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Exponential stability for higher-order impulsive fractional neutral stochastic integro-delay . . .
E∥(Φy)(ς) − (Φ¯y)(ς)∥ p
Z ς Z ϑ
T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , σ 2 (ϑ, τ,
ϑ R 2 = E
Z
n
≤ 3 p−1 E
g(ς, y ς + z ς , σ 1 (ϑ, τ, y τ + z τ )dτ) 0 0
y τ + z τ )dτ) − f(ϑ, ¯y ϑ + z ϑ ,
0
ϑ
ϑ Z
Z
p p
− g(ς, ¯y ς + z ς , σ 1 (ϑ, τ, ¯y τ + z τ )dτ)
σ 2 (ϑ, τ, ¯y τ + z τ )dτ) dϑ∥
0
0
ς ϑ
ς ϑ Z Z
Z Z
T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , σ 2 (ϑ, τ, y τ + z τ )dτ)
≤ E
T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , σ 2 (ϑ, τ,
+ E
0 0
0 0
y τ + z τ )dτ) − f(ϑ, ¯y ϑ + z ϑ ,
ϑ
Z
− f(ϑ, ¯y ϑ + z ϑ , σ 2 (ϑ, τ, ¯y τ + z τ )dτ) dϑ∥ p Z ϑ
σ 2 (ϑ, τ, ¯y τ + z τ )dτ) dϑ∥ p
0
ς
Z 0
T α (ς − ϑ)[G(ς, ϑ, y ϑ + z ϑ ) ≤ M ς L (1 + M )E∥y ϑ − ¯y ϑ ∥ .
p p p p p
+ E
ς f σ 2
0
o
− G(ς, ϑ, ¯y ϑ + z ϑ )]dw(ϑ)∥ p By using Lemma 4, we have
3
X
≤ 3 p−1 R i . (5)
i=1 p−1 p p p p h p
≤ 2 M ς L (1 + M ) ∥y 0 ∥
ς f σ 2
We have to compute the R.H.S of (5). By (H 2 ), p p p
we get the following. + l sup E∥y(ϑ)∥ − ∥¯y 0 ∥
ϑ∈[0,ς 1 ]
i
− l p sup E∥¯y(ϑ)∥ p
ϑ
Z ϑ∈[0,ς 1 ]
p
R 1 = E
g(ς, y ς + z ς , σ 1 (ϑ, τ, y τ + z τ )dτ) ≤ 2 p−1 M ς L (1 + M ) l p
p
p
p
ς
f
σ 2
p
0 × sup E∥y(ϑ) − ¯y(ϑ)∥ .
ϑ ϑ∈[0,ς 1 ]
Z
p
− g(ς, ¯y ς + z ς , σ 1 (ϑ, τ, ¯y τ + z τ )dτ)
0 By Lemma 1, and assumptions (H 1 ), (H 4 ), we
h
≤ M g p E∥y ς + z ς − ¯y ς − z ς ∥ p get
ς
Z
i
+ E∥ [σ 1 (ϑ, τ, y τ + z τ ) − σ 1 (ϑ, τ, ¯y τ + z τ )]dτ∥ p
ς
Z
0
h
h i R 3 = E
T α (ς − ϑ) G(ς, ϑ, y ϑ + z ϑ )
p
p
≤ M g p E∥y ς − ¯y ς ∥ + M E∥y ς − ¯y ς ∥ p
σ 1
0
p
p
p
≤ M (1 + M )E∥y ς − ¯y ς ∥ . − G(ς, ϑ, ¯y ϑ + z ϑ )dw(ϑ)∥ p i
g
σ 1
By Lemma 4, we have Z ς
h
p ˜
≤ M C p E∥G(ς, ϑ, y ϑ + z ϑ )
ς
h 0
p
p
p
≤ 2 p−1 M (1 + M ) ∥y 0 ∥ + l p sup E∥y(ϑ)∥ p 2 i 2
g
σ 1
ϑ∈[0,ς 1 ] p p p
− G(ς, ϑ, ¯y ϑ + z ϑ )∥ dϑ
i
p
− ∥¯y 0 ∥ − l p sup E∥¯y(ϑ)∥ p p ˜ p p p
ϑ∈[0,ς 1 ] ≤ M C p ς 2 L E∥y ϑ − ¯y ϑ ∥
G
ς
p
p
p
p
p
p ˜
p
≤ 2 p−1 M (1 + M ) l p sup E∥y(ϑ) − ¯y(ϑ)∥ . ≤ 2 p−1 M C p ς 2 L l p sup E∥y(ϑ) − ¯y(ϑ)∥ .
g σ 1 ς G
ϑ∈[0,ς 1 ] ϑ∈[0,ς 1 ]
By (H 1 ) and (H 3 ), we have the following estima-
tion: Using these equations in (5), we get
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