Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 206, pp. 1–29.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ASYMPTOTIC STABILITY OF FRACTIONAL IMPULSIVE
NEUTRAL STOCHASTIC PARTIAL INTEGRO-DIFFERENTIAL
EQUATIONS WITH STATE-DEPENDENT DELAY
ZUOMAO YAN, HONGWU ZHANG
Abstract. In this article, we study the asymptotical stability in p-th moment
of mild solutions to a class of fractional impulsive partial neutral stochastic integro-differential equations with state-dependent delay in Hilbert spaces.
We assume that the linear part of this equation generates an α-resolvent operator and transform it into an integral equation. Sufficient conditions for
the existence and asymptotic stability of solutions are derived by means of
the Krasnoselskii-Schaefer type fixed point theorem and properties of the αresolvent operator. An illustrative example is also provided.
1. Introduction
Partial stochastic differential equations have attracted the considerable attention
of researchers and many qualitative theories for the solutions of this kind have been
derived; see [7, 8] and the references therein. In particular, the stability theory
of stochastic differential equations has been popularly applied in variety fields of
science and technology. Several authors have established the stability results of
mild solutions for these equations by using various techniques; see, for example,
Govindan [13] considered the existence and stability for mild solution of stochastic partial differential equations by applying the comparison theorem. Caraballo
and Liu [5] proved the exponential stability for mild solution to stochastic partial
differential equations with delays by utilizing the well-known Gronwall inequality.
The exponential stability of the mild solutions to the semilinear stochastic delay
evolution equations have been discussed by using Lyapunov functionals in Liu [16].
The author [17] considered the exponential stability for stochastic partial functional differential equations by means of the Razuminkhin-type theorem. Liu and
Truman [18] investigated the almost sure exponential stability of mild solution for
stochastic partial functional differential equation by using the analytic technique.
Taniguchi [32] discussed the exponential stability for stochastic delay differential
equations by the energy inequality. Using fixed point approach, Luo [20] studied
the asymptotic stability of mild solutions of stochastic partial differential equations
with finite delays. Further, Sakthivel et al. [22, 25, 26] established the asymptotic
2000 Mathematics Subject Classification. 34A37, 60H15, 35R60, 93E15, 26A33.
Key words and phrases. Asymptotic stability; impulsive neutral integro-differential equations;
stochastic integro-differential equations; α-resolvent operator.
c
2013
Texas State University - San Marcos.
Submitted April 28, 2013. Published September 18, 2013.
1
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Z. YAN, H. ZHANG
EJDE-2013/206
stability and exponential stability of second-order stochastic evolution equations in
Hilbert spaces.
Impulsive differential and integro-differential systems are occurring in the field
of physics where it has been very intensive research topic since the theory provides
a natural framework for mathematical modeling of many physical phenomena [3].
Moreover, various mathematical models in the study of population dynamics, biology, ecology and epidemic can be expressed as impulsive stochastic differential
equations. In recent years, the qualitative dynamics such as the existence and
uniqueness, stability for first-order impulsive partial stochastic differential equations have been extensively studied by many authors; for instance, Sakthivel and
Luo [23, 24] studied the existence and asymptotic stability in p-th moment of mild
solutions to impulsive stochastic partial differential equations through fixed point
theory. Anguraj and Vinodkumar [1] investigated the existence, uniqueness and
stability of mild solutions of impulsive stochastic semilinear neutral functional differential equations without a Lipschitz condition and with a Lipschitz condition.
Chen [6], Long et al. [19] discussed the exponential p-stability of impulsive stochastic partial functional differential equations. He and Xu [14] studied the existence,
uniqueness and exponential p-stability of a mild solution of the impulsive stochastic neutral partial functional differential equations by using Banach fixed point
theorem.
On the other hand, fractional differential equations play an important role in
describing some real world problems. This is caused both by the intensive development of the theory of fractional calculus itself and by applications of such
constructions in various domains of science, such as physics, mechanics, chemistry,
engineering, etc. For details, see [21] and references therein. The existence of
solutions of fractional semilinear differential and integrodifferential equations are
one of the theoretical fields that investigated by many authors [11, 30, 33, 34].
Recently, much attention has been paid to the differential systems involving the
fractional derivative and impulses. This is due to the fact that most problems in a
real life situation to which mathematical models are applicable are basically fractional order differential equations rather than integer order differential equations.
Consequently, there are many contributions relative to the solutions of various impulsive semilinear fractional differential and integrodifferential systems in Banach
spaces; see [2, 9, 31]. The qualitative properties of fractional stochastic differential
equations have been considered only in few publications [7, 12, 27, 28, 35]. More
recently, Sakthivel et al. [29] studied the existence and asymptotic stability in p-th
moment of a mild solution to a class of nonlinear fractional neutral stochastic differential equations with infinite delays in Hilbert spaces. However, up to now the
existence and asymptotic stability of mild solutions for fractional impulsive neutral
partial stochastic integro-differential equations with state-dependent delay have not
been considered in the literature. In order to fill this gap, this paper studies the
existence and asymptotic stability of the following nonlinear impulsive fractional
stochastic integro-differential equation of the form
Z t
c α
D N (xt ) = AN (xt ) +
R(t − s)N (xs )ds + h(t, x(t − ρ2 (t)))dt
0
(1.1)
dw(t)
, t ≥ 0, t 6= tk ,
+ f (t, x(t − ρ3 (t)))
dt
x0 (·) = ϕ ∈ BF0 ([m̃(0), 0], H), x0 (0) = 0,
(1.2)
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ASYMPTOTIC STABILITY
∆x(tk ) = Ik (x(t−
k )),
t = tk , k = 1, . . . , m,
3
(1.3)
where the state x(·) takes values in a separable real Hilbert space H with inner
product h·, ·iH and norm k · kH , c Dα is the Caputo fractional derivative of order
α ∈ (1, 2), A, (R(t))t≥0 are closed linear operators defined on a common domain
which is dense in (H, k · kH ), and Dtα σ(t) represents the Caputo derivative of order
α > 0 defined by
Z t
dn
α
Dt σ(t) =
ηn−α (t − s) n σ(s)ds,
ds
0
where n is the smallest integer greater than or equal to α and ηβ (t) := tβ−1 /Γ(β),
t > 0, β ≥ 0. Let K be another separable Hilbert space with inner product h·, ·iK
and norm k · kK . Suppose {w(t) : t ≥ 0} is a given K-valued Wiener process
with a covariance operator Q > 0 defined on a complete probability space (Ω, F, P )
equipped with a normal filtration {Ft }t≥0 , which is generated by the Wiener process
w; and N (xt ) = x(0) + g(t, x(t − ρ1 (t))), x ∈ H, and g, h : [0, ∞) × H → H, f :
[0, ∞) × H → L(K, H), are all Borel measurable; Ik : H → H(k = 1, . . . , m), are
given functions. Moreover, the fixed moments of time tk satisfies 0 < t1 < · · · <
−
tm < limk→∞ tk = ∞, x(t+
k ) and x(tk ) represent the right and left limits of x(t)
+
at t = tk , respectively; ∆x(tk ) = x(tk ) − x(t−
k ), represents the jump in the state
x at time tk with Ik determining the size of the jump; let ρi (t) ∈ C(R+ , R+ )(i =
1, 2, 3) satisfy t − ρi (t) → ∞ as t → ∞, and m̃(0) = max{inf s≥0 (s − ρi (s)), i =
1, 2, 3}. Here BF0 ([m̃(0), 0], H) denote the family of all almost surely bounded,
F0 - measurable, continuous random variables ϕ(t) : [m̃(0), 0] → H with norm
kϕkB = supm̃(0)≤t≤0 Ekϕ(t)kH .
To the best of our knowledge, most of the previous research on the existence and
stability investigation for impulsive stochastic systems was based upon a Lipschitz
condition. This condition turns out to be restrictive. In this paper, we establish sufficient conditions for the existence and asymptotic stability in p-th moment of mild
for problem (1.1)–(1.3) by using Krasnoselskii-Schaefer type fixed point theorem [4]
with the α-resolvent operator. The obtained result can be seen as a contribution
to this emerging field.
This article is organized as follows. In Section 2, we give some preliminaries.
Section 3 aims to prove the main results. An example is presented in the last
section.
2. Preliminaries
Let K and H be two real separable Hilbert spaces with inner product h·, ·iK and
h·, ·iH , their inner products and by k · kK , k · kH their vector norms, respectively.
Let (Ω, F, P ; F)(F = {F}t≥0 ) be a complete probability space satisfying that F0
contains all P -null sets. Let {ei }∞
i=1 be a complete orthonormal basis of K. Suppose
that {w(t) : t ≥ 0} is a cylindrical
P∞ K-valued Brownian motion with a trace class
operator Q, denote Tr(Q) √
= i=1 λi = λ < ∞, which satisfies that Qei = λi ei .
P∞
So, actually, w(t) = i=1 λi wi (t)ei , where {wi (t)}∞
i=1 are mutually independent
one-dimensional standard Brownian motions. Then, the above K-valued stochastic
process w(t) is called a Q-Wiener process. Let L(K, H) denote the space of all
bounded linear operators from K into H equipped with the usual operator norm
k · kL(K,H) and we abbreviate this notation to L(H) when H = K. For ς ∈ L(K, H)
4
Z. YAN, H. ZHANG
EJDE-2013/206
we define
kςk2L0 = Tr(ςQς ∗ ) =
2
∞ p
X
k λn ςen k2 .
n=1
kςk2L0
2
If
< ∞, then ς is called a Q-Hilbert-Schmidt operator, and let L02 (K, H)
denote the space of all Q-Hilbert-Schmidt operators ς : K → H. For a basic
reference, the reader is referred to [8].
Let Y be the space of all F0 -adapted process ψ(t, w̃) : [m̃(0), ∞) × Ω → R which
is almost certainly continuous in t for fixed w̃ ∈ Ω. Moreover ψ(s, w̃) = ϕ(s) for
s ∈ [m̃(0), 0] and Ekψ(t, w̃)kpH → 0 as t → ∞. Also Y is a Banach space when it is
equipped with a norm defined by
kψkY = sup Ekψ(t)kpH .
t≥0
Now, we give knowledge on the α-resolvent operator which appeared in [30].
Definition 2.1. A one-parameter family of bounded linear operators (Rα (t))t≥0
on H is called an α-resolvent operator for
Z t
c α
D x(t) = Ax(t) +
R(t − s)x(s)ds,
(2.1)
0
x0 (0) = 0,
x(0) = x0 ∈ H,
(2.2)
if the following conditions are satisfied
(a) The function Rα (·) : [0, ∞) → L(H) is strongly continuous and Rα (0)x = x
for all x ∈ H and α ∈ (1, 2);
(b) For x ∈ D(A), we have Rα (·)x ∈ C([0, ∞), [D(A)]) ∩ C 1 ((0, ∞), H),
Z t
Dtα Rα (t)x = ARα (t)x +
R(t − s)Rα (s)x ds,
0
Z t
Dtα Rα (t)x = Rα (t)Ax +
Rα (t − s)R(s)x ds
0
for every t ≥ 0.
In this work we use the following assumptions:
(P1) The operator A : D(A) ⊆ H → H is a closed linear operator with [D(A)]
dense in H. Let α ∈ (1, 2). For some φ0 ∈ (0, π2 ], for each φ < φ0 there is
a positive constant C0 = C0 (φ) such that λ ∈ ρ(A) for each
λ ∈ Σ0,αϑ = {λ ∈ C, λ 6= 0, | arg(λ)| < αϑ},
0
where ϑ = φ + π2 and kR(λ, A)k ≤ C
|λ| for all λ ∈ Σ0,αϑ .
(P2) For all t ≥ 0, R(t) : D(R(t)) ⊆ H → H is a closed linear operator,
D(A) ⊆ D(R(t)) and R(·)x is strongly measurable on (0, ∞) for each
x ∈ D(A). There exists b(·) ∈ L1loc (R+ ) such that bb(λ) exists for Re(λ) > 0
and kR(t)xkH ≤ b(t)kxk1 for all t > 0 and x ∈ D(A). Moreover, the operab : Σ0,π/2 → L([D(A)], H) has an analytical extension
tor valued function R
b to Σ0,ϑ such that kR(λ)xk
b
b
(still denoted by R)
H ≤ kR(λ)kH kxk1 for all
b
x ∈ D(A), and kR(λ)kH = O(1/|λ|), as |λ| → ∞.
(P1) There exists a subspace D ⊆ D(A) dense in [D(A)] and a positive constant
e such that A(D) ⊆ D(A), R(λ)(D)
b
b
e
C
⊆ D(A), and kAR(λ)xk
H ≤ CkxkH
for every x ∈ D and all λ ∈ Σ0,ϑ .
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ASYMPTOTIC STABILITY
5
In the sequel, for r > 0 and θ ∈ ( π2 , ϑ),
Σr,θ = {λ ∈ C, |λ| > r, | arg(λ)| < θ},
for
Γr,θ , Γir,θ ,
i = 1, 2, 3, are the paths
Γ1r,θ = {teiθ : t ≥ r},
Γ2r,θ = {teiξ : |ξ| ≤ θ},
Γ3r,θ = {te−iθ : t ≥ r},
and Γr,θ = ∪3i=1 Γir,θ oriented counterclockwise. In addition, ρα (Gα ) are the sets
−1
b
ρα (Gα ) = {λ ∈ C : Gα (λ) := λα−1 (λα I − A − R(λ))
∈ L(X)}.
We now define the operator family (Rα (t))t≥0 by
(
R
1
λt
2πi Γr,θ e Gα (λ)dλ, t > 0,
Rα (t) :=
I,
t = 0.
Lemma 2.2 ([30]). Assume that conditions (P1)–(P3) are fulfilled. Then there
exists a unique α-resolvent operator for problem (2.1)-(2.2).
Lemma 2.3 ([30]). The function Rα : [0, ∞) → L(H) is strongly continuous and
Rα : (0, ∞) → L(H) is uniformly continuous.
Definition 2.4 ([30]). Let α ∈ (1, 2), we define the family (Sα (t))t≥0 by
Z t
Sα (t)x :=
gα−1 (t − s)Rα (s)ds
0
for each t ≥ 0.
Lemma 2.5 ([30]). If the function Rα (·) is exponentially bounded in L(H), then
Sα (·) is exponentially bounded in L(H).
Lemma 2.6 ([30]). If the function Rα (·) is exponentially bounded in L([D(A)]),
then Sα (·) is exponentially bounded in L([D(A)]).
α
Lemma 2.7 ([30]). If R(λα
0 , A) is compact for some λ0 ∈ ρ(A), then Rα (t) and
Sα (t) are compact for all t > 0.
Definition 2.8. A stochastic process {x(t), t ∈ [0, T ]}(0 ≤ T < ∞) is called a mild
solution of (1.1)-(1.3) if
(i) x(t) is adapted to Ft , t ≥ 0.
(ii) x(t) ∈ H has càdlàg paths on t ∈ [0, T ] a.s and for each t ∈ [0, T ], x(t)
satisfies the integral equation

Rα (t)[ϕ(0) − g(0, ϕ(−ρ1 (0)))] + g(t, x(t − ρ1 (t)))


Rt



+ 0 Sα (t − s)h(s, x(s − ρ2 (s)))ds


Rt



+ 0 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),
t ∈ [0, t1 ],




−
−
+
+


Rα (t − t1 )[x(t1 ) + I1R(x(t1 )) − g(t1 , x(t1 − ρ1 (t1 )))]


t
+g(t, x(t − ρ1 (t))) + t1 Sα (t − s)h(s, x(s − ρ2 (s)))ds
x(t) =
Rt

+ t1 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),
t ∈ (t1 , t2 ],





...




−
+
+

Rα (t − tm )[x(t−

m ) + Im (x(tm )) − g(tm , x(tm − ρ1 (tm )))]

Rt


+g(t, x(t − ρ1 (t))) + t Sα (t − s)h(s, x(s − ρ2 (s)))ds

m

 Rt
+ tm Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),
t ∈ (tm , T ].
(2.3)
6
Z. YAN, H. ZHANG
EJDE-2013/206
Definition 2.9. Let p ≥ 2 be an integer. Equation (2.3) is said to be stable in the
p-th moment, if for any ε > 0, there exists a δ̃ > 0 such that kϕkB < δ̃ guarantees
that
E sup kx(t)kpH < ε.
t≥0
Definition 2.10. Let p ≥ 2 be an integer. Equation (2.3) is said to be asymptotically stable in p-th moment if it stable in the p-th moment and for any ϕ ∈
BF0 ([m̃(0), 0], H),
lim E sup kx(t)kpH = 0.
T →+∞
t≥T
Lemma 2.11 ([8]). For any p ≥ 1 and for arbitrary L02 (K, H)-valued predictable
process φ(·) such that
Z
Z t
p
s
2p
sup E φ(v)dw(v) ≤ Cp
(Ekφ(s)k2p0 )1/p ds , t ∈ [0, ∞),
s∈[0,t]
0
H
L2
0
where Cp = (p(2p − 1))p . Next, we state a Krasnoselskii-Schaefer type fixed point
theorem.
Lemma 2.12 ([4]). Let Φ1 , Φ2 be two operators such that:
(a) Φ1 is a contraction, and
(b) Φ2 is completely continuous.
Then either
(i) the operator equation Φ1 x + Φ2 x has a solution, or
(ii) the set Υ = {x ∈ H : λΦ1 ( λx ) + λΦ2 x = x} is unbounded for λ ∈ (0, 1).
3. Main results
In this section we present our result on asymptotic stability in the p-th moment
of mild solutions of system (1.1)-(1.3). for this, we state the following hypotheses:
(H1) The operator families Rα (t) and Sα (t) are compact for all t > 0, and
there exist constants M > 0, δ > 0 such that kRα (t)kL(H) ≤ M e−δt and
kSα (t)kL(H) ≤ M e−δt for every t ≥ 0.
(H2) The function g : [0, ∞) × H → H is continuous and there exists Lg > 0
such that
Ekg(t, ψ1 ) − g(t, ω2 )kpH ≤ Lg kψ1 − ψ2 kpH ,
Ekg(t, ψ)kpH
≤
Lg kψkpH ,
t ≥ 0, ω1 , ψ2 ∈ H;
t ≥ 0, ψ ∈ H.
(H3) The function h : [0, ∞) × H → H satisfies the following conditions:
(i) The function h : [0, ∞) × H → H is continuous.
(ii) There exist a continuous function mh : [0, ∞) → [0, ∞) and a continuous nondecreasing function Θh : [0, ∞) → (0, ∞) such that
Ekh(t, ψ)kpH ≤ mh (t)Θh (EkψkpH ),
t ≥ 0, ψ ∈ H.
(H4) The function f : [0, ∞) × H → L(K, H) satisfies the following conditions:
(i) The function f : [0, ∞) × H → L(K, H) is continuous.
(ii) There exist a continuous function mf : [0, ∞) → [0, ∞) and a continuous nondecreasing function Θf : [0, ∞) → (0, ∞) such that
Ekf (t, ψ)kpH ≤ mf (t)Θf (EkψkpH ), t ≥ 0, ψ ∈ H,
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ASYMPTOTIC STABILITY
with
Z
7
∞
1
ds = ∞.
(3.1)
Θ
(s)
+
Θf (s)
h
1
(H5) The functions Ik : H → H are completely continuous and that there are
constants djk , k = 1, 2, . . . m, j = 1, 2, such that EkIk (x)kpH ≤ d1k EkxkpH +
d2k , for every x ∈ H.
In the proof of the existence theorem, we need the following lemmas.
Lemma 3.1. Assume that conditions (H1), (H3) hold.
defined by: for each x ∈ Y,
R t

S (t − s)h(s, x(s − ρ2 (s)))ds,


R0t α



 t1 Sα (t − s)h(s, x(s − ρ2 (s)))ds,
(Φ1 x)(t) = . . .

Rt



 tm Sα (t − s)h(s, x(s − ρ2 (s)))ds,

. . .
Let Φ1 be the operator
t ∈ [0, t1 ],
t ∈ (t1 , t2 ],
(3.2)
t ∈ (tm , tm+1 ],
Then Φ1 is continuous and maps Y into Y.
Proof. We first prove that Φ1 is continuous in the p-th moment on [0, ∞). Let
x ∈ Y, t̃ ≥ 0 and |ξ| be sufficiently small. Then for t̃ ∈ [0, t1 ], by using Hölder’s
inequality, we have
Ek(Φ1 x)(t̃ + ξ) − (Φ1 x)(t̃)kpH
Z
t̃
p
≤ 2p−1 E [Sα (t̃ + ξ − s) − Sα (t̃ − s)]h(s, x(s − ρ2 (s)))dsH
0
+ 2p−1 E t̃+ξ
Z
t̃
≤ 2p−1 E
t̃
hZ
p
Sα (t̃ + ξ − s)h(s, x(s − ρ2 (s)))dsH
ip
k[Sα (t̃ + ξ − s) − Sα (t̃ − s)]h(s, x(s − ρ2 (s)))kH ds
0
+ 2p−1 M p E
hZ
t̃+ξ
ip
e−δ(t̃+ξ−s) kh(s, x(s − ρ2 (s)))kH ds
t̃
≤ 2p−1
t̃
hZ
0
Z
×
0
+2
(p/p−1)
kSα (t̃ + ξ − s) − Sα (t̃ − s)kL(H)
ip−1
ds
t̃
Ekh(s, x(s − ρ2 (s)))kpH ds
p−1
M
p
t̃+ξ
hZ
ip−1
e−(pδ/p−1)(t̃+ξ−s) ds
t̃
Z
×
t̃
t̃+ξ
Ekh(s, x(s − ρ2 (s)))kpH ds → 0
as ξ → ∞.
Similarly, for any t̃ ∈ (tk , tk+1 ], k = 1, 2, . . ., we have
Ek(Φ1 x)(t̃ + ξ) − (Φ1 x)(t̃)kpH
h Z t̃
ip−1
−(p/p−1)
≤ 2p−1
kSα (t̃ + ξ − s) − Sα (t̃ − s)kL(H)
ds
tk
8
Z. YAN, H. ZHANG
Z
t̃
Ekh(s, x(s − ρ2 (s)))kpH ds
×
tk
+2
EJDE-2013/206
p−1
M
p
hZ
t̃+ξ
ip−1
e−(pδ/p−1)(t̃+ξ−s) ds
t̃
Z
t̃+ξ
Ekh(s, x(s − ρ2 (s)))kpH ds → 0
×
t̃
as
ξ → ∞.
Then, for all x(t̃) ∈ Y, t̃ ≥ 0, we have
Ek(Φ1 x)(t̃ + ξ) − (Φ1 x)(t̃)kpH → 0
as ξ → ∞.
Thus Φ1 is continuous in the p-th moment on [0, ∞).
Next we show that Φ1 (Y) ⊂ Y. By using (H1), (H3) and Hölder’s inequality, we
have for t ∈ [0, t1 ]
Z
t
p
Ek(Φ1 x)(t)kpH ≤ E Sα (t − s)h(s, x(s − ρ2 (s)))dsH
0
hZ t
ip
≤E
kSα (t − s)h(s, x(s − ρ2 (s)))kH ds
0
hZ t
ip
p
≤M E
e−δ(t−s) kh(s, x(s − ρ2 (s)))kH ds
0
hZ t
ip
= M pE
e−(δ(p−1)/p)(t−s) e−(δ/p)(t−s) kh(s, x(s − ρ2 (s)))kH ds
0
hZ t
ip−1 Z t
≤ Mp
e−δ(t−s) ds
e−δ(t−s) Ekh(s, x(s − ρ2 (s)))kpH ds
0
0
Z t
≤ M p δ 1−p
e−δ(t−s) mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds.
0
Similarly, for any t ∈ (tk , tk+1 ], k = 1, 2, . . ., we have
Z
t
p
p
Ek(Φ1 x)(t)kH ≤ E
Sα (t − s)h(s, x(s − ρ2 (s)))dsH
tk
≤ M p δ 1−p
Z
t
tk
e−δ(t−s) mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds.
Then, for all x(t) ∈ Y, t ∈ [m̃(0), ∞), we have
Z t
p
p p−1
Ek(Φ1 x)(t)kH ≤ M δ
e−δ(t−s) mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds.
(3.3)
0
However, for any any ε > 0, there exists a τ̃1 > 0 such that Ekx(s − ρ2 (s))kpH < ε
for t ≥ τ̃1 . Thus, we obtain
Ek(Φ1 x)(t)kpH
≤ M p δ 1−p e−δt
Z
≤ M p δ 1−p e−δt
Z
0
0
t
eδs mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds
τ̃1
eδs mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds + M p δ 1−p Lh Θh (ε),
EJDE-2013/206
ASYMPTOTIC STABILITY
9
Rt
where Lh = supt≥0 τ̃1 e−δ(t−s) mh (s)ds. As e−δt → 0 as t → ∞ and, there exists
τ̃2 ≥ τ̃1 such that for any t ≥ τ̃2 we have
Z τ̃1
eδs mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds < ε − M p δ p−1 Lh Θh (ε).
M p δ p−1 e−δt
0
From the above inequality, for any t ≥ τ̃2 , we obtain Ek(Φ1 x)(t)kpH < ε. That is to
say Ek(Φ1 x)(t)kpH → 0 as t → ∞. So we conclude that Φ1 (Y) ⊂ Y.
Lemma 3.2. Assume that conditions (H1), (H4) hold. Let Φ2 be the operator
defined by: for each x ∈ Y,
R t

 0 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s), t ∈ [0, t1 ],


R

 t Sα (t − s)f (s, x(s − ρ3 (s)))dw(s), t ∈ (t1 , t2 ],

t1
(Φ2 x)(t) = . . .

Rt




 tm Sα (t − s)f (s, x(s − ρ3 (s)))dw(s), t ∈ (tm , tm+1 ],

...
Then Φ2 is continuous on [0, ∞) in the p-th mean and maps Y into itself.
Proof. We first prove that Φ2 is continuous in the p-th moment on [0, ∞). Let
x ∈ Y, θ̃ ≥ 0 and |ξ| be sufficiently small. Then for θ̃ ∈ [0, t1 ], by using Hölder’s
inequality and Lemma 2.11, we have
Ek(Φ2 x)(θ̃ + ξ) − (Φ2 x)(θ̃)kpH
Z
θ̃
p
p−1 ≤2 E
[Sα (θ̃ + ξ − s) − Sα (θ̃ − s)]f (s, x(s − ρ3 (s)))dw(s)H
0
θ̃+ξ
Z
+ 2p−1 E θ̃
≤ 2p−1 Cp
hZ
θ̃
ip/2
(Ek[Sα (θ̃ + ξ − s) − Sα (θ̃ − s)]f (s, x(s − ρ3 (s)))kpH )2/p ds
0
+ 2p−1 Cp
p
Sα (θ̃ + ξ − s)f (s, x(s − ρ3 (s)))dw(s)H
hZ
t̃+ξ
θ̃
ip/2
(EkSα (θ̃ + ξ − s)f (s, x(s − ρ3 (s)))kpH )2/p ds
→0
as ξ → ∞. Similarly, for any θ̃ ∈ (tk , tk+1 ], k = 1, 2, . . ., we have
Ek(Φ2 x)(θ̃ + ξ) − (Φ2 x)(θ̃)kpH
h Z θ̃
ip/2
≤ 2p−1 Cp
(Ek[Sα (θ̃ + ξ − s) − Sα (θ̃ − s)]f (s, x(s − ρ3 (s)))kpH )2/p ds
tk
+ 2p−1 Cp
hZ
θ̃
t̃+ξ
ip/2
(EkSα (θ̃ + ξ − s)f (s, x(s − ρ3 (s)))kpH )2/p ds
→0
as ξ → ∞. Then, for all x(θ̃) ∈ Y, θ̃ ≥ 0, we have
Ek(Φ2 x)(θ̃ + ξ) − (Φ2 x)(θ̃)kpH → 0
as ξ → ∞.
Thus Φ2 is continuous in the p-th moment on [0, ∞).
Next we show that Φ2 (Y) ⊂ Y. By using (H1), (H4) and Hölder’s inequality, for
t ∈ [0, t1 ], we have
Ek(Φ2 x)(t)kpH
10
Z. YAN, H. ZHANG
≤ E
Z
t
p
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H
0
hZ
EJDE-2013/206
t
ip/2
(EkSα (t − s)f (s, x(s − ρ3 (s)))kpH )(2/p) ds
0
hZ t
ip/2
p
[e−pδ(t−s) (Ekf (s, x(s − ρ3 (s)))kpH )]2/p ds
≤ Cp M
0
hZ t
ip/2
≤ Cp M p
[e−pδ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
0
hZ t
ip/2
[e−(p−1)δ(t−s) e−δ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
= Cp M p
0
hZ t
ip/2−1
2(p−1)
e−[ p−2 ]δ(t−s) ds
≤ Cp M p
0
Z t
×
e−δ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )ds
0
h 2δ(p − 1) i1−p/2 Z t
≤ Cp M p
e−δ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )ds.
p−2
0
≤ Cp
Similarly, for any t ∈ (tk , tk+1 ], k = 1, 2, . . ., we have
Ek(Φ2 x)(t)kpH
Z
t
p
≤ E
Sα (t − s)f (s, x(s − ρ3 (s))dw(s)H
tk
≤ Cp M
p
h 2δ(p − 1) i1−p/2 Z
p−2
t
tk
e−δ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )ds.
Then, for all x(t) ∈ Y, t ∈ [m̃(0), ∞), we have
Ek(Φ2 x)(t)kpH
h 2δ(p − 1) i1−p/2 Z t
e−δ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )ds.
≤ Cp M p
p−2
0
(3.4)
However, for any any ε > 0, there exists a θ̃1 > 0 such that Ekx(s − ρ3 (s))kpH < ε
for t ≥ θ̃1 . Thus from (3.4) we obtain
Ek(Φ2 x)(t)kpH
Z t
h
i1−p/2
p 2δ(p − 1)
−δt
≤ Cp M
e
eδs mf (s)Θf (Ekx(s − ρ3 (s))kpH )ds
p−2
0
Z t̃1
h 2δ(p − 1) i1−p/2
≤ Cp M p
e−δt
eδs mf (s)Θf (Ekx(s − ρ3 (s))kpH )ds
p−2
0
h
i1−p/2
p 2δ(p − 1)
+ Cp M
Lf Θf (ε),
p−2
Rt
where Lf = supt≥0 t̃1 e−δ(t−s) mf (s)ds. As e−δt → 0 as t → ∞ and, there exists
θ̃2 ≥ θ̃1 such that for any t ≥ θ̃2 we have
Z t̃1
p 2δ(p − 1) 1−p/2
]
e−δ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )ds
Cp M [
p−2
0
EJDE-2013/206
ASYMPTOTIC STABILITY
< ε − Cp M p [
11
2δ(p − 1) 1−p/2
]
Lf Θf (ε).
p−2
From the above inequality, for any t ≥ θ̃2 , we obtain Ek(Φ2 x)(t)kpH < ε. That is
to say Ek(Φ2 x)(t)kpH → 0 as t → ∞. So we conclude that Φ2 (Y) ⊂ Y.
Now, we are ready to present our main result.
Theorem 3.3. Assume the conditions (H1)-(H5) hold. Let p ≥ 2 be an integer.
Then the fractional impulsive stochastic differential equations (1.1)–(1.3) is asymptotically stable in the p-th moment, provided that
max {12p−1 M p (1 + d1k + 2p−1 Lg ) + 8p−1 Lg } < 1.
1≤k≤m
(3.5)
Proof. We define the nonlinear operator Ψ : Y → Y as (Ψx)(t) = ϕ(t) for t ∈
[m̃(0), 0] and for t ≥ 0,

Rα (t)[ϕ(0) − g(0, ϕ(−ρ1 (0)))] + g(t, x(t − ρ1 (t)))



Rt


+ 0 Sα (t − s)h(s, x(s − ρ2 (s)))ds


Rt



+ 0 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),





t ∈ [0, t1 ],





−
+
+

Rα (t − t1 )[x(t−

1 ) + I1 (x(t1 )) − g(t1 , x(t1 − ρ1 (t1 )))]

Rt



+g(t, x(t − ρ1 (t))) + t1 Sα (t − s)h(s, x(s − ρ2 (s)))ds

 Rt

+ t1 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),
(Ψx)(t) =
(3.6)


t ∈ (t1 , t2 ],




...




+
+
−

Rα (t − tm )[x(t−

m ) + Im (x(tm )) − g(tm , x(tm − ρ1 (tm )))]

Rt



x(t − ρ1 (t))) + tm Sα (t − s)h(s, x(s − ρ2 (s)))ds

+g(t,
Rt



+ tm Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),



 t ∈ (tm , tm+1 ],



...
Using (H2)–(H5), and the proofs of Lemmas 3.1 and 3.2, it is clear that the nonlinear
operator Ψ is well defined and continuous in p-th moment on [0, ∞). Moreover, for
all t ∈ [0, t1 ] we have
Ek(Ψx)(t)kpH
≤ 4p−1 EkRα (t)[ϕ(0) − g(0, ϕ(−ρ1 (0)))]kpH + 4p−1 Ekg(t, x(t − ρ1 (t)))kpH
Z
t
p
p−1 +4 E
Sα (t − s)h(s, x(s − ρ2 (s)))dw(s)H
0
Z t
p
+ 4p−1 E Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H .
0
Similarly, for any t ∈ (tk , tk+1 ], k = 1, 2, . . ., we have
p
−
+
+
Ek(Ψx)(t)kpH ≤ 4p−1 EkRα (t − tk )[x(t−
k ) + Ik (x(tk )) − g(tk , x(tk − ρ1 (tk )))]kH
+ 4p−1 Ekg(t, x(t − ρ1 (t)))kpH
Z
t
p
+ 4p−1 E Sα (t − s)h(s, x(s − ρ2 (s)))dsH
tk
12
Z. YAN, H. ZHANG
+ 4p−1 E Z
t
tk
EJDE-2013/206
p
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H .
Then, for all t ≥ 0, we have
Ek(Ψx)(t)kpH ≤ 4p−1 EkRα (t)[ϕ(0) − g(0, ϕ(−ρ1 (0)))]kpH
p
−
+
+
+ 4p−1 EkRα (t − tk )[x(t−
k ) + Ik (x(tk )) − g(tk , x(tk − ρ1 (tk )))]kH
+ 4p−1 Ekg(t, x(t − ρ1 (t)))kpH
Z
p
t
p−1 Sα (t − s)h(s, x(s − ρ2 (s)))dsH
+4 E
0
Z t
p
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H .
+ 4p−1 E 0
By (H1)–(H5), Lemmas 3.1 and 3.2 again, we obtain
4p−1 EkRα (t)[ϕ(0) − g(0, ϕ(−ρ1 (0)))]kpH
≤ 8p−1 M p e−pδt [Ekϕ(0)kpH + Lg Ekϕ(−ρ1 (0))kpH ] → 0
as
t → ∞,
p
−
+
+
4 EkRα (t − tk )[x(t−
k ) + Ik (x(tk )) − g(tk , x(tk − ρ1 (tk )))]kH
p
p
−
≤ 12p−1 M p e−pδt [Ekx(t−
k )kH + EkIk (x(tk ))kH
p
+
+ Lg Ekx(t+
k − ρ1 (tk )))kH ] → 0 as t → ∞,
4p−1 Ekg(t, x(t − ρ1 (t)))kpH ≤ 4p−1 Lg Ekx(t − ρ1 (t)))kpH → 0 as t →
p−1
4p−1 E Z
0
4p−1 E Z
0
t
t
p
Sα (t − s)h(s, x(t − ρ2 (t)))dsH → 0
p
Sα (t − s)f (s, x(t − ρ3 (t)))dw(s)H → 0
∞,
as t → ∞,
as
t → ∞.
That is to say Ek(Ψx)(t)kpH → 0 as t → ∞. So Ψ maps Y into itself.
Next we prove that the operator Ψ has a fixed point, which is a mild solution
of the problem (1.1)-(1.3). To see this, we decompose Ψ as Ψ1 + Ψ2 for t ∈ [0, T ],
where


−Rα (t)g(0, ϕ(−ρ1 (0))) + g(t, x(t − ρ1 (t))),
t ∈ [0, t1 ],




+
+
t ∈ (t1 , t2 ],
(Ψ1 x)(t) = −Rα (t − t1 )g(t1 , x(t1 − ρ1 (t1 ))) + g(t, x(t − ρ1 (t))),


.
.
.



+
−Rα (t − tm )g(tm , x(t+
m − ρ1 (tm ))) + g(t, x(t − ρ1 (t))), t ∈ (tm , T ],
EJDE-2013/206
ASYMPTOTIC STABILITY
13
and

Rt
Rα (t)ϕ(0) + 0 Sα (t − s)h(s, x(s − ρ2 (s)))ds



Rt


+ 0 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),
t ∈ [0, t1 ],





−
−

Rα (t − t1 )[x(t1 ) + I1 (x(t1 ))]


Rt



+ t1 Sα (t − s)h(s, x(s − ρ2 (s)))ds
R
(Ψ2 x)(t) = + t Sα (t − s)f (s, x(t − ρ3 (t)))dw(s),
t ∈ (t1 , t2 ],
t1




.
.
.




−
−

RαR(t − tm )[x(tm ) + Im (x(tm ))]


t

+ tm Sα (t − s)h(s, x(s − ρ2 (s)))ds


 Rt
+ tm Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),
t ∈ (tm , T ].
To use Lemma 2.12, we will verify that Ψ1 is a contraction while Ψ2 is a completely
continuous operator. For better readability, we break the proof into a sequence of
steps.
Step1. Ψ1 is a contraction on Y. Let t ∈ [0, t1 ] and x, y ∈ Y. From (H2), we have
Ek(Ψ1 x)(t) − (Ψ1 y)(t)kpH ≤ Ekg(t, x(t − ρ1 (t))) − g(t, y(t − ρ1 (t)))kpH
≤ Lg Ekx(t − ρ1 (t)) − y(t − ρ1 (t))kpH
≤ Lg kx − ykY .
Similarly, for any t ∈ (tk , tk+1 ], k = 1, . . . , m, we have
Ek(Ψ1 x)(t) − (Ψ1 y)(t)kpH
p
+
+
+
≤ 2p−1 EkRα (t − tk )[−g(tk , x(t+
k − ρ1 (tk ))) + g(tk , y(tk − ρ1 (tk )))]kH
+ 2p−1 Ekg(t, x(t − ρ1 (t))) − g(t, y(t − ρ1 (t)))kpH
p
+
+
+
≤ 2p−1 M p Lg Ekx(t+
k − ρ1 (tk )) − y(tk − ρ1 (tk ))kH
+ 2p−1 Lg Ekx(t − ρ1 (t)) − y(t − ρ1 (t))kpH
≤ 2p−1 Lg (M p + 1)kx − ykY .
Thus, for all t ∈ [0, T ],
Ek(Ψ1 x)(t) − (Ψ1 y)(t)kpH ≤ 2p−1 Lg (M p + 1)kx − ykY .
Taking supremum over t,
kΨ1 x − Ψ1 ykY ≤ L0 kx − ykY ,
where L0 = 2p−1 Lg (M p + 1) < 1. By (3.5), we see that L0 < 1. Hence, Ψ1 is a
contraction on Y.
Step 2. Ψ2 maps bounded sets into bounded sets in Y. Indeed, it is sufficient
to show that there exists a positive constant L such that for each x ∈ Br = {x :
kxkY ≤ r} one has kΨ2 xkY ≤ L. Now, for t ∈ [0, t1 ] we have
Z t
(Ψ2 x)(t) = Rα (t)ϕ(0) +
Sα (t − s)h(s, x(t − ρ2 (t)))ds
0
(3.7)
Z t
+
Sα (t − s)f (s, x(t − ρ3 (t)))dw(s).
0
14
Z. YAN, H. ZHANG
EJDE-2013/206
If x ∈ Br , from the definition of Y, it follows that
Ekx(s − ρi (s))kpH ≤ 2p−1 kϕkpB + 2p−1 sup Ekx(s)kpH
s∈[0,T ]
p−1
≤2
kϕkpB
+2
p−1
r := r∗ ,
i = 1, 2, 3.
By (H1)-(H4), from (3.7) and Hölder’s inequality, for t ∈ [0, t1 ], we have
Ek(Ψ2 x)(t)kpH
Z
p
t
≤ 3p−1 EkRα (t)ϕ(0)kpH + 3p−1 E Sα (t − s)h(s, x(t − ρ2 (t)))dsH
0
Z t
p
Sα (t − s)f (s, x(t − ρ3 (t)))dw(s)H
+ 3p−1 E 0
hZ t
ip
≤ 3p−1 M p Ekϕ(0)kpH + 3p−1 M p E
e−δ(t−s) kh(s, x(t − ρ2 (t)))kH ds
0
hZ t
ip/2
+ 3p−1 Cp M p
[e−pδ(t−s) (Ekf (s, x(t − ρ3 (t)))kpH )]2/p ds
0
hZ t
ip−1
p−1
p
≤ 3 M Ekϕ(0)kpH + 3p−1 M p
e−δ(t−s) ds
0
Z t
−δ(t−s)
×
e
Ekh(s, x(t − ρ2 (t)))kpH ds
0
hZ t
ip/2
p−1
p
+ 3 Cp M
[e−pδ(t−s) mf (s)Θf (Ekx(s))kpH )]2/p ds
0
≤ 3p−1 M p Ekϕ(0)kpH + 3p−1 M p δ 1−p
Z t
×
e−δ(t−s) mh (s)Θh (Ekx(t − ρ2 (t))kpH )ds + 3p−1 Cp M p
0
h 2δ(p − 1) i1−p/2 Z t
×
e−δ(t−s) mf (s)Θf (Ekx(t − ρ3 (t))kpH )ds
p−2
0
Z t1
≤ 3p−1 M p Ekϕ(0)kpH + 3p−1 M p δ 1−p Θh (r∗ )
e−δ(t−s) mh (s)ds
0
h 2δ(p − 1) i1−p/2 Z t1
+ 3p−1 Cp M p Θf (r∗ )
e−δ(t−s) mf (s)ds := L0 .
p−2
0
Similarly, for any t ∈ (tk , tk+1 ], k = 1, . . . , m, we have
(Ψ2 x)(t)
= Rα (t − tk )[x(t−
k ) + Ik (x(tk ))] +
Z
Z
t
Sα (t − s)h(s, x(t − ρ2 (t)))ds
tk
(3.8)
t
Sα (t − s)f (s, x(t − ρ3 (t)))dw(s).
+
tk
By (H1)–(H5), from (3.8) and Hölder’s inequality, we have for t ∈ (tk , tk+1 ], k =
1, . . . , m,
Ek(Ψ2 x)(t)kpH
p
−
≤ 3p−1 EkRα (t − tk )[x(t−
k ) + Ik (x(tk ))]kH
EJDE-2013/206
ASYMPTOTIC STABILITY
+ 3p−1 E Z
t
tk
t
+ 3p−1 E Z
tk
15
p
Sα (t − s)h(s, x(t − ρ2 (t)))dsH
p
Sα (t − s)f (s, x(t − ρ3 (t)))dw(s)H
p
p
−
≤ 6p−1 M p [Ekx(t−
k )kH + EkIk (x(tk ))kH ]
hZ t
ip
+ 3p−1 M p E
e−δ(t−s) kh(s, x(t − ρ2 (t)))kH ds
tk
+ 3p−1 Cp M p
hZ
t
tk
ip/2
[e−pδ(t−s) (Ekf (s, x(t − ρ3 (t)))kpH )]2/p ds
p
2
p−1
≤ 6p−1 M p (r + d1k Ekx(t−
Mp
k )kH + dk ) + 3
Z
t
ip−1
e−δ(t−s) ds
tk
t
×
tk
+3
hZ
e−δ(t−s) Ekh(s, x(t − ρ2 (t)))kpH ds
p−1
Cp M
p
hZ
t
tk
ip/2
[e−pδ(t−s) mf (s)Θf (Ekx(t − ρ3 (t))kpH )]2/p ds
≤ 6p−1 M p (r + d1k r + d2k ) + 3p−1 M p δ p−1
Z t
×
e−δ(t−s) mh (s)Θh (Ekx(t − ρ2 (t))kpH )ds
tk
+3
p−1
Cp M
p
h 2δ(p − 1) i1−p/2 Z
p−2
t
tk
e−δ(t−s) mf (s)Θf (Ekx(t − ρ3 (t))kpH )ds
≤ 6p−1 M p (r + dk ) + 3p−1 M p δ 1−p Θh (r∗ )
Z
tk+1
tk
tk+1
h 2δ(p − 1) i1−p/2 Z
+ 3p−1 Cp M p Θf (r∗ )
p−2
tk
e−δ(t−s) mh (s)ds
e−δ(t−s) mf (s)ds := Lk .
Take L = max0≤k≤m Lk , for all t ∈ [0, T ], we have Ek(Ψ2 x)(t)kpH ≤ L. Then for
each x ∈ Br , we have kΨ2 xkY ≤ L.
Step 3. Ψ2 : Y → Y is continuous. Let {xn (t)}∞
n=0 ⊆ Y with xn → x(n → ∞)
in Y. Then there is a number r > 0 such that Ekxn (t)kpH ≤ r for all n and a.e.
t ∈ [0, T ], so xn ∈ Br and x ∈ Br . By the assumption (H3) and Ik , k = 1, 2, . . . , m,
are completely continuous, we have
Ekh(s, xn (s − ρ2 (s))) − h(s, x(s − ρ2 (s)))kpH → 0
Ekf (s, xn (s − ρ3 (s))) − f (s, x(s −
ρ3 (s)))kpH
→0
as n → ∞,
as n → ∞
for each s ∈ [0, t], and since
Ekh(s, xn (s − ρ2 (s))) − h(s, x(s − ρ2 (s)))kpH ≤ 2mh (t)Θh (r∗ ),
Ekf (s, xn (s − ρ3 (s))) − f (s, x(s − ρ3 (s)))kpH ≤ 2mf (t)Θf (r∗ ).
Then by the dominated convergence theorem, for t ∈ [0, t1 ], we have
Ek(Ψ2 xn )(t) − (Ψ2 x)(t)kpH
Z
t
p
p−1 ≤2 E
Sα (t − s)[h(s, xn (s − ρ2 (s))) − h(s, x(s − ρ2 (s)))]dsH
0
16
Z. YAN, H. ZHANG
+ 2p−1 E Z
t
0
hZ
EJDE-2013/206
p
Sα (t − s)[f (s, xn (s − ρ3 (s))) − f (s, x(s − ρ3 (s)))]dw(s)H
t
ip
e−δ(t−s) kh(s, xn (s − ρ2 (s))) − h(s, x(s − ρ2 (s)))kH ds
0
hZ t
p−1
p
(EkSα (t − s)[f (s, xn (s − ρ3 (s)))
+ 2 Cp M
≤2
p−1
p
M E
0
ip/2
− f (s, x(s − ρ3 (s)))]kpH )2/p ds
Z t
p−1
p 1−p
e−δ(t−s) Ekh(s, xn (s − ρ2 (s))) − h(s, x(s − ρ2 (s)))kpH ds
≤2 M δ
0
hZ t
+ 2p−1 Cp M p
e−2δ(t−s) (Ekf (s, xn (s − ρ3 (s)))
0
ip/2
− f (s, x(s − ρ3 (s)))kpH )2/p ds
→0
as n → ∞.
Similarly, for any t ∈ (tk , tk+1 ], k = 1, 2, . . . , m, we have
Ek(Ψ2 xn )(t) − (Ψ2 x)(t)kpH
p
−
−
−
≤ 3p−1 EkRα (t − tk )[xn (t−
k ) − x(tk ) + Ik (xn (tk )) − Ik (x(tk ))]kH
Z
t
p
+ 3p−1 E Sα (t − s)[h(s, xn (s − ρ2 (s))) − h(s, x(s − ρ2 (s)))]ds
tk
t
+ 3p−1 E Z
tk
H
p
Sα (t − s)[f (s, xn (s − ρ3 (s))) − f (s, x(s − ρ3 (s)))]dw(s)H
p
− p
−
−
M [Ekxn (t−
k ) − x(tk )kH + EkIk (xn (tk )) − Ik (x(tk ))kH ]
Z t
+ 3p−1 M p δ 1−p
e−δ(t−s) Ekh(s, xn (s − ρ2 (s))) − h(s, x(s − ρ2 (s)))kpH ds
0
hZ t
p−1
p
+ 3 Cp M
e−2δ(t−s) (Ekf (s, xn (s − ρ3 (s)))
p−1
≤6
p
tk
ip/2
→0
− f (s, x(s − ρ3 (s)))kpH )2/p ds
as n → ∞.
Then, for all t ∈ [0, T ] we have
kΨ2 xn − Ψ2 xkY → 0
as n → ∞.
Therefore, Ψ2 is continuous on Br .
Step 4. Ψ2 maps bounded sets into equicontinuous sets of Y.
Let 0 < τ1 < τ2 ≤ t1 . Then, by using Hölder’s inequality and Lemma 2.11, for
each x ∈ Br , we have
Ek(Ψ2 x)(τ2 ) − (Ψ2 x)(τ1 )kpH
≤ 7p−1 Ek[Rα (τ2 ) − Rα (τ1 )]ϕ(0)kpH
Z
τ1 −ε
p
p−1 +7 E
[Sα (τ2 − s) − Sα (τ1 − s)]h(s, x(s − ρ2 (s)))dsH
Z0
τ1
p
p−1 +7 E
[Sα (τ2 − s) − Sα (τ1 − s)]h(s, x(s − ρ2 (s)))ds
τ1 −ε
H
EJDE-2013/206
ASYMPTOTIC STABILITY
+ 7p−1 E τ2
Z
p
Sα (τ2 − s)h(s, x(s − ρ2 (s)))dsH
τ1
τ1 −ε
+ 7p−1 E +7
p−1
E
Z
Z0 τ1
τ1 −ε
τ2
+ 7p−1 E Z
17
p
[Sα (τ2 − s) − Sα (τ1 − s)]f (s, x(s − ρ3 (s)))dw(s)H
p
[Sα (τ2 − s) − Sα (τ1 − s)]f (s, x(s − ρ3 (s)))dw(s)H
p
Sα (τ2 − s)f (s, x(s − ρ3 (s)))dw(s)H
τ1
≤ 7p−1 Ek[Rα (τ2 ) − Rα (τ1 )]ϕ(0)kpH
h Z τ1 −ε
ip
p−1
+7 E
kSα (τ2 − s) − Sα (τ1 − s)kL(H) kh(s, x(s − ρ1 (s)))kH ds
0
h Z τ1
ip
kSα (τ2 − s) − Sα (τ1 − s)kL(H) kh(s, x(s − ρ1 (s)))kH ds
+ 7p−1 E
τ1 −ε
h Z τ2
ip
+ 7p−1 E
kSα (τ2 − s)kL(H) kh(s, x(s − ρ1 (s)))kH ds
τ1
+ 7p−1 Cp
τ1 −ε
hZ
0
[kSα (τ2 − s) − Sα (τ1 − s)kpL(H)
ip/2
× Ekf (s, x(s − ρ3 (s)))kpH ]2/p ds
h Z τ1
+ 7p−1 Cp
[kSα (τ2 − s) − Sα (τ1 − s)kpL(H)
τ1 −ε
ip/2
× Ekf (s, x(s − ρ3 (s)))kpH ]2/p ds
h Z τ2
ip/2
[kSα (τ2 − s)kpL(H) (Ekf (s, x(s − ρ3 (s)))kpH )]2/p ds
+ 7p−1 Cp
τ1
Ek[Rα (τ2 ) − Rα (τ1 )]ϕ(0)kpH
Z τ1 −ε
p−1 p
+7 T
kSα (τ2 − s) − Sα (τ1 − s)kpL(H) mh (s)Θh (Ekx(s − ρ1 (s)))kpH )ds
0
h Z τ1
ip−1 Z τ1
p−1
p
−δ(τ1 −s)
+ 14 M
e
ds
e−δ(τ1 −s) mh (s)
p−1
≤7
τ1 −ε
τ1 −ε
× Θh (Ekx(s − ρ1 (s)))kpH )ds
h Z τ2
ip−1 Z
p−1
p
−δ(τ2 −s)
+7 M
e
ds
+ 4p−1 Cp
hZ
τ1
τ1 −ε
0
τ2
τ1
e−δ(τ2 −s) mh (s)Θh (Ekx(s − ρ2 (s)))kpH )ds
[kSα (τ2 − s) − Sα (τ1 − s)kpL(H)
ip/2
× mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
h Z τ1
ip/2
p−1
p
+ 14 Cp M
[e−pδ(τ1 −s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
τ1 −ε
h Z τ2
ip/2
+ 7p−1 Cp M p
[e−pδ(τ2 −s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
τ1
18
Z. YAN, H. ZHANG
EJDE-2013/206
≤ 7p−1 Ek[Rα (τ2 ) − Rα (τ1 )]ϕ(0)kpH
Z τ1 −ε
p−1 p
∗
kSα (τ2 − s) − Sα (τ1 − s)kpL(H) mh (s)ds
+ 7 T Θh (r )
0
Z τ1
p−1
p
∗ 1−p
e−δ(τ1 −s) mh (s)ds
+ 14 M Θh (r )δ
τ1 −ε
Z τ2
p−1
p
∗ 1−p
e−δ(τ2 −s) mh (s)ds
+ 7 M Θh (r )δ
hZ
+ 7p−1 Cp Θf (r∗ )
τ1
τ1 −ε
ip/2
[kSα (τ2 − s) − Sα (τ1 − s)kpL(H) mf (s)]2/p ds
0
h 2δ(p − 1) i1−p/2 Z τ1
+ 14p−1 Cp M p Θf (r∗ )
e−δ(τ1 −s) mf (s)ds
p−2
τ1 −ε
h 2δ(p − 1) i1−p/2 Z τ2
+ 4p−1 Cp M p Θf (r∗ )
e−δ(τ2 −s) mf (s)ds.
p−2
τ1
Similarly, for any τ1 , τ2 ∈ (tk , tk+1 ], τ1 < τ2 , k = 1, . . . , m, we have
(Ψ2 x)(t)
= Rα (t − tk )[x̄(t−
k ) + Ik (x(tk ))] +
Z
Z
t
Sα (t − s)h(s, x(s − ρ2 (s)))ds
tk
t
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s).
+
tk
Then
Ek(Ψ2 x)(τ2 ) − (Ψ2 x)(τ1 )kpH
p
−
≤ 7p−1 Ek[Rα (τ2 ) − Rα (τ1 )][x(t−
k ) + Ik (x(tk ))]kH
Z
τ1 −ε
p
+ 7p−1 E [Sα (τ2 − s) − Sα (τ1 − s)]h(s, x(s − ρ2 (s)))ds
H
tk
τ1
+ 7p−1 E Z
τ1 −ε
τ2
+7
p−1
E
Z
p
[Sα (τ2 − s) − Sα (τ1 − s)]h(s, x(s − ρ2 (s)))dsH
p
Sα (τ2 − s)h(s, x(s − ρ2 (s)))dsH
τ1
τ1 −ε
+ 7p−1 E Z
tk
τ1
+ 7p−1 E Z
τ1 −ε
τ2
+ 7p−1 E Z
τ1
p
[Sα (τ2 − s) − Sα (τ1 − s)]f (s, x(s − ρ3 (s)))dw(s)H
p
[Sα (τ2 − s) − Sα (τ1 − s)]f (s, x(s − ρ3 (s)))dw(s)H
p
Sα (τ2 − s)f (s, x(s − ρ3 (s)))dw(s)H
p
−
Ek[Rα (τ2 ) − Rα (τ1 )][x(t−
k ) + Ik (x(tk ))]kH
Z τ1 −ε
+ 7p−1 T p Θh (r∗ )
kSα (τ2 − s) − Sα (τ1 − s)kpL(H) mh (s)ds
tk
Z τ1
p−1
p
∗ 1−p
+ 14 M Θh (r )δ
e−δ(τ1 −s) mh (s)ds
≤4
p−1
τ1 −ε
(3.9)
EJDE-2013/206
ASYMPTOTIC STABILITY
+ 7p−1 M p Θh (r∗ )δ 1−p
hZ
Z
τ2
19
e−δ(τ2 −s) mh (s)ds
τ1
τ1 −ε
ip/2
[kSα (τ2 − s) − Sα (τ1 − s)kpL(H) mf (s)]2/p ds
tk
h
i1−p/2 Z τ1
p−1
p
∗ 2δ(p − 1)
+ 8 Cp M Θf (r )
e−δ(t−s) mf (s)ds
p−2
τ1 −ε
h 2δ(p − 1) i1−p/2 Z τ2
+ 4p−1 Cp M p Θf (r∗ )
e−δs mf (s)ds.
p−2
τ1
+ 4p−1 Cp Θf (r∗ )
The fact of Ik , k = 1, 2, . . . , m, are completely continuous in H and the compactness
of Rα (t), Sα (t) for t > 0 imply the continuity in the uniform operator topology.
So, as τ2 − τ1 → 0, with ε is sufficiently small, the right-hand side of the above
inequality is independent of x ∈ Br and tends to zero. The equicontinuities for the
cases τ1 < τ2 ≤ 0 or τ1 ≤ 0 ≤ τ2 ≤ T are very simple. Thus the set {Ψ2 x : x ∈ Br }
is equicontinuous.
Step 5. The set W (t) = {(Ψ2 x)(t) : x ∈ Br } is relatively compact in H. To this
end, we decompose Ψ2 by Ψ2 = Γ1 + Γ2 , where
R t
S (t − s)h(s, x(s − ρ2 (s)))ds

0R α


t


+ 0 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s), t ∈ [0, t1 ],



Rt



 t1 Sα (t − s)h(s, x(s − ρ2 (s)))ds
R
(Γ1 x)(t) = + t Sα (t − s)f (s, x(s − ρ3 (s)))dw(s), t ∈ (t1 , t2 ],
t1



. . .


Rt



Sα (t − s)h(s, x(s − ρ2 (s)))ds


 tmR t
+ tm Sα (t − s)f (s, x(s − ρ3 (s)))dw(s), t ∈ (tm , T ],
and

t ∈ [0, t1 ],

Rα (t)ϕ(0),

R (t − t )[x(t− ) + I (x(t− ))],
t
∈ (t1 , t2 ],
α
1
1
1
1
(Γ2 x)(t) =
. . .



−
Rα (t − tm )[x(t−
m ) + Im (x(tm ))], t ∈ (tm , T ].
We now prove that Γ1 (Br )(t) = {(Γ1 x)(t) : x ∈ Br } is relatively compact for
every t ∈ [0, T ]. Let 0 < t ≤ s ≤ t1 be fixed and let ε be a real number satisfying
0 < ε < t. For x ∈ Br , we define
Z t−ε
(Γε1 x)(t)(t) =
Sα (t − s)h(s, x(s − ρ2 (s)))ds
0
Z t−ε
+
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s).
0
Using the compactness of Sα (t) for t > 0, we deduce that the set Uε (t) = {(Γε1 x)(t) :
x ∈ Br } is relatively compact in H for every ε, 0 < ε < t. Moreover, by using
Hölder’s inequality, we have for every x ∈ Br
Ek(Γ1 x)(t)(t) − (Γε1 x)(t)(t)kpH
Z
t
p
≤ 2p−1 E Sα (t − s)h(s, x(s − ρ2 (s)))dsH
t−ε
20
Z. YAN, H. ZHANG
+ 2p−1 E Z
t
t−ε
t
≤ 2p−1 M p E
EJDE-2013/206
p
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H
hZ
ip
e−δ(t−s) kh(s, x(s − ρ2 (s)))kH ds
t−ε
+ 2p−1 Cp M p
≤ 2p−1 M p δ 1−p
hZ
t
t−ε
t
Z
t−ε
t
+ 2p−1 Cp M p
ip/2
[e−pδ(t−s) Ekf (s, x(s − ρ3 (s)))kpH ]2/p ds
e−δ(t−s) mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds
hZ
t−ε
ip/2
[e−pδ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
≤ 2p−1 M p δ 1−p Θh (r∗ )
Z
t
e−δ(t−s) mh (s)ds
t−ε
h 2δ(p − 1) i1−p/2 Z t
e−δ(t−s) mf (s)ds.
+ 2p−1 Cp M p Θf (r∗ )
p−2
t−ε
Similarly, for any t ∈ (tk , tk+1 ], k = 1, . . . , m. Let tk < t ≤ s ≤ tk+1 be fixed and
let ε be a real number satisfying 0 < ε < t. For x ∈ Br , we define
Z t−ε
Z t−ε
ε
(Γ1 x)(t) =
Sα (t−s)h(s, x(s−ρ2 (s)))ds+
Sα (t−s)f (s, x(s−ρ3 (s)))dw(s).
tk
tk
Using the compactness of Sα (t) for t > 0, we deduce that the set Uε (t) = {(Γε1 x)(t) :
x ∈ Br } is relatively compact in H for every ε, 0 < ε < t. Moreover, for every
x ∈ Br we have
Ek(Γ1 x)(t)(t) − (Γε1 x)(t)(t)kpH
Z
t
p
≤ 2p−1 E Sα (t − s)h(s, x(s − ρ2 (s)))dsH
t−ε
+ 2p−1 E Z
t
t−ε
p
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H
≤ 2p−1 M p δ 1−p Θh (r∗ )
Z
t
e−δ(t−s) mh (s)ds
t−ε
+ 2p−1 Cp M p Θf (r∗ )
h 2δ(p − 1) i1−p/2 Z
p−2
t
e−δ(t−s) mf (s)ds.
t−ε
There are relatively compact sets arbitrarily close to the set W (t) = {(Γ1 x)(t) :
x ∈ Br }, and W (t) is a relatively compact in H. It is easy to see that Γ1 (Br )
is uniformly bounded. Since we have shown Φ1 (Br ) is equicontinuous collection,
by the Arzelá-Ascoli theorem it suffices to show that Γ1 maps Br into a relatively
compact set in H.
Next, we show that Γ2 (Br )(t) = {(Γ2 x)(t) : x ∈ Br } is relatively compact for
every t ∈ [0, T ]. For all t ∈ [0, t1 ], since (Γ2 x)(t) = Rα (t)ϕ(0), by (H1), it follows
that {(Γ2 x)(t) : t ∈ [0, t1 ], x ∈ Br } is a compact subset of H. On the other hand,
for t ∈ (tk , tk+1 ], k = 1, . . . , m, and x ∈ Br , and that the interval [0, T ] is divided
into finite subintervals by tk , k = 1, 2, . . . , m, so that we need to prove that
−
W = {Rα (t − tk )[x(t−
k ) + Ik (x(tk ))],
t ∈ [tk , tk+1 ], x ∈ Br }
EJDE-2013/206
ASYMPTOTIC STABILITY
21
is relatively compact in C([tk , tk+1 ], H). In fact, from (H1) and (H5), it follows that
−
the set {Rα (t − tk )[x(t−
k ) + Ik (x(tk ))], x ∈ Br } is relatively compact in H, for all
t ∈ [tk , tk+1 ], k = 1, . . . , m. Also, we see that the functions in W are equicontinuous
due to the compactness of Ik and the strong continuity of the operator Rα (t), for all
t ∈ [0, T ]. Now an application of the Arzelá-Ascoli theorem justifies the relatively
compactness of W . Therefore, we conclude that operator Γ2 is also a compact map.
Step 6. We shall show the set Υ = {x ∈ Y : λΨ1 ( λx ) + λΨ2 (x) = x for some λ ∈
(0, 1)} is bounded on [0, T ]. To do this, we consider the nonlinear operator equation
x(t) = λΨx(t), 0 < λ < 1,
(3.10)
where Ψ is already defined. Next we gives a priori estimate for the solution of the
above equation. Indeed, let x ∈ Y be a possible solution of x = λΨ(x) for some
0 < λ < 1. This implies by (3.10) that for each t ∈ [0, T ] we have

λRα (t)[ϕ(0) − g(0, ϕ(−ρ1 (0)))] + λg(t, x(t − ρ1 (t)))


Rt



+λ
S (t − s)h(s, x(s − ρ2 (s)))ds


R0t α



+λ
S (t − s)f (s, x(s − ρ3 (s)))dw(s),
t ∈ [0, t1 ],

0 α



−
+
+


λRα (t − t1 )[x(t−

1 ) + I1 (x(t1 )) − g(t1 , x(t1 − ρ1 (t1 )))]

Rt

+λg(t, x(t − ρ1 (t))) + λ t1 Sα (t − s)h(s, x(s − ρ2 (s)))ds
x(t) =
Rt

t ∈ (t1 , t2 ],

+λ t1 Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),



...




+
+
−
λRα (t − tm )[x(t−

m ) + Im (x(tm )) − g(tm , x(tm − ρ1 (tm )))]

R

t


+λg(t, x(t − ρ1 (t))) + λ tm Sα (t − s)h(s, x(s − ρ2 (s)))ds


Rt

+λ tm Sα (t − s)f (s, x(s − ρ3 (s)))dw(s),
t ∈ (tm , T ].
(3.11)
By using Hölder’s inequality and Lemma 2.11, we have for t ∈ [0, t1 ]
Ekx(t)kpH
≤ 4p−1 EkRα (t)[ϕ(0) − g(0, ϕ(−ρ1 (0)))]kpH + 4p−1 Ekg(t, x(t − ρ1 (t)))kpH
Z
t
p
p−1 +4 E
Sα (t − s)h(s, x(s − ρ2 (s)))dsH
0
Z t
p
+ 4p−1 E Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H
0
≤ 8p−1 M p [Ekϕ(0)kpH + Ekg(0, ϕ(−ρ1 (0)))kpH ] + 4p−1 Ekg(t, x(t − ρ1 (t)))kpH
hZ t
ip
+ 4p−1 M p E
e−δ(t−s) kh(s, x(s − ρ2 (s)))kH ds
0
hZ t
ip/2
+ 4p−1
[e−pδ(t−s) (Ekf (s, x(s − ρ3 (s)))kpH )]2/p ds
0
≤ 8p−1 M p [Ekϕ(0)kpH + Lg Ekϕ(−ρ1 (0))kpH ] + 4p−1 Lg Ekx(t − ρ1 (t)))kpH
hZ t
ip−1 Z t
p−1
p
−(pδ/p−1)(t−s)
+4 M
e
ds
mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds
0
0
hZ t
ip/2
+ 4p−1 Cp M p
[e−pδ(t−s) mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
.
0
22
Z. YAN, H. ZHANG
EJDE-2013/206
Similarly, for any t ∈ (tk , tk+1 ], k = 1, . . . , m, we have
Ekx(t)kpH
p
−
+
+
≤ 4p−1 EkRα (t − tk )[x(t−
k ) + Ik (x(tk )) − g(tk , x(tk − ρ1 (tk )))]kH
+ 4p−1 Ekg(t, x(t − ρ1 (t)))kpH
Z
p
t
p−1 Sα (t − s)h(s, x(s − ρ2 (s)))dsH
+4 E
tk
t
+ 4p−1 E Z
tk
≤ 12
p−1
p
Sα (t − s)f (s, x(s − ρ3 (s)))dw(s)H
p
p
− p
+
+
M [Ekx(t−
k )kH + EkIk (x(tk )kH + Ekg(tk , x(tk − ρ1 (tk )))kH ]
p
+ 4p−1 Ekg(t, x(t − ρ1 (t)))kpH
ip
hZ t
p−1
p
e−δ(t−s) kh(s, x(s − ρ2 (s)))kH ds
+4 M
tk
+ 4p−1
hZ
t
tk
≤ 12
p−1
ip/2
[e−pδ(t−s) (Ekf (s, x(s − ρ3 (s)))kpH )]2/p ds
p
p
− p
+
+
1
2
M [Ekx(t−
k )kH + dk Ekx(tk )kH + dk + Lg Ekx(tk − ρ1 (tk ))kH ]
p
+ 4p−1 Lg Ekx(t − ρ1 (t))kpH
hZ t
ip−1 Z t
p−1
p
−(pδ/p−1)(t−s)
+4 M
e
ds
mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds
tk
+ 4p−1 Cp M p
tk
hZ
t
tk
ip/2
.
[e−pδ(t−s) mf (s)Θf (Ekx(s − ρ3 (s)))kpH )]2/p ds
Then, for all t ∈ [0, T ], we have
Ekx(t)kpH
f + 12p−1 M p [Ekx(t− )kp + d1k Ekx(t− )kp
≤M
H
H
k
k
p
+
p−1
Lg Ekx(s − ρ1 (s))kpH
+ Lg Ekx(t+
k − ρ1 (tk ))kH ] + 4
hZ t
ip−1 Z t
+ 4p−1 M p
e−(pδ/p−1)(t−s) ds
mh (s)Θh (Ekx(s − ρ2 (s))kpH )ds
0
0
hZ t
ip/2
p−1
p
−pδ(t−s)
+ 4 Cp M
[e
mf (s)Θf (Ekx(s − ρ3 (s))kpH )]2/p ds
,
0
where
˜
f = max{8p−1 M p [Ekϕ(0)kp + Lg Ekϕ(−ρ1 (0))kp ], 12p−1 M p d},
M
H
H
d˜ = max1≤k≤m d2k . By the definition of Y, it follows that
Ekx(s − ρi (s))kpH ≤ 2p−1 kϕkpB + 2p−1 sup kx(s)kpH , i = 1, 2, 3.
s∈[0,t]
If µ(t) =
2p−1 kϕkpB
+
2p−1 sups∈[0,t] Ekx(s)kpH ,
we obtain that
f + 12p−1 M p [µ(t) + d1k µ(t) + 2p−1 Lg µ(t)] + 8p−1 Lg µ(t)
µ(t) ≤ 2p−1 kϕkpB + 2p−1 M
Z
h t
ip−1 Z t
p−1
p
−(pδ/p−1)(t−s)
+8 M
e
ds
mh (s)Θh (µ(s))ds
0
0
EJDE-2013/206
ASYMPTOTIC STABILITY
+ 8p−1 Cp M p
hZ
t
ip/2
[e−pδ(t−s) mf (s)Θf (µ(s))]2/p ds
.
0
Note that
hZ
t
ip/2
[e−pδ(t−s) mf (s)Θf (µ(s))]2/p ds
0
hZ t
ip/2−1 Z t
2p
≤
e−[ p−2 ]δ(t−s) ds
mf (s)Θf (µ(s))ds
0
0
Z
2δp 1−p/2 t
mf (s)Θf (µ(s))ds.
≤[
]
p−2
0
So, we obtain
f + 12p−1 M p [µ(t) + d1k µ(t) + 2p−1 Lg µ(t)]
µ(t) ≤ 2p−1 kϕkpB + 2p−1 M
Z
pδ 1−p t
]
mh (s)Θh (µ(s))ds
+ 8p−1 Lg µ(t) + 8p−1 M p [
p−1
0
Z t
p−1
p 2δp 1−p/2
+ 8 Cp M [
]
mf (s)Θf (µ(s))ds.
p−2
0
e = max1≤k≤m {12p−1 M p (1 + d1 + 2p−1 Lg ) + 8p−1 Lg } < 1, we obtain
Since L
k
Z t
h
1
f + 8p−1 M p [ pδ ]1−p
2p−1 kϕkpB + 2p−1 M
mh (s)Θh (µ(s))ds
µ(t) ≤
e
p−1
1−L
0
Z t
i
p−1
p 2δp 1−p/2
]
mf (s)Θf (µ(s))ds .
+ 8 Cp M [
p−2
0
Denoting by ζ(t) the right-hand side of the above inequality, we have
µ(t) ≤ ζ(t) for all t ∈ [0, T ],
1
f],
[2p−1 kϕkpB + 2p−1 M
ζ(0) =
e
1−L
1 h p−1 p pδ 1−p
] mh (t)Θh (µ(t))
ζ 0 (t) =
8 M [
e
p−1
1−L
i
2δp 1−p/2
+ 8p−1 Cp M p [
]
mf (t)Θf (µ(t))
p−2
1 h p−1 p pδ 1−p
≤
8 M [
] mh (t)Θh (ζ(t))
e
p−1
1−L
i
2δp 1−p/2
+ 8p−1 Cp M p [
]
mf (t)Θf (ζ(t))
p−2
≤ m∗ (t)[Θh (ζ(t)) + Θf (ζ(t))],
where
m∗ (t) = max
n
pδ 1−p
1
8p−1 M p [
] mh (t),
e
p−1
1−L
o
1
2δp 1−p/2
]
mf (t) .
8p−1 Cp M p [
e
p−2
1−L
This implies that
Z
ζ(t)
ζ(0)
du
≤
Θh (u) + Θf (u)
Z
0
T
m∗ (s)ds < ∞.
23
24
Z. YAN, H. ZHANG
EJDE-2013/206
e such that ξ(t) ≤ K,
e t ∈ [0, T ], and
This inequality shows that there is a constant K
e where K
e depends only on M, δ, p, Cp and on the functions
hence kxkY ≤ ζ(t) ≤ K,
mf (·), Θf (·). This indicates that Υ is bounded on [0, T ]. Consequently, by Lemma
2.12, we deduce that Ψ1 + Ψ2 has a fixed point x(·) ∈ Y, which is a mild solution of
the system (1.1)-(1.3) with x(s) = ϕ(s) on [m̃(0), 0] and Ekx(t)kpH → 0 as t → ∞.
This shows that the asymptotic stability of the mild solution of (1.1)-(1.3). In fact,
let ε > 0 be given and choose δ̃ > 0 such that δ̃ < ε and satisfies
[16p−1 M p + 8p−1 M p [δ 1−p Lh + Cp (
2δ(p − 1) 1−2/p
e < ε.
)
Lf ]]δ̃ + Lε
p−2
If x(t) = x(t, ϕ) is mild solution of (1.1)-(1.3), with kϕkpB + Lg Ekϕ(−ρ1 (0))kpH < δ̃,
then (Ψx)(t) = x(t) and satisfies Ekx(t)kpH < ε for every t ≥ 0. Notice that
Ekx(t)kpH < ε on t ∈ [m̃(0), 0]. If there exists t̃ such that Ekx(t̃)kpH = ε and
Ekx(s)kpH < ε for s ∈ [m̃(0), t̃]. Then (3.4) show that
Ekx(t)kpH
h
2δ(p − 1) 1−2/p i
e < ε,
Lf δ̃ + Lε
≤ 16p−1 M p e−pδt̃ + 8p−1 M p δ 1−p Lh + Cp
p−2
which contradicts the definition of t̃. Therefore, the mild solution of (1.1)-(1.3) is
asymptotically stable in p-th moment. The proof is complete.
Remark 3.4. It is well known that the study on nonlocal problems are motivated
by physical problems. For example, it is used to determine the unknown physical
parameters in some inverse heat conduction problems [10]. Due to the importance
of nonlocal conditions in different fields, there has been an increasing interest in
study of the fractional impulsive stochastic differential equations involving nonlocal
conditions (see [27]). In this remark, we will try to make some simulations about
the above results and study the asymptotical stability in p-th moment of mild solutions to a class of fractional impulsive partial neutral stochastic integro-differential
equations with nonlocal conditions in Hilbert spaces
Z t
dw(t)
c α
D N (x(t)) = AN (x(t)) +
R(t − s)N (x(s))ds + h(t, x(t)) + f (t, x(t))
,
dt
0
t ≥ 0, t 6= tk ,
(3.12)
x0 + G(x) = x0 ,
∆x(tk ) =
Ik (x(t−
k )),
x0 (0) = 0,
t = tk , k = 1, . . . , m,
(3.13)
(3.14)
where c Dα , A, Q, w are defined as in (1.1)-(1.3). Here N (x) = x(0) + g(t, x), x ∈ H,
and g : [0, ∞) × H → H, f : [0, ∞) × H → L(K, H), are all Borel measurable;
Ik : H → H(k = 1, . . . , m), G : Y → H are given functions, where Y be the space of
all F0 -adapted process ψ(t, w̃) : [0, ∞)×Ω → R which is almost certainly continuous
in t for fixed w̃ ∈ Ω. Moreover ψ(0, w̃) = x0 and Ekψ(t, w̃)kpH → 0 as t → ∞. Also
Y is a Banach space when it is equipped with a norm defined by
kψkY = sup Ekψ(t)kpH .
t≥0
To prove the Asymptotic stability result, we assume that the following condition
holds.
EJDE-2013/206
ASYMPTOTIC STABILITY
25
(H6) The functions G : Y → H are completely continuous and that there is a
constant c such that EkG(x)kpH ≤ c for every x ∈ Y.
Further, the mild solution of the Fractional impulsive stochastic system (3.12)(3.14) can be written as

Rα (t)[x0 − G(x) − g(0, x(0))] + g(t, x(t))


Rt



+ 0 Sα (t − s)h(s, x(s))ds


R

t


+ 0 Sα (t − s)f (s, x(s))dw(s),
t ∈ [0, t1 ],




−
−
+


Rα (t − t1 )[x(tR1 ) + I1 (x(t1 )) − g(t1 , x(t1 ))]


t
+g(t, x(t)) + t1 Sα (t − s)h(s, x(s))ds
x(t) =
Rt

+ t1 Sα (t − s)f (s, x(s))dw(s),
t ∈ (t1 , t2 ],





...




+

) + Im (x(t−
Rα (t − tm )[x(t−

m )) − g(tm , x(tm ))]

R tm


+g(t, x(t)) + t Sα (t − s)h(s, x(s))ds

m

 Rt
+ tm Sα (t − s)f (s, x(s))dw(s),
t ∈ (tm , T ].
One can easily prove that by adopting and employing the method used in Theorem 3.3, the fractional impulsive stochastic differential equations (3.12)-(3.14) is
asymptotically stable in p-th moment.
4. Example
Consider the fractional impulsive partial stochastic neutral integro-differential
equation
Z t
∂ 2 N (zt )(x)
∂ 2 N (zt )(x)
∂ α N (zt )(x)
=
+
ds
(t − s)σ e−µ(t−s)
α
2
∂t
∂x
∂x2
0
(4.1)
dw(t)
,
+ ς(t, x, z(t − ρ2 (t), x)) + $(t, x, z(t − ρ3 (t), x))
dt
t ≥ 0, 0 ≤ x ≤ π, t 6= tk ,
z(t, 0) = z(t, π) = 0,
zt (0, x) = 0,
t ≥ 0,
0 ≤ x ≤ π,
z(τ, x) = ϕ(τ, x), τ ≤ 0, 0 ≤ x ≤ π,
Z π
+
−
4z(tk , x) = z(tk , x) − z(tk , x) =
ηk (s, z(tk , x))ds, k = 1, 2, . . . , m,
(4.2)
(4.3)
(4.4)
(4.5)
0
α
∂
where (tk )k ∈ N is a strictly increasing sequence of positive numbers, Dtα = ∂t
α
is a Caputo fractional partial derivative of order α ∈ (1, 2), σ, and µ are positive
numbers and w(t) denotes a standard cylindrical Wiener process in H defined on
a stochastic space (Ω, F, P ). In this system, ρi (t) ∈ C(R+ , R+ ), i = 1, 2, 3, and
N (zt )(x) = z(t, x) − ϑ(t, x, z(t − ρ1 (t), x)).
Let H = L2 ([0, π]) with the norm k·kH and define the operators A : D(A) ⊆ H → H
by Aω = ω 00 with the domain
D(A) := {ω ∈ H : ω, ω 0 are absolutely continuous, ω 00 ∈ H, ω(0) = ω(π) = 0}.
26
Z. YAN, H. ZHANG
EJDE-2013/206
Then
Aω =
∞
X
n2 hω, ωn iωn ,
ω ∈ D(A),
n=1
q
2
where ωn (x) =
π sin(nx), n = 1, 2, . . . is the orthogonal set of eigenvectors of
A. It is well known that A generates a strongly continuous semigroup T (t), t ≥ 0
which is compact, analytic and self-adjoint in H and A is sectorial of type and
(P1) is satisfied. The operator R(t) : D(A) ⊆ H → H, t ≥ 0, R(t)x = tσ e−ωt x00 for
x ∈ D(A). Moreover, it is easy to see that conditions (P2) and (P3) in Section 2
are satisfied with b(t) = tσ e−µt and D = C0∞ ([0, π]), where C0∞ ([0, π]) is the space
of infinitely differentiable functions that vanish at x = 0 and x = π.
Additionally, we will assume that
(i) The function ϑ : [0, ∞) × [0, π] × R → R is continuous and there exists a
positive constant Lϑ such that
|ϑ(t, x, u) − ϑ(t, x, v)| ≤ Lϑ |u − v|,
t ≥ 0, x ∈ [0, π], u, v ∈ R.
(ii) The function ς : [0, ∞) × [0, π] × R → R is continuous and there exists a
positive continuous function mς (·) : R × [0, π] → R such that
|ς(t, x, u)| ≤ mς (t, x)|u|,
t ≥ 0, x ∈ [0, π], u ∈ R.
(iii) The function ϑ : [0, ∞) × [0, π] × R → R is continuous and there exists a
positive continuous function m$ (·) : R × [0, π] → R such that
|$(t, x, u)| ≤ m$ (t, x)|u|,
t ≥ 0, x ∈ [0, π], u ∈ R.
(iv) The functions ηk : R2 → R, k ∈ N, are completely continuous and there
are positive continuous functions Lk : [0, π] → R(k = 1, 2, . . . , m) such that
|ηk (s, u)| ≤ Lk (s)|u|, s ∈ [0, π], u ∈ R.
We can define N : H → H, g, h : [0, ∞) × H → H, f : [0, ∞) × H → L(K, H)
and Ik : H → H respectively by
N (zt )(x) = ϕ(0, x) + g(t, z)(x),
g(t, z)(x) = ϑ(t, x, z(t − ρ1 (t), x)),
h(t, z)(x) = ς(t, x, z(t − ρ2 (t), x)),
f (t, z)(x) = $(t, x, z(t − ρ3 (t), x)),
Z π
Ik (z)(x) =
ηk (s, z(x))ds.
0
Then the problem (4.1)–(4.5) can be written as (1.1)–(1.3). Moreover, it is easy to
see that
Ekg(t, z1 ) − g(t, z2 )kpH ≤ Lg kz1 − z2 kpH , z1 , z2 ∈ H,
Ekh(t, z)kpH ≤ mh (t)kzkpH , z ∈ H,
Ekf (t, z)kpH ≤ mf (t)kzkpH , z ∈ H,
EkIk (z)kpH ≤ dk kzkpH , z ∈ H, k = 1, 2, . . . , m,
where Lg = Lpϑ , mh (t) = supx∈[0,π] mpτ (t, x), mf (t) = supx∈[0,π] mp$ (t, x), dk =
Rπ
[ 0 Lk (s)ds]p , k = 1, 2, . . . , m. Further, we can impose some suitable conditions on
EJDE-2013/206
ASYMPTOTIC STABILITY
27
the above-defined functions to verify the assumptions on Theorem 3.3, we can conclude that system (4.1)-(4.5) has at least one mild solution, then the mild solutions
is asymptotically stable in the p-th mean.
Conclusions. In this article, we are focused on the theory study on the asymptotical stability in the p-th moment of mild solutions to a class of fractional impulsive
partial neutral stochastic integro-differential equations with state-dependent delay. We derive some interesting sufficient conditions to guarantee the asymptotical
stability results for fractional impulsive stochastic evolution systems in infinite dimensional spaces. Our techniques rely on the fractional calculus, properties of the
α-resolvent operator, and Krasnoselskii-Schaefer type fixed point theorem. Our
methods not only present a new way to study such problems under the Lipschitz
condition not required in the paper, but also provide new theory results appeared
in paper previously are generalized to the fractional stochastic systems settings
and the case of state-dependent delay with impulsive conditions. An application is
provided to illustrate the applicability of the new result.
Our future work will try to make some the above results and study the exponential stability in p-th moment of mild solutions to fractional impulsive partial
neutral stochastic integro-differential equations with state-dependent delay.
Acknowledgments. The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions to improve the the
original manuscript.
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Equations Vol. 2013 (2013), No. 81, pp. 1-21.
Zuomao Yan
Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China
E-mail address: yanzuomao@163.com
EJDE-2013/206
ASYMPTOTIC STABILITY
Hongwu Zhang
Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China
E-mail address: zh-hongwu@163.com
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