Fixed point of multivalued mapping on polish space

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Fixed Point Of Multivalued Mapping On Polish Space Yogita R. Sharma School of Studies in Mathematics, Vikram University, Ujjain, M.P., India, yogitasharma17@yahoo.com

Abstract: is proved.

A general fixed point theorem, that incorporates several known random fixed point theorems, involving continuous random operators,

Keywords: Fixed point; Multivalued mapping; Polish space.

1.INTRODUCTION Random fixed point theorems are of fundamental importance in probabilistic functional analysis. They are stochastic generalization of classical fixed point theorems and are required for the theory of random equations. In a separable metric space random fixed point theorems for contractive mapping were proved by Spacek [1], Hans [2, 3], Mukherjee [4]. Itoh [5, 6] extended several fixed point theorems, i.e., for contraction, nonexpensive, mappings to the random case. Thereafter, various stochastic aspects of Schauder’s fixed point theorem have been studied by Sehgal and Singh [7] and Lin [8]. Afterwards Beg and Shahzad [9], Badshah and Sayyad [10] studied the structure of common random fixed points and random coincidence points of a pair of compatible random operators and proved the random fixed points theorems for contraction random operators in polish spaces.

2. PRELIMINARIES Let (X, d) be a Polish space; that is a separable, complete metric space and (, a) be a measurable space. Let 2X be a family of all

be a random operator and  n  a sequence of measurable mappings  :   X . The sequence is said to be asymptotically T-regular, if n

d  n (), T (,  n ())  0.

3. THEOREM Let X be a Polish space. Let T , S :   X  CB X  be two continuous random multivalued operators. If there exist measurable mappings , ,  :   0,1 such that (1) minH S , x , T , y , d x, S , x d  y, T , y ,

d  y, T , y   mind x, T , y , d  y, S , x 

 d x, S , x   d x, y d  y, T , y  for each x, y  X , where  , ,  are real numbers such that 0       1 and there exists a

measurable sequence

 n  which is asymptotically

subsets of X and CB(X) denote the family of all non empty closed bounded subsets of X. A mapping T :   2 X , is called measurable, if for any open subset C of X, T 1 (C )  {   : T ()  C  }  a .A mapping  :   X ,

regular with respect to S and T ,then there exists a common random fixed point of S and T. (H is Hausdorff metric on induced by metric d )

is said to be measurable selector of a measurable mapping T :   2 X , if  is measurable and for any   , ()  T (). A

3.1 Proof. Let 0 :   X bne an arbitrary measurable mapping and choose a measurable mapping 1 :   X such that

mapping f :   X  X is called a random operator if for any x  X , f (. , X ) is measurable. A mapping T :   X  CB ( X ), is called random multivalued operator if for every x  X , T (. , X ) is

measurable. A measurable mapping  :   X , is called the random fixed point of a random multivalued operator T :   X  CB( X ) ( f :   X  X ), if for every   ,

()  T (, ()) ( f (, ())  ()). Let T :   X  CB( X )

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1   S ,  0  for each    . Then for each   

 H S ,  0 , T , 1 ,    min d  0 , S ,  0 d 1 , T , 1 ,  d  , T ,    1 1      mind  0 , T , 1 , d 1 , S ,  0 

 d  0 , S ,  0    d  0 , 1  d 1 ,T , 1 .

Vol. 1 Issue II, September2013 ISSN: 2321-9653

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I N T E R N A T I O N A L J O U R N A L F O R R E S E A R C H I N A P P L I E D S C I E N C E AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) It further implies [lemma 2.3 of Beg (1993a)], that there exists a measurable mapping  :   X such that for any 2

   ,  2   T , 1  and for x   n 1  and y   n  by

condition (1), we have  H S ,  n1 , T ,  n ,    min d  n1 , S ,  n1 d  n , T ,  n ,  d  , T ,    n n  

 d (),  2 n  2 () 

 H (T (,  2 n 1 ()), S (, ())) ,

Or, d (), S (, ())   d (), S (, ())    d ,  2 n1   d   2 n1 , T ,  2 n1  ,

 n   S ,  n1  .

which tends to zero.

So,  2 n1 is asymptotically T-regular.

We have mind  n ,  n1 , d  n1 ,  n d  n ,  n1 

 d  n1 ,  n   d  n1 ,  n d  n ,  n1 

Letting n   we have d (), S (, ())   0 .

Hence, ()  S (, ()) for    .

and it follows that mind  n ,  n1 , d  n1 ,  n d  n ,  n1 

    d  n1 ,  n d  n ,  n1  Since d  n 1 ,  n d  n ,  n 1        d  n 1  ,  n  d  n  ,  n 1   is not possible (as 0         1 ).

Similarly, for

d (), T (, ())   d ,  2 n1   H S ,  2 n , T ,  d (), T (, ())   0 .

We have d  n ,  n1      d  n1 ,  n d  n ,  n1  Or d  n  ,  n 1    kd  n 1  ,  n  d  n  ,  n 1   where k     , 0  k 1

Hence, ()  T (, ()) for each    .

3.2 Corollary. Let X be a Polish space. Let T :   X  CB X  be a continuous random multivalued operator. If there exist measurable mappings , ,  :   0,1 such that minH T , x , T , y , d x, T , x d  y, T , y , d  y, T , y 

Similarly proceeding in the same way, by induction we produce a sequence of measurable mapping  n :   X such that for n  0 and any     2 n 1 ()  S ,  2 n () ,  2 n  2 ()  T ,  2 n 1 ()  d  n (),  n 1 ()   k d  n 1 (),  n ()   ...  k n d  o (), 1 () 

d  n (),  m ()   d  n (),  n1 ()   ...  d  m1 (),  m () 

 ...  k

d  n (),  m () 

References [1] A. Spacek, G. Zulfallige Czechoslovak Mathematical Journal (5) pp.462-466, 1955.

[2] O. Hans, R. Zufallige, Transformationen. Czechoslovak Mathematics Journal (7) 154-158, 1957.

[3] O.Hans, Random operator equations. Proceedings of the 4th Berkeley symposium in Mathematics and Statistical Probability, Vol. II: pp. 180-202, 1961.

 d  (),  () o

 d  x, T , x    d  x, y d  y, T , y , for each x, y  X and    where  , ,  are real numbers such

and converges to a random fixed point of T.

Further, for m > n,

 k  k

  mind  x, T , y , d  y, T , x 

that 0       1 then there exists a sequence  n  of measurable mappings  n :   X which asymptotically T-regular

and

m 1

sequence and there exists a measurable mapping :  X such that  n ()  () for each    . It implies that  2 n 1 ()  () and

 2 n 2 ()  () . Thus, we have for any    ,

Since d  , S ,    0 , n n 1

n 1

n   . It follows that  n () is a Cauchy

d (), S (, ())   d (),  2 n 2 ()   d  2 n 2 (), S (, ()) 

  mind  n1 , T ,  n , d  n , S ,  n1   d  n1 , S ,  n1    d  n1 ,  n  d  n , T ,  n 

n

which tends to zero as

1

 kn    d  o (), 1 ()  1  k 

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I N T E R N A T I O N A L J O U R N A L F O R R E S E A R C H I N A P P L I E D S C I E N C E AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) [4] A. Mukherjee, Random transformations of Banach spaces.

Ph. D. Dissertation. Wayne State University, Detroit, Michigan, USA.1968.

[5] S. Itoh, Random fixed point theorem with applications to random differential equations in Banach spaces. Journal of Mathematical Analysis and Application (67) 261-273.1979.

[6] V.M. Sehgal, & S.P. Singh, On random approximation and a random fixed point theorem for a set valued mappings.

Proceedings of the American Mathematical Society (95) 9194, 1985.

[7] T.C. Lin, Random approximations and random fixed point theorems for non-self maps. Proceedings of the American Mathematical Society (103) 1129-1135, 1988.

[8] I. Beg, & N.Shahzad, 1993. Random fixed points of random multivalued operators on Polish spaces. Nonlinear Analysis 20: 835-847.

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[9] V. H. Badshah, & F. Sayyad, Random fixed points of random multivalued operators on Polish spaces. Kuwait Journal of Science and Engineering, (27) 203-208, 2000.

[10] I. Beg, & N. Shahzad, Common random fixed points of random multivalued operators on metric spaces. Bollettino U.M.I. (7) 493-503, 1995.

[11] D. Turkoglu, & B. Fisher, Fixed point of multivalued mapping in Uniform spaces. Proc.Indian Acad.Sci. (Math.

Sci.) 113 (2): 183-187, 2003.

[12] Y.R. Sharma, Fixed Point and Weak Commuting Mapping, International Journal of Research in Engineering and Technology, 2 (1), pp.1-4, 2013.