Page 125 - IJOCTA-15-1

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

refers the complete orthogonal basis in H and
∞ √ n
P p−1 p−1 ¯ p p p p−1 p p p p p
w(ς) = λ n w n (ς)Z n , ς ≥ 0, λ n > 0, where 6 2 M (1 + M ) l + 2 M ς L (1 + M ) l
ς
g
f
σ 1
σ 2
n=1
{w n (ς)} n≥0 are one-dimensional standard Bm. p−1 p ˜ p p p p−1 p p ∞ 1 p q
X
+ 2 M C p ς 2 L l + 2 M l L k
ς
Define the operator A : D(A) ⊂ H → H by A = G ϑ
∂ 2 2 1 k=1
∂ς 2 , with domain D(A) = H ([0, π]) ∩ H ([0, π]), p−1 p p ∞ 2 p o
q
X
where + 2 M l L k < 1.
ς
k=1
n p
p
p ˜
p
p
¯ p
12 p−1 p M (1 + M ) + M ς ˆa 3 + M C p ς 2 ˆa 5
l
ς
g
ς
∂C σ 1
2
1
2
H ([0, π]) = {C ∈ L ([0, π]), ∈ L ([0, π]), ˆ a 7 π 2 ˆ a 8 π 4 o
∂Z + + < 1.
C(0) = C(π) = 0}, 6 90
∂C Then, we can easily obtain the result that the sys-
2
2
2
H ([0, π]) = {C ∈ L ([0, π]), ∈ L ([0, π])}. tem (12) has a mild solution. It is evident that
∂Z
all the assumptions are satisfied with a 1 = a 2 =
1
2
Then 1, β 1 = β 2 = π, L = ˆ a 7 2 , L = ˆ a 8 4 ,
k
k
k
k
n p 1−p p
p
¯ p
p
∞ 8 p−1 M (1 + M ) + [a β [L (1 + M )]
g
X 2 σ 1 2 2 f σ 2
Ax = −n (x, Z n ), x ∈ D(A), 1− p
p
˜
L
L G ] + a β
n=1 + a C p β 2 2 ¯ p 2 −p CHς pH−1 ˆ
2
2
∞ p ∞
q X q X p o
q
where Z n (ξ) = 2 sin(nξ) is the orthonormal + a p L 1 + a p L 2 < 1.
π 1 k 2 k
set of eigenfunctions of A. A is the infinitesimal k=1 k=1
generator of a strongly continuous cosine family p−1 n p 1−p p 1− p
8 ˆ a 1 + a β ˆ a 3 + a β 2 ˜
C p ˆa 5 + ˆa 6
{C(ς), ς ∈ R}, define as 2 2 2 2
p ˆ a 7 π 2 p ˆ a 8 π 4 o
+ a + a < 1.
∞ 1 6 2 90
X
C(ξ)Z = cosnξ < (x, Z n ), Z ∈ H,
Thus, all the assumptions (H 1 )−(H 7 ) and (H 9 ) of
n=1
Theorem 2 are satisfied. So we conclude that the
mild solution of the system (12) is exponentially
and the associated sine family is given by
stable.
∞ 6. Conclusion
X 1
S(ξ)Z = − sinnξ < (x, Z n ) >, Z ∈ H.
n The existence and uniqueness results have been
n=1
investigated for a class of higher-order impul-
sive FNSIDDEs with nonlocal conditions driven
Next, we give a special B-space. Let l(ϑ) =
0 by mixed fBm with infinite delays in a Hilbert
R
2ϑ
e , ϑ < 0 then, l 0 = l(ϑ)dϑ = 1 and define space. The well-posedness of mild solution for
2
−∞ (1) is achieved by using the BFPT along with
theories of the semigroup of linear operators and
0
Z stochastic processes. Some novel sufficient con-
1
p
∥φ∥ B = l(ϑ) sup (E∥φ(θ)∥ ) dϑ. ditions have been derived by using BFPT. Next,
p
ϑ≤θ≤0
−∞ the exponential stability of mild solution higher-
order impulsive FNSIDDEs with nonlocal con-
It follows from Hino, 36 that (B b , ∥·∥ B b ) is a Banach ditions driven by mixed fBm in stochastic set-
space. Thus, for (ς, φ) ∈ J × B b , where φ(θ)(x) = tings has been obtained by the integral inequality
φ(θ, x)(θ, x) ∈ (−∞, 0]×[0, π], Z(ς)(x) = Z(ς, x). technique. Furthermore, it was shown that sys-
tem is well-posed even if the system (1) is inte-
grated along with the nonlocal conditions. Mov-
Accordingly, all the assumptions of Theorem 1 are ing ahead in this direction, exponential stability
satisfied and in particular, ς = 1 , M ϑ = M ς = of non-instantaneous impulsive neutral stochastic
2
1
˜
= 0.01, M g = 0.02, α = , C p = 1, l = integro-differential equations of Sobolev type in-
4
1, M σ 1
0.5, ˆa 1 = 0.075, ˆa 2 = 0.003, ˆa 3 = 0.001, ˆa 5 = volving Poisson jumps can be considered in the
0.025, ˆa 7 = 0.002, ˆa 8 = 0.03, p = 2; future. The numerical example presented can be
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