Existence Theory for Second Order Nonlinear Functional Random Differential Equation in Banach Algebra

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. 1 (Jan - Feb. 2015), PP 06-12
www.iosrjournals.org
DOI: 10.9790/5728-11110612 www.iosrjournals.org 6 | Page
Existence Theory for Second Order Nonlinear Functional
Random Differential Equation in Banach Algebra
Mrs. M. K. Bhosale1
, Dr. R. N. Ingle2
1
Dept. of MathematicsShri. ChhatrapatiShivajiMaharaj College of Engineering
Dist. Ahmednagar/Pune University,Maharashtra. India.
2
Dept. of Mathematics BahirjiSmarak Mahavidyalaya, Basmathnagar /SRTMUN University Nanded,
Maharashtra. India.
Abstract: In this paper we prove the existence of the solution for the second order nonlinear functional
random differential equation in Banach Algebra under suitable condition.
2000 Mathematics Subject Classification: 47H10, 34F05.
Keywords and Phrases: functional Random differential equation, Existence theorem etc.
I. Introduction
Consider the second order nonlinear functional random differential equation (in short NFRDE)
x(t,ω)
f(t,x t,ω ,ω)
′′
= g(t, xt ω , ω) a.e. t∈ I
x0 ω = φ0 (ω)
x0
′
(ω) = φ1 (ω)
for all ωϵ Ω where
f:I × ℝ × Ω → ℝ; g:I × C × Ω → ℝ ; φ1,φ2 ∶ Ω → ℝ
We shall obtain the existence of the random solution of the above NFRDE in the space
x = C(J, ℝ)⋂C(I0, ℝ)⋂AC(J, ℝ) under some suitable condition.
II. Statement Of Problem
Let ℝ denote the real line and Let I0 = [−r, 0] and I = [0, a] be two closed and bounded interval in ℝ for some
r> 0 and a > 0. Let 𝐽 = I0 UI. Let C(I0,ℝ) denote the space of continuous ℝ valued function I0 . We equip the
space C = C(I0,ℝ) with a supremum norm ∥.∥c defined by
∥x∥c = suptϵI0
|x(t)|
Clearly C is a Banach Space which is also a Banach Algebra with respect to this norm.
For a given t ϵI define a continuous R-valued function.
xt: I0 → ℝ by
xt(θ) = t + θ , θ ϵI0
Let Ω, A be a measurable space. Given a random variable φ ∶ Ω → C
We consider a Nonlinear Functional Random Differential Equation (in short NFRDE)
x(t,ω)
f(t,x t,ω ,ω)
′′
= g(t, xt ω , ω) a.e. t∈ I
x0 ω = φ0 (ω) ---(1.1)
x0
′
(ω) = φ1 (ω)
for all ωϵ Ω where f:I × ℝ × Ω → ℝ; g:I × C × Ω → ℝ{0} ; φ1,φ2 ∶ Ω → ℝ.
Theorem 2.1 (Dhage 1) let X be a Banach algebra and let A,B,C: X →X be three operator such that
a) A and C are D- Lipschitzicians with D-functions φ and ψ respectively.
b) B is compact and continuous.
c) M φ r + ψ r < r , r > 0 where M= B(X) = sup {Bx: x ∈ X}.
Then
(i) The operator equation λ A(x/ λ) B x + λ C(x/ λ) = x has a solution for λ = 1 or
(ii) The solution set є={ u ∈ X / λ A(x/ λ) B x + λ C(x/ λ) = x ; 0< λ <1 } is unbounded.
Before going to the main result of this paper, we state the following two useful lemmas.
Lemma 2.1: ( Dhage 5): Assume that all the conditions of theorem 2.1 hold then map T: X → X define by
T x = Ax B x + C x is continuous on X.

Existence Theory For Second Order Nonlinear Functional Random Differential Equation In…
Lemma 2.2: ( Dhage 6): Assume that all the conditions of theorem 2.1 hold then the set
Fix(T) = { x∈ X / Ax B x + C x = x} is compact.
Theorem 2.2 : let X be a separable Banach Algebra and let A,B,C: Ω × X →X be three random operator
satisfying for each ωϵ Ω
a) A(ω) and C(ω) are D- Lipschitzicians with D-functions φA(ω)and φAc ω respectively .
b) B(ω) is compact and continuous.
c) M (ω)φA(ω) r + φC(ω) r < r , r > 0 for all ωϵ Ω where M(ω)=∥B ω (x) ∥
d) The set ε ={ u ∈ X / λ(ω)A(ω) (u/ λ) B(ω)u + λ(ω)C(ω) (u/ λ) = u } is bounded for all measurable λ:
Ω→ℝ with 0 < λ ω < 1 Then the random equation
A (ω) x B (ω) x + C (ω) x = x ---(2.1)
has a random solution
Corollary 2.1 : Let X be a separable Banach Algebra and let A,B,C: Ω × X →X be three random operator
a) A(ω) and C(ω) are D- Lipschitzicians with Lipschitz costant α(ω)and β(ω) respectively
c) α ω M ω + β ω < 1 for all ωϵ Ωwhere M(ω)=∥B ω (x) ∥
d) The set ε ={ u ∈ X / λA(ω) (u/ λ) B(ω)u + λ(ω)C(ω) (u/ λ) = u } is bounded for all 0< λ<1
Then the random equation (2.1) has a random solution and the set of such random solution is compact.
On taking C(ω) = 0 in theorem (2.2) we obtain
Theorem 2.3: Let X be a separable Banach Algebra and A,B: Ω × X →X be two random operator satisfying for
each ωϵ Ω
a) A(ω) is D- Lipschitzicians with D-functions φA(ω) .
c) M (ω)φA(ω) < r , r > 0 for all ωϵ Ω where M(ω)=∥B ω (x) ∥
d) The set є ={ u ∈ X / λ(ω)A(ω) (u/ λ) B(ω)u = u } is bounded for all measurable
λ: Ω→ℝ with 0 < λ ω < 1
Then the random equation
A(ω)x B(ω)x = x ---(2.2)
has a random solution.
Corollary 2.2 : Let X be a separable Banach Algebra and let A,B: Ω × X →X be two random operator
a) A(ω) is D- Lipschitzicians with Lipschitz costant α ω .
c) α ω M ω < 1 for all ωϵ Ωwhere M(ω)=∥B ω (x) ∥
d) The set є ={ u ∈ X / λ A(ω) (u/ λ) B(ω)u = u } is bounded for all 0< λ(ω)< 1
Then the random equation (2.2) has a random solution and the set of such random solution is compact.
In the following section we shall prove an existence of the random solution of a nonlinear functional random
differential equation (1.1) in Banach Algebra.
III. Existence Theory For Random Solution
Let M(𝐽,ℝ), B(𝐽,ℝ), BM(𝐽,ℝ)and C(𝐽,ℝ) denote respectively the space of all measurable, bounded,
bounded and measurable and continuous real-valued function on 𝐽. Notice that C(𝐽,ℝ)⊂ BM(𝐽,ℝ)⊂ M(𝐽,ℝ)
we shall obtain the existence of the random solution of the NFRDE (1.1) is the space X= C(𝐽,ℝ)⋂
C(I0 ,ℝ)⋂ AC(𝐽,ℝ)under some suitable condition .
Define a norm ∥.∥ in X by
∥x∥= maxtϵ J |x(t)| -- (3.1)
Clearly X is a separable Banach Algebra with this maximum norm. By L′
(𝐽, ℝ) we denote the space of all
Lebesgue integral real valued function on 𝐽 equipped with a norm ∥. ∥L′ given by

∥ x ∥L′ = x t ds
t1
t0
. --- (3.2)
Now the NFRDE(1.1) is equivalent to the functional Random Integral equation (in short FRIE)
f t, X t, ω , ω [ φ0 0, ω + φ1 0, ω t + g s, xs ω , ω ds ]
t
0
,tϵI
x(t,ω ) =
---(3.3)
φ t, ω if t є I0
i.e φ1 0, ω tf t, x t, ω , ω +
f t, x t, ω , ω [ φ0 0, ω + g s, xs ω , ω ds ]
t
0
,tϵI
x(t,ω) =
φ t, ω if t є I0
We need the following definition
Definition 3.1.:- A mapping β: 𝐽 × C× Ω → R is said to satisfy a condition of ω-Caratheodory or simply called
ω-Caratheodory if for each ωϵΩ
(i) t→ β t, x, ω is measurable for each x є C.
(ii) x→ β t, x, ω is continuous almost everywhere tϵI
Further a ω-Caratheodory function β is called Lω′ –Caratheodory if
(iii) there exist a function h: Ω→ L′
(𝐽,ℝ) such that
|β t, x, ω |≤ h(t,ω) a. e. t ϵI for all xϵ ℝ and ωϵΩ
We consider the following hypothesis in the sequel.
(H1) The function q:Ω → C(𝐽,ℝ) is measurable.
(H2)The function f:Ω → C(𝐽× ℝ,ℝ) is measurable and there exist a functionα1:Ω → B(I,ℝ) with bound ∥α1(ω)∥
satisfying for each ωϵΩ .
|f t, x, ω − f t, y, ω | ≤ α1 t, ω |x − y| a. e. t ϵI for all x, y є C
(H3) The function ω → g t, x, ω is measurable for all tϵI and
(H4) The function g t, x, ω is Lω′ –Caratheodory.
(H5) There exist function γ: Ω→ L′
(I,ℝ) with γ t, ω > 0 a.e. tϵI , for all ωϵΩ and conditions non decreasing
function ψ: 0, ∞ → (0, ∞) satisfying for each ωϵΩ.
|g t, x, ω | ≤ γ t, ω ψ( x ) a.e. tϵ𝐽 --- (3.4)
for all xє C.
Theorem 3.1: Assume that the hypothesis (H1) – (H5) holds. Suppose further that
ds
ψ(s)
∞
C1(ω)
> C2(ω) ∥ r ∥L′ ---(3.5)
where
𝐶1 𝜔 =
[1+𝐹 𝜔 ]∥𝜑(𝜔)∥ 𝐶
1−∥𝛼1 𝜔 ∥[∥𝜑 𝜔 ∥ 𝐶+∥𝑕 𝜔 ∥ 𝐿′]
𝐶2 𝜔 =
𝐹 𝜔
1−∥𝛼1 𝜔 ∥[∥𝜑 𝜔 ∥ 𝐶+∥𝑕 𝜔 ∥ 𝐿′]
Then the NFRDE (1.1) has a random solution on 𝐽.
Proof :- Let X = 𝐶(𝐽,ℝ) and define three mapping A, B,C: Ω×X→X by
f(t,x(t,𝜔),𝜔 ) if t є I
---- (3.6)

A(𝜔)x(t) =
1 if t є 𝐼0
and
𝜑0 0, 𝜔 + 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
0
if t є I
----(3.7)
B(𝜔)x(t) =
𝜑 𝑡, 𝜔 if t є 𝐼0
and
𝜑1 0, 𝜔 𝑡 𝑓(𝑡, 𝑥(𝑡, 𝜔), 𝜔 ) if t є I
----(3.8)
C(𝜔)x(t) =
𝜑1 𝑡, 𝜔 if t є 𝐼0
Then the FRIE (3.3) is transformed into the random operator equation
A(𝜔)x(t) B(𝜔)x(t) + C(𝜔)x(t) = x(t, 𝜔) ---(3.9)
for t є 𝐽 and 𝜔𝜖𝛺 .
We shall show that the operator A(𝜔), B(𝜔) and C(𝜔) satisfy all the conditions of corollary 2.1 on X.
This will be done in the following steps.
Step I :- First we show that A(𝜔) and B(𝜔) are random operator on X. Since the function f(t,x,𝜔) is measurable
in 𝜔 for all t є I and x є ℝ and since constant function is measurable on Ω the function 𝜔 →A(𝜔)x is
measurable for all x є X. Hence A(𝜔) is a random operator on X . Now by (H3) the function 𝜔 →g(t,x,𝜔) is
measurable for all t є I and x є C. We know that the Riemann integral in a limit of a finite sum of measurable
function, which is again measurable.
Therefore the function 𝜔 → 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
0
is measurable . Hence B(𝜔) is random operator on
X.
Similarly it is shown that C(𝜔) is a randome operator on X.
Again since the function
𝑓(𝑡, 𝑥(𝑡, 𝜔), 𝜔 ) if t є I
t→ A(𝜔)x(t) =
1 if t є 𝐼0
𝜑0 0, 𝜔 + 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
0
if t є I
t→ B(𝜔)x(t) =
𝜑1 0, 𝜔 𝑡 𝑓(𝑡, 𝑥(𝑡, 𝜔), 𝜔 ) if t є I
t→ C(𝜔)x(t) =
𝜑1 𝑡, 𝜔 if t є 𝐼0
are continuous. The function A(𝜔)x(t) ,B(𝜔)x(t) and C(𝜔)x(t) are continuous and hence bounded and
measurable on 𝐽 for each 𝜔𝜖𝛺 . Hence A(𝜔), B(𝜔), and C(𝜔) define the random operator A, B,C: Ω×X→X.
Step II: Next we show that A(𝜔) is Lipschitzian random operator on X. Let x, yє X. Then by (H1)
|A(𝜔)x(t) - A(𝜔)y(t) | =|𝑓(𝑡, 𝑥(𝑡, 𝜔), 𝜔 ) - 𝑓(𝑡, 𝑦(𝑡, 𝜔), 𝜔 )| ≤∥ 𝛼1 ∥∥ 𝑥(𝜔) − 𝑦(𝜔) ∥
for all t є I.

Similarly
| A(𝜔)x(t) - A(𝜔)y(t) | =0 ≤∥ 𝛼1 ∥∥ 𝑥(𝜔) − 𝑦(𝜔) ∥
for all t є 𝐼0.
Thus
|A(𝜔)x(t) - A(𝜔)y(t) | ≤∥ 𝛼1(𝜔) ∥∥ 𝑥(𝜔) − 𝑦(𝜔) ∥ for all t𝜖𝐽 and 𝜔𝜖𝛺 .
Taking the maximum over t in the above inequality. We obtain
|A(𝜔)x(t) - A(𝜔)y(t) | ≤∥ 𝛼1 ∥∥ 𝑥(𝜔) − 𝑦(𝜔) ∥
This shows that A (𝜔) is a Lipschitzian random operator on X with Lipschitz constant ∥𝛼1 (𝜔)∥.
Similarly it is shown that C(𝜔) is a Lipschitzian random operator on X with Lipschitz constant ∥𝛽1 𝜔 ∥.
Step III.: Next we show that B(𝜔) is a continuous and compact random operator on X. Using the standard
argument as in Granas et.al[9] it is shown that B(𝜔) is a continuous random operator on X. To show that B(𝜔)
is compact. It is sufficient to show that B(𝜔)(x) is uniformly bounded and equi-contiuous set in X for each
𝜔𝜖𝛺. First we show that B(𝜔)(x) is uniformly bounded for each 𝜔𝜖𝛺. Let xє X be arbitrary . Thus
𝜑0 0, 𝜔 + 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
0
if t є I
B(𝜔)x(t) =
for all 𝜔𝜖𝛺 Since g is 𝐿 𝑥
′
(𝜔)- Caratheodeory
We have
|𝐵(𝜔)𝑥(𝑡) | ≤ ∥ 𝜑0 𝜔 ∥ 𝐶+ 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
0
= ∥ 𝜑0 𝜔 ∥ 𝐶 +∥ 𝑕 𝜔 ∥ 𝐿′
Taking the maximum over t, one obtains ∥ 𝐵 𝜔 𝑥 ∥≤ K for all x𝜖 𝑋 where
K= ∥ 𝜑0 𝜔 ∥ 𝐶 +∥ 𝑕 𝜔 ∥ 𝐿′
This shows that 𝐵 𝜔 (𝑥) is a uniformly bounded subset of X for each. Secondly we show that 𝐵 𝜔 (𝑥) is an
equicontinuous set in X for each 𝜔𝜖𝛺.
Now there are three cases
Case I :- Let t,𝜏 ∈ 𝐼 Then for any x∈ 𝑋 we have by (3.7)
|𝐵 𝜔 𝑥 𝑡 - 𝐵 𝜔 𝑥 (𝜏)| ≤ | 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠 −
𝑡
0
𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝜏
0
|
≤ | 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
𝜏
|
≤ | 𝑕 𝑠, 𝜔 𝑑𝑠
𝑡
𝜏
|
=| 𝑝 𝑡, 𝜔 − 𝑝(𝜏, 𝜔) |
Where p(t) = 𝑕 𝑠, 𝜔 𝑑𝑠
𝑡
0
Now p is a continuous function on a compact interval I. So it is uniformly continuous there and hence
|𝐵 𝜔 𝑥 𝑡 - 𝐵 𝜔 𝑥 (𝜏)| → 0 as t → 𝜏 for each 𝜔𝜖𝛺
Case II :- Again let 𝜏𝜖𝐼0 and t ϵ I then we have
|𝐵 𝜔 𝑥 𝑡 - 𝐵 𝜔 𝑥 (𝜏)| ≤ |𝜑0 0, 𝜔 -𝜑0(𝜏, 𝜔)| + 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
𝜏
≤ |𝜑0 0, 𝜔 -𝜑0(𝜏, 𝜔)| + |𝑝 𝑡, 𝜔 -𝑝(𝜏, 𝜔)|
Where the function p defined above. Again 𝜑0 is a continuous on compact interval I0
And the function p is continuous on compact interval I, so they are uniformly continuous and hence
|𝐵 𝜔 𝑥 𝑡 - 𝐵 𝜔 𝑥 (𝜏)| → 0 as t→ 𝜏
Case III :- similarly t, 𝜏 ∈ 𝐼0 Thus we have
|𝐵 𝜔 𝑥 𝑡 - 𝐵 𝜔 𝑥 (𝜏)| = |𝜑 𝑡, 𝜔 - 𝜑(𝜏, 𝜔)| → 0 as t→ 𝜏

Thus in all three case we have
|𝐵 𝜔 𝑥 𝑡 - 𝐵 𝜔 𝑥 (𝜏)| → 0as t→ 𝜏 for t, 𝜏 ∈ 𝐼0 and 𝜔𝜖𝛺.
Hence 𝐵 𝜔 (𝑥) is equicontinuous set in X for each 𝜔𝜖𝛺. This further in view of Arzela Ascolli Theorem
implies that 𝐵 𝜔 (𝑥) is compact for each 𝜔𝜖𝛺. Hence 𝐵 𝜔 is a continuous and compact random operator on
X
Step IV :- Here
M 𝜔 = ∥ B 𝜔 (𝑥)∥
= sup { ∥ B 𝜔 (𝑥)∥ : xϵX }
= 𝑠𝑢𝑝𝑥∈𝑋 𝑚𝑎𝑥𝑡∈𝐽 𝐵 𝜔 𝑥 𝑡
≤∥ 𝜑 𝜔 ∥ 𝐶+ 𝑠𝑢𝑝𝑥∈𝑋 𝑚𝑎𝑥𝑡∈𝐽 𝑔 𝑠, 𝑥 𝑠 𝜔 , 𝜔 𝑑𝑠
𝑡
0
= ∥ 𝜑 𝜔 ∥ 𝐶 + ∥ 𝑕 𝜔 ∥ 𝐿′
Therefore
∥ 𝛼1 𝜔 ∥M(𝜔) + ∥ 𝛽1(𝜔) ∥ = ∥ 𝛼1 𝜔 ∥ [∥ 𝜑 𝜔 ∥ 𝐶 + ∥ 𝑕 𝜔 ∥ 𝐿′ ] +∥ 𝛽1 𝜔 ∥ 𝐶 for all 𝜔𝜖𝛺
Step V :- Finally we show that condition (d) of corollary (2.1) is satisfied . Let u𝜖 E be arbitrary. Then we have
for all 𝜔𝜖𝛺.
λu 𝑡, 𝜔 = 𝐴 𝜔 𝑢 𝑡 𝐵 𝜔 𝑢 𝑡 + 𝐶 𝜔 𝑢 𝑡
𝜑1 0, 𝜔 𝑡𝑓 𝑡, 𝑢 𝑡, 𝜔 , 𝜔 +
𝑓 𝑡, 𝑢 𝑡, 𝜔 , 𝜔 [ 𝜑0 0, 𝜔 + 𝑔 𝑠, 𝑢 𝑠 𝜔 , 𝜔 𝑑𝑠 ]
𝑡
0
,tϵI
=
𝜑 𝑡, 𝜔 if t𝜖𝐼0
For some real number λ > 1
Therefore
𝑢 𝑡, 𝜔 < 𝜆−1
𝜑 𝑡, 𝜔 + 𝜆−1
[𝜑1 0, 𝜔 𝑡𝑓 𝑡, 𝑢 𝑡, 𝜔 , 𝜔 +
𝑓 𝑡, 𝑢 𝑡, 𝜔 , 𝜔 [ 𝜑0 0, 𝜔 + 𝑔 𝑠, 𝑢 𝑠 𝜔 , 𝜔 𝑑𝑠 ]
𝑡
0
≤∥ 𝜑 𝜔 ∥ 𝐶 + 𝜆−1
|𝑓 𝑡, 𝑢 𝑡, 𝜔 , 𝜔 | + | 𝑓 𝑡, 𝑢 𝑡, 𝜔 , 𝜔 |[ |𝜑0 0, 𝜔 | + 𝑕 𝑠, 𝜔 𝑑𝑠 ]
𝑡
0
≤C1 𝜔 + C2 𝜔 + 𝛾(𝑠,
𝑡
0
𝜔) 𝛹(∥ 𝑢 𝑠 𝜔 ∥ 𝐶) ds ---(3.10)
Where
C1 𝜔 =
1+𝐹 𝜔 ∥𝜑 𝜔 ∥ 𝐶+𝐹 𝜔
1− ∥𝛼1 𝜔 ∥ ∥𝜑 𝜔 ∥ 𝐶 + ∥𝑕 𝜔 ∥ 𝐿′ −∥𝛽1 𝜔 ∥ 𝐶
And
C2 𝜔 =
𝐹 𝜔
1− ∥𝛼1 𝜔 ∥ ∥𝜑 𝜔 ∥ 𝐶 + ∥𝑕 𝜔 ∥ 𝐿′
Let m (t,𝜔) = 𝑠𝑢𝑝𝑡∈[−𝑟,𝑡]|u(t,𝜔)| Then one has |u(t,𝜔)|≤ m(t) and ∥ 𝑢𝑡 𝜔 ∥ 𝐶≤ m(t,𝜔) for all 𝑡𝜖𝐼 and 𝜔𝜖𝛺.
Then there is a 𝑡∗
∈[-r,t] such that Let m(t,𝜔) = |u(𝑡∗
,𝜔)| for all 𝜔𝜖𝛺
Hence from inequality (3.10) it follows that
m (t,𝜔) = |u(𝑡∗
,𝜔)|
≤ C1 𝜔 + C2 𝜔 𝛾(𝑠,
𝑡
0
𝜔) 𝛹(∥ 𝑢 𝑠 𝜔 ∥ 𝐶) ds
≤ C1 𝜔 + C2 𝜔 𝛾(𝑠,
𝑡
0
𝜔) 𝛹(m(s,𝜔)) ds
Put w (t, 𝜔)= C1 𝜔 + C2 𝜔 𝛾(𝑠,
𝑡
0
𝜔) 𝛹(m(s,𝜔)) ds
w’(t,𝜔) = C2 𝜔 𝛾 𝑡, 𝜔 𝛹(m(t,𝜔))
w(0,𝜔)= C1 𝜔
This further implies that
w’(t ,𝜔) ≤ C2 𝜔 𝛾 𝑡, 𝜔 𝛹(m(t,𝜔))

w(0,𝜔)= C1 𝜔
OR , ----(3.11)
𝑤’(𝑡 ,𝜔)
𝛹(𝑚 𝑡,𝜔 )
≤ C2 𝜔 𝛾 𝑡, 𝜔
w(0,𝜔)= C1 𝜔
Integrating from 0 to t yield that
𝑤’(𝑡 ,𝜔)
𝛹(𝑚(𝑡,𝜔)
𝑑𝑠 ≤
𝑡
0
C2 𝜔 γ s, ω ds
𝑡
0
By changing the variable formula we get
ds
Ψ(s)
w(t ,ω)
C1 ω
≤C2 ω γ s, ω ds
t
0
≤C2 ω γ s, ω ds
a
0
= C2 ω ∥ γ ω ∥L′
<
ds
Ψ(s)
∞
C1 ω
Now by an application of mean value theorem yield that there is a constant M >0 such that w(t , ω) ≤ m for all
t ϵ I and ωϵ Ω.
This further implies that |u(t,ω) | ≤ M for all t ϵ I and ωϵΩ.
Hence the set ε is bounded and condition (d) of corollary (2.1) yield
Hence the random operator equation (3.9) and consequently by the FRDE (1.1) has a random solution .
This completes the proof.
References
[1]. B.C. Dhage, Random Fixed Point theorems in Banach Algebras with applications to random integral equations, Tamkang J. Math.
34 (2003), pp. 29-43.
[2]. B.C. Dhage, A random version of Schaefer’s fixed point theorem with applications to functional integral equations. Tamkang J.
Math 35 (3) (2004), pp. 197-205.
[3]. C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), pp 53-72.
[4]. D.W. Boyd and J.S.W. Wong, On nonlinear contractions Proc. Amer. Math. Soc. 20 (1969), pp. 456-464
[5]. B.C. Dhage, On existence theorem for nonlinear integral equations in Banach algebras via fixed point technique, East Asian Math.J.
17 (1) (2001), pp. 33-45.
[6]. B.C. Dhage, Some algebraic and topological random fixed point theorem with applications to nonlinear random integral equations,
Tamkang J. Math. 35 (4) (2004).
[7]. S. Itoh, Random fixed point theorems for a multivalued contraction mappings, Pac. Jour. Math. Vol. 68. No. 1 (1977), pp. 85-90.
[8]. K. Kuratowskii and C. Ryll-Nardzewskii, A general theorem on Selectors. Bull. Acad. Polons, Sci. Ser. Math Sci. A str. Phys. 13
(1965), pp. 397-403.
[9]. A.Granas,R.B.Guenther,andJ.W.Lee,Some general existence principles in the Caratheodory theory of nonlinear differential
equations, J.Math.Pure.et. Appl.70(1991), pp. 153-196.

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