Fixed points theorem on a pair of random generalized non linear contractions

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
326
Fixed Points Theorem on A Pair of Random Generalized Non-
Linear Contractions
R.N.Yadava
Director, Patel Group Of Institutions Bhopal (M.P.), India
(dryadava@gmail.com)
Rajesh Shrivastava
Department of Mathematics
Govt Science & commerce college Benazir, Bhopal(M.P.), India
(Rajeshraju0101@rediffmail.com)
S.K.Pandey
Department of Mathematics
Millennium Group of Institutions Bhopal(M.P.), India
(Sudeeppandey1976@gmail.com)
ABSTRACT
A fixed point theorem for a pair of random generalized non-linear contraction mappings involving four points of
the space under consideration is proven. It is shown that this result includes the result of Lee and Padgett [1].
Also an application of the result is given.
Key words : Complete probability measure space, δ-algebras, Borel subsets, random operator, separable Banach
space, upper semi-continuous functions. Bochner integral,
Mathematics Subject classification: Primary 60H99, 47H10, and Secondary 60H 20.
1. Introduction and preliminaries
The idea of fixed points plays a very important role in solving deterministic operator equations. Recently the
idea of random fixed point theorems which are the stochastic generalization of the classical fixed point theorems
has become a very important part of the theory of some operator equations which can be regarded as random
operator equations. Many interesting results have been established by various authors (see for example Bharucha
– Reid [2], Hans [3], Padgett [4], Tsokos [5], Tsokas and padgett [6], Lee and padgett [7] in this area.
Recently, a fixed point theorem for a pair of generalized non-linear contraction mappings involving four
points of the space under consideration, which includes many well known results as special cases has been
established by Achari [8] (see also Achari [9], Pittanuer [10]).
The object of this paper is to study a stochastic version of a pair of generalized non-linear contraction
mappings of Achari [8]. Also it has been shown that this result generalizes the result of Lee and Padgett [1]. It is
interesting to note that with suitable modification of the conditions of the theorem, we can easily obtain
stochastic generalizations of the results of different classical fixed points. Finally, we apply theorem 2 to prove
the existence of a solution in a Banach space of a random non-linear integral equation of the form
x (t; w) = h (t; w) + ∫s k (t, s; w) ƒ(s, x (s; w) d, µ (s) ...... (1.1)
where s is a locally compact metric space with metric d defined on S X S, µ is a complete δ-finite measure
defined on the collection of Borel subsets of S and the integral is a Bochner integral.
In this section, we state some definitions as used by Lee and padgett [1]. Let (Ω, S. P) be complete probability
measure space, and Let (X, β) and (Y, C) be two measurable spaces, where X and Y are Banach spaces and B
and C are δ-algebras of Borel subsets of X and Y, respectively. First, we state the usual definitions of a Banach
space-valued random variable and of a random operator.
Definition 1.1 : A function V : Ω → X is said to be an X – valued random variable (Random element in X, or
generalized random variable)
if { ω ∈ Ω : v (w) ∈ B} ∈ S for each B ∈ β.
Definition 1.2 : A mapping T (w) : Ω x X → Y is said to be a random operator if y (w) = T (w) x is a Y-valued
random variable for every x ε X.
Definition 1.3 : Any X-valued random variable x (w) which satisfies the condition.
P ( { W : T (w) x (w) – y (w) } ) = 1
is said to be a random solution of the random operator equation
T (w) x = y (w).
Definition 1.4 : Let yj (w), j = 1, 2, ........, n be second order real valued random variables on a probability space
(Ω, S, P), that is yj (w) ∈ L2 (Ω, S, P). The collection of all n-component random vectors y' (w) = (y1 (w), ....., yn
(w)) constitutes a linear vector space if all equivalent random vectors are identified. Define the norm of y by

327
( )½|||||||||| 2max
12
max
12 ∫ Ω<<<< == dpyyy jnjLjnj
n
L
The space of all n-component random vectors y with second order components and norm given by
n
2L
||.|| above
is separable Banach space and will be denoted by
n
2L (Ω, s, p) or
n
2L . Let S be locally compact metric space
with metric d defined on S X S and let µ be a complete δ-finite measure defined on the Borel subsets of S.
Definition 1.5 : We define the space C (S,
n
2L (Ω, s, p)) to be the space of all continuous functions from S into
n
2L (Ω, s, p) with the topology of uniform convergence on compacta. It may be noted that C (S,
n
2L (Ω, s, p)) is
locally convex space whose topology is defined by a countable family of saminorms given by –
|| x (t; w) ||j =
jct
sup
∈
|| x (t; w)
n
2L
|| , j = 1, 2, ......
Definition 1.6 : Let B and D be Banach spaces. The pair (B, D) is said to be admissible with respect to a random
operator U (w) if U (w) (B) ⊂D.
Definition 1.7 : A random operator T (w) on a Banach space x with domain D (T(w)) is said to be a random
generalized nonlinear contraction if there exists non-negative real-valued upper semi-continuous functions Ψi
(w), i=1, 2, ......, 7. Satisfying Ψi (w) (r) <
7
r
for r > 0, Ψi (w) (0) and such that
|| T (w) x1 – T (w) x2 || < Ψ1 (||x1-x2||) + Ψ2 (||x1 – T(w)x1 ||)
+ Ψ3 (||x2-T(w)x2||) + Ψ4 (||x1-T (w)x2 ||)
+ Ψ5 (||x2-T (w)x1||)
+ Ψ6 [||x1-T (w) x1 ||) + (||x1-T(w)x2||)]
1+[x1-T(w)x1||) (||x1-T(w)x2||)]
+ Ψ7 [||x2-T (w) x2 ||) + (||x2-T(w)x1||)]
1+[x2-T(w)x2||) (||x2-T(w)x1||)]
For all x1, x2 ∈ D (T (w)
2.Main results
A fixed point theorem for a pair of random generalized non-linear contraction.
Theorem 2.1 :- suppose A1 (w) and A2 (w) are a pair of random operators from a separable banach space x into
itself such that
||A1(w) x1-A2 (w) x2 ||
< Ψ1 (||x1-x2||) + Ψ2 (||x1-A1 (w)x3||) +Ψ3 (||x2-A2 (w) x4||
+ Ψ4 (||x1-A2 (w)x4||) + Ψ5 (||x2-A1 (w) x3||)
+ Ψ6
||)])((||||)([(||1
||)])((||||))([(||
421311
421311
xwAxxwAx
xwAxxwAx
−−+
−+−
+ Ψ7
||)]x)w(Ax(||||x)w(Ax[(||1
||)]x)w(Ax(||||)x)w(Ax[(||
312422
312422
−−+
−+−
............(2.1)
Where Ψi (w), i = 1, 2, ........... 7, are non-negative real-valued upper semi-continuous functions
satisfying Ψi (w) (r) ,
7
r
< for r > 0, Ψi = 0 (w) (o) and for all x1, x2, x3, x4 ∈ X. Then there exists an x – valued
random variable η (w) which is the unique common fixed point of A1 (w) and A2 (w).
Proof : Let x, y ∈ x and we define.
x1 = A2 (w)y, x2 = A1 (w) x, x3 = x, x4 = y
Then (2.1) takes the form
||A1 (w) A2 (w) y – A2 (w) A1 (w) x ||
< Ψ1 A1 (w) x - A2 (w) y ||)+ Ψ2 (|| A1 (w) x – A2 (w) y ||)
+ Ψ3 (||A1 (w) x – A2 (w) y ||)+ Ψ4 (||A2 (w) y – A2 (w) y||)
+ Ψ5 (||A1 (w) x – A1 (w) x ||)
+ Ψ6
||)]y)w(Ay)w(A[(||||]y)w(Ax)w(A[(||1
||)]y)w(Ay)w(A(||||)y)w(2Ax)w(A[(||
2121
221
−−+
−+−

328
+ Ψ7
||)]x)w(Ax)w(A[(||||]y)w(Ax)w(A[(||1
||)]x)w(Ax)w(A(||||)y)w(Ax)w(A[(||
1121
1121
−−+
−+−
< Ψ1 (||A1 (w) x–A2 (w)y|| + Ψ2 [||A1(w) x – A2 (w) y ||)
+ Ψ3 (|| A1 (w) x - A2 (w) y||)+ Ψ6 (||A1 (w) x - A2 (w) y ||)
+ Ψ7 (||A1 (w) x- A2 (w) y ||) ..... (2.2)
Let x0 ∈ X be arbitrary and construct a sequence {xn) defined by
A1(w) xn-1 = xn, A2 (w) xn = xn+1, A1 (w) xn+1 = xn+2, n=1,2....
Let us put x = xn-1 and y = xn in (2.2), then we have
A1 (w) A2 (w) xn-A2(w) A1 (w) xn-1||< Ψ1 (||A1 (w) xn-1-A2 (w) xn||)
+ Ψ2 (||A1 (w) xn-1 - A2 (w) xn ||
+ Ψ3 (||A1 (w) xn-1 - A2 (w) xn||)
+ Ψ6 (||A1 (w) xn-1 - A2 (w) xn ||
+ Ψ7 (||A1 (w) xn-1 - A2 (w) xn||)
or
||xn+2 – xn+1|| < Ψ1 (||xn – xn+1||) + Ψ2 (||xn-xn+1||)
+ Ψ3 (||xn-xn+1||) + Ψ2 (||xn-xn+1||)
+ Ψ7 (||xn – xn+1 ||) ...... (2.3)
We take n to be even and set αn = ||xn-1 – xn||.
Then αn+2 = || xn+1 – xn+2||
< Ψ1 (||xn-xn+1||) + Ψ2 (||xn-xn+1)|| + Ψ3 (||xn – xn+1) ||
+ Ψ6 (|| xn-xn+1||)+ Ψ7 (||xn – xn+1||)
< Ψ1 (αn+1) + Ψ2 (αn+1) + Ψ3 (αn+1)
+ Ψ6 (αn+1) + Ψ7 (αn+1) ......... (2.4)
From (2.4) it is clear αn decrease with n and hence αn → α as n → ∞.
Let α > 0. Then since Ψ1 is upper semi-continuous, we obtain in the
limit as n → ∞
α < Ψ1 (α) + Ψ2 (α) + Ψ3 (α) + Ψ6 (α) + Ψ7 (α) < α
7
5
which is impossible unless α = 0
Now, we shall show that {xn} is a cauchy sequence. If not, then there is an ∈ > 0 and for all positive integers K,
there exist {m (k)} and {n (k)}with m (k) > n (k) > k, such that
dk = ||xm(k) – xn(k) || > ∈, .......... (2.5)
We may assume that - ||xm(k)-1
–
xn(k) || < ∈
by choosing m (k) to be the smallest number exceeding n (k) for which (2.5) holds. Then we have
dk < || xm(k) – xm(k)-1 || + ||xm(k)-1 – xn(k) ||
< αm(k) + ∈ < αk + ∈
which implies that dk → ∈ as k → ∞. Now the following cases are to be considered.
(i) m is even and n is odd,
(ii) m and n are both odd,
(iii) m is odd and n is even,
(iv) m and n are both even
i.e. (i) :
dk = ||xm-xn|| < ||xm – xm+1||+||xm+1 – xn+1|| + || xn-xn+1||
< αn+1 + αn+1 + ||A1 (w) xm – A2 (w) xn||
By putting x1 = xn, x2 = xm, x3 = xn-1, x4 = xm-1 in (2.1), we have
< αm+1 + αn+1 + Ψ1 (||xm-xn||) + Ψ2 (||xn-A1 (w) xn-1 ||)
+ Ψ3 (||xm-A2(w) xm-1||) + Ψ4 (|| xn-A2(w) xm-1||)
+ Ψ5 (||xm-A1(w) xn-1||)
+ Ψ6
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax||||)x)w(Ax[(||
1m2n1n1n
1m2n1n1n
−−
−−
−−+
−+−
+ Ψ7
||)])([(||||)])([(||1
||))((||||))([(||
1112
1112
−−
−−
−−+
−+−
nmmm
nmmm
xwAxxwAx
xwAxxwAx

329
< αm+1 + αn+1 +Ψ1 (dk) +Ψ4 (dk) + Ψ5 (dk) +Ψ6 (dk) + Ψ7 (dk) letting k → ∞ we have
∈ <
7
5
∈
This is contradiction if ∈ > 0. In the case (ii), we have
dk = || xm-xn|| < || xm - xm+1|| + ||xm+2 - xm+1||+||xm+2 - xn+1||+||xn-xn+1||
< αm+2 + αm+1 + αn+1 + || A2 (w) xm+1 – A1 (w) xn ||
By putting x1 = xn, x2 = xm+1, x3 = xn-1, x4 = xm in (3.1), we get
< αm+2 + αm+1+αn+1+Ψ1 (||xm+1 – xn||) + Ψ2 (||xn-A1 (w) xn-1|| )
+ Ψ3 (||xm+1-A2(w) xm||) + Ψ4 (|| xn-A2(w) xm ||)
+ Ψ5 (||xm-A1(w) xn-1||)
+ Ψ6
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax||||)x)w(Ax[(||
m2n1n1n
m2n1n1n
−−+
−+−
−
−
+ Ψ7
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax(||||)x)w(Ax[(||
1n1mm21m
1n1mm21m
−+
−+
−−+
−+−
< αm+2 + αm+1 + αn+1 + Ψ1 (dk+αm+1) + Ψ4 (dk + αm+1)
+ Ψ5 (dk+αm+1) +Ψ6 (dk+αm+1) + Ψ7 (dk+αm+1).
Letting k → ∞ in the above inequality we obtain ∈ <
7
5
∈, which is a contradiction if ∈ > 0. Similarly, the
cases (iii) and (iv) may be disposed of. This Leads us to conclude that {xn} is a cachy sequence. Let η (w) be the
limit of the sequence. We shall now show that
A1 (w) η (w) = η (w) = A2 (w) η (w). Putting x1 = xn-1, x2 = η (w), x3 = xn+1, x4 = xn, in (2.1), we get
|| A1 (w) xn-1 – A2 (w) η (w) ||
< Ψ1 (||xn-1 - η (w) ||) + Ψ2 (||xn-1 – A1 (w) xn+1||)
+ Ψ3 (|| η (w) – A2(w) xn ||) + Ψ4 (||xn-1 – A2 (w) xn||)
+ Ψ5 (|| η (w)–A1 (w) xn+1||) +
Ψ6
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax(||||)x)w(Ax[(||
n21n1n11n
n21n1n11n
−−+
−+−
−+−
−+−
+ Ψ7
||)]x)w(A)w((||(||)x)w(A)w([(||1
||)]x)w(A)w((||||)x)w(A)w([(||
1n1n2
1n1n2
+
+
−η−η+
−η+−η
Letting n → ∞, we get || η(w)- A2 (w) η (w) || < 0 which is a contradiction and hence η (w) = A2 (w) η (w). In
the same, way, it is possible to show that η (w) = A1 (w) η (w). Thus η (w) is a common fixed point of A1 (w)
and A2 (w). Suppose there is another fixed point ξ (w) ≠ η (w) of A1 (w) and A2 (w). Then putting x1 = x4 = η (w)
and x2 = x4 = ξ (w) in (2.1), we have
||η (w) - ξ (w)|| < Ψ1 || η(w) - ξ (w) || + Ψ2 (||η (w) - ξ (w) ||
+ Ψ3 (||η(w)-ξ (w)|| + Ψ6 (||η(w)|| - ξ (w)||) + Ψ7 || (||η (w) - ξ (w)||)
<
7
5
|| η (w) - ξ (w) ||
which is contradiction. Hence η (w) = ξ (w). This completes the proof. If in theorem 2.1, we put A1 (w) = A2 (w)
= A (w) and x1 = x3 = x, x2 = x4 = y then we have the following theorem which we only state without proof.
Theorem 2.2 : If A (w) is a random generalized non-linear contraction from a separable Banach space x into
itself, then there exists an x – valued random variable which is the unique fixed point of A (w).
We now have the following corollary of theorem 2.2
Corollary 2.3 : If Ab
(w) is a random generalized contraction from x into itself for some positive integer b, then
A (w) has a unique fixed point η (w) which is an x – valued random variable.
Proof : Since Ab
(w) is a random generalized non-linear contraction operator on X, by theorem 2.2, there exists a
unique x – valued random variable η (w) such that
Ab
(w) η (w) = η (w)
We claim that A (w) η (w) = η (w). If not, consider
||Ab+1
(w) η (w) – Ab
(w) η (w) ||
< Ψ1 (||A (w) η (w) - η (w) ||)+ Ψ2 (||A (w) η (w) – Ab+1
(w) η (w) ||)
+ Ψ3 (||η (w) – Ab
(w) η (w)||)+ Ψ4 (||A (w) η (w) – Ab
(w) η (w)||)

330
+ Ψ5 (||η (w) - Ab+1
(w) η (w)||)
+ Ψ6
||])w()w(A)w()w(A[||||])w()w(A)w()w(A[(||1
||])w()w(A)w()w(A(||||))w()w(A)w()w(A[(||
b1b
b1b
η−ηη−η+
η−η+η−η
+
+
+ Ψ7
||])w()w(A)w([||||])w()w(A)w([(||1
||])w()w(A)w((||||))w()w(A)w([(||
1bb
1bb
η−ηη−η+
η−η+η−η
+
+
.....(2.31)
Moreover, the left hand side of (2.31) is
|| Ab+1
(w) η (w) – Ab
(w) η (w)|| = ||A (w) η (w) - η (w)|| .... (2.32)
From (2.31) and (2.32), we have
||A (w) η (w) - η (w) ||
< Ψ1 (|| A (w) η (w) - η (w) || + Ψ4 (||A (w) η (w) - η (w)||
+ Ψ5 (||A (w) η (w) - η (w)||) + Ψ6 (||A (w) η (w) - η (w)||)
+ Ψ7 (||A (w) η (w) - η (w) ||)
which is, a contradiction and hence A (w) η (w) = η (w)
we remark that under the conditions A1 (w) = A2 (w) = A (w) and
x1 = x3 = x, x2 = x4 = y, Ψi = Ψ, Ψj (w) (r) = 0, j = 2, 3, 4, 5, 6, 7,
the theorem 2.1 reduces to the following corollary.
Corollary 2.4 : (Lee and padgett [1]) :- If A (w) is a random non-linear contraction operator from a separable
Banach space x into itself, then there exists an x – valued random variable η (w) which is the unique random
fixed point of A (w).
2.4 Application to a Random Non-linear integral Equation :-
In this section we give an application of a random non-linear integral equation. To do so we have
followed the steps of Lee and padgett [1] with necessary modifications as required for the more general settings.
We shall assume the following conditions concerning the random kernel k (t, s; w). The function k (., .; .) : S x S
x Ω → R is such that
(i) K (t, s; w) : S x S → L∞ (Ω, S, P) such that || k (t, s; w) || . || x (s; w) ||
n
2L is µ-integrable with respect to
s∈S for each t∈S and x ∈ C (s,
n
2L (Ω, s, p) where for each (t, S) ∈ s x S
||| k (t, s; w) ||| = || k (t, s; w) || L∞ (Ω,S,P) is the norm in L∞ (Ω,S,P);
(ii) For each s ∈ S, k (t, s; w) is continuous in t ∈ S from S into L∞ (Ω,S, P); for each t ∈ S, k (t, s; w) is
continuous in s ∈ S from s into L∞ (Ω, S, P) and
(iii) There exists a positive real-valued function H on S such that H (S) || x (s; w) ||
n
2L
is µ-integrable for x
∈ c (s,
n
2L (Ω, S, P) and such that for each t, s ∈ S.
||| K (t, u; w) – K (s, u; w) ||| . || x (u; w) ||
n
2L
< H (U) || x (u; w) || L
n
2
Thus, for each (t, s) ∈ S x S, we have K (t, s; w) x (s; w) ∈ L
n
2
(Ω, S, P). we now define the random integral
operator T (w) on c (s, L
n
2
(Ω, S, P) by
[T (w) x ] (t; w) = ∫S
k (t, s; w) x (s; w) d µ (s) .......... (2.41)
where the integral is a Bochner Integral. Moreover, we have that for each t ∈ S, [T (w) x] (t; w) ∈ L
n
2
(Ω, S, P) and that is a continuous linear operator from c (s, L
n
2
(Ω, S, P)) into itself. We now have the
following theorem.
Theorem 2.5 : We consider the stochastic integral equation (1.1) subject to the following conditions :
(a) B and D are Banach spaces stronger (cf. [1]) then C (s, L
n
2
(Ω, S, P) such that (B, D) is admissible
with respect to the integral operator defined by (4.1);
(b) x ( t; w) → f (t, x (t; w)) is an operator from the set Q (ρ) = { x (t; w) : x (t; w) ∈ D, || x (t; w) ||D < ρ }
into the space B satisfying.
|| f (t, x (t; w) – f (t, y (t; w)) ||B
< Ψ1 (w) (|| x (t; w) – y (t; w) ||D )+ Ψ2 (w) (|| x (t; w) – f(t, x (t; w))||D)

331
+ Ψ3 (w) (||y (t; w) – t (t, y (t; w))||D)+ Ψ4 (w) (||x (t; w) – f (t, y (t; w))||D)
+ Ψ5 (w) (||y (t; w) – f (t, x (t; w))||D)
+ Ψ6
]||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||1
]||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||
DD
DD
−−+
−+−
+ Ψ7
]||))w;t(x,t(f)w;t(y(||)||))w;t(x,t(f)w;t(y[(||1
]||))w;t(x,t(f)w;t(y(||)||))w;t(y,t(f)w;t(y[(||
DD
DD
−−+
−+−
for x (t; w), y (t; w) ∈ Q (ρ), where Ψi (w), i = 1, 2, ......, 7 are non-negative real-valued upper semi-
continuous functions satisfying Ψi (w) (r) <
7
r
for r > 0 and Ψi (w) (0) = O;
(c) h (t; w) ∈ D.
Then there exists a unique random solution of (1.1) in Q (ρ), provided c (w) < 1 and ||h (t; w) ||D + 2c (w) || f (t;
o)||B < ρ (1- c (w))
where c (w) is the norm of T (w).
Proof : Define the operator U (w) from Q (ρ) into D by
[U (w) x] (t; w) = h (t; w) + ∫S
k (t, s; w) f (s; w) dµ (s).
Now ||[U(w) x ] (t; w)||D < ||h (t; w) ||D + c (w)|| f (t, x (t; w) ||B
< ||(t;w)||D + c (w) ||f (t; 0)||B + c (w)|| f (t, x (t; w) – f (t; 0) ||B.
Then from the conditions of the theorem.
c (w) || f (t, x (t; w)) – f (t, 0)||B
< C (w) [Ψ1 (w) (||x (t; w)||D)]+ Ψ2 (w) (||x(t;w) – f (t, x (t; w)||D)
+ Ψ3 (w) ||f (t; o)||D)+ Ψ4 (w)(||x(t;w)||D) + Ψ5 (w) (|| f (t, x (t; w))||D)
+Ψ6
)]||)w;t(x[(||]||)w;t(x,t(f)w;t(x[(||1
)]||)w;t(x(||)||)w;t(x,t(f)w;t(x[(||
DD
DD
−+
+−
+Ψ7
)]||)w;t(x,t(f(||)||)0;t(f[(||1
)]||)w;t(x,t(f(||)||)0;t(f[(||
DD
DD
+
+
i.e.
7
5
cw|| f(t, x (t; w))–f (t; o)||B <
7
5
c (w) ρ +
7
5
c (w)||f (t; 0)||B
Hence
|| [ U (w)x ] (t; w)||D < ||h (t; w)||D + 2 c (w)|| f (t; 0)||B + c (w) ρ
< ρ (1- c(w)) + c (w) ρ
< ρ
Hence, [ U (w) x ] (t; w) ∈ Q (ρ)
Now, for x (t; w), y (t; w) ∈ Q (ρ) We have by condition (b)
|| [U (w) x ] (t; w) – [U (w)y] (t; w) ||D
= || ∫S
K (t, s; w) [f s, (x, w)) – f (s, y (s; w))] dµ (s) ||D
< c (w)|| f (t, x (t; w)) – f (t, y (t;w))||B
< c (w) [Ψ1 (|| x (t; w) – y (t; w)||D]+ Ψ2 (w) (|| x (t; w) – f (t, x(t, w)||D)
+ Ψ3 (w) (||y (t;w) – f (t, y (t; w)) ||D)+ Ψ4 (w) (||x (t; w) – f (t, y (t;w))||D)
+ Ψ5 (w) (||y (t; w) – f (t, x (t; w))||D)
+ Ψ6
)]||))w;t(y,t(f)w;t(x[(||)]||))w;t(x,t(f)w;t(x[(||1
)||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||
DD
DD
−−+
−+−
+ Ψ7
)]||))w;t(x,t(f)w;t(y[(||)]||))w;t(y,t(f)w;t(y[(||1
)||))w;t(x,t(f)w;t(y(||)||))w;t(y,t(f)w;t(y[(||
DD
DD
−−+
−+−
< Ψ1 (w) (||x (t; w) – y (t; w))||D) + Ψ2 (w) (||x (t; w)-f (t, x (t; w))||D)
+ Ψ3 (w) (||y (t; w) – f (t, y (t;w)||D) + Ψ4 (w) (||x (t;w) – f (t, y (t;w))||D)
+ Ψ5 (w) (||y (t;w) – f (t, x (t; w))||D)

332
+ Ψ6
)]||))w;t(y,t(f)w;t(x[(||)]||))w;t(x,t(f)w;t(x[(||1
)||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||
DD
DD
−−+
−+−
+Ψ7
)]||))w;t(x,t(f)w;t(y[(||)]||))w;t(y,t(f)w;t(y[(||1
)||))w;t(x,t(f)w;t(y(||)||))w;t(y,t(f)w;t(y[(||
DD
DD
−−+
−+−
Since c (w) < 1. Thus U (w) is a random non-linear contraction operator on Q (ρ). Hence, by theorem 2.2 there
exists a unique x – valued random variable x* (t; w) ∈ Q (ρ) which is a fixed point of U (w), that is x * (t; w) is
the unique random solution of the equation (1.1).
REFERENCES
1. Lee, Arthur C.H., Padgett, W.J. “On random non-linear contraction”, math. systems theory 11 (1977),
77-84.
2. Bharucha – Reid, A.T. “Random Integral Equations”, Academic Press, New York (1972).
3. Hans, O. “Random fixed point-theorems”, Trans. First prague conf. on information theory, statist.
decision functions on random process, (1957), 105-125.
4. Padgett, W.J. “On a non-linear stochastic integral equations of Hammerstein type”, proc. Amer. Math.
Soc. 38 (1973), 625-631.
5. Tsokos, C.P. “On a stochastic integral equation of voltera type, math”, systems theory 3 (1969), 222-
231.
6. Tsokos, C.P. , Padgett, W.J. “Random Integral equations with applications in life science and
engineering”, Academic Press, New York (1974).
7. Lee, A.C.H. , Padgett, W.J. “On heavily non-linear stochastic integral equations”, utilitas mathematics 9
(1976), 128, 138.
8. Achari, J. “A result on fixed points, kyungpook math”. Journal 19 (1979), 245-248.
10. Achari, J. “On a pair of Random generalized non-linear contractions”, Internal J. Math. and math. sci.
vol. 6 No. 3 (1983) 467-475.
11. Pittnaur, F. “A fixed point theorem in complete metric spaces, to appear in periodica math”. Hungarica.

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