![IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 05, Issue 09 (September. 2015), ||V2|| PP 01-09
International organization of Scientific Research 1 | P a g e
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
])}(,{),()}(,{),([
)}(,{),()}(,{),(
gEhdhEgd
hEhdgEgd
)}(,{),()}(,{),(])(),([
2
)}(,{),()}(,{),()(),(
)}(,{),()}(,{),()}(,{),(
)})(,{)},(,{(
hEhdhEgdhgd
hEhdgEhdhgd
hEgdhEhdgEgd
hEgEd](https://image.slidesharecdn.com/a05920109-151006071724-lva1-app6892/85/A05920109-1-320.jpg)
![Random Fixed Point TheoremInMetric Spaces
International organization of Scientific Research 2 | P a g e
])(),([
)(),(
)}(,{),()}(,{),(
hgd
hgd
hEhdgEgd
(3.1.1)
Forall )(g ) and Xh )( with )()( hg , where )1,0[:,,,,
R are such that
122 , then E has a unique fixed point in X.
Proof:Let {gn}be a sequence, define d as follows
)}(
1
,{}{ g n
Eg n
, n = 1,2,3,4…
If }{
1
}{ g ng n
for some n then the result follows immediately.
So let )()( 1 nn gg for all n then
)}(,{)},(,{)(),( 11 nnnn gEgEdggd
)}(,{),()}(,{),()(),(
)}(,{),()}(,{),()(),(
)}(,{),()}(,{),()}(,{),(
1
2
1
11
111
nnnnnn
nnnnnn
nnnnnn
gEgdgEgdggd
gEgdgEgdggd
gEgdgEgdgEgd
)}(,{),()}(,{),(
)}(,{),()}(,{),(
11
11
nnnn
nnnn
gEgdgEgd
gEgdgEgd
)(),(.
)(),(
)}(,{),()}(,{),(
1
1
11
nn
nn
nnnn
ggd
ggd
gEgdgEgd
)(),()(),()(),(
)(),()(),()(),(
)(),()(),()(),(
111
2
1
11
1111
nnnnnn
nnnnnn
nnnnnn
ggdggdggd
ggdggdggd
ggdggdggd
)(),()(),(
)(),()(),(
11
11
nnnn
nnnn
ggdggd
ggdggd
)(),(.
)(),(
)(),()(),(
1
1
11
nn
nn
nnnn
ggd
ggd
ggdggd
)(),()(),(
)(),()(),()(),(
111
1111
nnnn
nnnnnn
ggdggd
ggdggdggd
)(),()(),(
)(),()(),(
11
11
nnnn
nnnn
ggdggd
ggdggd
)(),(.
)(),(.
1
1
nn
nn
ggd
ggd
](https://image.slidesharecdn.com/a05920109-151006071724-lva1-app6892/85/A05920109-2-320.jpg)
![Random Fixed Point TheoremInMetric Spaces
International organization of Scientific Research 3 | P a g e
)(),(.
)(),(.)(),()(),(
)(),()(),()(),(
1
111
111
nn
nnnnnn
nnnnnn
ggd
ggdggdggd
ggdggdggd
)(),()()(),().( 11 nnnn ggdggd
)(),().()(),()1( 11 nnnn ggdggd
)(),(
)1(
).(
)(),( 11
nnnn ggdggd
)(),(
)1(
).(
............
10
ggd
Thus by triangle inequality, we have for nm
)(),()....(
)(),(.......)(),()(),()(),(
10
121
1211
ggdssss
ggdggdggdggd
mnnn
mmnnnnmn
Where 1
)1(
).(
s since 122
Therefore
nmggd
s
s
ggd
n
mn ,as,0)(),(
1
)(),( 10 . Hence the sequence {gn}is a Cauchy sequence, X
being complete, there exist some Xp such that
)()(lim)}(,{.lim)(lim))(,( 1 uggEgEuE 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 )()( vu then from
(3.1.1) we have
)}(,{)},(,{)(),( vEuEdvud
)}(,{),()}(,{),(])(),([
)}(,{),()}(,{),()(),(
)}(,{),()}(,{),()}(,{),(
2
vEvdvEudvud
vEvduEvdvud
vEudvEvduEud
)}(,{),()}(,{),(
)}(,{),()}(,{),(
uEvdvEud
vEvduEud
)(),(.
)(),(
)}(,{),()}(,{),(
vud
vud
vEvduEud
)(),()2(
)(),(.)(),()(),(
vud
vuduvdvud
)(),()2()(),( vudvud
Since 12122
)(),()(),( vudvud
This is contradiction, so )()( vu .
This completes the proof of the theorem (3.1.1). We now prove another theorem.](https://image.slidesharecdn.com/a05920109-151006071724-lva1-app6892/85/A05920109-3-320.jpg)
![Random Fixed Point TheoremInMetric Spaces
International organization of Scientific Research 6 | P a g e
)}(,{),(.)}(,{),(])(),([
)}(,{),(.)}(,{),(.)(),(
)}(,{),(.)}(,{),(.)}(,{),(
2
hFhdhFgdhgd
hFhdgEhdhgd
hFgdhFhdgEgd
)}(,{),()}(,{),(
)}(,{),()}(,{),(
gEhdhFgd
hFhdgEgd
)(),(.
)(),(
)}(,{),()}(,{),(
hgd
hgd
hFhdgEgd
(3.4.1)
For all )(g ), Xh )( with )()( hg ,where )1,0[:,,,,
R are such that 122
,if E & F are continuous on X , then E& F have a unique fixed point in X.
Proof:Let ng be a continuous sequence defined as
{
evenisn)(,
oddisn)(,
1
1
)(
n
n
gE
gF
ng and }{}{ 1 nn gg for all n.
Now )}(,{)},(,{)(),( 212122 nnnn gFgEdggd
.)}(,{),(.)}(,{),()(),(
)}(,{),()}(,{),(.)(),(
)}(,{),(.)}(,{),(.)}(,{),(
22212
2
212
22122212
212221212
nnnnnn
nnnnnn
nnnnnn
gFgdgFgdggd
gFgdgEgdggd
gFgdgFgdgEgd
)}(,{),()}(,{),(
)}(,{),(.)}(,{),(
122212
221212
nnnn
nnnn
gEgdgEgd
gFgdgEgd
)(),(.
)(),(
)}(,{),(.)}(,{),(
212
212
221212
nn
nn
nnnn
ggd
ggd
gFgdgEgd
)(),(.)(),()(),(
)(),()(),(.)(),(
)(),(.)(),(.)(),(
1221212
2
212
12222212
1212122212
nnnnnn
nnnnnn
nnnnnn
ggdggdggd
ggdggdggd
ggdggdggd
)(),()(),(
)(),()(),(
221212
122212
nnnn
nnnn
ggdggd
ggdggd
)(),(.
)(),(
)(),(.)(),(
212
212
122212
nn
nn
nnnn
ggd
ggd
ggdggd
)(),()(),(
)(),()(),(
)(),(
122212
122212
212
nnnn
nnnn
nn
ggdggd
ggdggd
gg](https://image.slidesharecdn.com/a05920109-151006071724-lva1-app6892/85/A05920109-6-320.jpg)
![Random Fixed Point TheoremInMetric Spaces
International organization of Scientific Research 9 | P a g e
REFERENCES
[1]. Beg, I. and Shahzad, N. “Random approximations and random fixed point theorems,” J. Appl. Math.
Stochastic Anal. 7(1994). No.2, 145-150.
[2]. Bharucha-Reid, A.T. “Fixed point theorems in probabilistic analysis,” Bull. Amer. Math. Soc. 82(1976),
641-657.
[3]. 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.
[4]. 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.
[5]. O’Regan,D. “A continuous type result for random operators,” Proc. Amer. Math, Soc. 126(1998), 1963-
1971.
[6]. SehgalV.M. andWaters,C. “Some random fixed point theorems for condensing operators,” Proc. Amer.
Math. Soc. 90(1984). No.3, 425-429.
[7]. SushantkumarMohanta “Random fixed point in Banach spaces” Inter. J of Math Analysis Vol. 5 (2011)
No. 10, 451-461.
[8]. B. R wadkar,Ramakant Bhardwaj, Rajesh Shrivastava, “Some NewResult In Topological Space For Non-
Symmetric RationalExpression Concerning Banach Space” ,International Journal of Theoretical and
Applied Science 3(2): 65-78(2011).](https://image.slidesharecdn.com/a05920109-151006071724-lva1-app6892/85/A05920109-9-320.jpg)
A05920109
- 1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 05, Issue 09 (September. 2015), ||V2|| PP 01-09 International organization of Scientific Research 1 | P a g e 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) [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) [email protected] 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 ])}(,{),()}(,{),([ )}(,{),()}(,{),( gEhdhEgd hEhdgEgd )}(,{),()}(,{),(])(),([ 2 )}(,{),()}(,{),()(),( )}(,{),()}(,{),()}(,{),( )})(,{)},(,{( hEhdhEgdhgd hEhdgEhdhgd hEgdhEhdgEgd hEgEd
- 2. Random Fixed Point TheoremInMetric Spaces International organization of Scientific Research 2 | P a g e ])(),([ )(),( )}(,{),()}(,{),( hgd hgd hEhdgEgd (3.1.1) Forall )(g ) and Xh )( with )()( hg , where )1,0[:,,,, R are such that 122 , then E has a unique fixed point in X. Proof:Let {gn}be a sequence, define d as follows )}( 1 ,{}{ g n Eg n , n = 1,2,3,4… If }{ 1 }{ g ng n for some n then the result follows immediately. So let )()( 1 nn gg for all n then )}(,{)},(,{)(),( 11 nnnn gEgEdggd )}(,{),()}(,{),()(),( )}(,{),()}(,{),()(),( )}(,{),()}(,{),()}(,{),( 1 2 1 11 111 nnnnnn nnnnnn nnnnnn gEgdgEgdggd gEgdgEgdggd gEgdgEgdgEgd )}(,{),()}(,{),( )}(,{),()}(,{),( 11 11 nnnn nnnn gEgdgEgd gEgdgEgd )(),(. )(),( )}(,{),()}(,{),( 1 1 11 nn nn nnnn ggd ggd gEgdgEgd )(),()(),()(),( )(),()(),()(),( )(),()(),()(),( 111 2 1 11 1111 nnnnnn nnnnnn nnnnnn ggdggdggd ggdggdggd ggdggdggd )(),()(),( )(),()(),( 11 11 nnnn nnnn ggdggd ggdggd )(),(. )(),( )(),()(),( 1 1 11 nn nn nnnn ggd ggd ggdggd )(),()(),( )(),()(),()(),( 111 1111 nnnn nnnnnn ggdggd ggdggdggd )(),()(),( )(),()(),( 11 11 nnnn nnnn ggdggd ggdggd )(),(. )(),(. 1 1 nn nn ggd ggd
- 3. Random Fixed Point TheoremInMetric Spaces International organization of Scientific Research 3 | P a g e )(),(. )(),(.)(),()(),( )(),()(),()(),( 1 111 111 nn nnnnnn nnnnnn ggd ggdggdggd ggdggdggd )(),()()(),().( 11 nnnn ggdggd )(),().()(),()1( 11 nnnn ggdggd )(),( )1( ).( )(),( 11 nnnn ggdggd )(),( )1( ).( ............ 10 ggd Thus by triangle inequality, we have for nm )(),()....( )(),(.......)(),()(),()(),( 10 121 1211 ggdssss ggdggdggdggd mnnn mmnnnnmn Where 1 )1( ).( s since 122 Therefore nmggd s s ggd n mn ,as,0)(),( 1 )(),( 10 . Hence the sequence {gn}is a Cauchy sequence, X being complete, there exist some Xp such that )()(lim)}(,{.lim)(lim))(,( 1 uggEgEuE 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 )()( vu then from (3.1.1) we have )}(,{)},(,{)(),( vEuEdvud )}(,{),()}(,{),(])(),([ )}(,{),()}(,{),()(),( )}(,{),()}(,{),()}(,{),( 2 vEvdvEudvud vEvduEvdvud vEudvEvduEud )}(,{),()}(,{),( )}(,{),()}(,{),( uEvdvEud vEvduEud )(),(. )(),( )}(,{),()}(,{),( vud vud vEvduEud )(),()2( )(),(.)(),()(),( vud vuduvdvud )(),()2()(),( vudvud Since 12122 )(),()(),( vudvud This is contradiction, so )()( vu . This completes the proof of the theorem (3.1.1). We now prove another theorem.
- 4. Random Fixed Point TheoremInMetric Spaces International organization of Scientific Research 4 | P a g e 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 ng be sequence which converges to some. Xu . Therefore its subsequence kn g also converges to u also )()(lim)(,lim)(lim))(,( 1 ugg p Eg p Eu p E kkk n k n k n k Therefore )(u is a fixed point of E p , we now show that )())(,( uuE . Let m be the smallest positive integer such that 11),(and)())(,( mquEuuE qm , If m>1 then by (3.1.1) we get, )}(,{)}),(,{( )}(,{)},(,{)}(,{),( 1 uEuEEd uEuEduEud m m )}(,{),()}(,{)},(,{)()},(,{ )}(,{),()}(,{),(.)()},(,{ )}(,{)},(,{.)}(,{),(.)}(,{)},(,{ 121 1 11 uEuduEuEduuEd uEuduEuduuEd uEuEduEuduEuEd mm mm mmm )}(,{),()}(,{)},(,{ )}(,{),()}(,{)},(,{ 1 1 uEuduEuEd uEuduEuEd mm mm )()},(,{ )()},(,{ )}(,{),()}(,{)},(,{ 1 1 1 uuEd uuEd uEuduEuEd m m mm )}(,{),()}(,{)},(,{)()},(,{ )}(,{),(.)(),(.)()},(,{ )}(,{)},(,{.)}(,{),(.)()},(,{ 121 1 11 uEuduEuEduuEd uEuduuduuEd uEuEduEuduuEd mm m mm )(),()}(,{)},(,{ )}(,{),()()},(,{ 1 1 uuduEuEd uEuduuEd m m )()},(,{ )()},(,{ )}(,{),(.)()},(,{ 1 1 1 uuEd uuEd uEuduuEd m m m )()},(,{ 1 uuEd m )}(,{),(.)(()},(,{ )}(,{),()()},(,{ 1 1 uEududuEd uEuduuEd m m )()},(,{ )}(,{),( 1 uuEd uEud m
- 5. Random Fixed Point TheoremInMetric Spaces International organization of Scientific Research 5 | P a g e )}(,{),()()()},(,{)( 1 uEuduuEd m )()},(,{)()}(,{),()1( 1 uuEduEud m )()},(,{ )1( )( )}(,{),( 1 uuEduEud m )()},(,{.)}(,{),( 1 uuEdsuEud m where 1 )1( )( s Further, )}(,{),( ........................ )}(,{)},(,{. )}(,{)},(,{)()},(,{ 1 21 11 uEuds uEuEds uEuEduuEd m mm mmm Therefore )}(,{),( )}(,{),()}(,{),( uEud uEudsuEud m Which is contradiction, therefore )(u is fixed point of E.That is )}(,{)( uEu .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 hEhdhEgdhgd hEhdgEhdhgd hEgdhEhdgEgd hEgEd mm mm mmm mm )}(,{),()}(,{),( )}(,{),()}(,{),( gEhdhEgd hEhdgEgd mm mm )(),( )(),( )}(,{),()}(,{),( hgd hgd hEhdgEgd mm (3.3.1) For all )(g ) and Xh )( with )()( hg , where )1,0[:,,,, R are such that 122 ,if for some positive integer m,Em is continuous, then E has a unique fixed point in X. Proof: m E has unique fixed point )(u in X follows from theorem (3.2). )}(,{)))(,(()}(,{ uEEuEEuE mm , Which implies that )}(,{ uE is fixed point of m E but has unique fixed point )(u , so )()}(,{ uuE . Since any fixed point of E is also a fixed point of m E . 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: )}(,{)},(,{ hFgEd
- 6. Random Fixed Point TheoremInMetric Spaces International organization of Scientific Research 6 | P a g e )}(,{),(.)}(,{),(])(),([ )}(,{),(.)}(,{),(.)(),( )}(,{),(.)}(,{),(.)}(,{),( 2 hFhdhFgdhgd hFhdgEhdhgd hFgdhFhdgEgd )}(,{),()}(,{),( )}(,{),()}(,{),( gEhdhFgd hFhdgEgd )(),(. )(),( )}(,{),()}(,{),( hgd hgd hFhdgEgd (3.4.1) For all )(g ), Xh )( with )()( hg ,where )1,0[:,,,, R are such that 122 ,if E & F are continuous on X , then E& F have a unique fixed point in X. Proof:Let ng be a continuous sequence defined as { evenisn)(, oddisn)(, 1 1 )( n n gE gF ng and }{}{ 1 nn gg for all n. Now )}(,{)},(,{)(),( 212122 nnnn gFgEdggd .)}(,{),(.)}(,{),()(),( )}(,{),()}(,{),(.)(),( )}(,{),(.)}(,{),(.)}(,{),( 22212 2 212 22122212 212221212 nnnnnn nnnnnn nnnnnn gFgdgFgdggd gFgdgEgdggd gFgdgFgdgEgd )}(,{),()}(,{),( )}(,{),(.)}(,{),( 122212 221212 nnnn nnnn gEgdgEgd gFgdgEgd )(),(. )(),( )}(,{),(.)}(,{),( 212 212 221212 nn nn nnnn ggd ggd gFgdgEgd )(),(.)(),()(),( )(),()(),(.)(),( )(),(.)(),(.)(),( 1221212 2 212 12222212 1212122212 nnnnnn nnnnnn nnnnnn ggdggdggd ggdggdggd ggdggdggd )(),()(),( )(),()(),( 221212 122212 nnnn nnnn ggdggd ggdggd )(),(. )(),( )(),(.)(),( 212 212 122212 nn nn nnnn ggd ggd ggdggd )(),()(),( )(),()(),( )(),( 122212 122212 212 nnnn nnnn nn ggdggd ggdggd gg
- 7. Random Fixed Point TheoremInMetric Spaces International organization of Scientific Research 7 | P a g e )(),(.)(),(. 212122 nnnn ggdggd )(),()()(),()( 122212 nnnn ggdggd )(),()()(),()1( 212122 nnnn ggdggd )(),( )1( )( )(),( 212122 nnnn ggdggd )(),(.)(),( 212122 nnnn ggdsggd 1 )1( )( where s Similarly )(),( ............................... ............................ )(),(.)(),( 10 .2 1222 2 122 ggds ggdsggd n nnnn Hence )(),()(),( 10 .12 2212 ggdsggd n nn Hence the sequence { n g } is a Cauchy sequence in X and X being complete, therefore there exist )(u in X such that )()(lim ug n n the subsequence )(ug nk Now, if EF is continuous on X then )()(lim)}(lim{))(,( 1 uggEFuEF kk n k n k Thus )()}(,{ uuEF i.e. )(u is fixed point of EF. Now we show that )()}(,{ uuF . If )()}(,{ uuF , then )}(,{)},(,{)}(,{),( uFuEFduFud )}(,{),()}(,{)},(,{)()},(,{ )(,{),()}(,{),()()},(,{ )}(,{)},(,{)}(,{),()}(,{)},(,{ 2 uFuduFuFduuFd uFuduEFuduuFd uFuFduFuduEFuFd )(,{),()}(,{)},(,{ )}(,{),()}(,{)},(,{ uEFuduFuFd uFuduEFuFd )()},(,{ )()},(,{ )}(,{),()}(,{)},(,{ uuFd uuFd uFuduEFuFd )}(,{),()}(,{)},(,{)()},(,{ )(,{),()}(),()()},(,{ )}(,{)},(,{)}(,{),()()},(,{ 2 uFuduFuFduuFd uFuduuduuFd uFuFduFuduuFd )(),()}(,{)},(,{ )}(,{),()()},(,{ uuduFuFd uFuduuFd
- 8. Random Fixed Point TheoremInMetric Spaces International organization of Scientific Research 8 | P a g e )()},(,{ )()},(,{ )}(,{),()()},(,{ uuFd uuFd uFuduuFd 2 )()},(,{ )}(,{)},(,{)}(,{),()()},(,{ uuFd uFuFduFuduuFd )()},(,{)()},(,{.0)()},(,{2 uuFduuFduuFd )()},(,{2 uuFd )()},(,{ )()},(,{2)()},(,{ uuFd uuFduuFd Since 122 implies that 12 Hence )()}(,{ uuF . Further )()},(,{)()},(,{ uuEduuEd implies that )()}(,{ uuE . So u is common fixed point of E & F Uniqueness: Let )(v is another common fixed point of E & F we have )}(,{)},(,{)(),( vFuEdvud )}(,{),()}(,{),()(),( )}(,{),()}(,{),()(),( )}(,{),()}(,{),()}(,{),( 2 vFvdvFudvud vFvduEvdvud vFudvFvduEud )(),( )(),( )}(,{),()}(,{),( )}(,{),()}(,{),( )}(,{),()}(,{),( vud vud vFvduEud uEvdvFud vFvduEud )(),()(),()(),( )(),()}(),()(),()(),()(),()(),( 2 vvdvudvud vvduvdvudvudvvduud )(),()(),( )(),()(),( uvdvud vvduud )(),( )(),( )(),()(),( vud vud vvduud )(),(2 vud )(),( )(),(2)(),( vud vudvud Because 12 . This implies )()( vu .This completes the proof of (3.4)
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