Some Fixed Point Theorems for Convex Metris Space taking Random Operator

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 12, December 2015)

155

Some Fixed Point Theorems for Convex Metris Space taking

Random Operator

Nidhi Gargav

1

, Rajesh Shrivastava

2

, Rizwana Jamal

3 1_{Research Scholar Saifia Science College, Bhopal (M.P.), India}

2_{Professor and Head Govt. Science and Commerce College Benazeer Bhopal, India} 3_{Professor and Head, Department of Mathematics, Saifia Science College, Bhopal, India}

Abstract-- In the present paper, some common fixed point theorems are obtained for random operators in convex matrix spaces. We consider the compatible mappings for this purpose. The obtained results include some well known results for special cases.

Keywords-- Convex metric space, Common fixed point, Random operators

AMS Classifications: 47H10

I. INTRODUCTION

Probabilistic functional analysis has emerged as one of the important mathematical disciplines in view of its role in analyzing probabilistic models in the applied sciences. The study of fixed points of random operators forms a central topic in this area. The Prague school of probabilistic initiated its study in the 1950. However, the research in this area flourished after the publication of the survey article of Bharucha-Reid [4]. Since then many interesting random fixed point results and several applications have appeared in the literature; for example the work of Beg and Shahazad [2, 3], Lin [10], O'Regan [11].

In recent years, the study of random fixed points have attracted much attention some of the recent literatures in random fixed points may be noted in [1,2,3,5,12,15].In particular ,random iteration schemes leading to random fixed point of random operators have been discussed in [5,6,7]. Jungck introduced the concept of compatible mappings on metric spaces, as a generalization of weakly commuting mappings, which have been a useful tool for obtaining more comprehensive fixed point theorems .On the other hand, since Takahashi ([15]) introduced a notion of convex metric spaces, many authors have discussed the existence of fixed point and the convergence of iterative processes for nonexpansive mappings in this kind of spaces. The purpose of this paper is to give some fixed point theorems for compatible mappings in convex metric spaces for random operator. Our main results are generalizations of some known results of Choudhary [5, 6, 7] , Nan-Jing Huang and Hong Xu Li [12]

II. PRELIMINARIES

Definition 2.1. The pair ( ) of self-mappings of a metric space ( ) is said to be compatible on if whenever * + is a sequence in such that then ( )

Definition 2.2. Let ( ) be a metric space and , -. A mapping is called a convex structure on if for each ( ) and ,

( ( )) ( ) ( ) ( )

A metric space together with a convex structure is called a convex metric space.

Definition 2.3.A nonempty subset of a convex metric space ( ) with a convex structure is said to be convex if for all ( ) , ( )

Definition 2.4. Let be a convex subset of a convex metric space ( ) with a convex structure . A mapping is said to be W-affine if for all ( ) ( ) ( )

Throughout this paper,





,





denotes a measurable space, C is non empty subset of K

Definition 2.5 Measurable function: A function

C

f

:





is said to be measurable if









)

(

1

C

B

f

foe every Borel subset B ofX.

Definition 2.6Random operator: A function

C

f

:







is said to be random operator, if

C

X

F

(.,

)

:





is measurable for every X



C

Definition 2.7 Continuous Random operator: A random operator

f

:





C



c

is said to be continuous if for

fixed

t





,

f

(

t

,.)

:

C



C

is

continuous

Definition 2.8. Random fixed point: A measurable function

C

g

:





is said to be random fixed point of the

random operator





  

 

 C C if f t g t g t t

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International Journal of Emerging Technology and Advanced Engineering

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Definition 2.9: Let (X, d) be a metric space and





,





is a measurable space, J= [0,1]. A mapping ,is called a convex structure on X for random operator if for each

( ( ) ( ) ) and u(t)

( ( ) ( ( ( ) ( ) )) ≤ ( ( ) ( )) ( ) ( ( ) ( ))

A metric space X together with a convex structure w and random operator is called a convex random metric space

Definition 2.10: A nonempty subset K of a convex random metric space (X,d) with a convex structure w is said to be

convex if for all

K )] y(t), (x(t), [t, w J, Kx x K ) y(t),

(x(t),









Definition 2.11: Let K be a convex subset of a convex random metric space (X,d) with a convex structure W. A mapping

R

:

K



K

is said to be w-affine of for all

)] (t), Ry (t), (Rx (t, W ) (t), y (t), (x RW . J x K x K ) (t), y (t), (x



 

III. MAIN RESULTS

Throughout this section, (Ω, ) measurable spaces t .We assume that ( ) is a complete convex metric space with a convex structure W and K is a nonempty closed convex subset of .

Theorem 3.1.Let the pair ( ) be compatible on K such that for all ,

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/ { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/} { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/ . ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)_} (3.1)

Where are nonnegative real numbers such that and ( )

If ( ) ( ) and is W-affine and continuous, then there exists a unique common random fixed point of and , and is continuous at .

Proof. It is clear that ( ) ( ) implies , since and are nonegative real numbers.

Let be an arbitrary point in K and choose point and in K such that

( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) . (3.2)

This can be done since ( ) ( ). For by (3.1), (3.2) and the triangle inequality, we have

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{ . (

( )) ( ( ))/ . ( ( )) ( ( ))/

( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/) }

If . ( ( )) ( ( ))/ . ( ( )) ( ( ))/, it follows from the above inequalities that

. ( ( )) ( ( ))/ ( )

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/,

Which is a contradiction and therefore

. ( ( )) ( ( ))/ ( ( ( ))

( ( ))) (3.3)

From (3.1)-(3.3), we get

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/}

{

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)_}

( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/) . ( ( )) ( ( ))/

{ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/)}

( ) . ( ( )) ( ( ))/

Let . /. Then we know that and

( ( )) . ( ( )) ( ( )) / . ( ( )) ( ( )) /.

By (3.2)-(3.4), we obtain

. ( ( )) ( ( ))/ ( ( ( )) . ( ( )) ( ( )) /)

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. ( ( )) ( ( ))/. (3.5)

. ( ( )) ( ( ))/ ( ( ( )) . ( ( )) ( ( )) /)

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ (3.6)

. ( ( )) ( ( ))/ ( ( ( )) . ( ( )) ( ( )) /)

( ( ( )) ( ( )) . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ . (3.7)

It follows from (3.1)-(3.7) that

. ( ( )) ( ( ))/

( ( ( )) . ( ( )) ( ( )) /)

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ { . ( ( )) ( ( ))/}

{ . ( ( )) (

( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)}

. ( ( )) ( ( ))/ { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/}

{

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/

( . ( ( )) ( ( ))/ . ( ( )) ( _{( ))/)}}

( )

. ( ( )) ( ( ))/

{ . ( ( )) ( ( ))/ . ( ( )) ( ( ))/}

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. ( ( )) ( ( ))/ { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/}

{ . ( ( )) ( ( ))/ . ( ( )) ( ( ))/}

If ( ( ( )) ( ( )) . ( ( )) ( ( ))/, then the above inequalities imply that

. ( ( )) ( ( ))/ . ( ) ( )/ . ( ( )) ( ( ))/

( ( )) . ( ( )) ( ( ))/.

This is a contradiction, since ( )

( ) and imply . Hence, we have

. ( ( )) ( ( ))/ ( ( )) . ( ( )) ( ( ))/ . (3.8)

Letting ( ), we know that . Since is an arbitrary point in K, it follows from (3.8) that there exists a sequence * + in K such that

. ( ( )) ( ( ))/ ( ( ( )) ( ( ))

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/,

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

Which yield that . ( ( )) ( ( ))/ . ( ( )) ( ( ))/ and so we have

. ( ( )) ( ( ))/ . (3.9)

Setting

{ . ( ( )) ( ( ))/ }

Then (3.9) implies that and Obviously,

̅̅̅̅̅ and ̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅̅

For any , it follows from (3.1) that

. ( ( )) ( ( ))/

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{

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)_}

( . ( ( )) ( ( ))/) ( . ( ( )) ( ( ))/)

( ) ( ) . ( ( )) ( ( ))/,

Which yields

. ( ( )) ( ( ))/ ( ).

Therefore, we have

( ̅̅̅̅̅) ( ) ,

Where ( ̅̅̅̅̅) denotes the diameter of ̅̅̅̅̅. By the Cantor’s theorem, there exists a point such that * + . Since , for each there exists a point such that . ( ( ))/ ,

and so ( ( )) .

Further, since , we have

. ( ( )) ( ( ))/ _and₍ _{( ))}_. Since R is continuous and the pair (F,R) is compatible, we

can induce easily that

( ( )) ( ( )) ( ( )) ( ( )). Now, (3.1) yields that

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/}

* . ( ( )) ( ( ))/ . ( ( )) ( ( ))/

( ( ( )) ( ( )) ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)}

. ( ( )) ( ( ))/.

Letting , we have

. ( ( )) ( ( ))/ ( ) . ( ( )) ( ( ))/,

Which leads to . ( ( )) ( ( ))/ and so ( ( )) ( ( )). Hence , ( ( )) ( ( )) ( ( )) ( ( )), since the pair (F,R) is compatible. Using (3.1), we get

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

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International Journal of Emerging Technology and Advanced Engineering

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{

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)_}

( ) . ( ( )) ( ( ))/

This implies that ( ( ( )) ( ( )) , since and . Therefore, ( ( )) ( ( )).

Letting ( ( )), then is a common fixed point of F and R. if is another common fixed point of F and R, then ( ) . It follows from (3.1) that

( )

. ( ( )) ( ( ))/

. ( ( )) ( ( ))/ { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/}

{

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)_}

( ) ( ),

Which is a contradiction since . Hence, is the unique common fixed point of F and R.

Now, we prove that F is continuous at . Let * + and . Since R is continuous, ( ( )) ( ( )). By (3.1), we have

. ( ( )) ( ( ))/

. ( ( )) ( ( ))/ { . ( ( )) ( ( ))/ . ( ( )) ( ( ))/ }

{

. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

. ( ( )) ( ( ))/ ( . ( ( )) ( ( ))/ . ( ( )) ( ( ))/)_}

This leads to

. ( ( )) ( ( ))/ ( ) . ( ( )) ( ( ))/

And so . ( ( )) ( ( ))/ since . Therefore, F is continuous at . This completes the proof.

Form Theorem 3.1, we have the following result.

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. ( ( )) ( ( ))/ . ( ( )) ( ( ))/

( ) { ( ( ( )) ( ( )) . ( ( )) ( ( ))/)}

For all , where is a constant. If ( ) ( ) and R is W-affine and continuous, then F and R have a unique common random fixed point and F is continuous at .

REFERENCES

[1] Beg, I. and Shahzad, N. “Random fixed points of random multivalued operators on Polish spaces” Nonlinear Anal. 20 (1993) no. 7, 835-847.

[2] Beg, I. and Shahzad, N. “Random approximations and random fixed point theorems” J. Appl. Math. Stochastic Anal. 7 (1994). No 2, 145-150.

[3] Beg, I. and Shahzad, N. “Random fixed points of weakly inward operators in conical shells” J. Appl. Math, Stoch. Anal. 8(1995) 261-264.

[4] Bharucha –Reid, A.T. Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc.82 (1976) 641-657.

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

[6] Choudhary, B.S. and Upadhyay, A. “An iteration leading to random solutions and fixed points of operators” Soochow J. Math.25 (1999). No 4,395-400.

[7] Choudhary, B.S. “A common unique fixed point theorem for two random operators in Hilbert spaces” I. J. M.M. S. 32 (2002) 177-182.

[8] Dhagat, V.B., Sharma, A. and Bhardwaj, R.K. “Fixed point theorem for random operators in Hilbert spaces” International Journal of Math. Analysis 2(2008) no. 12, 557-561.

[9] Himmelbeg, C.J. “Measurable relations” Fund. Math.87 (1975) 53-72.

[10] Lin, T.C. “Random approximations and random fixed point theorems for continuous 1-set- contractive random amps” Proc. Amer. Math. Soc. 123(1995) 1167-1176.

[11] O’Regan, D. “A continuous type result for random operators” Proc. Amer. Math. Soc. 126 (1998) 1963-1971

[12] Nan-Jing Huang and Hong-Xu Li, “Fixed point theorems of compatible mappings in convex Metric spaces” Soochow Journal of Mathematics. 22 (1996), no. 3, 439-447.

[13] Plubtieng, S., Kumam, P. and Wangkeeree, R. “Approximation of a common fixed point for a finite family of random operators” International J. of Math. And Mathematical sciences (2007) 1-12. [14] Sehgal, V.M. and Waters, C. “Some random fixed point theorems

for condensing operators” Proc. Amer. Math. Soc. 90(1984), no.3, 425-429

[15] W. Takahashi, “A convexity in metric spaces and nonexpensive mappings” Kodai Math. Sem. Rep.22 (1970) 142-149