Page 113 - IJOCTA-15-1

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

random process independent of the Wiener pro-
0
∥f(x) − f(y)∥ ≤ L∥x − y∥, 0 < L < 1. cess. Here, L (K, H) refers the space of all Q-
Q
Hilbert Schmidt operators from K into H.
In this manuscript, we define the phase space B
3. Model description as follows:
Let l : J 0 = (−∞, 0] → (0, +∞) is continuous
In this paper, we consider the following second R 0
order FNSIDEs with impulses and nonlocal con- functions with l = q(ς)dς, for any τ > 0,
−∞
ditions driven by fBm described as follows: n
we define a Banach space (B, ∥ · ∥ B ). B := ϕ :
1
p
ς J 0 → H : (E∥ϕ(ς)∥ ) p is bounded and measur-
Z
C α able function on [−τ, 0],
D +[x(ς) + g(ς, x ς , σ 1 (ς, ϑ, x ϑ )dϑ)]
0
0
0
ς Z o
Z p 1
q(ϑ) sup (E∥ϕ(θ)∥ ) dϑ < ∞ .
p
= A[x(ς) + g(ς, x ς , σ 1 (ς, ϑ, x ϑ )dϑ)] ϑ≤θ≤0
−∞
0
ς Here B is endowed with the norm
Z
dw(ς)
+ f(ς, x ς , σ 2 (ς, ϑ, x ϑ )dϑ) + G(ς, ϑ, x ϑ )
dς Z 0
0 p 1
∥ϕ∥ B = q(ϑ) sup (E∥ϕ(θ)∥ ) dϑ.
p
H
+ σ(ς)dB (ς), ς k ̸= ς ∈ J := [0, T], −∞ ϑ≤θ≤0
Q
1
k
∆x(ς k ) = I (x ς k ), k = 1, 2, . . . , m := 1, m; For J T = (−∞, T], we consider the space B T :=
n
′ 2 +
k
k
∆ x(ς k ) = I (x ς k ), k = 1, m, x : J T → H ∋ x k ∈ C(J k , H) ∃ x(ς ) and
) = x 0 = φ ∈ B, −
k
x(ς) + ˜p(x ς 1 , x ς 2 , x ς n x(ς ) with
′
x (ς) = x 1 ∈ H, for a.e ς ∈ (−∞, 0], (1)
−
k
x(ς k ) = x(ς ), x(0) − ˜p(x ς 1 , x ς 2 , . . . , x ς n ) = φ(0) ∈ B,
o
the state variable x(ς) : J → H takes values in a k = 1, m ,
real and separable Hilbert space H with the inner
product ⟨·, ·⟩ H and the norm ∥ · ∥ H . where x k is the restriction of x on J k :
(ς k , ς k+1 ], k = 1, m. Set ∥ · ∥ T as a semi-norm
of B T defined by
α
Let C D + denotes the Caputo fractional deriva-
0
tive of order 1 < α < 2. A : D(A) ⊂ H → H 1

is the infinitesimal generator of a strongly con- ∥x∥ T = ∥φ∥ B + sup E∥x(ϑ)∥ p p , x ∈ B T .
tinuous semi-group of bounded linear operators ϑ∈J
{S α (ς)} and {T α (ς)} on the Hilbert space 44,45
ς≥0 ς≥0 Lemma 4. Assume that x ∈ B T , then for
H. The history x ς : J → H, x ς (θ) = x(ς + θ) for ς ∈ J, x ς ∈ B. Moreover,
ς ≥ 0 belongs to the phase space B which will be
described later in section 2. The non-linear maps 1 1

f, g : J ×B×H → H, σ i : J ×J ×B → H, i = 1, 2, l E∥x(ς)∥ p p ≤ ∥x ς ∥ B ≤ ∥x 0 ∥ B +l sup E∥x(ϑ)∥ p p ,
0
0
G : J × J × B → L (K, H), σ : J → L (K, H) ϑ∈J
Q
Q
n
2
1
and I , I : B → H, k := 1, m and ˜p : B → B are 0
k k where l = R q(ϑ)dϑ < ∞.
appropriate continuous functions to be specified
−∞
in the sequel. Furthermore, let 0 = ς 0 < ς 1 <
ς 2 < ς 3 , . . . , ς m < ς m+1 = T be prefixed points Definition 5. An F ς -valued stochastic process
−
+
and ∆x(ς k ) = x(ς ) − x(ς ) denotes the left and x(ς), ς ∈ (−∞, 0] is known as a mild solution of
k k
+
−
′
′
′
right limits of x(ς). Also ∆x (ς k ) = x (ς )−x (ς ) the system (1) if
k
k
′
denotes the left and right limits of x (ς) at ς = ς k , (1) x(ς) is F ς -measurable with cadlag path on
respectively. Let K be the another separable ς ≥ 0 a.s.
Hilbert space with inner product ⟨·, ·⟩ K and norm (2) For ς ∈ (−∞, 0], x(ς) = φ(ς) +
′
∥ · ∥ K . Here {w(ς) : ς ∈ J} is a standard Wiener ˜ p(x ς 1 , x ς 2 , . . . , x ς n ), x (ς) = x 1 ∈ H
process on a real and separable Hilbert space. Set (3) For each ς ∈ J, x(ς) satisfies the following
φ ∈ L p (Ω, H), x 1 are F ς -measurable H-valued integral equation:
107