Fixed Point Theorm In Probabilistic Analysis

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. VI (Jan - Feb. 2015), PP 16-18
www.iosrjournals.org
DOI: 10.9790/5728-11161618 www.iosrjournals.org 16 | Page
Fixed Point Theorm In Probabilistic Analysis
Dr.Ayaz Ahmad
Head, Department Of Mathematics Millat College, L.N. Mithila Universitydarbhanga
Pin: 846004 (India)
Probabilistic operator theory is the branch of probabilistic analysis which is concerned with the study of
operator-valued random variables and their properties. The development of a theory of random operators is of
interest in its own right as a probabilistic generalization of (deterministic) operator theory and just as operator
theory is of fundamental importance in the study of operator equations, the development of probabilistic operator
theory is required for the study of various classes of random equations.
Defenition.1.1.Any -valued random variable x() which satisfies the condition ({ : T ()  () = y
()}) = 1 is said to be a random solution of the random operator equation T() x () = y ().
Defenition.1.2: An -valued random variable () is said to be a fixedpoint of the random operator T() if
() is a random solution of the equation T()() = ().
The study of fixed point theorems for random operators was initiated by Špaček and Hanš1
. The first
systematic investigation of random fixed point theorems was carried out by Hanš1
. Because of the wide
applicability of Banach’s contraction mapping theorem in the study of deterministic operator equations, Špaček
and Hanš directed their attention to probabilistic versions of Banach’s theorem and used their results to prove the
existence, uniqueness, and measurability of solutions of integral equations with random kernels.
Defenitin.1.3 :A random operator T() on a Banach space with domain D(T()) is said to be a random
contraction operator if there exists a nonnegative real-valued random variable such that k() < 1, and such that
as, T()x1 – T()x2 k()x1 – x2for all x1, x2  D(T()). If k() = k (a- constant) for all  
, then T() is called a uniform random contraction operator.
Theorem. 1.4: Let be a separable Banach space, and let. T() be a continuous random operator on to
itself such that
= 1,
Where for every   , x  , and n = 1, 2, …., we put T1
()x = T()x, and Tn+1
()x = T()[Tn
()x]. Then.
There exists an -valued random variable () which is the unique fixed point of T(); that is if () is
another fixed point, then ()=().
PROOF. Let E denote those elements of  belonging to the set
for with T () is continuous. Clearly E  P, and by hypothesis, (E) = 1. Let the mapping () :   be
defined as follows : For every   E, () is equal to the unique fixed point of T(); and for every   - E,
put () =  (the null element of ). Then T()() = ().
Now,we proceed to establish the measurability of the fixed point (). Let x0() be an arbitrary -
valued random variable, and put x1() = T() x0 (). x1() is an -valued random variable and a sequence of
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-valued random variables can be defined as follows : xn() = T()xn-1(), n = 1, 2,… Now, since T() is
continuous, the sequence xn() converges almost surely to (); hence () is an -valued random variable.
The uniqueness of the fixed point follows from the uniqueness of () for every  E.
Theorem.1.5: Let T() be a continuous random operator on a separable Banach space to itself, and let k()
be a nonnegative real-valued random variable such that k() < 1.and T() x1 – T ()x2  k()x1 –
x2 for every pair of elements x1, x2  . Then there exists an -valued random variable () which is the
unique fixed point of T().
PROOF. Let E = {: k() < 1}, F = { : T() x is continuous in x}, and
Gx1,x2 = {:T()x1 – T()x2 k()x1 – x2}.
Since is separable, the intersections in the expression
can be replaced by intersections over a countable dense set of . Therefore the condition of Theorem (1.4) is
satisfied with n = 1.
Random contraction mapping theorems are of fundamental importance in the theory of random
equations in that they can be used to establish the existence, uniqueness, and measurability of solutions of
random operator equations.
Theorem.1.6. Let T() be a random contraction operator on a separable Banach space , and let k() be a
nonnegative real-valued random variable which is bounded. Then, for every real  0 such that k() < .
there exists a random operator S() which is the inverse of T()- I.
Proof. Since   0, T()- I is invertible whenever the random operator (I/) T()– I is invertible, and vice
versa. However, for every y  the random operator Ty() defined, for every    and x  , by
Ty()[x] = (1/) T() x– y is a random contraction operator. Therefore, by Theorem (1.6) there exists a unique
random fixed point y() satisfying the relation y() = (1/) T() y() – y a.s. However, the above
statement is equivalent to the invertibility of the random operator (1/) T()– I, and therefore the invertibility
of the random operator T()- I.
Theorem 1.7: Let ( ) be an atomic probability measure space, and let E be a compact (or closed and
bounded) convex subset of a separable Banach space . Let T() be a compact random operator mapping E
into itself. Then, there exists an E-valued random variable () such that T()() = () a.s. n, such that
T(n)n = n. Put () = n for   Cn, and 0 otherwise. Then T()()= ().
Theorem 1.8. Let E be a compact convex subset of a Banach space and T() be a continuous random operator
mapping E into itself. Then there exists an E-valued random variable () such that T()() = ().
PROOF. Let A() = {x  E : T()x = x}. Then by Theorem (D) for each  the set A() is nonempty.
Furthermore, for any closed subset F of E
{ : A()  F is nonempty} = { : A()x = x for somex in F}
Where the xi’s form a dense sequence in F. It is therefore clear that set { : A()  F is nonempty} is
measurable for every closed subset F of E. To prove the theorem, it is sufficient to find an E-valued random
variable () such that ()  A(). It is known that we can associate with the space E a sequence of triples
(Cn, Pn, n) (n a nonnegative integer) such that
(i) Each Cn is a countable set and Pn maps Cn+1 onto Cn;
(ii) n maps Cn into a class of nonempty closed subsets of E of diameter 2-n
.
(iii) E = UcC0 0(c);
(iv) for each n and for each c in Cn,
Without any loss of generality. We assume that C0 and each Pn
-1
(c) with c in Cn are naturally linearly ordered
such that only finitely many elements can precede any element in this order. Now, for each n, we intend to find a
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suitable partition of . We proceed inductively as follows: For each c in C0, we define c by   c if and only
if A ()  0(c) is nonempty and A()  0(c) is empty for c  C0, c < c. Then the cs arepair wise
disjoint measurable sets with union . Suppose now that we have found a partition of  corresponding to the
elements of Ck. To do this for Ck+1, we define for c in Ck+1 the set c by   c if and only   pk(c) and A()
 k+1(c) is nonempty, but A()  k+1(c) is empty for c in Pk
-1
(Pk(c)) and c < c.
For any positive n and each c  Cn we choose an element xn(c)  n(c) and define n() = xn(c) for 
 c where the cs are members of the partition of  corresponding to the elements of Cn. Then each n( is
measurable and
n()- n+1 () 2-n
, d (n(), A())  2-n
.
Therefore if () = lim n(), then ()  A() and the theorem follows.
Theorem. 1.10: Let T() be a stochastically continuous random operator on a separable Banch space  to
itself. Suppose that for each   , {x : T()x = x}  Ø (the null set). Then there exists a measurable multi-
valued map () :   2X
such that ()= (x : T()x = x}.
In a number of applications of fixed point theorems in probabilistic analysis, it is assumed that a random
operator T() satisfies the hypotheses of Schauder’s theorem for each   . Then, if T() is a continuous
random operator it is also stochastically continuous and separable.
Theorem.1.11: Let (, P, ) be a probability measure space, and let E be a compact and convex subset of a
separable Banach space . Let T()x1 + T()x2  E for all x1, x2  E and   , (ii) there exists a nonnegative
real-valued random variable k() such that S()x1 – S()x2 k()x1 – x2 for all x1, x2  E and
k() < 1 a.s. Then there exists an X– valued random variable () such that S()() + T()() = ()
for all   .
The proof of the above theorem follows easily from Theorem 1.8 observing that the operator [I – S()]-
1
T() is a well-defined continuous mapping on E into itsel
References
[1]. O. Hanŝ, Reduzierende zufällige Transformationen, Czechoslovak Math. J. 7 (82) (1957), 154-158. MR 19, 777.
[2]. R. Kannan and H. Salehi, Measurability du point fixe d’une transformation aleatoire separable, C. R. Acad. Sci. Paris Ser A-B 281
(1975), A663-A664.
[3]. Mukherjea, Random transformations on Banach spaces, Ph. D. Dissertation, Wayne State Univ., Michigan, 1966.
[4]. L. S. Prakasa Rao, Stochastic integral equations of mixed type. II, J. Mathematical and Physical Sci. 7 (1973), 245-260. MR 50 #
14933.

Fixed Point Theorm In Probabilistic Analysis

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