 (  ), g n (  )   d  g n (  ), g n  1 (  )    .d  g

n

  .d  g

n 1

(  ), g

n 1

(  ), g

n

( )  ( ) 

International organization of Scientific Research

2|Page

Random Fixed Point TheoremInMetric Spaces    d  g n  1 (  ), g n (  )     d  g n  1 (  ), g n (  )   d  g n (  ), g n  1 (  )     d  g n  1 (  ), g n (  )   d  g n (  ), g n  1 (  )    . d  g n (  ), g n  1 (  )    . d  g n  1 (  ), g n (  ) 

 (       ). d  g n 1 (  ), g n (  )   (      ) d  g n (  ), g n  1 (  ) 



(1       ) d  g n (  ), g n  1 (  )   (       ). d  g n 1 (  ), g n (  ) 



d  g n (  ), g

n 1

( )  

(        ).

(1       ) .......... ..



(       ).

(1       )

Thus by triangle inequality, we have for

d g

n 1

(  ), g n (  ) 

d  g 0 (  ), g 1 (  ) 

m  n

d  g n (  ), g m (  )   d  g n (  ), g n  1 (  )   d  g n  1 (  ), g n  2 (  )   .......  d  g m  1 (  ), g m (  )   (s

Where

n

s 

 s

n 1

 s

n2

 ....  s

(        ).

(1       )

m 1

) d  g 0 (  ), g 1 (  ) 

since

 1

  2   2      1

Therefore d  g n (  ), g m (  )  

s

n

1 s

d  g 0 (  ), g 1 (  )   0 , as m , n  

being complete, there exist some

p  X

. Hence the sequence {gn}is a Cauchy sequence, X

such that

  E (  , u (  ))  E  lim g n (  )   lim . E { , g n (  )}  lim g n  1 (  )  u (  ) n   n   n 

Therefore

u ( )

is a fixed point of E.

Uniqueness:Let if possible there exist another fixed point v (  ) of E in X, such that (3.1.1) we have

u ( )  v ( )

then from

d u (  ), v (  )   d  E { , u (  )}, E { , v (  )}   d  u (  ), E {  , u (  )} d  v (  ), E {  , v (  )} d  u (  ), E {  , v (  )}       d  u (  ), v (  )  d  v (  ), E {  , u (  )} d  v (  ), E {  , v (  )}      [ d  u (  ), v (  ) ] 2  d  u (  ), E {  , v (  )} d  v (  ), E {  , v (  )}      

  d u (  ), E { , u (  )}   d  v (  ), E { , v (  )}    d u (  ), E { , v (  )}   d  v (  ), E { , u (  )}  

d  u (  ), E {  , u (  )} d  v (  ), E {  , v (  )}  d  u (  ), v (  ) 

  . d  u (  ), v (  )    d u (  ), v (  )   d  v (  ), u (  )    . d u (  ), v (  )   ( 2    ) d u (  ), v (  ) 

d u (  ), v (  )   ( 2    ) d u (  ), v (  ) 

Since 

  2   2      1  2    1

d  u (  ), v (  )   d  u (  ), v (  ) 

This is contradiction, so u ( )  v ( ) .

This completes the proof of the theorem (3.1.1). We now prove another theorem.

International organization of Scientific Research

3|Page

Random Fixed Point TheoremInMetric Spaces

Theorem (3.2): Let Ebe a self-mapping of a complete metric space (X, d), if for some positive integer p, E p is continuous, then E has unique fixed point in X. Proof:Let g n  be sequence which converges to some. u  X . Therefore its subsequence g n  also converges k

to u also p

E

Therefore

p

(  , u (  ))  E

u ( )

lim g n (  )  lim E

k 

k

p

k 

 , g n

k



(  )  lim g n k 

is a fixed point of E p , we now show that E (  , u ( ))

Let m be the smallest positive integer such that E by (3.1.1) we get,

m

k 1

( )  u ( )

 u ( )

(  , u (  ))  u (  ) and

E

. q

 u (  ), 1  q  m  1

, If m>1 then

 { , u ( )}, E { , u ( )}   d E ( E { , u (  )}), E { , u (  )} 

d u (  ), E {  , u (  )}   d E

m

m 1







 d E m  1 {  , u (  )}, E m { , u (  )} . d  u (  ), E {  , u (  )} . d E m  1 {  , u (  )}, E { , u (  )}    d E m  1 { , u (  )}, u (  ) . d u (  ), E m { , u (  )} d u (  ), E {  , u (  )}   2 m 1 m 1  d E {  , u (  )}, u (  )  d E {  , u (  )}, E {  , u (  )} d u (  ), E {  , u (  )}   

 

    d  E

m 1

  d E



d E  

 

 d E

m 1

{  , u (  )}, E



d E

m 1

m

m 1

  

m 1

{  , u (  )}, E

m









{  , u (  )}  d  u (  ), E {  , u (  )} 





{  , u (  )}, E {  , u (  )}  d u (  ), E



{  , u (  )}, u (  )







m 1

d E  

 

 d E

    d  E   d  E

  d E











 E { , u (  )}   d  u (  ), u (  )  

{  , u (  )}, u (  )  d u (  ), E { , u (  )} 

{  , u (  )},

m 1



{  , u (  )}, u (  ) . d  u (  ), E {  , u (  )}    m 1 d E {  , u (  )}, u (  ) 



m 1



 d E m  1 {  , u (  )}, u (  ) . d  u (  ), E { , u (  )} . d E m  1 { , u (  )}, E {  , u (  )}      d E m  1 { , u (  )}, u (  ) . d  u (  ), u (  ) . d  u (  ), E {  , u (  )}      2 m 1 m 1  d E {  , u (  )}, u (  )  d E {  , u (  )}, E {  , u (  )} d u (  ), E {  , u (  )}      

 

m 1







    d E

{  , u (  )}

     

{  , u (  )} d  u (  ), E {  , u (  )}    {  , u (  )}, u (  ) 



  d E

m





{  , u (  )}, u (  )





 {  , u (  )}, u (  )   d  u (  ), E {  , u (  )}  {  , u (  )}, d ( u (  ) .  d  u (  ), E {  , u (  )} 

m 1

{  , u (  )}, u (  )

m 1

m 1

   d  u (  ), E {  , u (  )} 

 

 d E

m 1

{  , u (  )}, u (  )



International organization of Scientific Research

4|Page

Random Fixed Point TheoremInMetric Spaces



 (       ) d E

m 1



{  , u (  )}, u (  )  (      ) d  u (  ), E {  , u (  )} 



(1       ) d  u (  ), E {  , u (  )}   (        ) d E

(       )



d u (  ), E { , u (  )}  



d  u (  ), E {  , u (  )}   s . d E



Further,



d E

m 1



m 1



m

{ , u (  )}, u (  )  d E



 s .d E ..........  s

Therefore d  u (  ), E {  , u (  )}  

s

m



d E

(1       )

{ , u (  )}, E m 1

{  , u (  )}, u (  )

{ , u (  )}, u (  )

{  , u (  )}, u (  )

 where

s 

{ , u (  )}



m 1

{ , u (  )}, E

..........

m 1

m 1

m 1

m2

{  , u (  )}





(       ) (1       )

1



....

d  u (  ), E { , u (  )} 

d  u (  ), E {  , u (  )} 

 d  u (  ), E {  , u (  )}  Which is contradiction, therefore u (  ) is fixed point of E.That is u ( )

 E { , u (  )}

.This completes the proof.

Theorem (3.3): Let E be a self-mapping of a complete metric space X such that for some positive integer m, E

m

Satisfies



d E

m

{  , g (  )}, E

m

{  , h (  )}





 

 



 d g (  ), E m {  , g (  )} . d h (  ), E m {  , h (  )} . d g (  ), E m {  , h (  )}      d  g (  ), h (  ) . d h (  ), E m {  , g (  )} . d h (  ), E m {  , h (  )}    . 2 m m   d  g (  ), h (  )   d g (  ), E {  , h (  )} . d h (  ), E {  , h (  )}     



    d  g (  ), E

m

  d g (  ), E



m

 d g (  ), E  

 





 



 {  , h (  )}   d h (  ), E {  , g (  )}  {  , g (  )} d h (  ), E {  , h (  )}   



{  , g (  )}  d h (  ), E

m

{  , h (  )}

m

m

m

 

d  g (  ), h (  ) 

(3.3.1)

   d  g (  ), h (  )  

For all g (  ) ) and

h ( )  X

  2   2      1

Proof: E

m

with g (  )

 h ( )

, where

 ,  ,  , ,  : R



 [ 0 ,1 )

are such that

m

,if for some positive integer m,E is continuous, then E has a unique fixed point in X.

has unique fixed point u (  ) in X follows from theorem (3.2).

E {  , u (  )}  E ( E

m

(  , u (  )))  E

m

E {  , u (  )}

unique fixed point u (  ) , so E { , u ( )}

 u ( )

, Which implies that

E { , u (  )}

is fixed point of E

m

but has

.

Since any fixed point of E is also a fixed point of E . It follows that u (  ) is unique fixed point of E. This completes the proof of (3.3).We now prove another theorem. m

Theorem 3.4: Let E & F be a pair of self-mappings of a complete metric space X, satisfying the following conditions: d  E { , g (  )}, F { , h (  )} 

International organization of Scientific Research

5|Page

Random Fixed Point TheoremInMetric Spaces  d  g (  ), E {  , g (  )} . d  h (  ), F {  , h (  )} . d  g (  ), F {  , h (  )}       d  g (  ), h (  ) . d  h (  ), E {  , g (  )} . d  h (  ), F {  , h (  )}      [ d  g (  ), h (  ) ] 2  d  g (  ), F {  , h (  )} . d  h (  ), F {  , h (  )}      

  d  g (  ), E { , g (  )}   d  h (  ), F { , h (  )}    d  g (  ), F { , h (  )}   d  h (  ), E { , g (  )} 



d  g (  ), E {  , g (  )} d  h (  ), F {  , h (  )}  d  g (  ), h (  ) 

(3.4.1)

  . d  g (  ), h (  ) 

For all g (  ) ),

h ( )  X

with g (  )

 h ( )

,where

 ,  ,  , ,  : R



 [ 0 ,1 )

are such that

  2   2      1

,if E & F are continuous on X , then E& F have a unique fixed point in X. Proof:Let g n  be a continuous sequence defined as

g n ( )  Now

{

E  , g

n 1

( )

n is even

F  , g

n 1

( )

n is odd

and g n { } 

g n  1 { }

for all n.

d  g 2 n (  ), g 2 n  1 (  )   d  E { , g 2 n  1 (  )}, F { , g 2 n (  )}   d  g 2 n  1 (  ), E {  , g 2 n  1 (  )} . d  g 2 n (  ), F {  , g 2 n (  )} . d  g 2 n  1 (  ), F {  , g 2 n (  )}       d  g 2 n  1 (  ), g 2 n (  ) . d  g 2 n (  ), E {  , g 2 n  1 (  )} d  g 2 n (  ), F {  , g 2 n (  )}    2    d  g 2 n  1 (  ), g 2 n (  )   d  g 2 n  1 (  ), F {  , g 2 n (  )} . d  g 2 n (  ), F {  , g 2 n (  )} .    

  d  g 2 n  1 (  ), E { , g 2 n  1 (  )} .  d  g 2 n (  ), F { , g 2 n (  )}    d  g 2 n  1 (  ), E { , g 2 n (  )}   d  g 2 n (  ), E { , g 2 n  1 (  )}  

d  g 2 n  1 (  ), E {  , g 2 n  1 (  )} . d  g 2 n (  ), F {  , g 2 n (  )}  d  g 2 n  1 (  ), g 2 n (  ) 

  . d  g 2 n  1 (  ), g 2 n (  ) 

 d  g 2 n  1 (  ), g 2 n (  ) . d  g 2 n (  ), g 2 n  1 (  ) . d  g 2 n  1 (  ), g 2 n  1 (  )       d  g 2 n  1 (  ), g 2 n (  ) . d  g 2 n (  ), g 2 n (  ) d  g 2 n (  ), g 2 n  1 (  )    2    d  g 2 n  1 (  ), g 2 n (  )   d  g 2 n  1 (  ), g 2 n  1 (  ) . d  g 2 n (  ), g 2 n  1 (  )     

  d  g 2 n  1 (  ), g 2 n (  )   d  g 2 n (  ), g 2 n  1 (  )    d  g 2 n  1 (  ), g 2 n  1 (  )   d  g 2 n (  ), g 2 n (  ) 



d  g 2 n  1 (  ), g 2 n (  ) . d  g 2 n (  ), g 2 n  1 (  )  d  g 2 n  1 (  ), g 2 n (  ) 

  . d  g 2 n  1 (  ), g 2 n (  )   

 g 2 n  1 (  ),

g 2 n (  ) 

   d  g 2 n  1 (  ), g 2 n (  )   d  g 2 n (  ), g 2 n  1 (  )     d  g 2 n  1 (  ), g 2 n (  )   d  g 2 n (  ), g 2 n  1 (  ) 

International organization of Scientific Research

6|Page

Random Fixed Point TheoremInMetric Spaces   .d  g 2 n (  ), g 2 n  1 (  )    .d  g 2 n 1 (  ), g 2 n (  ) 

 (       ) d  g 2 n  1 (  ), g 2 n (  )   (      ) d  g 2 n (  ), g 2 n  1 (  )  (1       ) d  g 2 n (  ), g 2 n  1 (  )   (       ) d  g 2 n 1 (  ), g 2 n (  )  d  g 2 n (  ), g 2 n  1 (  )  



(       ) (1       )

d  g 2 n  1 (  ), g 2 n (  ) 

d  g 2 n (  ), g 2 n  1 (  )   s .d  g 2 n  1 (  ), g 2 n (  ) 

where

s 

(       ) (1       )

1

Similarly d  g 2 n (  ), g 2 n  1 (  )   s . d  g 2 n  2 (  ), g 2 n  1 (  )  2

..........

..........

........

..........

..........

..........

 s

2 n.

.

d  g 0 (  ), g 1 (  ) 

2 n 1 . d  g 0 (  ), g 1 (  )  Hence d  g 2 n  1 (  ), g 2 n  2 (  )   s

Hence the sequence { g n } is a Cauchy sequence in X and X being complete, therefore there exist such that

lim g n (  )  u (  )

n 

the subsequence

u ( )

in X

g nk  u (  )

Now, if EF is continuous on X then EF (  , u (  ))  EF { lim k 

g n (  )}  lim k

k 

gn

k 1

( )  u ( )

Thus EF {  , u (  )}  u (  ) i.e. u (  ) is fixed point of EF. Now we show that F { , u ( )}  u ( ) . If F {  , u (  )}  u (  ) , then d u (  ), F { , u (  )}   d  EF { , u (  )}, F { , u (  )}   d  F {  , u (  )}, EF {  , u (  )} d  u (  ), F {  , u (  )} d  F {  , u (  )}, F {  , u (  )}       d  F {  , u (  )}, u (  ) d  u (  ), EF {  , u (  )} d u (  ), F {  , u (  )     2    d  F {  , u (  )}, u (  )   d  F {  , u (  )}, F {  , u (  )} d  u (  ), F {  , u (  )}     

  d  F { , u (  )}, EF { , u (  )}   d u (  ), F { , u (  )}    d  F { , u (  )}, F { , u (  )}   d u (  ), EF { , u (  ) 



d  F {  , u (  )}, EF {  , u (  )} d  u (  ), F {  , u (  )}  d  F {  , u (  )}, u (  ) 

   d  F {  , u (  )}, u (  ) 

d  F {  , u (  )}, u (  ) d  u (  ), F {  , u (  )} d  F {  , u (  )}, F {  , u (  )}       d  F {  , u (  )}, u (  ) d  u (  ), u (  )} d  u (  ), F {  , u (  )      2   d  F {  , u (  )}, u (  )   d  F {  , u (  )}, F {  , u (  )} d u (  ), F {  , u (  )}     

  d  F { , u (  )}, u (  )   d u (  ), F { , u (  )}    d  F { , u (  )}, F { , u (  )}   d u (  ), u (  ) 

International organization of Scientific Research

7|Page

Random Fixed Point TheoremInMetric Spaces 

d  F {  , u (  )}, u (  ) d  u (  ), F {  , u (  )}  d  F {  , u (  )}, u (  ) 

   d  F {  , u (  )}, u (  )   d  F {  , u (  )}, u (  ) d  u (  ), F {  , u (  )} d  F {  , u (  )}, F {  , u (  )}       d  F { , u (  )}, u (  ) 2  

 2  d  F { , u (  )}, u (  )    0    .d  F { , u (  )}, u (  )    d  F { , u (  )}, u (  ) 

  2      d  F { , u (  )}, u (  )  d  F { , u (  )}, u (  )    2      d  F { , u (  )}, u (  )   d  F { , u (  )}, u (  ) 

Since   2   2   Hence F { , u ( )}  u ( ) .

  1

implies that  2 

Further d  E { , u ( )}, u ( )   d  E { , u ( )}, So u is common fixed point of E & F

 

u (  )  implies

1

that E { , u (  )}

 u ( )

.

Uniqueness: Let v (  ) is another common fixed point of E & F we have d u (  ), v (  )   d  E { , u (  )}, F { , v (  )}   d  u (  ), E {  , u (  )}  d  v (  ), F {  , v (  )} d  u (  ), F {  , v (  )}       d  u (  ), v (  )  d  v (  ), E {  , u (  )} d  v (  ), F {  , v (  )}     2  d u (  ), v (  )   d u (  ), F { , v (  )} d  v (  ), F { , v (  )}          d  u (  ), E {  , u (  )}   d  v (  ), F {  , v (  )}     d  u (  ), F {  , v (  )}   d  v (  ), E {  , u (  )}  

d  u (  ), E {  , u (  )}  d  v (  ), F {  , v (  )}  d  u (  ), v (  ) 

   d  u (  ), v (  )   d  u (  ), u (  ) d  v (  ), v (  ) d  u (  ), v (  )   d  u (  ), v (  )  d  v (  ), u (  )}  d  v (  ), v (  )       d u (  ), v (  ) 2  d u (  ), v (  ) d  v (  ), v (  )   

  d u (  ), u (  )   d  v (  ), v (  )    d u (  ), v (  )   d  v (  ), u (  )  

d  u (  ), u (  )  d  v (  ), v (  )  d  u (  ), v (  ) 

   d  u (  ), v (  ) 

  2    d u (  ), v (  )  d u (  ), v (  )    2    d u (  ), v (  )   d u (  ), v (  ) 

Because  2 



  1 . This implies u ( )

 v ( )

.This completes the proof of (3.4)

International organization of Scientific Research

8|Page

Random Fixed Point TheoremInMetric Spaces REFERENCES [1].

[2].

[3].

[4].

[5].

[6].

[7].

[8].

Beg, I. and Shahzad, N. “Random approximations and random fixed point theorems,” J. Appl. Math.

Stochastic Anal. 7(1994). No.2, 145-150.

Bharucha-Reid, A.T. “Fixed point theorems in probabilistic analysis,” Bull. Amer. Math. Soc. 82(1976), 641-657.

Choudhary B.S. and Ray,M. “Convergence of an iteration leading to a solution of a random operator equation,”J. Appl. Stochastic Anal. 12(1999). No. 2, 161-168.

Dhagat V.B., Sharma A. and BhardwajR.K. “Fixed point theorems for random operators in Hilbert spaces,” International Journal of Math. Anal.2(2008).No.12,557-561.

O’Regan,D. “A continuous type result for random operators,” Proc. Amer. Math, Soc. 126(1998), 19631971.

SehgalV.M. andWaters,C. “Some random fixed point theorems for condensing operators,” Proc. Amer.

Math. Soc. 90(1984). No.3, 425-429.

SushantkumarMohanta “Random fixed point in Banach spaces” Inter. J of Math Analysis Vol. 5 (2011) No. 10, 451-461.

B. R wadkar,Ramakant Bhardwaj, Rajesh Shrivastava, “Some NewResult In Topological Space For NonSymmetric RationalExpression Concerning Banach Space” ,International Journal of Theoretical and Applied Science 3(2): 65-78(2011).

International organization of Scientific Research

9|Page