Page 123 - IJOCTA-15-1

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

ς
Z p−1 p−1 p −pβ 1 ς p

p M 1 = 6 [3 a e [E∥φ∥
1

¯ ¯
B 5 ≤ E
T α (ς − ϑ)G(ς, τ, x τ )dw(τ)
p p−1 ¯ p
+ L ˜p E∥x ς 1 , x ς 2 , . . . , x ς n ∥ + 2 M g (E∥φ∥ + L ˜p )]]
0
p
p
a e
ς M 2 = 6 p−1 [2 p−1 p −pβ 2 ς [E∥x 1 ∥ + E∥η∥ ]
Z 2 i p 2
h
p ˜ −β 2 (ς−ϑ) p p 2
p
L].
≤ a C p e E∥G(ς, τ, x τ )∥ dϑ + a β −p CHς pH−1 ˆ
2
2 2
0
ς
Z
p ˜ 1− p 2 ¯ −β 2 (ς−ϑ) p
≤ a C p β L G e sup E∥x ϑ ∥ dϑ.
2 2 By Lemma 5 and equation (10), we have
ϑ∈[0,ς 1 ]
0
˜
p
By Lemma 2 and (H 9 ), we have E∥x(ς)∥ ≤ M 1 e −ως , ς ≥ −r, ω ∈ (0, σ 1 Λ σ 2 ),
˜

where M 1 = max M 1 + M 2 , M 3 := 6 p−1 ¯ p
M g (1 +
ς p p−1 p 1−p p p
Z ), M 4 := 6 a β )], M 5 :=
f
M σ 1 2 2 [L (1 + M σ 2
H
B 6 = E∥ T α (ς − ϑ)σ(ϑ)dB (ϑ)∥ p 1− p
Q
a C p β
6 p−1 p 2 ˜ 2 2 ¯
L G .
0
ς
Z 1 1
p −p pH−1 p For p ≥ 2 and 1 < q ≤ 2 with p + q = 1, we have
≤ a β CHς E∥σ(ϑ)∥ dϑ
2 2 the following estimations:
0
p −p pH−1 ˆ
≤ a β CHς L.
X
p
2 2
1
k
S α (ς − ς k )I (x ς k
B 7 = E
)
From the above estimates together with (9), we 0<ς k <ς
∞ p
have p X 1 q X 1 −β1(ς−ς k ) p
≤ a L L e E∥x ς k ∥
1 k k
k=1 0<ς k <ς
n p
X
p
p
E∥x(ς)∥ ≤ 6 p−1 3 p−1 a e −pβ 1 ς [E∥φ∥ p
2
1 B 8 = E
T α (ς − ς k )I (x ς k )
k
¯ p p−1 ¯ 0<ς k <ς
∞ p X
+ L ˜p E∥x ς 1 , x ς 2 , . . . x ς n ∥ + 2 M g
X
¯
p
p
2 −β 2 (ς−ς k )
a e
× (E∥φ∥ + L ˜p )] + 2 p−1 p −pβ 2 ς [E∥x 1 ∥ p ≤ a p L 2 q L e ∥ .
2 2 k k E∥x ς k
p
p
¯ p
+ E∥η∥ + M (1 + M ) sup E∥x ϑ ∥ p k=1 0<ς k <ς
g σ 1 n
ϑ∈[0,ς 1 ] p p−1 p−1 p −pβ 1 ς p
E∥x(ς)∥ ≤ 8 3 a e [E∥φ∥
1
p 1−p p p
+ [a β [L (1 + M )]
2 2 f σ 2 ¯ ∥ p
+ L ˜p E∥x ς 1 , x ς 2 , . . . , x ς n
p ˜ 1− p 2 ¯
+ a C p β 2 L G ]ϑ + 2 p−1 ¯ p ¯
M g (E∥φ∥ + L ˜p )]
2
ς
+ 2 a e [E∥x 1 ∥ + E∥η∥ ]
Z p−1 p −pβ 2 ς p p
2
p
× e −β 2 (ς−ϑ) sup E∥x ϑ ∥ dϑ
ϑ∈[0,ς 1 ] ¯ p p p
0 + M (1 + M ) sup E∥x ϑ ∥
g
σ 1
p −p pH−1 ˆ ϑ∈[0,ς 1 ]
+ a β 2 CHς L}. (10) p
2
p
p
p
p
˜
+ [a β 1−p [L (1 + M )] + a C p β 1− 2 ¯
L G ]
2 2 f σ 2 2 2
Then, the integral term in the above inequality ς
Z
(10) will be reduced to −β 2 (ς−ϑ) p
× e sup E∥x ϑ ∥ dϑ
ϑ∈[0,ς 1 ]
0
h p
p
E∥x(ς)∥ ≤ 6 p−1 3 p−1 a e −pβ 1 ς [E∥φ∥ p ∞ p
X
1 p −p pH−1 ˆ p 1 q
+ a β 2 CHς L + a 1 L k
2
¯ ∥ p
+ L ˜p E∥x ς 1 , x ς 2 , . . . , x ς n k=1
1 −β1(ς−ς k )
+ 2 p−1 ¯ p ¯ × X L e E∥x ς k ∥ p
M g (E∥φ∥ + L ˜p )]
k
p−1 p −pβ 2 ς p p 0<ς k <ς
+ 2 a e [E∥x 1 ∥ + E∥η∥ ]
2
∞ p
i X X o
p −p pH−1 ˆ p 2 q 2 −β 2 (ς−ς k ) p
+ a β CHς L + a 2 L k L e E∥x ς k ∥ .
k
2 2
k=1 0<ς k <ς
≤ M 1 e −pβ 1 ς + M 2 e −pβ 2 ς , β 1 , β 2 ≥ 0, (11)
where
117