Fixed point theorems for random variables in complete metric spaces
0 likes204 views
This document presents two fixed point theorems for random variables in complete metric spaces. Theorem 1 proves that if a self-mapping E on a complete metric space satisfies certain rational inequalities involving distances between random variables, then E has a fixed point. Theorem 2 proves a similar result for a self-mapping E satisfying alternative rational inequalities, assuming E is onto. Both theorems use properties of complete metric spaces and rational inequalities to show the existence of fixed points for random variables under the given conditions.
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
70
Fixed Point Theorems for Random Variables in Complete Metric
Spaces
Deepak Singh Kaushal
Abstract
In this paper we prove some fixed point theorem for random variables with one and two self maps satisfying
rational inequality in complete metric spaces
Keywords: Random fixed point,Complete metric space, Common fixed points.
Mathematics Subject Classification: 47H10
1.INTRODUCTION
Many theorems of fixed point have been proved using rational inequality in ordinary metric space. Some of the
noteworthy contributions are by Bhardwaj, Rajput and Yadava [3], Jaggi and Das[9], Das and Gupta[5] Fisher
[7,8] who obtained some fixed point theorems using rational inequality in complete metric spaces.
Random fixed point theory has received much attention in recent years.Some of the recent results in random
fixed points have been proved by Beg and Shahzad [1, 2], Choudhary and Ray [4], Dhagat, Sharma and
Bhardwaj [6]. In particular random iteration schemes leading to random fixed point of random
operator.Throughout this paper ( )∑Ω, denotes a measurable space, (X,d) be a complete metric space and C is
non empty subset of X.
2.PRELIMINARIES
Definition 2.1:A function :f C CΩ × → is said to be random operator, if ( ) CXf →Ω:,. is measurable
for every .CX ∈
Definition 2.2: A function :f CΩ → is said to be measurable if ( ) ∑∈∩ CBf ' for every Borel subset B
of X.
Definition 2.3:A measurable function Cg →Ω: is said to be random fixed point of the random operator
CCf →×Ω: , if ( )( ) ( ) .,, Ω∈∀= ttgtgtf
Definition 2.4:A random operator CCf →×Ω: is said to be continuous if for fixed
( ) CCtft ×Ω∈ :,., is continuous.
3.MAIN RESULTS
Theorem3.1: Let E be a self mapping on a complete metric space
( ),X d satisfying the condition
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ), ( , ( )) . ( ), ( , ( ))
( , ( )), ( , ( ))
( ), ( )
( ), ( , ( )) . ( ), ( , ( ))
( ), ( )
( ), ( , ( )) ( ), ( , ( ))
( ), ( )
d g E g d h E h
d E g E h
d g h
d g E h d h E g
d g h
d g E g d h E h
d g h
ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ α
ξ ξ
ξ ξ ξ ξ ξ ξ
β
ξ ξ
γ ξ ξ ξ δ ξ ξ ξ
η ξ ξ
≥
+
+ +
+
For all ( ), ( )g h Xξ ξ ∈ with ( ) ( )g hξ ξ≠ ,where , , , , 0α β γ δ η > and 1α β γ δ η+ + + + > and
1α γ+ < and E is onto.Then E has a fixed point in .X
Proof:- Let 0 ( ) .g Xξ ∈ Since E is onto there is an element 1( )g ξ satisfying
1
1 0( ) ( , ( ))g E gξ ξ ξ−
∈ By the
same way we can choose
1
1( ) ( , ( ))n ng E gξ ξ ξ−
−∈ where 2,3,4,...n =
If 1( ) ( )m mg gξ ξ− = for some m then ( )mg ξ is a fixed point of E.](https://image.slidesharecdn.com/fixedpointtheoremsforrandomvariablesincompletemetricspaces-130617072911-phpapp01/85/Fixed-point-theorems-for-random-variables-in-complete-metric-spaces-1-320.jpg)
Fixed point theorems for random variables in complete metric spaces
- 1. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 70 Fixed Point Theorems for Random Variables in Complete Metric Spaces Deepak Singh Kaushal Abstract In this paper we prove some fixed point theorem for random variables with one and two self maps satisfying rational inequality in complete metric spaces Keywords: Random fixed point,Complete metric space, Common fixed points. Mathematics Subject Classification: 47H10 1.INTRODUCTION Many theorems of fixed point have been proved using rational inequality in ordinary metric space. Some of the noteworthy contributions are by Bhardwaj, Rajput and Yadava [3], Jaggi and Das[9], Das and Gupta[5] Fisher [7,8] who obtained some fixed point theorems using rational inequality in complete metric spaces. Random fixed point theory has received much attention in recent years.Some of the recent results in random fixed points have been proved by Beg and Shahzad [1, 2], Choudhary and Ray [4], Dhagat, Sharma and Bhardwaj [6]. In particular random iteration schemes leading to random fixed point of random operator.Throughout this paper ( )∑Ω, denotes a measurable space, (X,d) be a complete metric space and C is non empty subset of X. 2.PRELIMINARIES Definition 2.1:A function :f C CΩ × → is said to be random operator, if ( ) CXf →Ω:,. is measurable for every .CX ∈ Definition 2.2: A function :f CΩ → is said to be measurable if ( ) ∑∈∩ CBf ' for every Borel subset B of X. Definition 2.3:A measurable function Cg →Ω: is said to be random fixed point of the random operator CCf →×Ω: , if ( )( ) ( ) .,, Ω∈∀= ttgtgtf Definition 2.4:A random operator CCf →×Ω: is said to be continuous if for fixed ( ) CCtft ×Ω∈ :,., is continuous. 3.MAIN RESULTS Theorem3.1: Let E be a self mapping on a complete metric space ( ),X d satisfying the condition ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( , ( )) . ( ), ( , ( )) ( , ( )), ( , ( )) ( ), ( ) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( , ( )) ( ), ( ) d g E g d h E h d E g E h d g h d g E h d h E g d g h d g E g d h E h d g h ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ ξ δ ξ ξ ξ η ξ ξ ≥ + + + + For all ( ), ( )g h Xξ ξ ∈ with ( ) ( )g hξ ξ≠ ,where , , , , 0α β γ δ η > and 1α β γ δ η+ + + + > and 1α γ+ < and E is onto.Then E has a fixed point in .X Proof:- Let 0 ( ) .g Xξ ∈ Since E is onto there is an element 1( )g ξ satisfying 1 1 0( ) ( , ( ))g E gξ ξ ξ− ∈ By the same way we can choose 1 1( ) ( , ( ))n ng E gξ ξ ξ− −∈ where 2,3,4,...n = If 1( ) ( )m mg gξ ξ− = for some m then ( )mg ξ is a fixed point of E.
- 2. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 71 Without loss of generality we can suppose that 1( ) ( )n ng gξ ξ− = ,for every .n So ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 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 n n n d g g d E g E g d 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 ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ ξ δ ξ ξ ξ η ξ − + + + + + + + + + = ≥ + + + + ( )1( )ng ξ+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 ( ), ( ) ( ), ( ) 1 ( ), ( ) ( ), ( ) n n n n n n n n d g g d g g d g g d g g α γ ξ ξ δ η ξ ξ α γ ξ ξ ξ ξ δ η − + + − − + ≥ + − + ≤ + It follows that ( ) ( )1 1( ), ( ) ( ), ( )n n n nd g g k d g gξ ξ ξ ξ+ −≤ where ( ) ( ) 1 1.k α γ δ η − + = < + By routine calculation the following inequality holds for .p n> ( ) ( )1( ), ( ) ( ), ( )n n p n i n id g g d g gξ ξ ξ ξ+ + − +≤∑ ( ) ( ) 1 0 1 0 1 ( ), ( ) ( ), ( ) 1 n i n k d g g k d g g k ξ ξ ξ ξ + − ≤ ≤ − Now making n → ∞we obtain ( )( ), ( ) 0n n pd g gξ ξ+ → Hence { }( )ng ξ is a Cauchy sequence. Since X is complete{ }( )ng ξ converges to ( )g ξ ,for some ( ) .g Xξ ∈ Since E is onto then there exists ( )h Xξ ∈ such that 1 ( ) ( , ( ))h E gξ ξ ξ− ∈ and for infinitely many , ( ) ( )nn g gξ ξ≠ for such .n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 ( ), ( ) ( , ( )), ( , ( )) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( , ( )) ( ), ( ) n n n n n n n n n n n d g g d E g E h d g E g d h E h d g h d g E h d h E g d g h d g E g d h E h d g h ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ ξ δ ξ ξ ξ η ξ ξ + + + + + + + + + + = ≥ + + + +
- 3. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 72 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 ( ), ( ) . ( ), ( ) ( ), ( ) ( ), ( ) . ( ), ( ) ( ), ( ) ( ), ( )) ( ), ( )) ( ), ( ) n n n n n n n n n d g g d h g d g h d g g d h g d g h d g g d h g d g h ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ δ ξ ξ η ξ ξ + + + + + + ≥ + + + + On taking limit as n → ∞we have ( ) ( )10 ( ), ( ) lim ( ), ( )n nd h g d g hδ ξ ξ η ξ ξ→∞ +≥ + Since ( ), 0, So ( ), ( ) 0d g hδ η ξ ξ> = and ( )1lim ( ), ( ) 0n nd g hξ ξ→∞ + = So in both cases we get ( ) ( )g hξ ξ= .Thus E has a fixed point in .X Theorem 3.2:Let E be a self mapping on a complete metric sapce ( ),X d satisfying the condition ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( , ( )), ( , ( )) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( , ( )) . ( ), ( , ( )) , ( ), ( ) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( , ( )) . ( ), ( , ( )) min , ( ), ( ) ( ), ( , ( )) , ( ), ( , d E g E h d g E g d h E h d g E h d h E g d g h d g E h d h E g d h E h d h E h d g h d h E h d g E g ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ + + ≥ ( ) ( )( )) , ( ), ( )d g hξ ξ ξ For all ( ), ( )g h Xξ ξ ∈ with ( ) ( )g hξ ξ≠ , where 1α > and E is onto. Then E has a fixed point in .X Proof:- Let 0 ( ) .g Xξ ∈ Since E is onto there is an element 1( )g ξ satisfying 1 1 0( ) ( , ( ))g E gξ ξ ξ− ∈ By the same way we can choose 1 1( ) ( , ( ))n ng E gξ ξ ξ− −∈ where 2,3,4,...n = If 1( ) ( )m mg gξ ξ− = for some mthen ( )mg ξ is a fixed point of E. Without loss of generality we can suppose that 1( ) ( )n ng gξ ξ− = for every .n So ( ) ( )1 1( ), ( ) ( , ( )), ( , ( ))n n n nd g g d E g E gξ ξ ξ ξ ξ ξ− += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 2 1 1 1 1 1 1 1 ( ), ( , ( )) . ( ), ( , ( )) ( ), ( , ( )) . ( ), ( , ( )) , ( ), ( ) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( , ( )) min , ( ), ( ) ( ), ( , ( )) n n n n n n n n n n n n n n n n n n n d g E g d g E 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 g g d g E g ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ + + + + + + + + + + + + + + ≥ ( ) ( )1, ( ), ( , ( )) , ( ), ( )n n n nd g E g d g gξ ξ ξ ξ ξ+
- 4. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 73 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 2 1 1 1 1 1 1 1 ( ), ( ) . ( ), ( ) ( ), ( )) . ( ), ( )) , ( ), ( ) ( ), ( ) . ( ), ( ) ( ), ( ) min , ( ), ( ) ( ), ( ) , ( ), ( )) , ( ), ( ) n n n n n n n n n n n n n n n n n n n n n n n n 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 g g d g g d g g d g g ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ − + + − + + − + + + − + + + = ( ) ( ){ }1 1min ( ), ( ) , ( ), ( )n n n nd g g d g gα ξ ξ ξ ξ− += ( ) ( )1 1 1 ( ), ( ) ( ), ( )n n n nd g g d g gξ ξ ξ ξ α − +≥ ( ) ( )1 1 1 ( ), ( ) ( ), ( )n n n nd g g d g gξ ξ ξ ξ α+ −≤ Where 1 1 α < ,therefore by well known way { }( )ng ξ is a Cauchy sequence in .X Since X is complete { }( )ng ξ converges to ( )g ξ ,for some ( ) .g Xξ ∈ So by continuity of E,we have 1( , ( )) ( ) ( , ( )), asn nE g g E g nξ ξ ξ ξ ξ−= → → ∞ Hence ( , ( )) ( )E g gξ ξ ξ= Thus E has a fixed point in .X Theorem3.3:Let E be a self mapping on a complete metric sapce ( ),X d satisfying the condition ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2( ), ( , ( )) . ( ), ( , ( )) ( ), ( , ( )) . ( ), ( , ( )) ( , ( )), ( , ( )) ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( ) d g E g d h E h d g E h d h E g E g E h d g E g d g h d g E h d g h ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + ≥ + + For each ( ), ( )g h Xξ ξ ∈ with ( ) ( )g hξ ξ≠ , where 1α > and E is onto.Then E has a fixed point in .X Proof:- Let 0 ( ) .g Xξ ∈ Since E is onto there is an element 1( )g ξ satisfying 1 1 0( ) ( , ( ))g E gξ ξ ξ− ∈ By the same way we can choose 1 1( ) ( , ( ))n ng E gξ ξ ξ− −∈ where 2,3,4,...n = If 1( ) ( )m mg gξ ξ− = for some mthen ( )mg ξ is a fixed point of E. Without loss of generality we can suppose that 1( ) ( )n ng gξ ξ− = for every .n So ( ) ( )1 1( ), ( )) ( , ( )), ( , ( ))n n n nd g g d E g E gξ ξ ξ ξ ξ ξ− += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 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 d g E g d g E g d g E g d g E g d g E g d g g d g E g d g g ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + + + + + + + + ≥ + +
- 5. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 74 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 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 d g g d g g d g g d g g d g g d g g d g g d g g ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ − + + − − + + + ≥ + + ( ) ( ) 1 2 1 12 ( ), ( ) ( ), ( )n n n nd g g d g gα ξ ξ ξ ξ− + ≥ ( ) ( )1 12 1 ( ), ( ) ( ), ( ) 2 n n n nd g g d g gξ ξ ξ ξ α+ −≤ Therefore by well known way { }( )ng ξ is a Cauchy sequence in .X Since X is complete { }( )ng ξ converges to ( )g ξ , for some ( ) .g Xξ ∈ Since E is onto there exists ( ) .h Xξ ∈ such that 1 ( ) ( , ( ))h E gξ ξ ξ− ∈ and for infinitely many n, ( ) ( )ng gξ ξ≠ for all n. ( ) ( )1( ), ( )) ( , ( )), ( , ( ))n nd g g d E g E hξ ξ ξ ξ ξ ξ+= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 1 1 1 1 11 1 ( ), ( , ( )) ( ), ( , ( )) ( ), ( , ( )) ( ), ( , ( )) ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( ) n n n n n n n n n d g E g d h E h d g E h d h E g d g E g d g h d g E h d g h ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + + + + + + + + + + ≥ + + On taking limit as n → ∞ ,we obtain ( )( ), ( )) 0nd g gξ ξ → So we have ( )( ), ( ) 0 ( ) ( )d g h g hξ ξ ξ ξ= ⇒ = Hence E has a fixed point in .X Theorem3.4: Let E and F be two self mappings on a complete metric space ( ),X d satisfying the condition ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( , ( )) . ( ), ( , ( )) ( , ( )), ( , ( )) ( ), ( ) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( , ( )) ( ), ( ) d g E g d h F h d E g F h d g h d g F h d h E g d g h d g E g d h F h d g h ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ ξ δ ξ ξ ξ η ξ ξ ≥ + + + + For all ( ), ( )g h Xξ ξ ∈ with ( ) ( )g hξ ξ≠ ,where , , , , 0α β γ δ η > and 1α β γ δ η+ + + + > and 1α γ+ < .Then E and F have a common fixed point. Proof: Let 0 ( )g ξ be any point of X, we define a sequence{ }( )ng ξ as follows
- 6. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 75 0 1 1 2 2 2 1 2 1 2 2 ( ) ( , ( )), ( ) ( , ( )), ............................... ( ) ( , ( )), ( ) ( , ( )) n n n n g E g g E g g E g g E g ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + + + = = = = Now if 2 1 2 2( ) ( )n ng gξ ξ+ +≠ then ( ) ( )2 2 1 2 1 2 2( ), ( ) ( , ( )), ( , ( ))n n n nd g g d E g F gξ ξ ξ ξ ξ ξ+ + += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 2 2 2 2 2 1 2 2 2 1 2 2 2 2 2 1 2 1 2 2 2 11 2 1 2 2 2 2 2 1 2 ( ), ( , ( )) . ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) . ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( , ( )) ( ), n n n n n n n n n n n n n n n n n n d g E g d g F g d g g d g F g d g E g d g g d g E g d g F g d g g ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ ξ δ ξ ξ ξ η ξ + + + + + + + + + + + + + + + + + ≥ + + + + ( )2 ( )ξ+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 2 1 2 2 2 1 2 2 2 2 1 2 1 2 2 ( ), ( ) . ( ), ( ) ( ), ( ) ( ), ( ) . ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) n n n n n n n n n n n n n n n n n n 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 g g ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ δ ξ ξ η ξ ξ + + + + + + + + + + + + + + + ≥ + + + + ( ) ( ) ( ) ( ) ( )2 2 1 2 1 2 2 1 2 2( ), ( ) ( ), ( ) ( ), ( )n n n n n nd g g d g g d g gξ ξ α γ ξ ξ δ η ξ ξ+ + + +≥ + + + ( ) ( ) ( ) ( )2 2 1 2 1 2 21 ( ), ( ) ( ), ( )n n n nd g g d g gα γ ξ ξ δ η ξ ξ+ + + − + ≥ + ( ) ( ) ( ) ( ) 2 1 2 2 2 2 1 1 1 ( ), ( ) ( ), ( ) ,Since 0 1n n n nd g g d g g α γ α γ ξ ξ ξ ξ δ η δ η+ + + − + − + ≤ < < + + It follow that { }( )ng ξ is a Cauchy sequence .By completeness of X there is some point ( )z ξ in X and { }( )ng ξ converge to ( )z ξ .By the condition there is another point ( )u ξ in X such that ( )( , ( ))E u zξ ξ ξ= Since we can suppose ( ) ( )2 2nu gξ ξ+≠ for many infinitely n we can write ( ) ( )2 1 2 2( ), ( ) ( , ( )), ( , ( ))n nd z g d E u F gξ ξ ξ ξ ξ ξ+ += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ), ( , ( )) ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( , ( )) ( ), ( ) ( ), ( , ( )) ( ), ( , ( )) ( ), ( ) n n n n n n n n n d u E u d g F g d u g d u F g d g E u d u g d u E u d g F g d u g ξ ξ ξ ξ ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ β ξ ξ γ ξ ξ ξ δ ξ ξ ξ η ξ ξ + + + + + + + + + ≥ + + + + On taking limit as n → ∞ we obtain
- 7. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 76 ( ) ( ) ( ) ( ) 0 ( ), ( , ( )) ( ), ( ) 0 ( ), ( ) d u E u d u z d u z γ ξ ξ ξ η ξ ξ γ η ξ ξ ≥ + ≥ + Which implies that ( ) ( ) ( , ( ))u z E uξ ξ ξ ξ= = i.e. ( ) ( , ( )) Similarly ( ) ( , ( )) z E u z F u ξ ξ ξ ξ ξ ξ = = Hence E and F have a common fixed point. REFERENCES 1. Beg, I. and Shahzad, N. “Random approximations and random fixed point theorems,” J. Appl. Math. Stochastic Anal. vol. 7 (2), 1994, 145-150. 2. Beg, I. and Shahzad, N. “Common Random fixed points of random multi-valued operators on metric spaces,” Bollettino U.M.I. vol. 7, 1995, 493-503. 3. Bhardwaj, R. K., Rajput, S. S., Yadava, R. N., “Some fixed point theorems in complete metric spaces,” International Journal of mathematical Science and Engineering Applications, vol.2( 2007), 193-198. 4. Choudhary, B. S. and Ray, M. “Convergence of an iteration leading to a solution of a random operator equation,” J. Appl. Stochastic Anal. vol. 12 (2), 1999, 161-168. 5. Dass, B.K.,Gupta, S. “ An extension of Banach contraction principles through rational expression” Indian Journal of Pure and Applied Mathematics, Vol.6(1975),1455-1458. 6. Dhagat, V. B., Sharma, A. and Bhardwaj, R. K. “Fixed point theorems for random operators in Hilbert spaces,” International Journal of Math. Anal. vol. 2 (12), 2008, 557-561. 7. Fisher, B., “Common fixed point and constant mapping satisfying a rational inequality,” Mathematical Seminar Notes, Kobe University, Vol.6 (1978), 29-35. 8. Fisher, B., “Mappings satisfying rational inequality,” Nanta Mathematica., Vol.12 (1979), 195-199. 9. Jaggi, D.S. and Das, B.K. “An extension of Banach’s fixed point theorem through rational expression” Bulletin of the Calcutta Mathematical Society, Vol.72 (1980), 261-264 10. Xu, H. K. “Some random fixed point theorems for condensing and non expansive operators,” Proc. Amer. Math. Soc. vol. 110, 1990, 2395-2400. Deepak Singh Kaushal Sagar Institute of Science,Technology & Research Bhopal (M.P.) email:[email protected]
- 8. This academic article was published by The International Institute for Science, Technology and Education (IISTE). The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe. The aim of the institute is Accelerating Global Knowledge Sharing. More information about the publisher can be found in the IISTE’s homepage: http://www.iiste.org CALL FOR PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world. There’s no deadline for submission. Prospective authors of IISTE journals can find the submission instruction on the following page: http://www.iiste.org/Journals/ The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. Printed version of the journals is also available upon request of readers and authors. IISTE Knowledge Sharing Partners EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open Archives Harvester, Bielefeld Academic Search Engine, Elektronische Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial Library , NewJour, Google Scholar