Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 769257, 9 pages
http://dx.doi.org/10.1155/2013/769257
Research Article
Asymptotic Stability of Fractional Stochastic Neutral Differential
Equations with Infinite Delays
R. Sakthivel,1 P. Revathi,2 and N. I. Mahmudov3
1
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
Department of Mathematics, Anna University of Technology, Coimbatore 641 047, India
3
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Mersin 10, Turkey
2
Correspondence should be addressed to R. Sakthivel; krsakthivel@yahoo.com
Received 11 September 2012; Accepted 8 November 2012
Academic Editor: Elena Braverman
Copyright © 2013 R. Sakthivel et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the existence and asymptotic stability in pth moment of a mild solution to a class of nonlinear fractional neutral stochastic
differential equations with infinite delays in Hilbert spaces. A set of novel sufficient conditions are derived with the help of semigroup
theory and fixed point technique for achieving the required result. The uniqueness of the solution of the considered problem is also
studied under suitable conditions. Finally, an example is given to illustrate the obtained theory.
1. Introduction
The stochastic differential equations have been widely applied
in science, engineering, biology, mathematical finance and in
almost all applied sciences. In the present literature, there are
many papers on the existence and uniqueness of solutions
to stochastic differential equations (see [1–4] and references
therein). More recently, Chang et al. [5] investigated the existence of square-mean almost automorphic mild solutions to
nonautonomous stochastic differential equations in Hilbert
spaces by using semigroup theory and fixed point approach.
Fu and Liu [2] discussed the existence and uniqueness of
square-mean almost automorphic solutions to some linear
and nonlinear stochastic differential equations and in which
they studied the asymptotic stability of the unique squaremean almost automorphic solution in the square-mean sense.
On the other hand, recently fractional differential equations
have found numerous applications in various fields of science
and engineering [6]. The existence and uniqueness results
for abstract stochastic delay differential equation driven by
fractional Brownian motions have been studied in [7]. In
particular the stability investigation of stochastic differential
equations has been investigated by several authors [8–15].
Let 𝐾 and 𝐻 be two real separable Hilbert spaces with
inner products ⟨⋅, ⋅⟩𝐾 and ⟨⋅, ⋅⟩𝐻, respectively. We denote
their norms by | ⋅ |𝐾 and | ⋅ |𝐻. To avoid confusion we just
use ⟨⋅, ⋅⟩ for the inner product and | ⋅ | for the norm. Let
{𝑒𝑖 }∞
𝑖=1 be an orthonormal basis of 𝐾. Throughout the paper,
we assume that (Ω, F, 𝑃; F) (F = {F𝑡 }𝑡≥0 ) is a complete
filtered probability space satisfying that F0 contains all 𝑃null sets of F. Suppose {𝑊(𝑡) : 𝑡 ≥ 0} is cylindrical 𝐾valued Brownian motion with a trace class operator 𝑄, denote
trac 𝑄 = ∑∞
𝑖=1 𝜆 𝑖 = 𝜆 < ∞, which satisfies that 𝑄𝑒𝑖 = 𝜆 𝑖 𝑒𝑖 .
∞
So, actually, 𝑊(𝑡) = ∑∞
𝑖=1 √𝜆 𝑖 𝑊𝑖 (𝑡)𝑒𝑖 , where {𝑊𝑖 (𝑡)}𝑖=1 are
mutually independent one-dimensional standard Brownian
motions. Define L(𝐾, 𝐻) as the set of all bounded linear
operators 𝐴 : 𝐾 → 𝐻 with the following norm:
∞
󵄨 󵄨2
|𝐴| = (∑󵄨󵄨󵄨𝐴𝑒𝑖 󵄨󵄨󵄨 )
1/2
< ∞.
(1)
𝑖=1
It is obvious that L(𝐾, 𝐻) is a Hilbert space with an inner
product induced by the above norm. Let 𝐴 ∈ L(𝐾, 𝐻) be
called a Hilbert-Schmidt operator. We further assume that
the filtration is generated by the cylindrical Brownian motion
𝑊(⋅) and augmented, that is,
F𝑡 = 𝜎 {𝑊 (𝑠) ; 0 ≤ 𝑠 ≤ 𝑡} ∨ N,
where N is the 𝑃-null sets.
(2)
2
Abstract and Applied Analysis
The qualitative properties of stochastic fractional differential equations have been considered only in few publications. El-Borai et al. [16] studied the existence uniqueness,
and continuity of the solution of a fractional stochastic
integral equation. Ahmed [17] derived a set of sufficient
conditions for controllability of fractional stochastic delay
equations by using a stochastic version of the well-known
Banach fixed point theorem and semigroup theory. Moreover,
theory of neutral differential equations is of both theoretical
and practical interests. For a large class of electrical networks containing lossless transmission lines, the describing
equations can be reduced to neutral differential equations.
However, to the author’s best knowledge no work has been
reported in the present literature regarding the existence,
uniqueness, and asymptotic stability of mild solutions for
neutral stochastic fractional differential equations with infinite delay in Hilbert spaces. Motivated by this consideration,
in this paper we consider the nonlinear fractional neutral
stochastic differential equations with infinite delays in the
following form:
𝑐
𝐷𝑡𝛼 [𝑋 (𝑡) + 𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))]
= 𝐴𝑋 (𝑡) + 𝑓 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))
+ 𝜎 (𝑡, 𝑋 (𝑡 − 𝜈 (𝑡)))
𝑑𝑊 (𝑡)
,
𝑑𝑡
(3)
𝑡 ≥ 0,
𝑋0 (⋅) = 𝜑 ∈ BF ([𝑚 (0) , 0] , 𝐻) ,
(4)
where 𝐴 is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operator 𝑆(𝑡), 𝑡 ≥ 0 in the
Hilbert space 𝐻, 𝑓 : 𝑅+ × 𝐻 → 𝐻, 𝜎 : 𝑅+ × 𝐻 → L(𝐾, 𝐻)
are two Borel measurable mappings, and 𝑔 : 𝑅+ × 𝐻 → 𝐻
is continuous mapping. The fractional derivative 𝑐 𝐷𝑡𝛼 , 𝛼 ∈
(0, 1) is understood in the Caputo sense. In addition, let
𝜏(𝑡), 𝜈(𝑡) ∈ 𝐶(𝑅+ , 𝑅+ ) satisfy 𝑡 − 𝜏(𝑡) → ∞, 𝑡 − 𝜈(𝑡) → ∞
as 𝑡 → ∞. Let 𝑚(0) = max{inf 𝑠≥0 (𝑠 − 𝜏(𝑠)), inf 𝑠≥0 (𝑠 −
𝜈(𝑠))}. Here BF0 ([𝑚(0), 0], 𝐻) denote the family of all
almost surely bounded, F0 -measurable, continuous random
variables 𝜑(𝑡) : [𝑚(0), 0] → 𝐻 with norm |𝜑|B =
sup𝑚(0)≤𝑡≤0 𝐸|𝜑(𝑡)|𝐻. Throughout this paper, we assume that
𝑝 ≥ 2 is an integer.
2. Preliminaries and Basic Properties
Let 𝐴 be the infinitesimal generator of an analytic semigroup
𝑆(𝑡) in 𝐻. Then, (𝐴−𝜂𝐼) is invertible and generates a bounded
analytic semigroup for 𝜂 > 0 large enough. Therefore, we can
assume that the semigroup 𝑆(𝑡) is bounded and the generator
𝐴 is invertible. It follows that (−𝐴)𝜂 , 0 < 𝜂 ≤ 1 can be
defined as a closed linear invertible operator with its domain
𝐷(−𝐴)𝜂 being dense in 𝐻. We denote by 𝐻𝜂 the Banach space
𝐷(−𝐴)𝜂 endowed with the norm |𝑥|𝜂 = |(−𝐴)𝜂 𝑥|, which is
equivalent to the graph norm of (−𝐴)𝜂 . For more details about
semigroup theory, one can refer [18].
Lemma 1 (see [18]). Suppose that the preceding conditions are
satisfied.
(a) Let 0 < 𝜂 ≤ 1, then 𝐻𝜂 is a Banach space.
(b) If 0 < 𝜈 ≤ 𝜂, then the embedding 𝐻𝜈 ⊂ 𝐻𝜂 is compact
whenever the resolvent operator of 𝐴 is compact.
(c) For every 𝜂 ∈ (0, 1], there exists a positive constant 𝐶𝜂
such that ‖𝐴𝜂 𝑆(𝑡)‖ ≤ 𝐶𝜂 /𝑡𝜂 , 𝑡 > 0.
Definition 2 (see [19]). The fractional integral of order 𝑞 with
the lower limit 0 for a function 𝑓 is defined as
𝐼𝛼 𝑓 (𝑡) =
𝑡
𝑓 (𝑠)
1
𝑑𝑠,
∫
Γ (𝛼) 0 (𝑡 − 𝑠)1−𝛼
𝑡 > 0, 𝑞 > 0
(5)
provided the righthand side is pointwise defined on [0, ∞),
where Γ(⋅) is the gamma function.
Definition 3 (see [19]). The Caputo derivative of order 𝛼 for a
function 𝑓 : [0, ∞) → 𝑅 can be written as
𝑐
𝑡
𝑓(𝑛) (𝑠)
1
𝑑𝑠 = 𝐼𝑛−𝛼 𝑓(𝑛) (𝑡) ,
∫
Γ (𝑛 − 𝛼) 0 (𝑡 − 𝑠)𝛼+1−𝑛
(6)
𝐷𝛼 𝑓 (𝑡) =
𝑡 > 0, 0 ≤ 𝑛 − 1 < 𝛼 < 𝑛.
If 𝑓 is an abstract function with values in 𝐻, then integrals
which appear in the above definitions are taken in Bochner’s
sense.
According to Definitions 2 and 3, it is suitable to rewrite
the stochastic fractional equation (3) in the equivalent integral equation
𝑋 (𝑡) = [𝜑 (0) + 𝑔 (0, 𝜑)] − 𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))
+
𝑡
1
∫ (𝑡 − 𝑠)𝛼−1 [𝐴𝑋 (𝑠) + 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))
Γ (𝛼) 0
+𝜎 (𝑠, 𝑋 (𝑠−𝜈 (𝑠)))
𝑑𝑊 (𝑠)
]𝑑𝑠.
𝑑𝑠
(7)
In view of [18, Lemma 3.1] and by using Laplace transform, we present the following definition of mild solution of
(3).
Definition 4. A stochastic process {𝑋(𝑡) : 𝑡 ∈ [0, 𝑇]}, 0 ≤
𝑇 < ∞ is called a mild solution of (3), if
(i) 𝑋(𝑡) is F𝑡 -adapted and is measurable, 𝑡 ≥ 0;
(ii) 𝑋(𝑡) ∈ 𝐻 has càdlàg paths on 𝑡 ∈ [0, 𝑇] almost surely
and for each 𝑡 ∈ [0, 𝑇], the function (𝑡 − 𝑠)𝛼−1 𝐴𝑇𝛼 (𝑡 −
𝑠)𝑔(𝑠, 𝑋(𝑠 − 𝜏(𝑠))) is integrable such that the following
integral equation is satisfied:
𝑋 (𝑡) = 𝑆𝛼 (𝑡) [𝜑 (0) + 𝑔 (0, 𝜑)] − 𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))
𝑡
− ∫ (𝑡 − 𝑠)𝛼−1 𝐴𝑇𝛼 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠
0
𝑡
+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠
0
𝑡
+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝜎 (𝑠, 𝑋 (𝑠 − 𝜈 (𝑠))) 𝑑𝑊 (𝑠) ,
0
(8)
Abstract and Applied Analysis
3
𝑝
(H4) Θ = [4(𝑝−1) |(−𝐴)(−𝛽) | 𝑀𝑔𝑝 + 4𝑝−1 𝑀𝑔𝑝 𝐾(𝛼, 𝛽)(𝑇𝛼𝛽 /
(iii) 𝑋0 (⋅) = 𝜑 ∈ BF ([𝑚(0), 0], 𝐻),
𝑝
∞
𝑆𝛼 (𝑡) 𝑋 = ∫ 𝜂𝛼 (𝜃) 𝑆 (𝑡𝛼 𝜃) 𝑋 𝑑𝜃,
0
𝐿󸀠 = ∫0 [𝑀𝑡−𝑠 (𝑡 − 𝑠)𝛼−1 ] 𝑑𝑠.
𝛼
𝑇𝛼 (𝑡) 𝑋 = 𝛼 ∫ 𝜃𝜂𝛼 (𝜃) 𝑆 (𝑡 𝜃) 𝑋 𝑑𝜃
0
In addition, in order to derive the stability of the
solution, we further assume that
with 𝜂𝛼 a probability density function defined on (0, ∞).
The following properties of 𝑆𝛼 (𝑡) and 𝑇𝛼 (𝑡) [18] are useful.
Lemma 5. Under previous assumptions on 𝑆(𝑡), 𝑡 ≥ 0 and 𝐴,
(i) 𝑆𝛼 (𝑡) and 𝑇𝛼 (𝑡) are strongly continuous;
𝐴𝑇𝛼 (𝑡) 𝑋 = 𝐴1−𝛽 𝑇𝛼 (𝑡) 𝐴𝛽 𝑋,
𝑡 ∈ [0, 𝑇] .
(10)
Definition 6. Let 𝑝 ≥ 2 be an integer. Equation (8) is said to
be stable in 𝑝th moment if for arbitrarily given 𝜖 > 0 there
exists a 𝛿 > 0 such that
𝐸 (sup|𝑋 (𝑡)|𝑝 ) < 𝜖,
𝑡≥0
󵄨 󵄨
when 󵄨󵄨󵄨𝜑󵄨󵄨󵄨B < 𝛿.
(11)
Definition 7. Let 𝑝 ≥ 2 be an integer. Equation (8) is said to
be asymptotically stable in 𝑝th moment if it is stable in 𝑝th
moment and, for any 𝜑 ∈ BF ([𝑚(0, 0)], 𝐻), it holds
lim 𝐸 (sup|𝑋 (𝑡)|𝑝 ) = 0.
𝑇→∞
𝑡≥𝑇
(12)
3. Main Result
In this section, we prove the existence, uniqueness, and
stability of the solution to fractional stochastic equation (3)
by using the Banach fixed point approach.
In order to obtain the existence and stability of the
solution to (3), we impose the following assumptions on (3).
(H1) There exist constants 𝑀 ≥ 1 and 𝑎 > 0 such that
|𝑆(𝑡)| ≤ 𝑀𝑒−𝑎𝑡 .
(H2) There exists a positive constant 𝐿 1 , for every 𝑡 ≥ 0 and
𝑥, 𝑦 ∈ 𝐻, such that
󵄨
󵄨
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑡, 𝑥) − 𝑓 (𝑡, 𝑦)󵄨󵄨󵄨 ≤ 𝐿 1 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ,
(13)
󵄨
󵄨󵄨
󵄨
󵄨
󵄨󵄨𝜎 (𝑡, 𝑥) − 𝜎 (𝑡, 𝑦)󵄨󵄨󵄨 ≤ 𝐿 1 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 .
(H3) There exist 0 < 𝛽 < 1 such that 𝑔 is 𝐻𝛽 -valued,
(−𝐴)𝛽 𝑔 is continuous and there exists a positive
constant 𝑀𝑔 such that
󵄨󵄨
󵄨
󵄨󵄨(−𝐴)𝛽 𝑔 (𝑡, 𝑥) − (−𝐴)𝛽 𝑔 (𝑡, 𝑦)󵄨󵄨󵄨 ≤ 𝑀𝑔 󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨
󵄨
󵄨
for every 𝑡 ≥ 0 and 𝑥, 𝑦 ∈ 𝐻.
(H5)
𝑔 (𝑡, 0) = 0,
𝑓 (𝑡, 0) = 0,
𝜎 (𝑡, 0) = 0.
(15)
It is obvious that (3) has a trivial solution when 𝜑 = 0
under the assumption (H5).
(ii) for any 𝑋 ∈ 𝐻, 𝛽 ∈ (0, 1) and 𝜃 ∈ (0, 1] one has
Γ (2 − 𝜃)
󵄨 𝛼𝐶𝜃
󵄨󵄨 𝜃
󵄨󵄨𝐴 𝑇𝛼 (𝑡)󵄨󵄨󵄨 ≤
,
󵄨
󵄨
𝑡𝛼𝜃 Γ (1 + 𝛼 (1 − 𝜃))
2
𝑇
(9)
∞
𝑝
𝛼𝛽)𝑝 + 4𝑝−1 𝛼𝑝 𝑀𝑝 𝐿 1 𝐿𝑝 + 4𝑝−1 𝐶𝑝 𝛼𝑝 𝑀𝑝 𝐿 1 𝐿󸀠(𝑝/2) ] <
=
(𝑝(𝑝 − 1))/2𝑝/2 , 𝑀𝑡
=
1, where 𝐶𝑃
𝛼𝜃
𝑡
𝑇
𝛼−1
−𝑎𝑡
∫0 𝜃𝜂𝛼 (𝜃)𝑒
𝑑𝜃, 𝐿 = ∫0 𝑀(𝑡−𝑠) (𝑡 − 𝑠) 𝑑𝑠 and
where
(14)
Lemma 8. Let 𝑝 ≥ 2, 𝑡 > 0 and let Φ be an L(𝐾, 𝐻)-valued,
𝑡
𝑝
predictable process such that 𝐸 ∫0 |Φ(𝑠)|L 𝑑𝑠 < ∞. Then,
󵄨󵄨 𝑠
󵄨󵄨󵄨𝑝
󵄨
sup 𝐸󵄨󵄨󵄨∫ Φ (𝑢) 𝑑𝑊 (𝑢)󵄨󵄨󵄨
󵄨󵄨
0≤𝑠≤𝑡 󵄨󵄨 0
𝑝/2
𝑝 (𝑝 − 1)
≤(
)
2
𝑡
(16)
𝑝/2
󵄨
𝑝 󵄨 2/𝑝
(∫ (𝐸 󵄨󵄨󵄨󵄨Φ(𝑠)L 󵄨󵄨󵄨󵄨) )
0
.
Theorem 9. Let 𝑝 ≥ 2 be an integer. Assume that the
conditions (H1)–(H4) hold, then the nonlinear fractional
neutral stochastic differential equation (3) is asymptotically
stable in the 𝑝th moment.
Proof. Denote by B the space of all F0 -adapted process
𝜙(𝑡, 𝑤) : [𝑚(0), 0] × Ω → 𝑅, which is almost surely
continuous in 𝑡 for fixed 𝑤 ∈ Ω and satisfies 𝜙(𝑡, 𝑤) = 𝜑(𝑡) for
𝑡 ∈ [𝑚(0), 0] and 𝐸|𝜙(𝑡, 𝑤)|𝑝 → 0 as 𝑡 → 0. It is then routine
to check that B is a Banach space when it is equipped with a
𝑝
norm defined by |𝜑|B = sup𝑡≥0 𝐸|𝜑(𝑡)|𝐻. Define the nonlinear
operator Ψ : B → B such that (Ψ𝑋)(𝑡) = 𝜑(𝑡), 𝑡 ∈ [𝑚(0), 0]
and, for 𝑡 ≥ 0,
(Ψ𝑋) (𝑡) = 𝑆𝛼 (𝑡) [𝜑 (0) + 𝑔 (0, 𝜑)] − 𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))
𝑡
− ∫ (𝑡 − 𝑠)𝛼−1 𝐴𝑇𝛼 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠
0
𝑡
+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠
0
𝑡
+∫ (𝑡−𝑠)𝛼−1 𝑇𝛼 (𝑡−𝑠) 𝜎 (𝑠, 𝑋 (𝑠−𝜈 (𝑠))) 𝑑𝑊 (𝑠) .
0
(17)
As mentioned in Luo [20], to prove the asymptotic stability
it is enough to show that the operator Ψ has a fixed point
in 𝐻. To prove this result, we use the contraction mapping
principle. To apply the contraction mapping principle, first
we verify the mean square continuity of Ψ on [0, ∞). Let
4
Abstract and Applied Analysis
𝑋 ∈ B, 𝑡1 ≥ 0 and let |ℎ| be sufficiently small, and observe
that
𝑡1
𝛼−1
+ 3𝑝−1 𝐸 (∫ [(𝑡1 + ℎ − 𝑠)
0
𝛼−1
− (𝑡1 − 𝑠)
]
󵄨
󵄨
× 󵄨󵄨󵄨󵄨(−𝐴)1−𝛽 𝑇𝛼 (𝑡1 + ℎ − 𝑠)󵄨󵄨󵄨󵄨
󵄨𝑝
󵄨
𝐸󵄨󵄨󵄨(Ψ𝑋) (𝑡1 + ℎ) − (Ψ𝑋) (𝑡1 )󵄨󵄨󵄨
𝑝
󵄨
󵄨
× 󵄨󵄨󵄨󵄨(−𝐴)𝛽 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨󵄨 𝑑𝑠)
(18)
5
󵄨
󵄨𝑝
≤ 5𝑝−1 ∑𝐸󵄨󵄨󵄨𝐹𝑖 (𝑡1 + ℎ) − 𝐹𝑖 (𝑡1 )󵄨󵄨󵄨 .
𝑖=1
𝑡1
+ 3𝑝−1 𝐸 (∫ (𝑡1 − 𝑠)
𝛼−1
0
Note that
󵄨󵄨
󵄨
󵄨󵄨(−𝐴)1−𝛽 󵄨󵄨󵄨
󵄨
󵄨
󵄨
󵄨
× 󵄨󵄨󵄨𝑇𝛼 (𝑡1 + ℎ − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)󵄨󵄨󵄨
󵄨𝑝
󵄨
𝐸󵄨󵄨󵄨𝐹1 (𝑡1 + ℎ) − 𝐹1 (𝑡1 )󵄨󵄨󵄨
𝑝
󵄨
󵄨𝑝
= 𝐸󵄨󵄨󵄨(𝑆𝛼 (𝑡1 + ℎ) − 𝑆𝛼 (𝑡1 )) [𝜑 (0) − 𝑔 (0, 𝜑)]󵄨󵄨󵄨 .
×
(19)
The strong continuity of 𝑆𝛼 (𝑡) [18] implies that the right hand
of (19) goes to 0 as |ℎ| → 0. In view of Lemma 5 and the
Holder’s inequality, the third term of (18) becomes
≤ 3𝑝−1 𝐾 (𝛼, 𝛽) 𝑀𝑔𝑝 (∫
󵄨󵄨 𝑡1 +ℎ
󵄨
𝛼−1
= 𝐸 󵄨󵄨󵄨󵄨∫
(𝑡1 + ℎ − 𝑠) (−𝐴) 𝑇𝛼 (𝑡1 + ℎ − 𝑠) 𝑔
󵄨󵄨 0
𝛼−1
− ∫ (𝑡1 − 𝑠)
×∫
𝑡1
×∫
(−𝐴) 𝑇𝛼 (𝑡1 − 𝑠) 𝑔
𝛼𝛽−1
(𝑡1 + ℎ − 𝑠)
𝑝−1
𝑑𝑠)
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
𝛼−1
𝑑𝑠)
𝛼(𝛽−1)
(𝑡1 + ℎ − 𝑠)
𝛼−1
(𝑡1 + ℎ − 𝑠)
𝑝−1
𝛼−1
− (𝑡1 − 𝑠)
𝛼−1
− (𝑡1 − 𝑠)
𝛼(𝛽−1)
(𝑡1 + ℎ − 𝑠)
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
𝑝−1
0
𝑡1
× ∫ (𝑡1 − 𝑠)
𝛼−1
0
𝑝−1
≤3
×∫
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
𝛼𝛽 𝑝−1
𝑝 ℎ
𝐾 (𝛼, 𝛽) 𝑀𝑔 (
)
𝛼𝛽
𝑡1 +ℎ
𝑡1
𝛼𝛽−1
(𝑡1 + ℎ − 𝑠)
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
+ 3𝑝−1 𝐾 (𝛼, 𝛽) 𝑀𝑔𝑝
× (∫
𝑡1
0
×∫
𝑡1
0
(𝑡1 + ℎ − 𝑠)
𝛼−1
𝛼(𝛽−1)
(𝑡1 + ℎ − 𝑠)
𝛼−1
(𝑡1 + ℎ − 𝑠)
𝑝
𝛼−1
− (𝑡1 − 𝑠)
𝛼(𝛽−1)
(𝑡1 + ℎ − 𝑠)
󵄨
󵄨𝑝 𝑡
+ 3𝑝−1 𝜖𝑝 𝑀𝑔𝑝 󵄨󵄨󵄨󵄨(−𝐴)1−𝛽 󵄨󵄨󵄨󵄨 ( 1 )
𝛼
𝑡1
× ∫ (𝑡1 − 𝑠)
0
𝛼−1
𝑝−1
𝛼−1
− (𝑡1 − 𝑠)
𝛼
󵄨󵄨
󵄨
󵄨󵄨(−𝐴)1−𝛽 𝑇𝛼 (𝑡1 + ℎ − 𝑠)󵄨󵄨󵄨
󵄨
󵄨
󵄨
󵄨
× 󵄨󵄨󵄨󵄨(−𝐴)𝛽 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨󵄨 𝑑𝑠)
𝑡1
𝛼−1
󵄨
󵄨𝑝 𝑡1
𝛼−1
+ 3𝑝−1 𝜖𝑝 𝑀𝑔𝑝 󵄨󵄨󵄨󵄨(−𝐴)1−𝛽 󵄨󵄨󵄨󵄨 (∫ (𝑡1 − 𝑠) 𝑑𝑠)
󵄨󵄨𝑝
󵄨
× (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
(𝑡1 + ℎ − 𝑠)
(𝑡1 + ℎ − 𝑠)
0
× [𝑇𝛼 (𝑡1 + ℎ − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)] 𝑔
𝑡1
𝑡1
0
󵄨󵄨 𝑡1 +ℎ
󵄨
𝛼−1
≤ 3𝑝−1 𝐸 󵄨󵄨󵄨󵄨∫
(𝑡1 + ℎ − 𝑠) (−𝐴) 𝑇𝛼 (𝑡1 + ℎ − 𝑠) 𝑔
󵄨󵄨 𝑡1
󵄨󵄨𝑝
󵄨
× (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡1
𝛼−1
𝛼−1
󵄨
+ 3𝑝−1 𝐸 󵄨󵄨󵄨∫ [(𝑡1 + ℎ − 𝑠) − (𝑡1 − 𝑠) ] (−𝐴) 𝑇𝛼
󵄨󵄨 0
󵄨󵄨𝑝
󵄨
× (𝑡1 + ℎ − 𝑠) 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡1
𝛼−1
󵄨
+ 3𝑝−1 𝐸 󵄨󵄨󵄨∫ (𝑡1 − 𝑠) (−𝐴)
󵄨󵄨 0
≤ 3𝑝−1 𝐸 (∫
𝑡1 +ℎ
× (∫
󵄨󵄨𝑝
󵄨
× (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨󵄨
󵄨󵄨
𝑡1 +ℎ
𝛼𝛽−1
(𝑡1 + ℎ − 𝑠)
𝑡1
× (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠
0
𝑡1 +ℎ
+ 3𝑝−1 𝐾 (𝛼, 𝛽) 𝑀𝑔𝑝
󵄨𝑝
󵄨
𝐸󵄨󵄨󵄨𝐹3 (𝑡1 + ℎ) − 𝐹3 (𝑡1 )󵄨󵄨󵄨
𝑡1
󵄨󵄨
󵄨
󵄨󵄨(−𝐴)𝛽 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨 𝑑𝑠)
󵄨
󵄨
𝑑𝑠)
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
𝑝−1
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠,
(20)
Abstract and Applied Analysis
𝑝
𝑝
5
𝑝
where 𝐾(𝛼, 𝛽) = 𝛼𝑝 Γ(1+𝛽) 𝐶1−𝛽 /Γ(1+𝛼𝛽) . Since 𝜖 is sufficiently
small, the right hand side of the above equation tends to zero
as |ℎ| → 0.
Next we consider
𝑡1
0
󵄨󵄨 𝑡1 +ℎ
󵄨
𝛼−1
= 𝐸 󵄨󵄨󵄨󵄨∫
(𝑡1 + ℎ − 𝑠) 𝑇𝛼 (𝑡1 + ℎ − 𝑠)
󵄨󵄨 0
𝛼−1
− (𝑡1 − 𝑠)
]𝐸
× |𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
+
󵄨𝑝
󵄨
𝐸󵄨󵄨󵄨𝐹4 (𝑡1 + ℎ) − 𝐹4 (𝑡1 )󵄨󵄨󵄨
𝛼−1
× ∫ 𝑀𝑡1 +ℎ−𝑠 [(𝑡1 + ℎ − 𝑠)
𝛼 𝑝−1
𝑝−1 𝑝 𝑝 𝑡1
3 𝜖 𝐿 1( )
𝛼
𝑡1
𝛼−1
× ∫ (𝑡1 − 𝑠)
0
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠.
(22)
× 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠
𝑡1
𝛼−1
− ∫ (𝑡1 − 𝑠)
0
󵄨󵄨𝑝
󵄨
𝑇𝛼 (𝑡1 − 𝑠) 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡1 +ℎ
󵄨
𝛼−1
≤ 3𝑝−1 𝐸 󵄨󵄨󵄨󵄨∫
(𝑡1 + ℎ − 𝑠) 𝑇𝛼 (𝑡1 + ℎ − 𝑠)
󵄨󵄨 𝑡1
󵄨󵄨󵄨𝑝
×𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡1
𝛼−1
𝛼−1
󵄨
+ 3𝑝−1 𝐸 󵄨󵄨󵄨∫ [(𝑡1 + ℎ − 𝑠) − (𝑡1 − 𝑠) ]
󵄨󵄨 0
󵄨
󵄨𝑝
𝐸󵄨󵄨󵄨𝐹5 (𝑡1 + ℎ) − 𝐹5 (𝑡1 )󵄨󵄨󵄨
󵄨󵄨 𝑡1 +ℎ
󵄨
𝛼−1
= 𝐸 󵄨󵄨󵄨󵄨∫
(𝑡1 + ℎ − 𝑠) 𝑇𝛼 (𝑡1 + ℎ − 𝑠)
󵄨󵄨 0
× 𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑊 (𝑠)
𝑡1
− ∫ (𝑡1 − 𝑠)
󵄨󵄨𝑝
󵄨
×𝑇𝛼 (𝑡1 + ℎ − 𝑠) 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
0
󵄨󵄨 𝑡1
𝛼−1
󵄨
+ 3𝑝−1 𝐸 󵄨󵄨󵄨∫ (𝑡1 − 𝑠) [𝑇𝛼 (𝑡1 + ℎ − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)] 𝑓
󵄨󵄨 0
󵄨󵄨󵄨𝑝
× (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
𝑡1
+ 3𝑝−1 𝐸 (∫ [(𝑡1 + ℎ − 𝑠)
𝛼−1
𝛼−1
− (𝑡1 − 𝑠)
0
]
󵄨
󵄨
󵄨󵄨
× 󵄨󵄨󵄨𝑇𝛼 (𝑡1 + ℎ − 𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨 𝑑𝑠)
𝑡1
+ 3𝑝−1 𝐸 (∫ (𝑡1 − 𝑠)
𝛼−1
0
Therefore, the right hand side of the above equation tends to
zero as |ℎ| → 0 and 𝜖 sufficiently small. Further, we have
𝑝
󵄨󵄨
󵄨
󵄨󵄨𝑇𝛼 (𝑡1 + ℎ − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)󵄨󵄨󵄨
𝑝
󵄨
󵄨
× 󵄨󵄨󵄨𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨 𝑑𝑠) .
(21)
󵄨󵄨 𝑡1 +ℎ
󵄨
𝛼−1
≤ 3𝑝−1 𝐸 󵄨󵄨󵄨󵄨∫
(𝑡1 + ℎ − 𝑠) 𝑇𝛼 (𝑡1 + ℎ − 𝑠)
󵄨󵄨 𝑡1
󵄨󵄨󵄨𝑝
×𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑊 (𝑠) 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡1
𝛼−1
𝛼−1
󵄨
+ 3𝑝−1 𝐸 󵄨󵄨󵄨∫ [(𝑡1 + ℎ − 𝑠) − (𝑡1 − 𝑠) ]
󵄨󵄨 0
󵄨󵄨𝑝
󵄨
×𝑇𝛼 (𝑡1 + ℎ − 𝑠) 𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑊 (𝑠) 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡1
𝛼−1
󵄨
+ 3𝑝−1 𝐸 󵄨󵄨󵄨∫ (𝑡1 − 𝑠) [𝑇𝛼 (𝑡1 + ℎ − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)]
󵄨󵄨 0
󵄨󵄨󵄨𝑝
×𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑊 (𝑠) 󵄨󵄨󵄨
󵄨󵄨
𝑡1 +ℎ
𝑡1
𝑝
≤ 3𝑝−1 𝐿 1 𝑀𝑝 𝛼𝑝 (∫
𝑡1 +ℎ
𝑡1
×∫
𝑡1 +ℎ
𝑡1
𝑀𝑡1 +ℎ−𝑠 (𝑡1 + ℎ − 𝑠)
𝛼−1
𝑀𝑡1 +ℎ−𝑠 (𝑡1 + ℎ − 𝑠)
𝛼−1
𝑝−1
𝑡1
+ 3𝑝−1 𝐶𝑝 𝐸 (∫ [(𝑡1 + ℎ − 𝑠)
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
0
󵄨󵄨2
󵄨󵄨𝑇𝛼 (𝑡1 + ℎ − 𝑠)󵄨󵄨
𝛼−1
𝑝/2
𝛼−1 2
− (𝑡1 − 𝑠)
󵄨
󵄨2
× 󵄨󵄨󵄨𝑇𝛼 (𝑡1 + ℎ − 𝑠)󵄨󵄨󵄨
𝑝
𝛼−1
× (∫ 𝑀𝑡1 +ℎ−𝑠 [(𝑡1 + ℎ − 𝑠)
0
2(𝛼−1) 󵄨󵄨
(𝑡1 + ℎ − 𝑠)
×|𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))|2 𝑑𝑠)
𝑑𝑠)
+ 3𝑝−1 𝐿 1 𝑀𝑝 𝛼𝑝
𝑡1
𝑇𝛼 (𝑡1 − 𝑠)
󵄨󵄨𝑝
󵄨
×𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑊 (𝑠) 󵄨󵄨󵄨󵄨
󵄨󵄨
≤ 3𝑝−1 𝐶𝑝 𝐸 (∫
By the Holder’s inequality, we obtain
𝛼−1
− (𝑡1 − 𝑠)
𝛼−1
𝑝−1
] 𝑑𝑠)
𝑝/2
×|𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))|2 𝑑𝑠)
]
6
Abstract and Applied Analysis
𝑡1
󵄨󵄨 𝑡
󵄨󵄨󵄨𝑝
󵄨
+ 6𝑝−1 𝐸󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨 0
󵄨󵄨
󵄨󵄨 𝑡
󵄨
+ 6𝑝−1 𝐸 󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠)
󵄨󵄨 0
󵄨󵄨󵄨𝑝
×𝜎 (𝑠, 𝑋 (𝑠 − 𝜈 (𝑠))) 𝑑𝑊 (𝑠) 󵄨󵄨󵄨 .
󵄨󵄨
(24)
2(𝛼−1)
+ 3𝑝−1 𝐶𝑝 𝐸 (∫ (𝑡1 − 𝑠)
0
󵄨
󵄨2
× 󵄨󵄨󵄨𝑇𝛼 (𝑡1 + ℎ − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)󵄨󵄨󵄨
×|𝜎 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))|2 𝑑𝑠)
𝑝/2
𝑝
≤ 3𝑝−1 𝐶𝑝 𝐿 1 𝑀𝑝 𝛼𝑝
× (∫
𝑡1 +ℎ
𝑡1
×∫
𝑡1 +ℎ
𝑡1
𝛼−1 𝑝/(𝑝−2)
(𝑀𝑡1 +ℎ−𝑠 (𝑡1 + ℎ − 𝑠)
)
Now, we estimate the terms on the right hand side of (24) by
using the assumptions (H1), (H3), and (H4). Now, we have
(𝑝−2)/2
𝑑𝑠)
󵄨𝑝
󵄨
6𝑝−1 𝐸󵄨󵄨󵄨𝑆𝛼 (𝑡) 𝜑 (0)󵄨󵄨󵄨
𝛼
󵄨 󵄨𝑝
≤ 6𝑝−1 𝑀𝑝 (∫ 𝜂𝛼 (𝜃) 𝑒−𝛼𝑡 𝜃 𝑑𝜃) 󵄨󵄨󵄨𝜑󵄨󵄨󵄨B
0
)
󳨀→ 0
𝑝
× 𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠 + 3𝑝−1 𝐶𝑝 𝐿 1 𝑀𝑝 𝛼𝑝
𝑡1
𝛼−1
𝑝
∞
𝛼
󵄨
󵄨𝑝
󵄨 󵄨𝑝
≤ 6𝑝−1 𝑀𝑝 (∫ 𝜂𝛼 (𝜃) 𝑒−𝛼𝑡 𝜃 𝑑𝜃) 󵄨󵄨󵄨󵄨(−𝐴)−𝛽 󵄨󵄨󵄨󵄨 𝑀𝑔𝑝 󵄨󵄨󵄨𝜑󵄨󵄨󵄨B
0
0
𝑡1
𝛼−1
𝛼−1
× ∫ (𝑀𝑡1 +ℎ−𝑠 [(𝑡1 + ℎ − 𝑠)
0
𝑝/(𝑝−2)
])
(𝑝−2)/2
󳨀→ 0
𝑑𝑠)
− (𝑡1 − 𝑠)
𝛼−1
𝑝
𝑡1
× ∫ (𝑡1 − 𝑠)
0
𝑝𝛼−2/𝑝−2
𝑝/2
󵄨
󵄨𝑝
≤ 6𝑝−1 󵄨󵄨󵄨󵄨(−𝐴)−𝛽 󵄨󵄨󵄨󵄨 𝑀𝑔𝑝 𝐸|𝑋 (𝑡 − 𝜏 (𝑡))|𝑝 .
])
(𝑝−2)/2
󵄨𝑝
󵄨
6𝑝−1 𝐸󵄨󵄨󵄨𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))󵄨󵄨󵄨 󳨀→ 0 as 𝑡 󳨀→ ∞.
(23)
As above, the right hand side of the above inequality tends
to zero. Similarly, we have 𝐹2 → 0 as ℎ → 0. Thus Ψ is
continuous in 𝑝th moment on [0, ∞).
Next we show that Ψ(B) ∈ B. Let 𝑋 ∈ B. From (18), we
have
󵄨󵄨𝑝
󵄨󵄨 𝑡
󵄨
󵄨
6𝑝−1 𝐸󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 (−𝐴) 𝑇𝛼 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 0
𝑡
󵄨
󵄨
≤ 6𝑝−1 𝐸 (∫ (𝑡 − 𝑠)𝛼−1 󵄨󵄨󵄨󵄨(−𝐴)1−𝛽 𝑇𝛼 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨
0
󵄨
󵄨
× 󵄨󵄨󵄨󵄨(−𝐴)𝛽 𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨󵄨 𝑑𝑠)
𝑝−1
+6
󵄨𝑝
󵄨
𝐸󵄨󵄨󵄨𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))󵄨󵄨󵄨
󵄨󵄨 𝑡
󵄨
+ 6𝑝−1 𝐸 󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 (−𝐴) 𝑇𝛼 (𝑡 − 𝑠)
󵄨󵄨 0
󵄨󵄨𝑝
󵄨
×𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
𝑝
𝑝−1
𝑡
󵄨𝑝
󵄨𝑝
󵄨
󵄨
≤ 6𝑝−1 𝐸󵄨󵄨󵄨𝑆𝛼 (𝑡) 𝜑 (0)󵄨󵄨󵄨 + 6𝑝−1 𝐸󵄨󵄨󵄨𝑆𝛼 (𝑡) 𝑔 (0, 𝜑)󵄨󵄨󵄨
(26)
For the fourth term of (24), we have
𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠.
𝐸|(Ψ𝑋) (𝑡)|𝑝
(25)
For 𝑋(𝑡) ∈ B and for any 𝜖 > 0 there exists a 𝑡1 > 0 such that
𝐸|𝑋(𝑡 − 𝜏(𝑡))|𝑝 ≤ 𝜖 for 𝑡 ≥ 𝑡1 .
Therefore,
𝑡1
)
𝑝𝛼 − 2/𝑝 − 2
𝑝(𝛼−1)/2
as 𝑡 󳨀→ ∞,
󵄨𝑝
󵄨
6𝑝−1 𝐸󵄨󵄨󵄨𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))󵄨󵄨󵄨
× 𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
+ 3𝑝−1 𝜖𝑝 𝐶𝑝 𝐿 1 (
as 𝑡 󳨀→ ∞,
󵄨
󵄨𝑝
6𝑝−1 𝐸󵄨󵄨󵄨𝑆𝛼 (𝑡) 𝑔 (0, 𝜑)󵄨󵄨󵄨
× (∫ (𝑀𝑡1 +ℎ−𝑠 [(𝑡1 + ℎ − 𝑠)
−(𝑡1 − 𝑠)
𝑝
∞
𝛼−1 𝑝/2
(𝑀𝑡1 +ℎ−𝑠 (𝑡1 + ℎ − 𝑠)
≤ 6𝑝−1 𝑀𝑔𝑝 𝐾 (𝛼, 𝛽) (∫ (𝑡 − 𝑠)𝛼𝛽−1 𝑑𝑠)
0
𝑡
× ∫ (𝑡 − 𝑠)𝛼𝛽−1 𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠
0
≤ 6𝑝−1 𝑀𝑔𝑝 𝐾 (𝛼, 𝛽) (
𝑝−1
𝑇𝛼𝛽
)
𝛼𝛽
𝑡
× ∫ (𝑡 − 𝑠)𝛼𝛽−1 𝐸|𝑋 (𝑠 − 𝜏 (𝑠))|𝑝 𝑑𝑠 󳨀→ 0 as 𝑡 󳨀→ ∞.
0
(27)
Abstract and Applied Analysis
7
󵄨󵄨 𝑡
󵄨
+ 4𝑝−1 sup 𝐸 󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠)
𝑡∈[0,𝑇] 󵄨󵄨 0
Also, we have
󵄨󵄨 𝑡
󵄨󵄨𝑝
󵄨
󵄨
6𝑝−1 𝐸󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠))) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨 0
󵄨󵄨
𝑝−1
≤6
𝑡
𝐸 (∫ (𝑡 − 𝑠)
𝛼−1
0
󵄨󵄨𝑝
󵄨
−𝜎 (𝑠, 𝑌 (𝑠 − 𝜈 (𝑠)))] 𝑑𝑊 (𝑠) 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨𝑇𝛼 (𝑡 − 𝑠)󵄨󵄨󵄨
󵄨
󵄨
× 󵄨󵄨󵄨𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨 𝑑𝑠)
𝑝
× [𝜎 (𝑠, 𝑋 (𝑠 − 𝜈 (𝑠)))
𝑡
󵄨
󵄨𝑝
≤ 4𝑝−1 󵄨󵄨󵄨󵄨(−𝐴)−𝛽 󵄨󵄨󵄨󵄨 𝑀𝑔𝑝 sup 𝐸|𝑋 (𝑡) − 𝑌 (𝑡)|𝑝
𝑝
≤ 6𝑝−1 𝛼𝑝 𝑀𝑝 𝐿 1 (∫ 𝑀𝑡−𝑠 (𝑡 − 𝑠)𝛼−1 𝑑𝑠)
𝑡∈[0,𝑇]
𝑡
󵄨
󵄨
+ 4𝑝−1 sup 𝐸 (∫ (𝑡 − 𝑠)𝛼−1 󵄨󵄨󵄨󵄨(−𝐴)1−𝛽 𝑇𝛼 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨
𝑝−1
0
𝑡∈[0,𝑇]
0
󵄨
× 󵄨󵄨󵄨󵄨(−𝐴)𝛽 [𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))
𝑡
× ∫ 𝑀𝑡−𝑠 (𝑡 − 𝑠)𝛼−1 𝐸|𝑋 (𝑠 − 𝜏 (𝑠) ))|𝑝 𝑑𝑠,
󵄨
−𝑔 (𝑠, 𝑌 (𝑠−𝜏 (𝑠)))] 󵄨󵄨󵄨󵄨 𝑑𝑠)
0
󵄨󵄨 𝑡
󵄨󵄨𝑝
󵄨
󵄨
6𝑝−1 𝐸󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝜎 (𝑠, 𝑋 (𝑠 − 𝜈 (𝑠))) 𝑑𝑊 (𝑠)󵄨󵄨󵄨
󵄨󵄨 0
󵄨󵄨
󵄨󵄨 𝑡
󵄨
󵄨2
󵄨
≤ 6𝑝−1 𝐶𝑝 𝐸 󵄨󵄨󵄨∫ (𝑡 − 𝑠)2(𝛼−1) 󵄨󵄨󵄨𝑇𝛼 (𝑡 − 𝑠)󵄨󵄨󵄨
󵄨󵄨 0
𝑡
󵄨
󵄨
+ 4𝑝−1 sup 𝐸 (∫ (𝑡 − 𝑠)𝛼−1 󵄨󵄨󵄨𝑇𝛼 (𝑡 − 𝑠)󵄨󵄨󵄨
0
𝑡∈[0,𝑇]
󵄨
× 󵄨󵄨󵄨𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))
󵄨󵄨𝑝/2
󵄨
×|𝜎 (𝑠, 𝑋 (𝑠 − 𝜈 (𝑠)))|2 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
𝑡
𝑝
≤ 6𝑝−1 𝐶𝑝 𝐿 1 𝛼𝑝 𝑀𝑝 (∫ [𝑀𝑡−𝑠 (𝑡 − 𝑠)𝛼−1 ]
𝑝/(𝑝−2)
0
𝑝
󵄨
−𝑓 (𝑠, 𝑌 (𝑠 − 𝜏 (𝑠)))󵄨󵄨󵄨 𝑑𝑠)
𝑝−2/𝑝
𝑑𝑠)
𝑡
󵄨2
󵄨
+ 4𝑝−1 𝐶𝑝 sup 𝐸 (∫ (𝑡 − 𝑠)2(𝛼−1) 󵄨󵄨󵄨𝑇𝛼 (𝑡 − 𝑠)󵄨󵄨󵄨
0
𝑡∈[0,𝑇]
𝑡
× ∫ 𝑀𝑡−𝑠 (𝑡 − 𝑠)𝛼−1 𝐸|𝑋 (𝑠 − 𝜈 (𝑠))|𝑝 𝑑𝑠.
0
× |𝜎 (𝑠, 𝑋 (𝑠 − 𝜈 (𝑠)))
(28)
By the same discussion as above, we have that (28) tends to
zero as 𝑡 → ∞. Thus 𝐸|Ψ𝑋(𝑡)|𝑝 → 0 as 𝑡 → ∞. We
conclude that Ψ(B) ∈ B.
Finally, we prove that Ψ has a unique fixed point. Indeed,
for any 𝑋, 𝑌 ∈ B, we have
𝑝
sup 𝐸|(Ψ𝑋) (𝑡) − (Ψ𝑌) (𝑡)|
𝑡∈[0,𝑇]
−𝜎 (𝑠, 𝑌 (𝑠 − 𝜈 (𝑠)))|2 𝑑𝑠)
𝑇𝛼𝛽
󵄨𝑝
󵄨󵄨(−𝐴)−𝛽 󵄨󵄨󵄨 𝑀𝑔𝑝 + 4𝑝−1 𝑀𝑔𝑝 𝐾 (𝛼, 𝛽) (
)
󵄨
󵄨
𝛼𝛽
𝑝−1 󵄨󵄨
≤ [4
≤4
󵄨󵄨𝑝
󵄨
sup 𝐸󵄨󵄨󵄨𝑔 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡))) − 𝑔 (𝑡, 𝑌 (𝑡 − 𝜏 (𝑡)))󵄨󵄨
× [𝑔 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))
󵄨󵄨𝑝
󵄨
−𝑔 (𝑠, 𝑌 (𝑠 − 𝜏 (𝑠)))] 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡
󵄨
+ 4𝑝−1 sup 𝐸 󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠)
𝑡∈[0,𝑇] 󵄨󵄨 0
× [𝑓 (𝑠, 𝑋 (𝑠 − 𝜏 (𝑠)))
󵄨󵄨𝑝
󵄨
−𝑓 (𝑠, 𝑌 (𝑠 − 𝜏 (𝑠)))] 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨
𝑝
𝑝
+4𝑝−1 𝐶𝑝 𝛼𝑝 𝑀𝑝 𝐿 1 𝐿󸀠
𝑝/2
] sup 𝐸|𝑋 (𝑡) − 𝑌 (𝑡)|𝑝 .
𝑡∈[0,𝑇]
𝑡∈[0,𝑇]
󵄨󵄨 𝑡
󵄨
+ 4𝑝−1 sup 𝐸 󵄨󵄨󵄨∫ (𝑡 − 𝑠)𝛼−1 (−𝐴) 𝑇𝛼 (𝑡 − 𝑠)
𝑡∈[0,𝑇] 󵄨󵄨 0
𝑝/2
+ 4𝑝−1 𝛼𝑝 𝑀𝑝 𝐿 1 𝐿𝑝
𝑝
𝑝−1
𝑝
(29)
Therefore, Ψ is a contradiction mapping and hence there
exists a unique fixed point, which is a mild solution of (3)
with 𝑋(𝑠) = 𝜑(𝑠) on [𝑚(0), 0] and 𝐸|𝑋(𝑡)|𝑝 → 0 as
𝑡 → ∞.
To show the asymptotic stability of the mild solution of
(3), as the first step, we have to prove the stability in 𝑝th
moment. Let 𝜖 > 0 be given and choose 𝛿 > 0 such that 𝛿 < 𝜖
satisfies 6𝑝−1 [𝑀𝑝 + 𝑀𝑝 |(−𝐴)−𝛽 |𝑝 𝑀𝑔𝑝 ]𝛿 + 6𝑝−1 [|(−𝐴)−𝛽 |𝑝 𝑀𝑔𝑝 +
𝑝
𝑝
𝑝
𝑀𝑔𝑝 𝐾(𝛼, 𝛽)(𝑇𝛼𝛽 /𝛼𝛽) + 𝛼𝑝 𝑀𝑝 𝐿 1 𝐿𝑝 + 𝐶𝑝 𝛼𝑝 𝑀𝑝 𝐿 1 𝐿󸀠𝑝/2 ]𝜖 <
𝜖.
𝑝
If 𝑋(𝑡) = 𝑋(𝑡, 𝜑) is mild solution of (3), with |𝜑|B < 𝛿,
then (Ψ𝑋)(𝑡) = 𝑋(𝑡) satisfies 𝐸|𝑋(𝑡)|𝑝 < 𝜖 for every 𝑡 ≥ 0.
Notice that 𝐸|𝑋(𝑡)|𝑝 < 𝜖 on 𝑡 ∈ [𝑚(0), 0]. If there exists 𝑡
8
Abstract and Applied Analysis
such that 𝐸|𝑋(𝑡)|𝑝 = 𝜖 and 𝐸|𝑋(𝑠)|𝑝 < 𝜖 for 𝑠 ∈ [𝑚(0), 𝑡].
Then (24) show that
󵄨𝑝
󵄨
𝐸󵄨󵄨󵄨𝑋 (𝑡)󵄨󵄨󵄨
𝑝
𝛼
≤ 6𝑝−1 [𝑀𝑝 (𝜂𝛼 (𝜃) 𝑒−𝛼𝑡 𝜃 )
+𝑀𝑝 (𝜂𝛼 (𝜃) 𝑒−𝛼𝑡
𝛼
𝐷 (𝐴) = {𝑤 ∈ 𝑋; 𝑤, 𝑤󸀠 are absolutely continuous,
𝑝󵄨
󵄨𝑝
𝜃 󵄨
) 󵄨󵄨󵄨(−𝐴)−𝛽 󵄨󵄨󵄨󵄨 𝑀𝑔𝑝 ] 𝛿
𝛼𝛽
𝑤󸀠󸀠 ∈ 𝑋, 𝑤 (0) = 𝑤 (𝜋) = 0} ,
𝑝
𝑇
󵄨𝑝
󵄨
+ 6𝑝−1 [󵄨󵄨󵄨󵄨(−𝐴)−𝛽 󵄨󵄨󵄨󵄨 𝑀𝑔𝑝 + 𝑀𝑔𝑝 𝐾 (𝛼, 𝛽) (
)
𝛼𝛽
𝑝
𝑝
+𝛼𝑝 𝑀𝑝 𝐿 1 𝐿𝑝 + 𝐶𝑝 𝛼𝑝 𝑀𝑝 𝐿 1 𝐿󸀠
𝑝/2
where 𝑊(𝑡) denotes a standard cylindrical Wiener process
and a standard one-dimensional Brownian motion. To write
the system (32) into the abstract form of (3), we consider the
space 𝐻 = 𝐾 = 𝐿2 [0, 𝜋] and define the operator 𝐴 : 𝐷(𝐴) ⊂
𝐻 → 𝐻 by 𝐴𝑤 = 𝑤󸀠󸀠 with domain
(30)
∞
𝐴𝑤 = ∑ 𝑛2 (𝑤, 𝑤𝑛 ) 𝑤𝑛 ,
where 𝑤𝑛 (𝑠) = √2 sin(𝑛𝑠), 𝑛 = 1, 2, . . . is the orthogonal set of
eigenvectors in 𝐴. It is well known that 𝐴 generates a compact,
analytic semigroup {𝑆(𝑡), 𝑡 ≥ 0} in 𝑋 and
]𝜖
∞
which contradicts the definition of 𝑡. Therefore, the mild
solution of (3) is asymptotically stable in 𝑝th moment.
In particular, when 𝑝 = 2 from Theorem 9 we have the
following.
Theorem 10. Suppose that the conditions (H1)–(H3) hold.
Then, the stochastic fractional differential equations (3)
are mean square asymptotically stable if 4𝑀𝑔2 [|(−𝐴)−𝛽 |2 +
2
𝑉(𝛼, 𝛽)(𝑇𝛼𝛽 /𝛼𝛽) ] + 4𝛼2 𝑀2 𝐿21 [𝐿2 + 𝐿󸀠 ] < 1, where 𝑉(𝛼, 𝛽) =
2
2
2
𝐶1−𝛽
/Γ(1+𝛼𝛽)
.
𝛼2 Γ(1+𝛽)
When 𝑔 ≡ 0, 𝑝 = 2, (3) reduces to
𝐷𝑡𝛼 𝑋 (𝑡) = 𝐴𝑋 (𝑡) + 𝑓 (𝑡, 𝑋 (𝑡 − 𝜏 (𝑡)))
+ 𝜎 (𝑡, 𝑋 (𝑡 − 𝜈 (𝑡)))
𝑑𝑊 (𝑡)
,
𝑑𝑡
𝑡 ≥ 0,
(31)
𝑋0 (⋅) = 𝜑 ∈ BF ([𝑚 (0) , 0] , 𝐻) .
From Theorems 9 and 10, we can easily get the following
result.
Corollary 11. Suppose the assumptions (H1) and (H2) hold.
Then, the stochastic equations (8) are mean square asymptotically stable if 2𝛼2 𝑀2 𝐿21 [𝐿2 + 𝐿󸀠 ] < 1.
Example 12. Consider the following stochastic nonlinear
fractional partial differential equation with infinite delay in
the following form
𝑐
̂ 𝑢 (𝑡 − 𝜏, 𝑦)]
𝐷𝑡𝛼 [𝑢 (𝑡, 𝑦) + 𝑔(𝑡,
=
𝜕2 𝑢 (𝑡, 𝑦) ̂
+ 𝑓 (𝑡, 𝑢 (𝑡 − 𝜏, 𝑦))
𝜕𝑦2
+ 𝜎̂ (𝑡, 𝑢 (𝑡 − 𝜏, 𝑦))
𝑑𝑊 (𝑡)
,
𝑑𝑡
𝑢 (𝑡, 0) = 𝑢 (𝑡, 𝜋) = 0,
𝑢 (𝑡, 𝑦) = 𝜙 (𝑡, 𝑦) ,
𝑦 ∈ [0, 𝜋] , 𝑡 ≤ 0,
(34)
𝑛=1
<𝜖
𝑐
𝑤 ∈ 𝐷 (𝐴) ,
(33)
(32)
2
𝑆 (𝑡) 𝑤 = ∑ 𝑒−𝑛 𝑡 (𝑤, 𝑤𝑛 ) 𝑤𝑛 .
(35)
𝑛=1
2
It is well known that |𝑆(𝑡)| ≤ 𝑒−𝜋 𝑡 . Take 𝑝 = 2. Since 𝑀 = 1,
2
we can get the inequality 4[|(−𝐴)−𝛽 |2 + 𝑉(𝛼, 𝛽)(𝜋𝛼𝛽 /𝛼𝛽) +
𝛼2 (𝐿2 + 𝐿󸀠 )] < 1. Further, if we impose suitable conditions
̂ and 𝜎̂ to verify assumptions of Theorem 10, then we
̂ 𝑓,
on 𝑔,
can conclude that the mild solution of (32) is mean square
asymptotically stable.
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