Page 115 - IJOCTA-15-1

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

Theorem 1. Assume that the assumptions (H 1 )
- (H 7 ) hold, then the Cauchy problem (1) has a
unique mild solution defined on J T , provided that
then z ς ∈ B T . If x(·) satisfies the system (1),
n p then we can decompose x(·) as x(ς) = y(ς)+z(ς),
p
p
p
p
p
6 p−1 2 p−1 M (1 + M ) l + 2 p−1 M ς L f which implies x ς = y ς + z ς for ς ∈ J iff y satisfies
ς
g
σ 1
′
′
p
p ˜
p
p
× (1 + M )l + 2 p−1 M C p ς 2 L p l p y 0 = 0, x (0) = x 1 , y (0) = y 1 .
σ 2 ς G
∞ p
X 1 q p−1 p p
p p
p−1
+ 2 M l L + 2 M l
ϑ k ς (Φy)(ς)
k=1 
∞  0, ς ∈ J 0

p o

X q  ), 0) + T α (ς)[y 1 + η]
× L 2 < 1.  S α (ς)g(0, φ + ˜p(y ς 1 , y ς 2 , . . . , y ς n

k 
 ς R

k=1  −g(ς, y ς + z ς , σ 1 (ς, τ, y τ + z τ )dτ)


 0



 ς R ϑ R
 + T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , σ 2 (ϑ, τ, y τ + z τ )dτ) dϑ



Proof. We consider the operator Φ : B T → B T  0 0

 ς R
defined by  + T α (ς − ϑ)G(ς, ϑ, y ϑ + z ϑ )dw(ϑ)




 0


 ς R
 H
(Φx)(ς)  + T α (ς − ϑ)σ(ϑ)dB (ϑ), ς ∈ [0, ς 1 ]

Q


 0

  .

), ς ∈ J 0  .
 .
φ(ς) + ˜p(x ς 1 , x ς 2 , . . . , x ς n


 
 ) ), 0) + T α (ς)[y 1 + η]
 S α (ς)[φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n S α (ς)g(0, φ + ˜p(y ς 1 , y ς 2 , . . . , y ς n
 =
 ς R
 ), 0)] 
 +g(0, φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n  −g(ς, y ς + z ς , σ 1 (ς, τ, y τ + z τ )dτ)

 
 +T α (ς)[x 1 + η]  0




 
 ς  ς R ϑ R

 R  + T α (ς − ϑ) f(ϑ, y ϑ + z ϑ , σ 2 (ϑ, τ, y τ + z τ )dτ) dϑ
 −g(ς, x ς , σ 1 (ς, τ, x τ )dτ) 



 0 0
 
 0 
  ς R

ς ϑ  + T α (ς − ϑ)G(ς, ϑ, y ϑ + z ϑ )dw(ϑ)



 R R 
 + T α (ς − ϑ) f(ϑ, x ϑ , σ 2 (ϑ, τ, x τ )dτ) dϑ  0





  ς R H

 0 0  + T α (ς − ϑ)σ(ϑ)dB (ϑ)
ς Q

 
 R 
 + T α (ς − ϑ)G(ς, τ, x τ )dw(τ)  0 P 1


 +


S α (ς − ς k )I (y ς k + z ς k )
  k
0 0<ς k <ς
 
 
 ς  P 2

 
R  + ),
k
 H T α (ς − ς k )I (y ς k + z ς k
 + T α (ς − ϑ)σ(ϑ)dB (ϑ), ς ∈ [0, ς 1 ]  0<ς k <ς

Q

 

0 

 ∀ ς ∈ (ς k , ς k+1 ], k = 1, m.
 .

 .

 .
= S α (ς)[φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n )

 ), 0)]
 +g(0, φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n 0 0
 Let B = y : y ∈ B T , y 0 = 0 , for any y ∈ B is
 ς T T

R
 +T α (ς)[x 1 + η] − g(ς, x ς , σ 1 (ς, τ, x τ )dτ) endowed with the norm



0


ς ϑ



R R
 + T α (ς − ϑ) f(ϑ, x ϑ , σ 2 (ϑ, τ, x τ )dτ) dϑ


1

0 0 p 0
 p
 ∥y∥ T = ∥y∥ 0 = ∥y 0 ∥ B +sup E∥y(ς)∥ , y ∈ B ;
 ς B T

R T

 + T α (ς − ϑ)G(ς, τ, x τ )dw(τ) ς∈J


0


 ς
 0
H
R then (B , ∥·∥ T ) is a Banach space. Set B b ⊆ {y ∈

 + T α (ς − ϑ)σ(ϑ)dB (ϑ) T

 Q 0 P 0
0
 B : ∥y∥ ≤ b} for some b > 0. Then B b ⊆ B

 P 1 T T T
 + )

S α (ς − ς k )I (x ς k is a closed, bounded and convex subset. For any
 k
0<ς k <ς


 y ∈ B b , we deduce from Lemma 4 that
P 2

 + T α (ς − ς k )I (x ς k ),

 k
0<ς k <ς



∀ ς ∈ (ς k , ς k+1 ], k = 1, m.
∥y ς + z ς ∥ p
B
p
≤ 2 p−1 ∥y ς ∥ + 2 p−1 ∥z ς ∥ p
In order to show the existence of mild solution B B
l (E∥y ϑ ∥ + ∥y 0 ∥) + 4
of the Cauchy problem (1), it is enough to prove ≤ 4 p−1 p p p−1 p p B
l (E∥z ϑ ∥ + ∥z 0 ∥)
B
that Φ has a unique fixed point. For φ ∈ B, we p−1 p p p p−1 p
≤ 4 l [b + M (E∥φ(0)∥ + L p )] + 4 ∥z 0 ∥
define ϑ B
∗
:= b .

)(ς) if ς ∈ J 0
φ(ς) + ˜p(x ς 1
z ς = 0 0
, x ς 2
, . . . , x ς n
T
T
S α (ς)[φ(ς) + ˜p(x ς 1 , x ς 2 , . . . , x ς n )(ς)] if ς ∈ J, Now, we define the operator Φ : B → B by
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