Page 117 - IJOCTA-15-1

Exponential stability for higher-order impulsive fractional neutral stochastic integro-delay . . .

ς ∗ p−1 p p p p−1 p
ϑ
Z b 4 l [b + M (E∥φ(0)∥ + L p )] + 4 ∥z 0 ∥
lim = lim
p
I 5 = E∥ T α (ς − ϑ)g(ϑ, τ, y τ + z τ )dw(τ)∥ b→∞ b b→∞ b
= 4 p−1 p
l .
0
ς
Z
≤ E ∥T α (ς − ϑ)g(ϑ, τ, y τ + z τ )dw(τ)∥ p
0 Thus,
ς
Z 2 i p
h
p ˜ p p 2
≤ M C p E∥g(ϑ, τ, y τ + z τ )∥ dϑ
ς
0
p
p ˜
≤ M C p ς 2 ¯ ∗ p−1 n p p p p p
L G b .
¯ p
ς
1 ≤ 24 M (1 + M ) + M ς [L (1 + M )]
g σ 1 ς f σ 2
By (H 1 ), (H 7 ) and Lemma 2, one can obtain p o
p ˜
+ M C p ς 2 ¯ p
L G l ,
ς
ς
Z
H
I 6 = E∥ T α (ς − ϑ)σ(ϑ)dB (ϑ)∥ p
Q
0 which contradicts (H 8 ). Hence the claim.
ς
Z
H
≤ E ∥T α (ς − ϑ)σ(ϑ)dB (ϑ)∥ p
Q
0
ς
Case (ii): For ς ∈ (ς k , ς k+1 ], k = 1, m. For this
Z
p
p
≤ M CHς pH−1 E∥σ(ϑ)∥ dϑ case, it is enough to find I 7 , I 8 , since other esti-
ς
0 mates are similar to that of as in case (i). Now,
p
≤ M CHς pH−1 ˆ by (H 1 ), (H 6 ) we get
L.
ς
Combining estimates I 1 − I 6 , we have
n p

p
¯
¯
p
b ≤ 6 p−1 2 p−1 M M g [E∥φ∥ + L p ]
X 1
ϑ
k + z ς k )
S α (ς − ς k )I (y ς k
I 7 = E
p
p
¯ p
p
p
+ 2 p−1 M [E∥y 1 ∥ + E∥η∥ ] + M (1 + M ) b ∗ 0<ς k <ς
ς
g
σ 1
X 1 p
p
p
+ M ς [L (1 + M )] b + M C p ς 2 ¯ ∗ ≤ E∥S α (ς − ς k )I (y ς k + z ς k )∥
∗
p ˜
p
p p
k
L G b
ς
ς
f
σ 2
0<ς k <ς
o
p
+ M CHς pH−1 ˆ ∞ p
L
ς p X q X
¯ 1 ∗
¯ 1
≤ M L L b .
n p ϑ k k
¯
p
¯
≤ 6 p−1 2 p−1 M M g [E∥φ∥ + L p ] + 2 p−1 M p k=1 0<ς k <ς
ϑ ς
p
p
p
¯ p
× [E∥y 1 ∥ + E∥η∥ ] + [M (1 + M )
g
σ 1
p
p
p
p ˜
p p
+ M ς [L (1 + M )] + M C p ς 2 ¯ ∗
L G ]b
ς
ς
f
σ 2
Similarly,
o
p
+ M CHς pH−1 ˆ (3)
L
ς
:= b 0 ,
where

p
X
2

k
T α (ς − ς k )I (y ς k + z ς k
I 8 = E
)
0<ς k <ς
n
p
¯
¯
p
b 0 = 6 p−1 2 p−1 M M g [E∥φ∥ + L p ] X
2
ϑ ≤ E∥T α (ς − ς k )I (y ς k + z ς k )∥ p
k
p
p
p
¯ p
p
+ 2 p−1 M [E∥y 1 ∥ + E∥η∥ ] + [M (1 + M ) 0<ς k <ς
g
ς
σ 1
p
p ˜
p
+ M ς [L (1 + M )] + M C p ς 2 ¯ ∗ p ∞ p q X
p
p p
L G ]b
¯ 2
X
¯ 2 ∗
ς f σ 2 ς ≤ M ς L k L b .
k
k=1 0<ς k <ς
o
p
L .
+ M CHς pH−1 ˆ
ς
Dividing both sides of (3) by b and letting b → ∞,
we have From these together with,
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