Page 116 - IJOCTA-15-1

D. Kasinathan et.al. / IJOCTA, Vol.15, No.1, pp.103-122 (2025)

(Φy)(ς) y τ + z τ )dτ) dϑ∥ p

 0, ς ∈ J 0 ς
Z


 ), 0) p
 S α (ς)g(0, φ + ˜p(y ς 1 , y ς 2 , . . . , y ς n + E∥ T α (ς − ϑ)G(ς, ϑ, y ϑ + z ϑ )dw(ϑ)∥

 +T α (ς)[y 1 + η]


ς 0


 R
 −g(ς, y ς + z ς , σ 1 (ς, τ, y τ + z τ )dτ) ς


 Z o
 0 H p
 + E∥ T α (ς − ϑ)σ(ϑ)dB (ϑ)∥
ς

 R Q
 + T α (ς − ϑ) f(ϑ, y ϑ + z ϑ ,


 0
 0


 ϑ 6
 R X
 p p−1
 σ 2 (ϑ, τ, y τ + z τ )dτ) dϑ E∥Φy(ς)∥ ≤ 6 I i . (2)


 0

ς i=1

 R
 + T α (ς − ϑ)G(ς, ϑ, y ϑ + z ϑ )dw(ϑ)



0

 We compute the terms on the R.H.S of (2). From
 ς

R H
 + T α (ς − ϑ)σ(ϑ)dB (ϑ), ς ∈ [0, ς 1 ] hypothesis (H 1 ), we have the following estima-


 Q
0

 tion:
 .


 .
 .

), 0)

= p
S α (ς)g(0, φ + ˜p(y ς 1
, . . . , y ς n
, y ς 2
 +T α (ς)[y 1 + η] I 1 = E∥S α (ς)g(0, φ + ˜p(y ς 1 , y ς 2 , . . . , y ς n ), 0)∥

 ς
 p p
R ), 0)∥
 −g(ς, y ς + z ς , σ 1 (ς, τ, y τ + z τ )dτ) ≤ ∥S α (ς)∥ E∥g(0, φ + ˜p(y ς 1 , y ς 2 , . . . , y ς n



0 p−1 p
 p
 ¯ ¯
 ς ≤ 2 M M g [E∥φ∥ + L p ]
 ϑ
R
 + T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , p


 I 2 = E∥T α (ς)[y 1 + η]∥
0



 ϑ p−1 p p p

 R ≤ 2 M [E∥y 1 ∥ + E∥η∥ ].
ς
 σ 2 (ϑ, τ, y τ + z τ )dτ) dϑ


 0

 ς

R
 By using hypothesis (H 2 ), we get the following
 + T α (ς − ϑ)G(ς, ϑ, y ϑ + z ϑ )dw(ϑ)


 estimation:
0

 ς

R
 H
 + T α (ς − ϑ)σ(ϑ)dB (ϑ)

 Q

 0
 P 1 ϑ
 + S α (ς − ς k )I (y ς k + z ς k ) Z


 k
0<ς k <ς p
 σ 1 (ϑ, τ, y τ + z τ )dτ)∥
 I 3 = E∥g(ς, y ς + z ς ,
P
 + ),
 2

k
T α (ς − ς k )I (y ς k + z ς k

 0
 0<ς k <ς

∀ ς ∈ (ς k , ς k+1 ], k = 1, m. ≤ M (1 + M )E∥y ς + z ς ∥
 ¯ p p p
g
σ 1
∗
¯ p
p
≤ M (1 + M ) b .
g
σ 1
To prove that Φ has a unique fixed point, it is By (H 1 ), (H 2 )(ii), (H 3 ) and H¨older’s inequality,
interesting to see that finding the fixed point of
Φ in H is same as finding the mild solutions of
(1) in H. Our proof will be divided into two steps. ς
Z

I 4 = E∥ T α (ς − ϑ) f(ϑ, y ϑ + z ϑ ,
Step 1: We claim that Φ(B b ) ⊂ B b . If it is not 0
true, then for each b > 0, ∃ a function y ∈ B b , Z ϑ

but Φ(B b ) ⊈ B b for ς ∈ J. σ 2 (ϑ, τ, y τ + z τ )dτ) dϑ∥ p
Case (i): For ς ∈ [0, ς 1 ],
0
b ≤ E∥(Φy)(ς)∥ p ς
Z
n ≤ E ∥T α (ς − ϑ) f(ϑ, y ϑ + z ϑ ,
≤ 6 p−1 E∥S α (ς)g(0, φ + ˜p(y ς 1 , y ς 2 , . . . , y ς n ), 0)∥ p
0
+ E∥T α (ς)[y 1 + η]∥ p ϑ
Z
p
ϑ
Z σ 2 (ϑ, τ, y τ + z τ )dτ) dϑ∥
+ E∥g(ς, y ς + z ς , σ 1 (ϑ, τ, y τ + z τ )dτ)∥ p
0
∗
p
p
p p
0 ≤ M ς [L (1 + M )] b .
ς
f
σ 2
ς ϑ
Z Z

+ E∥ T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , σ 2 (ϑ, τ, By Lemma 1, assumptions (H 1 ) and (H 4 ), we es-
timate
0 0
110