ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 05, Issue 09 (September. 2015), ||V2|| PP 01-09
Random Fixed Point TheoremsIn Metric Space
Balaji R Wadkar
1, Ramakant Bhardwaj
2, Basantkumar Singh
3 1(Dept. of Mathematics, “S R C O E” Lonikand, Pune & Research scholar of “AISECT University”, Bhopal, India)
[email protected]
2(Dept. of Mathematics, “TIT Group of Institutes (TIT &E)” Bhopal (M.P), India) [email protected] 3(Principal, “AISECT University”,
Bhopal-Chiklod Road, Near BangrasiaChouraha, Bhopal, (M.P), India)
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
( ), { , ( )} ( ), { , ( )}] [
)} ( , { ), ( )} ( , { ), (
g E h d h
E g d
h E h d g
E g d
)} ( , { ), ( )} ( , { ), ( ] ) ( ), (
[ 2
)} ( , { ), ( )} ( , { ), ( ) ( ), (
)} ( , { ), ( )} ( , { ), ( )} ( , { ), (
)}) ( , { )}, ( , { (
h E h d h E g d h
g d
h E h d g E h d h g d
h E g d h E h d g E g d
( ), ( )] [
) ( ), (
)} ( , { ), ( )} ( , { ), (
h g d
h g d
h E h d g E g d
(3.1.1)
Forallg()) and h()X withg() h(), where ,,,, :R [0,1) are such that 1
2
2
, then E has a unique fixed point in X.
Proof:Let {gn}be a sequence, define d as follows
)} ( 1 , { }
{ E gn
gn , n = 1,2,3,4…
If { }
1 }
{ gn
gn for some n then the result follows immediately.
So let gn() gn1()for all n then
gn(),gn 1() dE{,gn 1()}, E{,gn()}
d
)} ( , { ), ( )} ( , { ), ( )
( ), (
)} ( , { ), ( )} ( , { ), ( ) ( ), (
)} ( , { ), ( )}
( , { ), ( )} ( , { ), (
1 2
1
1 1
1 1
1
n n
n n
n n
n n
n n
n n
n n
n n
n n
g E g d g E g
d g
g d
g E g d g
E g d g g d
g E g
d g E g d g
E g d
( ), { , ( )} ( ), { , ( )}
)} ( , { ), ( )}
( , { ), (
1 1
1 1
n n
n n
n n
n n
g E g d g
E g
d
g E g d g
E g
d
( ), ( )
.
) ( ), (
)} ( , { ), ( )} ( , { ), (
1
1 1 1
n n
n n
n n
n n
g g d
g g
d
g E g d g
E g
d
) ( ), ( ) ( ), ( )
( ), (
) ( ), ( ) ( ), ( ) ( ), (
) ( ), ( )
( ), ( ) ( ), (
1 1
1 2
1
1 1
1 1
1 1
n n n
n n
n
n n n
n n
n
n n
n n n
n
g g d g
g d g
g d
g g d g g d g g
d
g g
d g
g d g g
d
( ), ( ) ( ), ( )
) ( ), ( )
( ), (
1 1
1 1
n n n
n
n n n
n
g g d g
g d
g g d g
g d
( ), ( )
.
) ( ), (
) ( ), ( ) ( ), (
1
1
1 1
n n
n n
n n n
n
g g d
g g d
g g d g g d
) ( ), ( ) ( ), (
) ( ), ( ) ( ), ( ) ( ), (
1 1
1
1 1
1 1
n n n
n
n n
n n n
n
g g d g g d
g g d g g d g g d
( ), ( ) ( ), ( )
) ( ), ( )
( ), (
1 1
1 1
n n n
n
n n n
n
g g d g
g d
g g d g
g d
( ), ( )
.
) ( ), ( .
1 1
n n
n n
g g
d
g g d
( ), ( )
.
) ( ), ( . ) ( ), ( )
( ), (
) ( ), ( )
( ), ( )
( ), (
1
1 1
1
1 1
1
n n
n n n
n n
n
n n n
n n
n
g g d
g g d g
g d g
g d
g g d g
g d g
g d
( ), ( ) ( ) ( ), ( )
).
( 1 1
d gn gn d gn gn
(1 )dgn(),gn1()( ).dgn1(),gn()
( ), ( )
) 1
(
). (
) ( ),
( 1 1
n n n
n g d g g
g
d
( ), ( )
) 1
(
). (
.. ...
1
0
g g d
Thus by triangle inequality, we have for m n
( ), ( )
) .... (
) ( ), ( ...
) ( ), ( )
( ), ( )
( ), (
1 0 1 2
1
1 2
1 1
g g d s s
s s
g g
d g
g d g
g d g
g d
m n
n n
m m
n n
n n m
n
Where 1
) 1
(
). (
s since 2 2 1
Therefore
d g g m n
s s g
g d
n
m
n ( ), ( ) 0,as ,
1 ) ( ),
( 0 1 . Hence the sequence {gn}is a Cauchy sequence, X
being complete, there exist some p X such that
) ( ) ( lim )} ( , { . lim ) ( lim ))
( ,
( u E g E g g 1 u
E n
n n 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 u()v()then from
(3.1.1) we have
u(),v() dE{,u()},E{,v()}
d
)} ( , { ), ( )} ( , { ), ( ] ) ( ), ( [
)} ( , { ), ( )} ( , { ), ( ) ( ), (
)} ( , { ), ( )} ( , { ), ( )} ( , { ), (
2
v E v d v E u d v
u d
v E v d u E v d v u d
v E u d v E v d u E u d
( ), { , ( )} ( ), { , ( )}
)} ( , { ), ( )} ( , { ), (
u E v d v
E u d
v E v d u
E u d
( ), ( )
.
) ( ), (
)} ( , { ), ( )} ( , { ), (
v u d
v u d
v E v d u E u d
( ), ( )
) 2 (
) ( ), ( . ) ( ), ( ) ( ), (
v u d
v u d u
v d v u d
u(),v() (2 )du(),v()
d
Since 2 2 1 2 1 d
u(),v()
d
u(),v()
This is contradiction, sou()v().
Theorem (3.2): Let Ebe a self-mapping of a complete metric space (X, d), if for some positive integer p, p E is
continuous, then E has unique fixed point in X.
Proof:Let gn be sequence which converges to some. u X . Therefore its subsequence
gnk also convergesto u also
, ( )
lim ( ) ( ) lim) ( lim ))
( , (
1
u E p g Ep g g u
p E
k k
k n
k n k
n k
Therefore u()is a fixed point ofE p , we now show thatE(,u()) u().
Let m be the smallest positive integer such thatEm(,u()) u() and Eq u(),1 q m 1 , If m>1 then by (3.1.1) we get,
( { , ( )}), { , ( )}
)} ( , { )}, ( , { )}
( , { ), (
1
u E u E E d
u E u E d u
E u d
m m
)} ( , { ), ( )} ( , { )}, ( , { )
( )}, ( , {
)} ( , { ), ( )} ( , { ), ( . ) ( )}, ( , {
)} ( , { )}, ( , { . )} ( , { ), ( . )} ( , { )}, ( , {
1 2
1 1
1 1
u E u d u E u E d u
u E d
u E u d u E u d u u E d
u E u E d u E u d u E u E d
m m
m m
m m
m
{ , ( )}, { , ( )} ( ), { , ( )}
)} ( , { ), ( )} ( , { )}, ( , {
1 1
u E u d u
E u E d
u E u d u
E u E d
m m
m m
{ , ( )}, ( )
) ( )}, ( , {
)} ( , { ), ( )} ( , { )}, ( , {
1
1 1
u u E d
u u E d
u E u d u E u E d
m
m m m
)} ( , { ), ( )} ( , { )}, ( , { )
( )}, ( , {
)} ( , { ), ( . ) ( ), ( . ) ( )}, ( , {
)} ( , { )}, ( , { . )} ( , { ), ( . ) ( )}, ( , {
1 2
1 1
1 1
u E u d u E u E d u
u E d
u E u d u u d u u E d
u E u E d u E u d u u E d
m m
m
m m
{ , ( )}, { , ( )} ( ), ( )
)} ( , { ), ( ) ( )}, ( , {
1 1
u u d u
E u E d
u E u d u u E d
m m
{ , ( )}, ( )
) ( )}, ( , {
)} ( , { ), ( . ) ( )}, ( , {
1
1 1
u u E d
u u E d
u E u d u u E d
m
m m
1{, ()}, ()
d Em u u
{ , ( )}, ( ( ). ( ), { , ( )}
)} ( , { ), ( ) ( )}, ( , {
1 1
u E u d u d u E d
u E u d u u E d
m m
{ , ( )}, ( )
)} ( , { ), (
1
u u E
d
u E u d
m
{ , ( )}, ( )
( ) ( ), { , ( )})
( d Em 1 u u d u E u
( ), { , ( )} ( )
{ , ( )}, ( )
)1
( d u E u d Em1 u u
{ , ( )}, ( )
) 1
(
) (
)} ( , { ),
( 1
E u d E u u
u
d m
du(),E{,u()} s.d
Em1{,u()}, u()
where 1 ) 1(
) (
s
Further,
( ), { , ( )}
.... ... ...
)} ( , { )}, ( , { .
)} ( , { )}, ( , { )
( )}, ( , {
1
2 1
1 1
u E u d s
u E u E d s
u E u E d u u E d
m
m m
m m
m
Therefore
( ), { , ( )}
)} ( , { ), ( )}
( , { ), (
u E u d
u E u d s u
E u
d m
Which is contradiction, therefore u()is fixed point of E.That isu() 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,
m
E Satisfies
.
)} ( , { ), ( . )} ( , { ), ( )
( ), (
)} ( , { ), ( . )} ( , { ), ( . ) ( ), (
)} ( , { ), ( . )} ( , { ), ( . )} ( , { ), (
)} ( , { )}, ( , {
2
h E h d h E g d h
g d
h E h d g E h d h g d
h E g d h E h d g E g d
h E g E d
m m
m m
m m
m m m
( ), { , ( )} ( ), { , ( )}
)} ( , { ), ( )} ( , { ), (
g E h d h
E g d
h E h d g
E g d
m m
m m
( ), ( )
) ( ), (
)} ( , { ), ( )} ( , { ), (
h g d
h g d
h E h d g E g
d m m
(3.3.1)
For allg()) and h()X withg() h(), where ,,,, :R [0,1) are such that 1
2
2
,if for some positive integer m,Em is continuous, then E has a unique fixed point in X.
Proof: Em has unique fixed point u()in X follows from theorem (3.2).
)} ( , { )))
( , ( ( )} ( ,
{ u E E u E E u
E m m , Which implies that E{,u()} is fixed point of
m
E but has
unique fixed pointu(), soE{,u()} u().
Since any fixed point of E is also a fixed point ofEm. It follows that u() is unique fixed point of E. This completes the proof of (3.3).We now prove another theorem.
Theorem 3.4: Let E & F be a pair of self-mappings of a complete metric space X, satisfying the following conditions:
E{,g()}, F{,h()}
)} ( , { ), ( . )} ( , { ), ( ] ) ( ), ( [ )} ( , { ), ( . )} ( , { ), ( . ) ( ), ( )} ( , { ), ( . )} ( , { ), ( . )} ( , { ), (
2
h F h d h F g d h g d h F h d g E h d h g d h F g d h F h d g E g d
( ), { , ( )} ( ), { , ( )}
)} ( , { ), ( )} ( , { ), ( g E h d h F g d h F h d g E g d ( ), ( ) . ) ( ), ( )} ( , { ), ( )} ( , { ), ( h g d h g d h F h d g E g d (3.4.1)For allg()), h() X withg()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 gn be a continuous sequence defined as
{
even is n ) ( , odd is n ) ( , 1 1)
(
n n g E g F ng
and } { }{ n1
n g
g for all n.
Now dg2n(),g2n1()dE{,g2n1()}, F{,g2n()}
( ), ( ) . ( ), ( )
. 2 2 1 2 1 2
d g n g n d g n g n
( ), ( ) ( ) ( ), ( )
)
( 2 1 2 2 2 1
d g n g n d g n g n
( ), ( ) ( ) ( ), ( )
) 1
( d g2n g2n1 d g2n1 g2n
( ), ( )
) 1
(
) (
) ( ),
( 2 1 2 1 2
2
n n n
n g d g g
g
d
dg2n(),g2n1()s.dg2n1(),g2n() 1 ) 1
(
) (
where
s
Similarly
( ), ( )
. ... ... ...
... ... ...
) ( ), ( .
) ( ), (
1 0 . 2
1 2 2 2 2 1
2 2
g g d s
g g
d s g
g d
n
n n
n n
Henced
g2n1(),g2n2()
s2n1.d
g0(),g1()
Hence the sequence {gn} is a Cauchy sequence in X and X being complete, therefore there exist u()in X
such thatlim gn() u()
n
the subsequence
) (
u gnk
Now, if EF is continuous on X then
) ( ) ( lim )} ( lim { )) ( , (
1
u EF g g u
EF
k
k n
k n
k
ThusEF{,u()} u() i.e.u() is fixed point of EF.
Now we show thatF{,u()} u(). IfF{,u()}u(), then
u(),F{,u()}
d
EF{,u()}, F{,u()}
d
)} ( , { ), ( )} ( , { )}, ( , { )
( )}, ( , {
) ( , { ), ( )} ( , { ), ( ) ( )}, ( , {
)} ( , { )}, ( , { )} ( , { ), ( )} ( , { )}, ( , {
2
u F u d u F u F d u
u F d
u F u d u EF u d u u F d
u F u F d u F u d u EF u F d
{ , ( )}, { , ( )} ( ), { , ( )
)} ( , { ), ( )} ( , { )}, ( , {
u EF u d u
F u F d
u F u d u
EF u F d
{ , ( )}, ( )
) ( )}, ( , {
)} ( , { ), ( )} ( , { )}, ( , {
u u F d
u u F d
u F u d u EF u F d
)} ( , { ), ( )} ( , { )}, ( , { )
( )}, ( , {
) ( , { ), ( )} ( ), ( ) ( )}, ( , {
)} ( , { )}, ( , { )} ( , { ), ( ) ( )}, ( , {
2
u F u d u F u F d u
u F d
u F u d u u d u u F d
u F u F d u F u d u u F d
{ , ( )}, { , ( )} ( ), ( )
)} ( , { ), ( ) ( )}, ( , {
u u d u
F u F d
u F u d u u F d
{ , ( )}, ( )
) ( )}, ( , {
)} ( , { ), ( ) ( )}, ( , {
u u F d
u u F d
u F u d u u F d
2
) ( )}, ( , {
)} ( , { )}, ( , { )} ( , { ), ( ) ( )}, ( , {
u u F d
u F u F d u F u d u u F d
{ , ( )}, ( )
0 .
{ , ( )}, ( )
{ , ( )}, ( )
2 d F u u d F u u d F u u
2 d F{,u()},u()
{ , ( )}, ( )
) ( )}, ( , { 2
) ( )}, ( , {
u u F d
u u F d u
u F d
Since 2 2 1 implies that 2 1
HenceF{,u()} u().
Further dE{,u()}, u()dE{,u()}, u()implies thatE{,u()} u() . So u is common fixed point of E & F
Uniqueness:
Let v()is another common fixed point of E & F we have
u(),v() dE{,u()}, F{,v()}
d
)} ( , { ), ( )} ( , { ), ( )
( ), (
)} ( , { ), ( )} ( , { ), ( ) ( ), (
)} ( , { ), ( )} ( , { ), ( )} ( , { ), (
2
v F v d v F u d v
u d
v F v d u E v d v u d
v F u d v F v d u E u d
( ), ( )
) ( ), (
)} ( , { ), ( )} ( , { ), (
)} ( , { ), ( )} ( , { ), (
)} ( , { ), ( )} ( , { ), (
v u d
v u d
v F v d u E u d
u E v d v
F u d
v F v d u
E u d
) ( ), ( ) ( ), ( )
( ), (
) ( ), ( )} ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), (
2
v v d v u d v
u d
v v d u v d v u d v u d v v d u u d
( ), ( ) ( ), ( )
) ( ), ( ) ( ), (
u v d v u d
v v d u u d
( ), ( )
) ( ), (
) ( ), ( ) ( ), (
v u d
v u d
v v d u u d
2 d u(),v()
( ), ( )
) ( ), ( 2
) ( ), (
v u d
v u d v
u d
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