Filomat 31:9 (2017), 2727–2748
https://doi.org/10.2298/FIL1709727Y
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Asymptotic Behavior of Second-Order Impulsive Partial Stochastic
Functional Neutral Integrodifferential Equations with Infinite Delay
Zuomao Yana , Xiumei Jiaa
a Department
of Mathematics, Hexi University, Zhangye, Gansu 734000, P.R. China
Abstract. In this paper, the existence and asymptotic stability in p-th moment of mild solutions to a class
of second-order impulsive partial stochastic functional neutral integrodifferential equations with infinite
delay in Hilbert spaces is considered. By using Hölder’s inequality, stochastic analysis, fixed point strategy
and the theory of strongly continuous cosine families with the Hausdorff measure of noncompactness, a
new set of sufficient conditions is formulated which guarantees the asymptotic behavior of the nonlinear
second-order stochastic system. These conditions do not require the the nonlinear terms are assumed to be
Lipschitz continuous. An example is also discussed to illustrate the efficiency of the obtained results.
1. Introduction
The theory of stochastic partial differential equations has attracted much attention since it plays a vital
role in many important areas such as insurance, finance, population dynamics (see [6, 7, 18, 19, 21, 24, 34, 36]).
Neutral stochastic partial differential equations arise in many areas of applied mathematics and for this
reason these equations have been investigated extensively in the last decades. The existence and uniqueness,
and stability for first-order neutral stochastic partial differential equations with delays and without delays
have been extensively studied by many authors; see [8, 12, 16, 22] and the references therein.
In many cases, it is advantageous to treat the second-order stochastic differential equations directly rather
than to convert them to first-order systems. The second-order stochastic partial differential equations are
the right model in continuous time to account for integrated processes that can be made stationary. We
know that it is useful for engineers to model mechanical vibrations or charge on a capacitor or condenser
subjected to white noise excitation through a second-order stochastic differential equations. Further, many
authors investigated the the qualitative analysis of solutions for those equations in Hilbert spaces by using
different techniques. Among them, Balasubramaniam and Muthukumar [4] established the approximate
controllability of second-order neutral stochastic evolution differential equations by using the Sadovskii
fixed-point theorem. Ren and Sun [25] discussed the existence and uniqueness of mild solutions for
second-order neutral stochastic evolution equations with infinite delay under Carathéodory conditions by
means of the successive approximation. Using the Banach contraction mapping principle, Mahmudov and
2010 Mathematics Subject Classification. Primary 34A37, 34F05, 35R60; Secondary 60H15, 35B35.
Keywords. Asymptotic stability, Impulsive partial stochastic functional neutral integrodifferential equations, Infinite delay, Strongly
continuous cosine families, Fixed point theorem.
Received: 03 September 2015; Accepted: 18 April 2016
Communicated by Svetlana Janković
Research supported by the National Natural Science Foundation of China (11461019), the President Fund of Scientific Research
Innovation and Application of Hexi University (xz2013-10).
Email addresses: [email protected] ( Zuomao Yan), [email protected] (Xiumei Jia)
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
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McKibben [23] derived the approximate controllability of abstract second-order neutral stochastic evolution
equations. Sakthivel et al. in [27, 31] studied the asymptotic stability of second-order stochastic partial
differential equations. Chen [9] also considered the exponential stability and the asymptotic stability for
mild solution to second-order neutral stochastic partial differential equations with infinite delay by using
Picard approximations and the elementary inequality.
However, in addition to stochastic effects, impulsive effects likewise exist in real systems. A lot of
dynamical systems have variable structures subject to stochastic abrupt changes, which may result from
abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections
of subsystems, sudden environment changes, etc. Therefore, it is necessary and important to consider the
existence, uniqueness and other quantitative and qualitative properties of solutions to stochastic systems
with impulsive effects [33, 35, 37]. Recently, based on the above stochastic analysis method, the existence,
uniqueness and stability of mild solutions for various first-order impulsive stochastic partial differential
equations and integrodifferential equations have been extensively studied. For example, Sakthivel and Luo
[29, 30], Anguraj and Vinodkumar [2], He and Xu [17], Chen et al. [10, 11], Long et al. [20]. Ren and Sun [26]
investigated the existence, uniqueness and stability of solution to second-order neutral impulsive stochastic
evolution equations with delay by using the successive approximation. Using the Banach contraction
mapping principle and the cosine function theory, Sakthivel et al. in [32] discussed the asymptotic stability
of second-order impulsive stochastic differential equations but the result is only in connection without
delay. Arthi et al. [3] proved the exponential stability of mild solution for the second-order neutral
stochastic partial differential equations with impulses.
In this paper we consider the existence and asymptotic stability of mild solutions for second-order
impulsive neutral partial stochastic functional integrodifferential equations with infinite delay in Hilbert
spaces of the form
Z t
0
d[x (t) − 1(t, x(t − ρ1 (t)))] = Ax(t) + h t, x(t − ρ2 (t)),
a(t, s, x(s − ρ3 (s)))ds dt
0
Z t
+ f t, x(t − ρ4 (t)),
b(t, s, x(s − ρ5 (s)))ds dw(t),
0
(1)
t ≥ 0, t , tk ,
∆x(tk ) = Ik (x(t−k )), t = tk , k = 1, . . . , m,
(2)
∆x0 (tk ) = Jk (x(t−k )), t = tk , k = 1, . . . , m,
(3)
x0 (·) = ϕ ∈ BF0 ([m̃(0), 0], H),
(4)
x0 (0) = φ,
where the state x(·) takes values in a separable real Hilbert space H with inner product h·, ·iH and norm k · kH .
The operator A : D(A) → H is the infinitesimal generator of a strongly continuous cosine family on H. 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 1 : [0, ∞) × H →
H, h : [0, ∞) × H × H → H, a, b : [0, ∞) × [0, ∞) × H → H, f : [0, ∞) × H × H → L(K, H), are all Borel measurable,
where L(K, H) denotes the space of all bounded linear operators from K into H; Ik , Jk : 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(t−k ) represent the right and left limits of x(t) at t = tk , respectively; ∆x(tk ) = x(t+k ) − x(t−k ), represents the
jump in the state x at time tk with Ik , Jk determining the size of the jump; let ρi (t) ∈ C(R+ , R+ )(i = 1, 2, 3, 4, 5)
satisfy t − ρi (t) → ∞ as t → ∞, and m̃(0) = max{infs≥0 (s − ρi (s)), i = 1, 2, 3, 4, 5}. 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 k ϕ kB = supm̃(0)≤t≤0 E k ϕ(t) kH .
To the best of the authors’ knowledge, no results about the existence and asymptotic stability of mild
solutions for second-order impulsive neutral partial stochastic functional integrodifferential equations with
infinite delay, which is expressed in the form (1)-(4). Although the papers [29, 30, 32] studied the asymptotic
stability for nonlinear impulsive stochastic differential and functional differential equations with delay and
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
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with infinite delay, besides the fact that [29, 30, 32] applies to the asymptotic stability of systems under
the Lipschitz conditions, the class of nonlinear impulsive stochastic systems is also different from the one
studied here. From a practical viewpoint, the second-order impulsive stochastic partial differential and
neutral functional differential equations with infinite delay deserve a study because they describe a kind of
system present in the real world. Hence, for a more realistic abstract model of the equation and for studying
the asymptotic stability property, this can be considered by introducing stochastic systems. Motivated by
the above consideration, we study this interesting problem, which is natural generalizations of the concepts
for impulsive equations well known in the theory of infinite dimensional systems.
The most common and easily verified conditions to guarantee the existence and stability of mild solutions
are the impulsive stochastic systems with the nonlinear function is a Lipschitz function. In this paper, we
discuss this problem by introducing a more appropriate concept for mild solutions. Then, using Hölder’s
inequality, stochastic analysis, the Darbo fixed point theorem and the theory of strongly continuous cosine
families combined with techniques of the Hausdorff measure of noncompactness, we get the existence and
asymptotic stability of mild solutions for system (1)-(4). Especially, as compared to the case for the previous
results, we no longer require the Lipschitz continuity of f, h and the compactness assumption on associated
operators. In fact, we assume that the nonlinear items f, h are continuous functions while the neutral item
1 satisfies the generally Lipschitz continuity condition, and some suitable conditions on the above-defined
functions, which can make the solution operator satisfies all conditions of the Darbo fixed point theorem.
Though the results are not must be limited to apply this theorem under our assumptions, for example, we
can choose the Krasnoselskii-Schaefer type fixed point theorem. However, the Darbo fixed point theorem
is first used to consider the stability for second-order impulsive partial stochastic functional differential
equations. The known results appeared in [9, 29, 30, 32] are generalized to the impulsive neutral stochastic
functional integrodifferential systems settings and the case of infinite delay.
The rest of this paper is organized as follows. In Section 2, we introduce some notations and necessary
preliminaries. In Section 3, we give our main results. In Section 4, an example is given to illustrate our
results. Finally, concluding remarks are given in Section 5.
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 }∞
be a complete orthonormal basis of K. Suppose that
i=1
P {w(t) : t ≥ 0} is a cylindrical K-valued Brownian
motion with a trace class operator Q, denote Tr(Q) = ∞
i=1 λi = λ < ∞, which satisfies that Qei = λi ei .
P √
∞
λ
w
(t)e
,
where
{w
(t)}
are
mutually
independent one-dimensional standard
So, actually, w(t) = ∞
i i
i
i
i=1
i=1
Brownian motions. Then, the above K-valued stochastic process w(t) is called a Q-Wiener process. Let
L(K, H) be the space of bounded linear operators mapping K into H equipped with the usual norm k · kH
and L(H) denotes the Hilbert space of bounded linear operators from H to H. For ψ̃ ∈ L(K, H) we define
k ψ̃ k2L0 = Tr(ψ̃Qψ̃∗ ) =
2
∞
X
k
p
λi ψ̃ei k2 .
i=1
If k ψ̃ k2L0 < ∞, then ς is called a Q-Hilbert–Schmidt operator, and let L02 (K, H) denote the space of all
2
Q-Hilber–Schmidt operators ψ̃ : K → H.
Let Y be the space of all F0 -adapted process ψ(t, w̃) : [m̃(0), ∞) × Ω → R which is almost certainly
p
continuous in t for fixed w̃ ∈ Ω. Moreover ψ(s, w̃) = ϕ(s) for s ∈ [m̃(0), 0] and E k ψ(t, w̃) kH → 0 as t → ∞.
Also Y is a Banach space when it is equipped with a norm defined by
p
p
k ψ kY = sup E k ψ(t) kH .
t≥0
The notation Br (x, H) stands for the closed ball with center at x and radius r > 0 in H.
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
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Definition 2.1. ([15]) The one parameter cosine family {C(t) : t ∈ R} ⊂ L(H) satisfying
(i) C(0) = I;
(ii) C(t)x is in continuous in t on R for all x ∈ R;
(iii) C(t + s) + C(t − s) = 2C(t)C(s) for all t, s ∈ R
is called a strongly continuous cosine family.
The corresponding strongly continuous sine family {S(t) : t ∈ R} ⊂ L(H) is defined by S(t)x =
C(s)xds, t ∈ R, x ∈ H. The generator A : H → H of {C(t) : t ∈ R} is given by Ax = (d2 /dt2 )C(t)x|t=0
0
for all x ∈ D(A) = {x ∈ H : C(·)x ∈ C2 (R, H)}. It is well known that the infinitesimal generator A is a closed,
densely defined operator on H. Such cosine and the corresponding sine families and their generators satisfy
the following properties.
Rt
Lemma 2.2. ([14]) Suppose that A is the infinitesimal generator of a cosine family of operators {C(t) : t ∈ R} Then,
the following holds:
(a) There exists M1 ≥ 1 and α ≥ 0 such that k C(t) kH ≤ M1 eαt and hence k S(t) kH ≤ M1 eαt .
Rr
(b) A s S(u)xdu = [C(r) − C(u)]x for all 0 ≤ s ≤ r < ∞.
Rs
(c) There exists M2 ≥ 1 such that k S(s) − S(r) kH ≤ M2 | r eα|θ| dθ| for all 0 ≤ r ≤ s < ∞.
Definition 2.3. A stochastic process {x(t), t ∈ [0, T]}(0 ≤ T < ∞) is called a mild solution of Eqs. (1)-(4) 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
t
Z
x(t) = C(t)ϕ(0) + S(t)[φ − 1(0, ϕ(−ρ1 (0)))] +
C(t − s)1(s, x(s − ρ1 (s)))ds
0
t
Z
+
Z s
S(t − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ ds
0
0
s
S(t − s) f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ2 (τ)))dτ dw(s)
0
X
X0
−
+
C(t − tk )Ik (x(tk )) +
S(t − tk )Jk (x(t−k )),
Z
t
+
0<tk <t
Z
(5)
0<tk <t
and
x0 (·) = ϕ ∈ BF0 ([m̃(0), 0], H),
x0 (0) = φ.
Definition 2.4. Let p ≥ 2 be an integer. Eq. (5) is said to be stable in p-th moment if for arbitrarily given ε > 0 there
exists a δ̃ > 0 such that k ϕ kB < δ̃ guarantees that guarantees that
p
E sup k x(t) kH < ε.
t≥0
Definition 2.5. Let p ≥ 2 be an integer. Eq. (5) is said to be asymptotically stable in p-th moment if it stable in p-th
moment and for any ϕ ∈ BF0 ([m̃(0), 0], H),
p
lim E sup k x(t) kH = 0.
T→+∞
t≥T
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
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Now, we introduce the Hausdorff measure of noncompactness χY defined by
χY (B) = inf{ε > 0; B has a finite ε − net in H},
for bounded set B in any Hilbert space Y. Some basic properties of χY (·) are given in the following lemma.
Lemma 2.6. ([5]) Let Y be a real Hilbert space and B, C ⊆ Y be bounded, the following properties are satisfied:
(1) B is pre-compact if and only if χY (B) = 0;
(2) χY (B) = χY (B) = χY (convB), where B and convB are the closure and the convex hull of B respectively;
(3) χY (B) ≤ χY (C) when B ⊆ C;
(4) χY (B + C) ≤ χY (B) + χY (C) where B + C = {x + y : x ∈ B, y ∈ C};
(5) χY (B ∪ C) = max{χY (B), χY (C)};
(6) χY (λB) ≤ |λ|χY (B) for any λ ∈ R;
(7) If the map Φ : D(Φ) ⊆ Y → Z is Lipschitz continuous with constant κ then χZ (ΦB) ≤ κχY (B) for any bounded
subset B ⊆ D(Φ), where Z is a Banach space;
Definition 2.7. ([28]) The map Φ : V ⊆ Y → Y is said to be a χY -contraction if there exists a positive constant
κ < 1 such that χY (Φ(B)) ≤ κχY (B) for any bounded close subset B ⊆ V where Y is a Banach space.
In this paper we denote by χC the Hausdorff’s measure of noncompactness of C([0, b], H) and by χY the
Hausdorff’s measure of noncompactness of Y.
Lemma 2.8. ([13]) For any p ≥ 1 and for arbitrary L02 (K, H)-valued predictable process φ(·) such that
Z s
w
w
Z t
p
2p
w
w
w
w
w
w
2p 1/p
p
w
w
w
sup Ew
φ(v)dw(v)
≤
(p(2p
−
1))
(E
k
φ(s)
k
)
ds
, t ∈ [0, ∞).
w
w
L02
w 0
wH
0
s∈[0,t]
In the rest of this paper, we denote by Cp = (p(p − 1)/2)p/2 .
Lemma 2.9. ([1] Darbo). If V ⊆ Y is closed and convex and 0 ∈ V, the continuous map Φ : V → V is a χY contraction, if the set {x ∈ V : x = λΦx} is bounded for 0 < λ < 1, then the map Φ has at least one fixed point in
V.
3. Main Results
In this section we present our main results on the existence and asymptotic stability in the p-th moment
of mild solutions of system (1)-(4). To do this, we make the following hypotheses:
(H1) A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ≥ 0} on H and the
corresponding sine family {S(t) : t ≥ 0} satisfy the conditions k C(t) kH ≤ Me−αt and k S(t) kH ≤ Me−βt , t ≥
0 for some constants M ≥ 1, α > 0 and β > 0.
(H2) The function 1 : [0, ∞) × H → H is continuous and there exists L1 > 0 such that
p
p
E k 1(t, ψ1 ) − 1(t, ω2 ) kH ≤ L1 E k ψ1 − ψ2 kH , t ≥ 0, ψ1 , ψ2 ∈ H,
and
p
p
E k 1(t, ψ) kH ≤ L1 E k ψ kH , t ≥ 0, ψ ∈ H
with Mp L1 α−p < 1.
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
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(H3) There exists a function continuous function ma : [0, ∞) → [0, ∞) such that
Z t
w
w
p
w
w
w
w
w
w
p
w
w
w
Ew
a(t, s, ψ)dsw
≤ ma (t)Θa (E k ψ kH )
w
w 0
wH
for a.e.t ≥ 0 and all ψ ∈ H, where Θa : [0, ∞) → (0, ∞) is a continuous and nondecreasing function.
(H4) The function h : [0, ∞) × H × H → H satisfies the following conditions:
(i) The function h : [0, ∞) × H × H → H is continuous.
(ii) There exist a continuous function mh : [0, ∞) → [0, ∞) and a continuous nondecreasing function
Θh : [0, ∞) → (0, ∞) such that
p
p
p
E k h(t, ψ, x) kH ≤ mh (t)Θh (E k ψ kH ) + E k x kH , t ≥ 0, ψ, x ∈ H.
(iii) The set {S(t − s)h(s, ψ,
Rs
0
a(s, τ, ψ)dτ) : t, s ∈ [0, b], ψ ∈ Br (0, H)} is relatively compact in H.
(H5) There exists a function continuous function mb : [0, ∞) → [0, ∞) such that
Z t
w
w
p
w
w
w
w
w
w
p
w
w
w
Ew
b(t,
s,
ψ)ds
≤ mb (t)Θb (E k ψ kH )
w
w
w 0
wH
for a.e.t ≥ 0 and all ψ ∈ H, where Θb : [0, ∞) → (0, ∞) is a continuous and nondecreasing function.
(H6) The function f : [0, ∞) × H × H → L(K, H) satisfies the following conditions:
(i) The function f : [0, ∞) × H × H → L(K, H) is continuous.
(ii) There exist a continuous function m f : [0, ∞) → [0, ∞) and a continuous nondecreasing function
Θ f : [0, ∞) → (0, ∞) such that
p
p
p
E k f (t, ψ, x) kH ≤ m f (t)Θ f (E k ψ kH ) + E k x kH , t ≥ 0, ψ, x ∈ H.
(iii) The set {S(t − s) f (s, ψ,
Rs
0
b(s, τ, ψ)dτ) : t, s ∈ [0, b], ψ ∈ Br (0, H)} is relatively compact in H.
( j)
(H7) The functions Ik , Jk : H → H are completely continuous and that there are constants dk , k =
p
(1)
p
(2)
p
(3)
p
(4)
1, 2, . . . m, j = 1, 2, 3, 4 such that E k Ik (x) kH ≤ dk E k x kH +dk , E k Jk (x) kH ≤ dk E k x kH +dk for
every x ∈ H.
In the proof of the main results, we need the following lemmas.
Lemma 3.1. Assume that conditions (H1), (H2) hold. Let Φ1 be the operator defined by: for each x ∈ Y,
t
Z
(Φ1 x)(t) =
C(t − s)1(s, x(s − ρ1 (s)))ds.
0
Then Φ1 is continuous on [0, ∞) in p-th mean and maps Y into itself.
(6)
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2733
Proof. We first prove that Φ1 is continuous in p-th moment on [0, ∞). Let x ∈ Y, t̃ ≥ 0 and |ξ| be sufficiently
small, we have
p
E k (Φ1 x)(t̃ + ξ) − (Φ1 x)(t̃) kH
w
w
p
w Z t̃
w
w
w
w
p−1 w
w
w
w
≤ 2 Ew
[C(t̃ + ξ − s) − C(t̃ − s)]1(s, x(s − ρ1 (s)))dsw
w
w 0
wH
Z t̃+ξ
w
w
p
w
w
w
w
w
w
w
w
+ 2p−1 Ew
C(t̃ + ξ − s)1(s, x(s − ρ1 (s)))dsw
w
w
w
w
H
t̃
Z
t̃
p
≤ 2p−1 E
k [C(t̃ + ξ − s) − C(t̃ − s)]1(s, x(s − ρ1 (s))) kH ds
0
t̃+ξ
Z
+ 2p−1 Mp E
e−α(t̃+ξ−s) k 1(s, x(s − ρ1 (s))) kH ds
p
t̃
p−1
t̃
Z
k C(t̃ + ξ − s) − C(t̃ − s)
≤2
0
+2
p−1
M
p
t̃+ξ
Z
e
−(pα/p−1)(t̃+ξ−s)
(p/p−1)
kH
ds
0
p
E k 1(s, x(s − ρ1 (s))) kH ds
t̃+ξ
p−1 Z
ds
t̃
t̃
t̃
p−1 Z
p
E k 1(s, x(s − ρ1 (s))) kH ds → 0
as ξ → ∞.
Thus Φ1 is continuous in p-th moment on [0, ∞).
Next we show that Φ1 (Y) ⊂ Y. By (H1) and (H2), from the equation (6), we have for t ∈ [m̃(0), ∞),
E k (Φ1 x)(t)
p
kH
t
Z
≤E
p
k C(t − s)1(s, x(s − ρ1 (s))) kH ds
0
t
Z
p
p
e−α(t−s) k 1(s, x(s − ρ1 (s))) kH ds
≤M E
≤ Mp
0
t
Z
e−α(t−s) ds
0
0
≤ Mp α1−p
0
p
e−α(t−s) E k x(s − ρ2 (s)) kH ds
t
= K1
p
e−α(t−s) E k 1(s, x(s − ρ1 (s))) kH ds
t
Z
Z
t
p−1 Z
p
e−α(t−s) E k x(s − ρ2 (s)) kH ds.
0
p
However, for any any ε > 0 there exists a t̃1 > 0 such that E k x(s − ρ1 (s)) kH < ε for t ≥ t̃1 . Thus, we obtain
p
t
Z
E k (Φ1 x)(t) kH ≤ K1
0
p
e−α(t−s) E k x(s − ρ1 (s)) kH ds
t̃1
Z
−αt
≤ K1 e
0
eαs E k x(s − ρ1 (s)) kH ds + K1 α−1 ε.
p
As e−αt → 0 as t → ∞ and, there exists t̃2 ≥ t̃1 such that for any t ≥ t̃2 we have
t̃1
Z
K1 e−αt
0
eαs E k x(s − ρ1 (s)) kH ds < ε − K1 α−1 ε.
p
p
p
From the above inequality, for any t ≥ t̃2 , we obtain E k (Φ1 x)(t) kH < ε. That is to say E k (Φ1 x)(t) kH → 0 as
t → ∞. So we conclude that Φ1 (Y) ⊂ Y.
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
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Lemma 3.2. Assume that conditions (H1), (H3), (H4)(i)-(ii) hold. Let Φ2 be the operator defined by: for each x ∈ Y,
Z s
Z t
a(s, τ, x(τ − ρ3 (τ)))dτ ds.
(7)
S(t − s)h s, x(s − ρ2 (s)),
(Φ2 x)(t) =
0
0
Then Φ2 is continuous and maps Y into itself.
Proof. We first prove that Φ2 is continuous in p-th moment on [0, ∞). Let x ∈ Y, t̃ ≥ 0 and |ξ| be sufficiently
small, we have
p
E k (Φ2 x)(t̃ + ξ) − (Φ2 x)(t̃) kH
Z s
w
w
p
w Z t̃
w
w
w
w
w
w
w
≤ 2p−1 Ew
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
dsw
[S(
t̃
+
ξ
−
s)
−
S(
t̃
−
s)]h
s,
x(s
−
ρ
(s)),
3
2
w
w
w 0
wH
0
Z s
w Z t̃+ξ
w
p
w
w
w
w
w
w
w
w
+ 2p−1 Ew
S(t̃ + ξ − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ dsw
w
w
w
wH
t̃
0
Z
Z
w
t̃ w
w
p
s
w
w
w
w
w
p−1
w
w
w
≤2 E
[S(t̃ + ξ − s) − S(t̃ − s)]h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ w
ds
w
w
w
wH
0
0
Z
w
w
Z t̃+ξ
s
p
w
w
w
w
w
w
w
w
e−β(t̃+ξ−s) w
h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ w
+ 2p−1 Mp E
ds
w
w
w
wH
t̃
0
Z s
w
w(p/p−1) p−1 Z t̃
Z t̃ w
w
p
w
w
w
w
w
w
w
w
w
w
w
w
w
w
≤ 2p−1
S(
t̃
+
ξ
−
s)
−
S(
t̃
−
s)
E
k
h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
ds
2
3
w
w
w
w
w
wH
H
0
0
0
p−1
Z t̃+ξ
p−1 p
e−(pβ/p−1)(t̃+ξ−s) ds
+2 M
t̃
t̃+ξ
Z
×
t̃
Z s
w
w
p
w
w
w
w
w
w
w
w
w
Ew
h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds → 0
2
3
w
w
w
wH
0
as ξ → ∞.
Thus Φ2 is continuous in p-th moment on [0, ∞).
Next we show that Φ2 (Y) ⊂ Y. By (H1), (H3) and (H4)(i)-(ii), from the equation (7), we have for
t ∈ [m̃(0), ∞),
p
E k (Φ2 x)(t) kH
Z s
Z t w
w
p
w
w
w
w
w
w
w
w
w
≤E
S(t − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ w
ds
w
w
w
wH
0
0
Z s
w
Z t
w
p
w
w
w
w
w
w
w
w
w
≤ Mp E
e−β(t−s) w
h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
2
3
w
w
w
wH
0
0
Z
w
Z t
p−1 Z t
w
s
p
w
w
w
w
w
w
w
w
w
≤ Mp
e−β(t−s) ds
e−β(t−s) Ew
h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
2
3
w
w
w
wH
0
0
0
Z t
p
p
p 1−p
≤M β
e−β(t−s) [mh (s)Θh (E k x(s − ρ2 (s)) kH ) + ma (s)Θa (E k x(s − ρ3 (s)) kH )]ds
0
t
Z
= K2
0
p
p
e−β(t−s) [mh (s)Θh (E k x(s − ρ2 (s)) kH ) + ma (s)Θa (E k x(s − ρ3 (s)) kH )]ds.
p
p
However, for any any ε > 0 there exists a τ̃1 > 0 such that E k x(s − ρ2 (s)) kH < ε and E k x(s − ρ3 (s)) kH < ε
for t ≥ τ̃1 . Thus, we obtain
Z t
p
p
p
E k (Φ2 x)(t) kH ≤ K2
e−β(t−s) [mh (s)Θh (E k x(s − ρ2 (s)) kH ) + ma (s)Θa (E k x(s − ρ3 (s)) kH )]ds
0
τ̃1
Z
≤ K2 e−βt
0
eβs [mh (s)Θh (E k x(s − ρ2 (s)) kH )
p
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2735
p
+ ma (s)Θa (E k x(s − ρ3 (s)) kH )]ds + K2 Lh,a [Θh (ε) + Θa (ε)],
Rt
where Lh,a = supt≥0 τ̃ e−β(t−s) [mh (s) + ma (s)]ds. As e−βt → 0 as t → ∞ and, there exists τ̃2 ≥ τ̃1 such that for
1
any t ≥ τ̃2 we have
Z τ̃1
p
p
−βt
K2 e
eβs [mh (s)Θh (E k x(s − ρ2 (s)) kH ) + ma (s)Θa (E k x(s − ρ3 (s)) kH )]ds < ε − K2 Lh,a [Θh (ε) + Θa (ε)].
0
p
p
From the above inequality, for any t ≥ τ̃2 , we obtain E k (Φ2 x)(t) kH < ε. That is to say E k (Φ2 x)(t) kH → 0 as
t → ∞. So we conclude that Φ2 (Y) ⊂ Y.
Lemma 3.3. Assume that conditions (H1), (H5), (H6)(i)-(ii) hold. Let Φ3 be the operator defined by: for each x ∈ Y,
Z s
Z t
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s).
(8)
S(t − s) f s, x(s − ρ4 (s)),
(Φ3 x)(t) =
0
0
Then Φ3 is continuous and maps Y into itself.
Proof. We first prove that Φ3 is continuous in p-th moment on [0, ∞). Let x ∈ Y, t̃ ≥ 0 and |ξ| be sufficiently
small, we have
p
E k (Φ3 x)(t̃ + ξ) − (Φ3 x)(t̃) kH
Z
Z s
w
w
p
w
w
w t̃
w
w
p−1 w
w
w
w
≤ 2 Ew
[S(
t̃
+
ξ
−
s)
−
S(
t̃
−
s)]
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
dw(s)
4
5
w
w
w
wH
0
0
Z
Z
w
w
s
t̃+ξ
p
w
w
w
w
w
w
w
w
S(t̃ + ξ − s) f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s)w
+ 2p−1 Ew
w
w
wH
w t̃
0
Z
Z t̃ w
w
s
p 2/p p/2
w
w
w
w
w
w
w
w
w
Ew
≤ 2p−1 Cp
[S(
t̃
+
ξ
−
s)
−
S(
t̃
−
s)]
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
4
5
w
w
w
wH
0
0
w
Z t̃+ξ w
w
w
p−1
w
Ew
+ 2 Cp
S(t̃ + ξ − s)
w
w
t̃
Z s
w
wp 2/p p/2
w
w
w
× f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ w
ds
→ 0 as ξ → ∞.
w
w
H
0
Thus Φ3 is continuous in p-th moment on [0, ∞).
Next we show that Φ3 (Y) ⊂ Y. By (H1),(H5) and (H6)(i)-(ii), from the equation (8), we have for t ∈
[m̃(0), ∞),
p
E k (Φ3 x)(t) kH
Z s
Z t w
w
p 2/p p/2
w
w
w
w
w
w
w
w
w
≤ Cp
Ew
S(t
−
s)
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
4
5
w
w
w
wH
0
0
Z s
w
Z t
w
p 2/p p/2
w
w
w
w w
p
−pβ(t−s) w
w
w
≤ Cp M
f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ w
ds
e
Ew
w
w
wH
w
0
0
Z
t
2/p p/2
p
p
≤ Cp Mp
e−pβ(t−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )] ds
0
≤ Cp Mp
t
Z
e−[
2(p−1)
p−2 ]β(t−s)
0
0
≤ Cp Mp
t
Z
= K3
0
p
0
p
e−β(t−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]ds
t
2β(p − 1) 1−p/2 Z
p−2
t
p/2−1 Z
ds
p
p
e−β(t−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]ds
p
p
e−β(t−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]ds.
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2736
p
p
However, for any any ε > 0 there exists a θ̃1 > 0 such that E k x(s − ρ4 (s)) kH < ε and E k x(s − ρ5 (s)) kH < ε
for t ≥ θ̃1 . Thus, we obtain
Z t
p
p
p
E k (Φ3 x)(t) kH ≤ K3
e−β(t−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]ds
0
θ̃1
Z
eβs [m f (s)Θ f (E k x(s − ρ4 (s)) kH )
p
≤ K3
0
p
+ mb (s)Θb (E k x(s − ρ5 (s)) kH )]ds + K3 L f,b [Θ f (ε) + Θb (ε)],
where L f,b = supt≥0
Rt
θ̃1
e−β(t−s) [m f (s) + mb (s)]ds. As e−βt → 0 as t → ∞ and, there exists θ̃2 ≥ θ̃1 such that for
any t ≥ θ̃2 we have
Z t1
p
p
K3 e−δt
eδs [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]ds < ε − K3 L f,b [Θ f (ε) + Θb (ε)].
0
p
p
From the above inequality, for any t ≥ θ̃2 , we obtain E k (Φ3 x)(t) kH < ε. That is to say E k (Φ3 x)(t) kH → 0 as
t → ∞. So we conclude that Φ3 (Y) ⊂ Y.
Now, we are ready to present our main result.
Theorem 3.4. Assume the conditions (H1)-(H7) hold. Let p ≥ 2 be an integer. Then the impulsive stochastic
P
(1)
(3)
differential equations (1)-(4) is asymptotically stable in p-th moment, provided that (14m)p−1 Mp m
k=1 (dk + dk ) < 1,
and
Z ∞
1
ds = ∞.
(9)
s
+
Θ
(s)
+
Θ
(s)
+ Θ f (s) + Θb (s)
a
h
1
Proof. We define the nonlinear operator Ψ : Y → Y as (Ψx)(t) = ϕ(t) for t ∈ [m̃(0), 0] and for t ≥ 0,
Z t
(Ψx)(t) = C(t)ϕ(0) + S(t)[φ − 1(0, ϕ(−ρ1 (0)))] +
C(t − s)1(s, x(s − ρ1 (s)))ds
0
t
Z
+
Z s
S(t − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ ds
0
0
s
S(t − s) f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s)
0
X
X0
−
+
C(t − tk )Ik (x(tk )) +
S(t − tk )Jk (x(t−k )).
Z
t
+
Z
0<tk <t
0<tk <t
Using (H1)-(H7), and the proof of the Lemmas 3.1-3.3, it is clear that the nonlinear operator Ψ is well
defined and continuous. Moreover, for each t ≥ 0 we have
p
p
p
E k (Ψx)(t) kH ≤ 7p−1 E k C(t)ϕ(0) kH +7p−1 E k S(t)[φ − 1(0, ϕ(−ρ1 (0)))] kH
w
w
p
wZ t
w
w
w
w
w
w
w
w
+7p−1 Ew
C(t
−
s)1(s,
x(s
−
ρ
(s)))ds
1
w
w
w 0
wH
Z
Z
w
w
t
s
p
w
w
w
w
w
p−1 w
w
w
+7 Ew
S(t − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ dsw
w
w
w 0
wH
0
Z t
Z s
w
w
p
w
w
w
w
w
w
w
w
w
+7p−1 Ew
S(t
−
s)
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
dw(s)
4
5
w
w
w
w
H
0
0
w
w
w
w
p
p
w
w
w
w
X
X
w
w
w
w
w
w
w
p−1 w
w
w
w
w
+7p−1 Ew
C(t − tk )Ik (x(t−k ))w
S(t − tk )Jk (x(t−k ))w
w + 7 Ew
w
w .
w
w
w
w
w
0<tk <t
H
0<tk <t
H
(10)
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2737
Using (H1) and (H2), we have
p
7p−1 E k C(t)ϕ(0) kH
p
≤ Mp e−pαt E k ϕ(0) kH → 0
as t → ∞,
p
p
p
7p−1 E k S(t)[φ − 1(0, ϕ(−ρ1 (0)))] kH ≤ 14p−1 Mp e−pβt [E k φ kH +L1 E k ϕ(−ρ1 (0)) kH ] → 0
as t → ∞.
By (H1)-(H7) and the proof of the Lemmas 3.1-3.3 again, we obtain
p−1
Z t
w
w
p
w
w
w
w
w
w
w
w
w
Ew
C(t − s)1(s, x(s − ρ1 (s)))dsw
→0
w
w 0
wH
p−1
Z s
Z t
w
w
p
w
w
w
w
w
w
w
w
w
Ew
a(s, τ, x(τ − ρ3 (τ)))dτ dsw
S(t − s)h s, x(s − ρ2 (s)),
→0
w
w 0
wH
0
7
7
as t → ∞,
as t → ∞,
Z t
Z s
w
w
p
w
w
w
w
w
w
w
w
w
7p−1 Ew
U(t,
s)
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
dw(s)
→0
4
5
w
w
w
wH
0
0
as t → ∞,
and
w
w
w
w
X
X
w
wp
w
p
− w
w
w
w
7p−1 Ew
S(t
−
t
)I
(x(t
))
≤ 7p−1
E k S(t − tk )Ik (x(t−k )) kH
k k
w
w
k
w
wH
0<tk <t
0<tk <t
p−1
≤7
p
Mp e−pαt E k Ik (x(t−k )) kH → 0
as t → ∞,
w
w
w
w
X
X
w
wp
w
p
− w
w
w
w
7p−1 Ew
S(t
−
t
)J
(x(t
))
≤ 7p−1
E k S(t − tk )Jk (x(t−k )) kH
k k
w
w
k
w
wH
0<tk <t
0<tk <t
p−1
≤7
p
Mp e−pβt E k Jk (x(t−k )) kH → 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)-(4). We
shall employ Lemma 2.9. For better readability, we break the proof into a sequence of steps.
Step 1. For 0 < λ < 1, set {x ∈ Y : x = λΨx} is bounded.
Let x ∈ Y be a possible solution of x = λΨ(x) for some 0 < λ < 1. Then, by (H1)-(H7), we have for each
t ∈ [0, T]
p
E k x(t) kH
p
p
≤ 7p−1 E k C(t)ϕ(0) kH +7p−1 E k S(t)[φ − 1(0, ϕ(−ρ1 (0)))] kH
Z t
w
w
p
w
w
w
w
w
w
w
w
w
+ 7p−1 Ew
C(t
−
s)1(s,
x(s
−
ρ
(s)))ds
1
w
w
w
w
H
0
Z
Z
w
w
s
p
w t
w
w
w
w
w
w
w
+ 7p−1 Ew
S(t
−
s)h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
dsw
2
3
w
w
w 0
wH
0
Z t
Z s
w
w
w
w
w
wp
w
p−1 w
w
w
+ 7 Ew
S(t − s) f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s)w
w
w
w 0
wH
0
w
w
w
w
p
p
w
w
w
w
X
X
w
w
w
w
w
w
w
w
w
w
w
w
+ 7p−1 Ew
C(t − tk )Ik (x(t−k ))w
+ 7p−1 Ew
S(t − tk )Jk (x(t−k ))w
w
w
w
w
w
w
w
w
H
0<tk <t
p−1 −αpt
≤7
e
M E k ϕ(0)
p
p
kH
+14
p−1 −βpt
e
p
M [E k
0<tk <t
p
φ kH +L1 E
H
p
k ϕ(−ρ1 (0)) kH ]
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2738
Z t
p
+ 7p−1 Mp E
e−α(t−s) k 1(s, x(s − ρ1 (s)) kH ds
0
t
Z s
w
w
p
w
w
w w
w
wh s, x(s − ρ2 (s)),
w
e
+7 M E
a(s, τ, x(τ − ρ3 (τ)))dτ w
ds
w
w
w
wH
0
0
Z s
w
Z t
w
p 2/p p/2
w
w
w
w
w
w
w
w
w
+ 7p−1 Cp
e−pβ(t−s) Ew
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
4
5
w
w
w
w
p−1
p
Z
−β(t−s) w
w
0
+ (7m)p−1 Mp
H
0
m
X
p
e−αp(t−tk ) E k Ik (x(t−k )) kH +(7m)p−1 Mp
k=1
m
X
p
e−βp(t−tk ) E k Jk (x(t−k )) kH
k=1
p
p
p
≤ 7p−1 e−αpt Mp E k ϕ(0) kH +14p−1 e−βpt Mp [E k φ kH +L1 E k ϕ(−ρ1 (0)) kH ]
Z t
p
+ 7p−1 Mp Tp−1 L1
e−αp(t−s) E k x(s − ρ1 (s)) kH ds
0
t
Z
p
0
t
Z
+7
p−1
p
Cp M T
0
+ (7m)p−1 Mp
p
e−βp(t−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]ds
p p/2−1
m
X
p
e−βp(t−s) [mh (s)Θh (E k x(s − ρ2 (s)) kH ) + ma (s)Θa (E k x(s − ρ3 (s)) kH )]ds
+ 7p−1 Mp Tp−1
p
(1)
(2)
e−αp(t−tk ) [dk E k x(t−k ) kH +dk ] + (7m)p−1 Mp
k=1
m
X
p
(3)
k=1
By the definition of Y, it follows that
p
p
p
E k x(s − ρi (s)) kH ≤ 2p−1 k ϕ kB +2p−1 sup k x(s) kH , i = 1, 2, 3, 4, 5.
s∈[0,t]
p
p
If µ(t) = 2p−1 k ϕ kB +2p−1 sups∈[0,t] k x(s) kH , we obtain that
p
p
p
p
µ(t) ≤ 2p−1 k ϕ kB +14p−1 e−αpt Mp E k ϕ(0) kH +28p−1 e−βpt Mp [E k φ kH +L1 E k ϕ(−ρ1 (0)) kH ]
Z t
p−1 p p−1
−αpt
+ 14 M T L1 e
eαps µ(s)ds
0
t
Z
+ 14p−1 Mp Tp−1 e−βpt
eβps [mh (s)Θh (µ(s)) + ma (s)Θa (µ(s))]ds
0
t
Z
+ 14p−1 Cp Mp Tp/2−1 e−βpt
eβps [m f (s)Θ f (µ(s)) + mb (s)Θb (µ(s))]ds
0
+ (14m)p−1 Mp e−αpt
m
X
eαptk [dk µ(t) + dk ] + (14m)p−1 Mp e−βpt
(1)
(2)
Since e
L = (14m)p−1 Mp
eγpt µ(t) ≤
1
1 −e
L
(1)
k=1 (dk
Pm
m
X
eβptk [dk µ(t) + dk ].
k=1
k=1
(3)
+ dk ) < 1, we obtain
Z t
e + 14p−1 Mp Tp−1 L1
M
eγps µ(s)ds
0
Z
t
+ 14p−1 Mp Tp−1
eγps [mh (s)Θh (µ(s)) + ma (s)Θa (µ(s))]ds
0
t
Z
+ 14p−1 Cp Mp Tp/2−1
0
(4)
e−βp(t−tk ) [dk E k x(t−k ) kH +dk ].
eγps [m f (s)Θ f (µ(s)) + mb (s)Θb (µ(s))]ds ,
(3)
(4)
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2739
where
e = 2p−1 k ϕ kp +14p−1 Mp E k ϕ(0) kp +28p−1 Mp [E k φ kp +L1 E k ϕ(−ρ1 (0)) kp ]
M
H
H
H
B
m
m
X
X
(2)
(4)
+ (14m)p−1 Mp
dk + (14m)p−1 Mp
dk , γ = min{α, β}.
k=1
k=1
Denoting by ζ(t) the right-hand side of the above inequality, we have
eγpt µ(t) ≤ ζ(t) for all t ∈ [0, T],
and ζ(0) =
1 e
M,
1−e
L
ζ0 (t) =
1
1 −e
L
14p−1 Mp Tp−1 L1 eγpt µ(t) + 14p−1 Mp Tp−1 eγpt [mh (t)Θh (µ(t)) + ma (t)Θa (µ(t))]
+ 14p−1 Cp Mp Tp/2−1 eγpt [m f (t)Θ f (µ(t)) + mb (t)Θb (µ(t))]
1
14p−1 Mp Tp−1 L1 ζ(t) + 14p−1 Mp Tp−1 eγpt [mh (t)Θh (e−γpt ζ(t)) + ma (t)Θa (e−γpt ζ(t))]
≤
e
1−L
+ 14p−1 Cp Mp Tp/2−1 eγpt [m f (t)Θ f (e−γpt ζ(t)) + mb (t)Θb (e−γpt ζ(t))] .
If ξ(t) = e−γpt ζ(t), then ξ(0) = ζ(0), ζ(t) ≤ ξ(t), and
1
14p−1 Mp Tp−1 L1 ξ(t) + 14p−1 Mp Tp−1 eγpt [mh (t)Θh (ξ(t)) + ma (t)Θa (ξ(t))]
ζ0 (t) ≤
e
1−L
+ 14p−1 Cp Mp Tp/2−1 eγpt [m f (t)Θ f (ξ(t)) + mb (t)Θb (ξ(t))] ,
and we have
ξ0 (t) = (−γp)e−γpt ζ(t) + e−γpt ζ0 (t)
1
≤ (−γp)ξ(t) +
14p−1 Mp Tp−1 L1 e−γpt ξ(t) + 14p−1 Mp Tp−1 [mh (t)Θh (ξ(t)) + ma (t)Θa (ξ(t))]
1 −e
L
+ 14p−1 Cp Mp Tp/2−1 [m f (t)Θ f (ξ(t)) + mb (t)Θb (ξ(t))]
≤ m∗ (t)[ξ(t) + Θh (ξ(t)) + Θa (ξ(t)) + Θ f (ξ(t)) + Θb (ξ(t))],
where
m∗ (t) = max (−γp) +
1
1 −e
L
1
1 −e
L
14p−1 Mp Tp−1 L1 e−γpt ,
14p−1 Cp Mp Tp/2−1 m f (t),
1
1 −e
L
1
1 −e
L
14p−1 Mp Tp−1 mh (t),
1
1 −e
L
14p−1 Mp Tp−1 ma (t),
14p−1 Cp Mp Tp/2−1 mb (t) .
This implies for each t ∈ [0, T] that
Z
ξ(t)
ξ(0)
du
≤
u + Θh (u) + Θa (u) + Θ f (u) + Θb (u)
T
Z
m∗ (s)ds < ∞.
0
e such that ξ(t) ≤ K,
e t ∈ [0, T], and hence k x kp ≤ µ(t) ≤ K,
e
This inequality shows that there is a constant K
Y
e depends only on p, γ, M, T and on the functions mh (·), Θa (·), m f (·), Θb (·). This indicates that x(·) are
where K
bounded on [0, T].
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2740
Step 2. Ψ : Y → Y is continuous.
p
Let {xn (t)}∞
⊆ Y with xn → x(n → ∞) in Y. Then there is a number r > 0 such that E k xn (t) kH ≤ r for all
n=0
p
n and a.e. t ∈ [0, T], so xn ∈ Br (0, Y) = {x ∈ Y :k x kY ≤ r} and x ∈ Br (0, Y). By the assumptions (H3)-(H7), we
have
Z s
w
w
w
w
w
Ew
a(s,
τ,
x
(τ
−
ρ
(τ)))dτ
h
s,
x
(s
−
ρ
(s)),
n
3
n
2
w
w
Z0 s
w
p
w
w
w
w
− h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ w
w → 0 as n → ∞,
w
H
0
Z s
w
w
w
w
w
f
s,
x
(s
−
ρ
(s)),
b(s,
τ,
x
(τ
−
ρ
(τ)))dτ
Ew
n
4
n
5
w
w
Z0 s
w
p
w
w
w
w
b(s, τ, x(τ − ρ5 (τ)))dτ w
− f s, x(s − ρ4 (s)),
→0
w
wH
0
as n → ∞
for each s ∈ [0, t], and since
Z s
Z s
w
w
p
w
w
w
w
w
w
w
w
w
Ew
h
s,
x
(s
−
ρ
(s)),
a(s,
τ,
x
(τ
−
ρ
(τ)))dτ
−
h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
n
2
n
3
2
3
w
w
w
w
0
0
H
≤ 2 max{Θh (r∗ ), Θa (r∗ )}[mh (s) + mh (s)],
Z s
Z s
w
w
p
w
w
w
w
w
w
w
w
w
Ew
f
s,
x
(s
−
ρ
(s)),
b(s,
τ,
x
(τ
−
ρ
(τ)))dτ
−
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
n
4
n
5
4
5
w
w
w
w
0
0
H
≤ 2 max{Θ f (r ), Θb (r )}[m f (s) + mb (s)].
∗
∗
Then by the dominated convergence theorem and Ik , Jk , k = 1, 2, . . . , m, are completely continuous, we have
for t ∈ [0, T],
p
E k (Ψxn )(t) − (Ψx)(t) kH
Z t
w
w
p
w
w
w
w
w
w
w
w
w
≤ 5p−1 Ew
C(t
−
s)[1(s,
x
(s
−
ρ
(s))
−
1(s,
x(s
−
ρ
(s))]ds
n
1
1
w
w
w 0
wH
Z
Z
w
s
w t
w
w
w
+ 5p−1 Ew
S(t − s) h s, xn (s − ρ2 (s)),
a(s, τ, xn (τ − ρ3 (τ)))dτ
w
w 0
0
Z s
w
p
w
w
w
w
− h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ dsw
w
wH
0
Z t
Z s
w
w
w
w
w
+ 5p−1 Ew
S(t
−
s)
f
s,
x
(s
−
ρ
(s)),
b(s,
τ,
x
(τ
−
ρ
(τ)))dτ
n
4
n
5
w
w
0
0
Z s
w
p
w
w
w
w
− f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s)w
w
w
H
0
w
w
p
w
w
X
w
w
w
w
w
w
+ 5p−1 Ew
C(t − tk )[Ik (xn (t−k )) − Ik (x(t−k ))]w
w
w
w
w
H
0<tk <t
w
w
w
w
X
wp
w
w
−
− w
w
w
w
+ 5p−1 Ew
S(t
−
t
)[J
(x
(t
))
−
J
(x(t
))]
k
k n k
k
w
w
k
w
wH
0<tk <t
Z t
p
≤ 5p−1 Tp−1
e−αp(t−s) E k 1(s, xn (s − ρ1 (s)) − 1(s, x(s − ρ1 (s)) kH ds
0
Z s
w
w
w
w
w
h
s,
x
(s
−
ρ
(s)),
+5 T
e
Ew
a(s,
τ,
x
(τ
−
ρ
(τ)))dτ
n
2
n
3
w
w
0
0
Z s
w
p
w
w
w
w
− h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ w
ds
w
wH
0
Z
p−1 p−1
t
−βp(t−s)
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2741
t
Z s
w
w
w
w
w
e−βp(t−s) Ew
f
s,
x
(s
−
ρ
(s)),
b(s,
τ,
x
(τ
−
ρ
(τ)))dτ
n
4
n
5
w
w
0
Z0 s
w
p
w
w
w
w
b(s, τ, x(τ − ρ5 (τ)))dτ w
− f s, x(s − ρ4 (s)),
ds
w
wH
0
m
X
p
+ (5m)p−1
e−α(t−tk ) E k Ik (xn (t−k )) − Ik (x(t−k )) kH
Z
+ 5p−1 Mp Cp Tp/2−1
k=1
m
X
+ (5m)p−1
p
e−β(t−tk ) E k Jk (xn (t−k )) − Jk (x(t−k )) kH → 0
as n → ∞.
k=1
Then, we have for all t ∈ [0, T],
p
k Ψxn − Ψx kY → 0
as n → ∞.
Therefore, Ψ is continuous.
Step 3. Ψ is χ-contraction.
To see this, we decompose Ψ as Ψ1 + Ψ2 for t ∈ [0, T], where
Z
(Ψ1 x)(t) = C(t)ϕ(0) + S(t)[φ − 1(0, ϕ(−ρ1 (0)))] +
t
C(t − s)1(s, x(s − ρ1 (s)))ds,
0
and
t
Z
Z s
S(t − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ ds
(Ψ2 x)(t) =
0
0
t
Z s
S(t − s) f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s)
0
X0
X
S(t − tk )Jk (x(t−k )).
C(t − tk )Ik (x(t−k )) +
+
Z
+
0<tk <t
0<tk <t
(1) Ψ1 is a contraction on Y.
Let t ∈ [0, T] and x, y ∈ Y. From (H1) and (H2), we have
p
E k (Ψ1 x)(t) − (Ψ1 y)(t) kH
Z t
w
w
p
w
w
w
w
w
w
w
w
= Ew
C(t − s)[1(s, x(s − ρ1 (s))) − 1(s, y(s − ρ1 (s)))]dsw
w
w
w 0
wH
p
Z t
e−α(t−s) k 1(s, x(s − ρ1 (s))) − 1(s, y(s − ρ1 (s))) kH ds
≤ Mp E
0
≤ Mp L1
t
Z
e−α(t−s) ds
p−1 Z
0
0
Z
≤ Mp L1 α1−p
0
≤ M L1 α
p
−p
t
p
e−α(t−s) E k x(s − ρ1 (s)) − y(s − ρ1 (s)) kH ds
t
p
e−α(t−s) ds sup E k x(s) − y(s) kH
kx−y
s∈[0,T]
p
kY
.
Taking supremum over t
p
p
k Ψ1 x − Ψ1 y kY ≤ L0 k x − y kY ,
where L0 = Mp L1 α−p < 1. Hence, Ψ1 is a contraction on Y.
(2) Ψ2 is a compact operator.
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2742
For this purpose, we decompose Ψ2 by Ψ2 = Υ1 + Υ2 , where
Z t
Z s
(Υ1 x)(t) =
S(t − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ ds
0
0
t
Z
+
Z s
S(t − s) f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s),
0
0
and
(Υ2 x)(t) =
X
0<tk <t
C(t − tk )Ik (x(t−k )) +
X
0<tk <t
S(t − tk )Ik (x(t−k )).
(i) Υ1 is a compact operator.
We now prove that Υ1 (Br (0, Y))(t) = {(Υ1 x)(t) : x ∈ Br (0, Y)} is relatively compact for every t ∈ [0, T]. If
x ∈ Br (0, Y), from the definition of Y, it follows that
p
p
p
E k x(s − ρi (s)) kH ≤ 2p−1 k ϕ kB +2p−1 sup E k x(s) kH
s∈[0,T]
p
≤ 2p−1 k ϕ kB +2p−1 r := r∗ , i = 1, 2, 3, 4, 5.
Rs
It follows from conditions (H4)(iii) and (H6)(iii) that the sets {S(t − s)h(s, ψ, 0 a(s, τ, ψ)dτ) : t, s ∈ [0, T], k
Rs
p
p
ψ kH ≤ r∗ } and {S(t − s) f (s, ψ, 0 b(s, τ, ψ)dτ) : t, s ∈ [0, T], k ψ kH ≤ r∗ } are relatively compact in H. Moreover,
for x ∈ Br (0, Y), from the mean value theorem for the Bochner integral, we can infer that
Z s
p
(Υ1 x)(t) ∈ tconv S(t − s)h s, ψ,
a(s, τ, ψ)dτ : t, s ∈ [0, T], k ψ kH ≤ r∗
0
Z s
1
p
2
+ t conv S(t − s) f s, ψ,
b(s, τ, ψ)dτ : t, s ∈ [0, T], k ψ kH ≤ r∗
0
for all t ∈ [0, T], and conv denotes the convex hull. As a result we conclude that the set {(Υ1 x)(t) : x ∈ Br (0, Y)
is relatively compact in H for every t ∈ [0, T].
Next we show that Υ1 maps bounded sets into equicontinuous sets of Y. Let 0 < ε < t < T. From
(Υ1 Br (0, Y))(t) is relatively compact for each t and by the strong continuity of S(t), we can choose 0 < δ < T −t
with
k S(t + ξ)x − S(t)x kH ≤ ε
for x ∈ (Φ2 Br (0, Y))(t) when 0 < ξ < δ. For any x ∈ Br (0, Y). Using (H1)-(H5) and Hölder’s inequality, it
follows that
p
E k (Υ1 x)(t + ξ) − (Υ1 x)(t) kH
Z t−ε
Z s
w
w
p
w
w
w
w
w
w
w
w
w
≤ 6p−1 Ew
[S(t
+
ξ
−
s)
−
S(t
−
s)]h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
2
3
w
w
w 0
wH
0
Z
Z
w
w
s
p
w
w t
w
w
w
w
w
w
w
+ 6p−1 Ew
[S(t
+
ξ
−
s)
−
S(t
−
s)]h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
2
3
w
w
w t−ε
wH
0
Z s
w Z t+ξ
w
p
w
w
w
w
w
w
w
w
+ 6p−1 Ew
S(t + ξ − s)h s, x(s − ρ2 (s)),
a(s, τ, x(τ − ρ3 (τ)))dτ dsw
w
w
w
w
H
t
0
Z
Z
w
w
t−ε
s
p
w
w
w
w
w
w
w
w
w
+ 6p−1 Ew
[S(t
+
ξ
−
s)
−
S(t
−
s)]
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
dw(s)
4
5
w
w
w 0
wH
0
Z
Z
w
w
t
s
w
w
w
wp
w
w
w
w
+ 6p−1 Ew
[S(t + ξ − s) − S(t − s)] f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s)w
w
w
w t−ε
wH
0
Z s
w Z t+ξ
wp
w
w
w
w
w
w
w
w
+ 6p−1 Ew
S(t + ξ − s) f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ dw(s)w
w
w
w
w
H
t
0
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2743
t−ε
Z s
w
w
p
w
w
w
w
w
w
w
w
w
Ew
[S(t
+
ξ
−
s)
−
S(t
−
s)]h
s,
x(s
−
ρ
(s)),
a(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
2
3
w
w
w
wH
0
0
Z s
w
Z t
w
p
w
w
w
w
w
w
p−1
w
w
+6 E
k S(t + ξ − s) − S(t − s) kH w
a(s, τ, x(τ − ρ3 (τ)))dτ w
h s, x(s − ρ2 (s)),
ds
w
w
w
wH
t−ε
0
Z s
w
w
Z t+ξ
p
w
w
w
w
w
w
w
w
+ 6p−1 E
k S(t + ξ − s) kH w
a(s, τ, x(τ − ρ3 (τ)))dτ w
w ds
wh s, x(s − ρ2 (s)),
w
w
H
t
0
Z s
Z t−ε w
w
p 2/p p/2
w
w
w
w
w
w
w
w
w
Ew
+ 6p−1 Cp
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
[S(t
+
ξ
−
s)
−
S(t
−
s)]
f
s,
x(s
−
ρ
(s)),
ds
5
4
w
w
w
wH
0
0
Z s
w
Z t w
w
w
w wp 2/p p/2
w
w
p
p−1
w
w
k S(t + ξ − s) − S(t − s) kH Ew
+ 6 Cp
b(s, τ, x(τ − ρ5 (τ)))dτ w
f s, x(s − ρ4 (s)),
ds
w
w
w
wH
t−ε
0
Z s
w
Z t+ξ w
w wp 2/p p/2
w
w
w
w
p
w
w
w
+ 6p−1 Cp
k S(t + ξ − s) kH Ew
f
s,
x(s
−
ρ
(s)),
b(s,
τ,
x(τ
−
ρ
(τ)))dτ
ds
4
5
w
w
w
wH
t
0
p−1 Z t
Z t
p
−β(t−s)
p−1
p p
p−1 p
≤ 6 (t − ε) ε + 12 M
e−β(t−s) [mh (s)Θh (E k x(s − ρ1 (s)) kH )
e
ds
Z
≤ 6p−1 (t − ε)p−1
t−ε
t−ε
p
+ ma (s)Θa (E k x(s − ρ2 (s)) kH )]ds
p−1 Z
Z t+ξ
e−β(t+ξ−s) ds
+ 6p−1 Mp
t+ξ
t
t
p
e−β(t+ξ−s) [mh (s)Θh (E k x(s − ρ1 (s)) kH )
p
+ ma (s)Θa (E k x(s − ρ2 (s)) kH )]ds
Z t−ε w
Z s
w
p
w
w
w
w
w
w
p−1
p/2−1
w
w
+ 6 Cp (t − ε)
Ew
S(t + ξ − s) − S(t − s)] f s, x(s − ρ4 (s)),
b(s, τ, x(τ − ρ5 (τ)))dτ w
ds
w
w
w
wH
0
0
Z
t
p/2
p
p
+ 12p−1 Cp Mp
[e−pβ(t−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]]2/p ds
+ 6p−1 Cp Mp
t−ε
t+ξ
Z
t
≤6
(t − ε) ε + 12
p p
p−1
M max{Θh (r ), Θa (r )}β
p
∗
∗
p/2
t
Z
p−1
p
p
[e−pβ(t+ξ−s) [m f (s)Θ f (E k x(s − ρ4 (s)) kH ) + mb (s)Θb (E k x(s − ρ5 (s)) kH )]]2/p ds
e−β(t−s) [mh (s) + ma (s)]ds
1−p
t−ε
t+ξ
Z
+ 6p−1 Mp max{Θh (r∗ ), Θa (r∗ )}β1−p
e−β(τ2 −s) [mh (s) + ma (s)]ds
t
+ 6p−1 Cp (t − ε)p/2 εp
+ 12
p−1
Cp M max{Θ f (r ), Θb (r )}
p
∗
∗
2β(p − 1) 1−p/2 Z
p−2
t
e−β(t−s) [m f (s) + mb (s)]ds
t−ε
t+ξ
2β(p − 1) 1−p/2 Z
e−β(t+ξ−s) [m f (s) + mb (s)]ds.
+ 6p−1 Cp Mp max{Θ f (r∗ ), Θb (r∗ )}
p−2
t
Then the right-hand side of the above inequality is independent of x ∈ Br and tends to zero as ξ → 0 and
sufficiently small positive number ε. Thus, the set {Υ1 x : x ∈ Br (0, Y)} is equicontinuous.
(ii) Υ2 is a compact operator.
To prove the compactness of Υ2 , note that
X
X
(Υ2 x)(t) =
S(t − tk )Ik (x(t−k )) +
S(t − tk )Jk (x(t−k ))
0<tk <t







=





0<tk <t
0,
C(t − t1 )I1 (x(t−1 )) + S(t − t1 )J1 (x(t−1 )),
·P· ·
Pm
m
−
−
k=1 S(t − tk )Jk (x(tk )),
k=1 C(t − tk )Ik (x(tk )) +
t ∈ [0, t1 ],
t ∈ (t1 , t2 ],
t ∈ (tm , T],
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2744
and that the interval [0, T] is divided into finite subintervals by tk , k = 1, 2, . . . , m, so that we only need to
prove that
W = {C(t − t1 )I1 (x(t−1 )) + S(t − t1 )J1 (x(t−1 )),
t ∈ [t1 , t2 ], x ∈ Br (0, Y)}
is relatively compact in C([t1 , t2 ], H), as the cases for other subintervals are the same. In fact, from (H1) and
(H7), it follows that the set {C(t − t1 )I1 (x(t−1 )) + S(t − t1 )J1 (x(t−1 )), x ∈ Br (0, Y)} is relatively compact in H for all
t ∈ [t1 , t2 ].
Next, using the semigroup property, we have for t1 ≤ s < t ≤ t2
p
E k [C(t − t1 ) − C(s − t1 )]I1 (x(t−1 )) + [S(t − t1 ) − S(s − t1 )]J1 (x(t−1 )) kH
p
p
≤ 2p−1 E k [C(t − t1 ) − C(s − t1 )]I1 (x(t−1 )) kH +2p−1 E k [S(t − t1 ) − S(s − t1 )]J1 (x(t−1 )) kH .
Thus, we see that the functions in W are equicontinuous due to the compactness of I1 , J1 and the strong
continuity of the operator C(t), S(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 Υ2 is also a compact map.
Let arbitrary bounded subset V ⊂ Y. Since the mapping Ψ2 is a compact operator, we get that χY (Ψ2 V) =
0. Consequently
χY (ΨV) = χY (Ψ1 V + Ψ2 V) ≤ χY (Ψ1 V) + χY (Ψ2 V) ≤ L0 χY (V) < χY (V).
Therefore, Ψ is χ−contraction. In view of Lemma 2.9, we conclude that Ψ has at least one fixed point
x∗ ∈ V ⊂ Y. Then, x is a fixed point of the operator Ψ, which is a mild solution of the system (1)-(4)
p
with x(s) = ϕ(s) on [m̃(0), 0] and E k x(t) kH → 0 as t → ∞. This shows that the asymptotic stability of
the mild solution of (1)-(4). In fact, let ε > 0 be given and choose γ̃ > 0 such that γ̃ < ε and satisfies
[14p−1 Mp + 14p−1 (K2 Lh,a + K3 L f,b ]γ̃ + (14p−1 K1 α−1 + e
L)ε < ε. If x(t) = x(t, ϕ) is mild solution of (1)-(4), with
p
p
p
p
k ϕ kB +E k φ kH +L1 E k ϕ(−ρ1 (0)) kH < γ̃, then (Ψx)(t) = x(t) and satisfies E k x(t) kH < ε for every t ≥ 0.
p
p
p
Notice that E k x(t) kH < ε on t ∈ [m̃(0), 0]. If there exists t̃ such that E k x(t̃) kH = ε and E k x(s) kH < ε for
s ∈ [m̃(0), t̃]. Then (10) show that
p
E k x(t) kH ≤ [14p−1 Mp e−pγt̃ + 14p−1 (K2 Lh,a + K3 L f,b )]γ̃ + (14p−1 K1 α−1 + e
L)ε < ε,
which contradicts the definition of t̃. Therefore, the mild solution of (1)-(4) is asymptotically stable in p-th
moment.
Remark 3.5. In [9, 32], the authors get the asymptotic stability results under the Lipschitz continuity of the nonlinear
items when A generates a strongly continuous cosine family, respectively. Here, the Darbo fixed point theorem is
effectively used to study the asymptotic stability of the system (1)-(4). The results are obtained by using the mixed
Lipschitz and continuous conditions, and some appropriate assumptions without the Lipschitz assumption on f, h.
Also, the results can generalize and improve the existing ones.
4. Example
Consider the following second-order impulsive partial stochastic neutral differential equations of the
form
Z t
∂
∂2
d z(t, x) − ϑ(t, z(t − ρ(t), x)) = 2 z(t, x)dt + ς t, z(t − ρ(t), x),
ς1 (t, s, z(s − ρ(s), x)ds dt
∂t
∂ t
0
Z t
+$ t, z(t − ρ(t), x),
$1 (t, s, z(s − ρ(s), x) dw(t),
(11)
0
t ≥ 0, 0 ≤ x ≤ π, t , tk ,
z(t, 0) = z(t, π) = 0, t ≥ 0,
(12)
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
z(t, x) = ϕ(t, x),
tk
Z
4z(tk , x) =
∂
z(0, x) = φ(x),
∂t
t ≤ 0, 0 ≤ x ≤ π,
tk
Z
ηk (tk − s)z(s, x)ds,
0
4z0 (tk , x) =
η̃k (tk − s)z(s, x)ds,
2745
(13)
(14)
0
where (tk )k ∈ N is a strictly increasing sequence of positive numbers, ρ(t) ∈ C(R+ , R+ ), and ηk , η̃k ∈
C(R+ , R+ ), k = 1, 2, . . . , m. w(t) denotes a one-dimensional standard Wiener process in H defined on a
stochastic space (Ω, F , P).
Let H = L2 ([0, π]) with the norm k · k and define the operator A by Aω = ω00 with the domain
D(A) := {ω(·) ∈ H : ω, ω0 are absolutely continuous, ω00 ∈ H, ω(0) = ω(π) = 0}.
It is well-known that A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R}
in H. Furthermore, A has a discrete spectrum with eigenvalues of the form −n2 , n ∈ N, and corresponding
√
normalized eigenfunctions given by en (x) = 2/π sin(nx). Then the following properties hold:
P
2
(a) The set {en : n ∈ N} is an orthonormal basis of H and Aω = − ∞
n=1 n hω, en ien for every x ∈ D(A).
P
P∞ 1
(b) For ω ∈ H, C(t)ω = ∞
n=1 cos(nt)hω, en ien , and the associated sine family is S(t)ω =
n=1 n sin(nt)hω, en ien .
−π2 t
−π2 t
Consequently, k C(t) k≤ e
, k S(t) k≤ e
for all t ∈ R and S(t) is compact for every t ∈ R.
(c) If Φ is the group of translations on H defined by Φ(t)ω(ξ) = ω̃(ξ + t), where ω̃ is the extension
of ω with period 2π, then C(t) = 12 (Φ(t) + Φ(−t)) and A = B2 , where B is the generator of Φ and
D = {ω ∈ H1 : ω(0) = ω(π)}(see [17] for details). In particular, we observe that the inclusion ι : D → H
is compact.
Additionally, we will assume that
(i) The function ϑ : [0, ∞) × R → R is continuous and there exists a positive constant Lϑ such that
ϑ(t, 0) = 0, and
|ϑ(t, u) − ϑ(t, v)| ≤ Lϑ |u − v|, t ≥ 0, u, v ∈ R.
(ii) The function ς : [0, ∞) × R × R → R is continuous and there exists a positive continuous function
mς (·) : [0, ∞) → R such that
|ς(t, u, v)| ≤ mς (t)|u| + 21−p |v|, t ≥ 0, u, v ∈ R.
(iii) The function ς1 : [0, ∞) × [0, ∞) × R → R is continuous and there exists a positive continuous function
mς1 (·) : [0, ∞) → R such that
Z t
ς1 (t, s, u)ds ≤ mς1 (t)|u|, t ≥ 0, u ∈ R.
0
(iv) The function ϑ : [0, ∞) × R × R → R is continuous and there exists a positive continuous function
m$ (·) : [0, ∞) → R such that
|$(t, u, v)| ≤ m$ (t)|u| + 21−p |v|, t ≥ 0, u, v ∈ R.
(v) The function ϑ1 : [0, ∞) × R → R is continuous and there exists a positive continuous function
m$1 (·) : [0, ∞) → R such that
Z t
$
(t,
s,
u)ds
1
≤ m$1 (t)|u|, t ≥ 0, u ∈ R.
0
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2746
Let z(s)(x) = z(s, x). We can define respectively 1 : [0, ∞) × H → H, h : [0, ∞) × H × H → H, f : [0, ∞) × H ×
H → L(K, H) and Ik , Jk : H → H by
1(t, z(t − ρ(t))(x) = ϑ(t, z(t − ρ(t), x)),
Z t
Z t
h t, z(t − ρ(t)),
a(t, s, z(s − ρ(s))ds (x) = ς t, z(t − ρ(t), x),
ς1 (t, s, z(s − ρ(s), x))ds ,
0
0
Z t
Z t
$1 (t, s, z(s − ρ(s), x))ds ,
b(t, s, z(s − ρ(s))ds (x) = $ t, z(t − ρ(t), x),
f t, z(t − ρ(t)),
0
0
Z
Ik (z)(x) =
π
π
Z
ηk (s)z(s, x)ds,
Jk (z)(x) =
η̃k (s)z(s, x)ds.
0
0
Then the problem (11)-(14) can be written as (1)-(4). Moreover, using (i) we can prove that
π
Z
E k 1(t, z1 ) − 1(t, z2 ) k = E
p
|ϑ(t, z1 (x)) − ϑ(t, z2 (x))| dx
2
21 p
0
π
Z
≤E
12 p
2
Lϑ |z1 (x) − z2 (x)| dx
0
p
≤ Lϑ E k z1 − z2 kp
p
p
for all (t, z j ) ∈ [0, +∞) × H, j = 1, 2, and E k 1(t, z) kH ≤ Lϑ k z kp for all (t, z) ∈ [0, +∞) × H. By assumptions (ii)
and (iii) we have
Z π
21 p
p
2
E k h(t, z, y) k = E
|ς(t, z(x), y(x))| dx
0
Z π
21 p
≤E
[Lς (t)|z(x)| + 21−p |y(x)|]2 dx
0
≤ 2p−1 [(Lς (t))p E k z kp +21−p E k y kp ]
= mh (t)E k z kp +E k y kp
for all (t, z, y) ∈ [0, +∞) × H × H, and
Z t
2 1 p
w
w
Z π Z t
p
w
w
2
w
w
w
w
w
w
Ew
a(t, s, z)dsw
ς1 (s, z(x))ds dx
w
w =E
w
w
0
0
0
Z π
21 p
2
≤E
|mς1 (t)z(x)| dx
0
≤ ma (t)E k z kp
p
p
for all (t, s, z) ∈ [0, +∞) × [0, +∞) × H, where mh (t) = mς (t), ma (t) = mς1 (t). Similarly, by using assumptions
(iv) and (v) we have
E k f (t, z, y) kp ≤ m f (t)E k z kp +E k y kp
for all (t, z, y) ∈ [0, +∞) × H × H, and
Z t
w
w
p
w
w
w
w
w
w
w
w
w
Ew
b(t,
s,
z)ds
≤ mb (t)E k z kp
w
w
w 0
w
p
p
for all (t, s, z) ∈ [0, +∞)×[0, +∞)×H, where m f (t) = m$ (t), mb (t) = m$1 (t). Therefore (H1)-(H6) are all satisfied
and condition (9) holds with Θh (s) = Θa (s) = Θ f (s) = Θb (s) = s. It is clear that Ik , Jk are bounded linear maps
with
E k Ik (z) kp ≤ dk E k z kp , E k Jk (z) kp ≤ d˜k E k z kp , z ∈ H, k = 1, 2, . . . , m,
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2747
Rπ
Rπ
where dk = ( 0 |ηk (s)|2 ds)p/2 , d˜k = ( 0 |η̃k (s)|2 ds)p/2 , k = 1, 2, . . . , m. Moreover, the map Ik , Jk are completely
P
˜
continuous. Further, suppose that (14m)p−1 m
k=1 (dk + dk ) < 1 holds. Then, from Theorem 3.4, we can
conclude that the mild solution of (11)-(14) is asymptotically stable in p-th mean.
5. Conclusion
This paper has investigated the existence and asymptotic stability in p-th moment of mild solutions
for a class of second-order impulsive partial stochastic functional neutral integrodidifferential equations
with infinite delay in Hilbert spaces. Firstly, we introduce a more appropriate concept for mild solutions.
Secondly, the existence and asymptotic stability of mild solutions is investigated by utilizing Hölder’s
inequality, stochastic analysis, the Darbo fixed point theorem and the theory of strongly continuous cosine
families combined with techniques of the Hausdorff measure of noncompactness. Here, the results are
obtained under the nonlinear items f, h are continuous functions and not the assumptions of compactness
on associated operators. Finally, an example is provided to show the effectiveness of the proposed results.
There are two direct issues which require further study. We will investigate the asymptotic stability for
second-order impulsive stochastic partial functional differential equations with not instantaneous impulses.
Also, we will be devoted to study the stability of pseudo almost periodic solutions to impulsive stochastic
partial functional differential equations.
Acknowledgments
The authors would like to thank the editor and the reviewers for their constructive comments and
suggestions.
References
[1] R. Agarwal, M. Meehan and D. O’Regan, Fixed point theory and applications, in: Cambridge Tracts in Mathematics, Cambridge
University Press, New York, 2001.
[2] A. Anguraj and A.Vinodkumar, Existence, uniqueness and stability results of impulsive stochastic semilinear neutral functional
differential equations with infinite delays, Electron. J. Qual. Theory Differ. Equ. 67 (2009) 1–13.
[3] G. Arthi, J.H. Park and H.Y. Jung, Exponential stability for second-order neutral stochastic differential equations with impulses,
Internat. J. Control 88 (2015) 1300–1309.
[4] P. Balasubramaniam and P. Muthukumar, Approximate controllability of second-order stochastic distributed implicit functional
differential systems with infinite delay, J. Optim. Theory Appl. 143 (2009) 225–244.
[5] J. Banas and K. Goebel, Measure of Noncompactness in Banach Space, in: Lecture Notes in Pure and Applied Matyenath, Dekker,
New York, 1980.
[6] T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch.
Anal. Appl. 17 (1999) 743–763.
[7] T. Caraballo and J. Real, Partial differential equations with delayed random perturbations: Existence, uniqueness and stability
of solutions, Stoch. Anal. Appl. 11 (1993) 497–511.
[8] T. Caraballo, J. Real and T. Taniguchi, The exponential stability of neutral stochastic delay partial differential equations, Discrete
Contin. Dyn. Syst. 18 (2007) 259–313.
[9] H. Chen, The asymptotic behavior for second-order neutral stochastic partial differential equations with infinite delay, Discrete
Dyn. Nat. Soc. 2011 (2011) 584510 1–15.
[10] H. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statist.
Probab. Lett. 80 ( 2010) 50–56.
[11] H. Chen, C. Zhu and Y. Zhang, A note on exponential stability for impulsive neutral stochastic partial functional differential
equations, Appl. Math. Comput. 227 (2014) 139–147.
[12] J. Cui, L.T. Yan and X.C. Sun, Exponential stability for neutral stochastic partial differential equations with delays and Poisson
jumps, Statist. Probab. Lett. 81 (2011) 1970–1977.
[13] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
[14] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North Holland Mathematics Studies Series No.
108 Elsevier Science, North-Holland, Amsterdam, The Netherlands, 1985.
[15] J.K. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.
[16] T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77
(2005) 139–154.
[17] D. He and L. Xu, Existence and stability of mild solutions to impulsive stochastic neutral partial functional differential equations,
Electron. J. Differential Equations 2013 (2013) 1–15.
Z. Yan, X. Jia / Filomat 31:9 (2017), 2727–2748
2748
[18] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman Hall, CRC, London, 2006.
[19] K. Liu and A. Truman, A note on almost exponential stability for stochastic partial differential equations, Statist. Probab. Lett. 50
(2000) 273–278.
[20] S. Long, L. Teng and D. Xu, Global attracting set and stability of stochastic neutral partial functional differential equations with
impulses, Statist. Probab. Lett. 82 (2012) 1699–1709.
[21] J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math.
Anal. Appl. 342 (2008) 753–760.
[22] J. Luo and T. Taniguchi, Fixed points and stability of stochastic neutral partial differential equations with infinite delays, Stoch.
Anal. Appl. 27 (2009) 1163–1173.
[23] N. I. Mahmudov and M. A. McKibben, Approximate controllability of second-order neutral stochastic evolution equations, Dyn.
Contin. Discrete Impuls. Syst. Ser. B 13 (2006) 619–634.
[24] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichestic, UK, 1997.
[25] Y. Ren, T. Hou, R. Sakthivel and X. Cheng, A note on the second-order non-autonomous neutral stochastic evolution equations
with infinite delay under Caratheodory conditions, Appl. Math. Comput. 232 (2014) 658–665.
[26] Y. Ren and D. Sun, Second-order neutral impulsive stochastic evolution equations with delay, J. Math. Phys. 50 (2009) 102709
1–14.
[27] P. Revathi, R. Sakthivel, D.-Y. Song, Y. Ren and P. Zhang, Existence and stability results for second-order stochastic equations
driven by fractional Brownian motion, Transport Theory Statist. Phys. 42 (2013) 299–317.
[28] Y.V. Rogovchenko, Nonlinear impulse evolution systems and applications to population models, J. Math. Anal. Appl. 207 (1997)
300–315.
[29] R. Sakthivel and J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math.
Anal. Appl. 356 (2009) 1–6.
[30] R. Sakthivel and J. Luo, Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett. 79
(2009) 1219–1223.
[31] R. Sakthivel and Y. Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps, Commun.
Nonlinear Sci. Numer. Simul. 17 (2012) 4517–4523.
[32] R. Sakthivel, Y. Ren and H. Kim, Asymptotic stability of second-order neutral stochastic differential equations, J. Math. Phys. 51
(2010) 052701 1–9.
[33] R. Sakthivel, Y. Ren and N. I. Mahmudov, Approximate controllability of second-order stochastic differential equations with
impulsive effects, Modern Phys. Lett. B 24 (2010) 1559–1572.
[34] R. Sakthivel, P. Revathi and S.M. Anthoni, Existence of pseudo almost automorphic mild solutions to stochastic fractional
differential equations, Nonlinear Anal. 75 (2012) 3339–3347.
[35] R. Sakthivel, P. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear
Anal. 81 (2013) 70–86.
[36] T. Taniguchi and K. Liu, A. Truman, Existence, uniqueness, and asymptotic behavior of mild solution to stochastic functional
differential equations in Hilbert spaces, J. Differential Equations 18 (2002) 72–91.
[37] Z. Yan and X. Yan, Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent
delay, Collect. Math. 64 (2013) 235–250.