Page 120 - IJOCTA-15-1

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

E∥(Φy)(ς) − (Φ¯y)(ς)∥ p R 4 = E
X T α (ς − ς k )

n 0<ς k <ς
p
p
p
p
≤ 3 p−1 2 p−1 M (1 + M ) l + 2 p−1 M ς p 2 2
p
g
ς
σ 1
k k
× [I (y ς k + z ς k ) − I (¯y ς k + z ς k )]
p p p p−1 p ˜ p p p o
× L (1 + M ) l + 2 M C p ς 2 L l ∞ p
f σ 2 ς G p−1 p p X 2 q X 2
≤ 2 M l L k L k
ς
× sup E∥y(ϑ) − ¯y(ϑ)∥ p 0<ς k <ς
ϑ∈[0,ς 1 ] k=1 p
sup
p
:= Θ 1 sup E∥y(ϑ) − ¯y(ϑ)∥ , × ϑ∈(ς k ,ς k+1 ] E∥y(ϑ) − ¯y(ϑ)∥ .
ϑ∈[0,ς 1 ]
These together with (5), we obtain
n
p
p
p
where Θ 1 := 6 p−1 p p p )+M ς ς L (1+
l
f
M g (1+M σ 1
p p ˜ p p o E∥(Φy)(ς) − (Φ¯y)(ς)∥ p
M σ 2 ) + M ς C p ς 2 L G .
n p
p
p
p
p
p
≤ 5 p−1 2 p−1 M (1 + M ) l + 2 p−1 M ς L f
g
ς
σ 1
p
p
Case (ii): For t ∈ (ς k , ς k+1 ], k = 1, m; × (1 + M ) l + 2 p−1 M C p ς 2 L l p
p
p ˜
p
σ 2 ς G
E∥(Φy)(ς) − (Φ¯y)(ς)∥ p ∞ p
X q X
p p
+ 2 p−1 M l L 1 L 1
n
ϑ Z ϑ k k
≤ 5 p−1 E g(ς, y ς + z ς , σ 1 (ϑ, τ, y τ + z τ )dτ) k=1 0<ς k <ς

∞ p
0 p−1 p p X 2 q X 2 o
+ 2 M l L k L k
ς
ϑ Z

p k=1 0<ς k <ς
− g(ς, ¯y ς + z ς , σ 1 (ϑ, τ, ¯y τ + z τ )dτ)
× sup E∥y(ϑ) − ¯y(ϑ)∥ p
0
ϑ∈(ς k ,ς k+1 ]

ς Z ϑ Z p
:= Θ 2 sup E∥y(ϑ) − ¯y(ϑ)∥ ,
T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , σ 2 (ϑ, τ, y τ + z τ )dτ)

ϑ∈(ς k ,ς k+1 ]
+ E
0 0
where
ϑ Z
p
− f(ϑ, ¯y ϑ + z ϑ , σ 2 (ϑ, τ, ¯y τ + z τ )dτ) dϑ∥
n
p
p
p
p
0 Θ 2 := 5 p−1 2 p−1 M (1 + M ) l + 2 p−1 M ς p
ς
g

ς Z σ 1 p
p p
T α (ς − ϑ)[G(ς, ϑ, y ϑ + z ϑ ) × L (1 + M ) l + 2 M C p ς 2 L l

p p p−1 p ˜ p
f σ 2 G
+ E
ς
0 ∞ p
X q X
p p
− G(ς, ϑ, ¯y ϑ + z ϑ )]dw(ϑ)∥ p + 2 p−1 M l L 1 L 1
ϑ k k

X 1 1
p k=1 0<ς k <ς
+ E
)]
k
k + z ς k ) − I (¯y ς k + z ς k
S α (ς − ς k )[I (y ς k
0<ς k <ς p−1 p p ∞ 2 p q X 2 o
X
+ 2 M l L L .
o

X
p ς k k
2 2
+ E
T α (ς − ς k )[I (y ς k + z ς k ) − I (¯y ς k + z ς k )]
.
k k k=1 0<ς k <ς
0<ς k <ς
Now, we take Θ = max{Θ 1 , Θ 2 } s.t
ς∈J
Since other estimates are similar to that of as in
ς ∈ [0, ς 1 ], it is enough to examine R 4 and R 5 . By p p
E∥(Φy)(ς)−(Φ¯y)(ς)∥ ≤ Θ max E∥y(ϑ)−¯y(ϑ)∥ .
(H 1 ) and (H 6 ), we estimate ϑ∈J
By Lemma 3, the operator Φ has a unique fixed

X
p point on H.
1 1
R 4 = E
S α (ς − ς k )[I (y ς k + z ς k ) − I (¯y ς k + z ς k )]
k k
0<ς k <ς
∞ p
p X 1 q X 1 p
≤ M L ∥
k
ϑ k L E∥y ς k − ¯y ς k
k=1 0<ς k <ς
∞ p X
X
p p
≤ 2 p−1 M l L 1 q L 1 4.1. Exponential stability
ϑ k k
k=1 0<ς k <ς
p
× sup E∥y(ϑ) − ¯y(ϑ)∥ . In this section, in order to evaluate the exponen-
ϑ∈(ς k ,ς k+1 ] tial stability for the mild solution of (1), addi-
tional assumption is imposed as follows:
Similarly,
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