Almost Periodic and Almost Automorphic Solutions to Some Classes of Stochastic Neutral ...

14 Pages·2014·0.12 MB·English

Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 2, 91 - 104
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/nade.2014.432
Almost Periodic and Almost Automorphic
Solutions to Some Classes of Stochastic
Neutral Evolution Equations
Yan Zhao
Department of Electronic and Information Engineering
Leshan Vocational Technical College
Leshan, 614000, P.R. China
Copyright(cid:2)c 2014YanZhao. ThisisanopenaccessarticledistributedundertheCreativeCommons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Abstract
We investigate some classes of stochastic neutral evolution equations in a real
separable Hilbert space. Sufficient conditions for the existence and uniqueness of
square-mean almost periodic and square-mean almost automorphic mild solutions
are derived with the help of the Banach fixed point theorem. An example is given
to illustrate the theory.
Keywords: Square-mean almost periodic; Square-mean almost automorphic; fixed
point; Exponentially stable semigroup
1 Introduction
The concept of an almost periodic function was introduced by Bohr for numerical valued
functions and extended by Bochner for functions with values in Polish spaces. In recent years,
theexistenceofalmostperiodicondifferentkindsofdeterministicdifferentialequationshasbeen
92 Yan Zhao
considerably investigated in lots of publications [1–8] becauseof its significance and applications
in physics, mechanics and mathematical biology.
In [9], Bezandry andDiaganaintroduced a newconcept ofa square-meanalmostperiodicity.
They established the existence and uniqueness of square-mean almost periodic mild solutions to
some stochastic differential equations and some functional integro-differential stochastic evolu-
tion equations in [9,10].
Theconceptofalmostautomorphy was firstinitiatedbyBochnerinhislandmarkpaper[11].
Sincethenthetheoryofalmostautomorphicfunctionshasfoundseveraldevelopmentsandappli-
cations in the theory of abstract differential equations, partial differential equations, functional-
differential equations, and integro-differential equations. For more on these and related issues,
we refer the reader to [12–16] and the references therein.
In a very recent paper [17], Fu and Liu introduced a new concept of a square-mean almost
automorphic stochastic process. They established the existence and uniqueness of square-mean
almost automorphic mild solutions to some stochastic differential equations.
Neutraldifferentialequationsariseinmanyareasofappliedmathematicsandforthisreason,
this type of equation has received much attention in recent years. The existence of almost peri-
odic and almost automorphic solutions to deterministic neutral differential equations has been
considerablyinvestigatedinlotsofpublications[1,4,13]. However, Theexistenceofsquare-mean
almost periodic and square-mean almost automorphic solutions to stochastic neutral differential
equations has not been considered. Motivated by the above discussion, We give in this work
some sufficient conditions for the existence and uniqueness of square-mean almost periodic and
square-mean almost automorphic mild solutions to some classes of stochastic neutral evolution
equations.
2 Preliminaries
For details of this section, we refer the reader to Da Prato and Zabczyk [18]. Throughout
this paper, H and K will denote real separable Hilbert spaces with respective norms (cid:3)·(cid:3) and
H
(cid:3)·(cid:3)K. Let (Ω,F,{Ft}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 satisfying
the usual conditions (i.e, it is right continuous and F0 contains all P-null sets). We denote by
L2(K;H) the space of all Hilbert-Schmidt operators acting between K and H equipped with
theHilbert-Schmidtnorm(cid:3)·(cid:3)2. ForasymmetricnonnegativeoperatorQ ∈ L2(K;H)with finite
trace we assume that W(t),t ∈ R is a Q−Wiener process defined on (Ω,F,{Ft}t≥0,P) and with
values in K.
The collection of all strongly measurable, square-integrable H−valued random variables,
(cid:2) (cid:3)
1
denotedby L2(P;H), isa Banach spaceequippedwith norm(cid:3)X(cid:3) = E(cid:3)X(cid:3) 2 2, where
(cid:4) L2(P;H) H
the expectation E is defined by E[g] = g(ω)dP (ω).
Ω
Almost periodic and almost automorphic solutions 93
1 0
Let K0 = Q2K and L2 = L2(K0;H) with respect to the norm
(cid:5) (cid:5)
(cid:5) (cid:5)2
(cid:3)Φ(cid:3)2L02 = (cid:5)ΦQ21(cid:5)2 = Tr(ΦQΦ∗).
We are concerned with the existence almost periodic and almost automorphic solutions to
the class of stochastic neutral evolution equations of the form
d[x(t)+f(t,Bx(t))] = (Ax(t)+g(t,Cx(t)))dt+σ(t,Dx(t))dW (t), t ∈ R, (1)
where A is the infinitesimal generator of a uniformly exponentially stable C0-semigroup acting
2 2
on L (P;H), B, C and D are densely defined closed linear operators on L (P;H). f, g :
(cid:6) (cid:7)
R×L2(P;H) → L2(P;H) and σ : R×L2(P;H)→ L2 P;L0 are jointly continuous functions.
2
One should point out that various stochastic differential equations, stochastic partial differ-
ential equations, stochastic integrodifferential equations, and stochastic functional differential
equations can be written of the form Eq. (1).
Let(B,(cid:3)·(cid:3) )beaBanach space. LetCUB(R;L2(P;B))denotethecollectionofallstochas-
B
tic processes X :R → L2(P;B) , which are continuous and uniformly bounded. It is then easy
2
to check that CUB(R;L (P;B)) is a Banach space when it is equipped with the norm:
(cid:2) (cid:3)
1
(cid:3)X(cid:3) = sup E(cid:3)X(t)(cid:3) 2 2.
B,∞ B
t∈R
(cid:6) (cid:7) (cid:6) (cid:7)
Let B1,(cid:3)·(cid:3) and B2,(cid:3)·(cid:3) be Banach spaces. L(B1;B2) and L(B1) stand respectively for
B1 B2
the class of bounded liner operator from B1 to B2 and the class of bounded liner operator from
B1 to itself.
Definition 2.1. A continuous stochastic process X : R → L2(P;B) is said to be square-mean
almost periodic if for each ε > 0 there exists l(ε) > 0 such that any interval of length l(ε)
contains at least a number τ for which
supE(cid:3)X(t+τ)−X(t)(cid:3)2 < ε.
B
t∈R
The collection of all stochastic processes X : R → L2(P;B) which are square-mean almost
(cid:6) (cid:7)
2
periodic is then denoted by AP R;L (P;B) .
The next lemma provides some properties of square-mean almost periodic processes.
(cid:6) (cid:7)
Lemma 2.1. [9] If X belongs to AP R;L2(P;B) , then
(i) the mapping t → E(cid:3)X(t)(cid:3)2 is uniformly continuous;
(ii) there exists a constant N > 0 such that E(cid:3)X(t)(cid:3)2 ≤ N, for all t ∈R.
(cid:6) (cid:7) (cid:6) (cid:7)
Lemma 2.2. [9] AP R;L2(P;B) ⊂ CUB R;L2(P;B) is a closed subspace.
94 Yan Zhao
(cid:6) (cid:7)
2
In view of the above, the space AP R;L (P;B) of square-mean almost periodic processes
equipped with the norm (cid:3)·(cid:3) is a Banach space.
B,∞
Definition 2.2. Afunction F : R×L2(P;B1)→ L2(P;B2),(t,Y)→ F (t,Y),whichisjointly
continuous, is said to be square-mean almost periodic in t ∈ R uniformly in Y ∈ K where
K ⊂ L2(P;B1) is a compact if for any ε > 0, there exists l(ε,K) > 0 such that any interval of
length l(ε,K) contains at least a number τ for which
supE(cid:3)F (t+τ,Y)−F (t,Y)(cid:3)2 < ε
B2
t∈R
for each stochastic process Y : R → K.
Lemma 2.3. [9]LetF : R×L2(P;B1) → L2(P;B2),(t,Y) → F (t,Y)besquare-meanalmost
periodic in t ∈ R uniformly in Y ∈ K where K ⊂ L2(P;B1) is a compact. Suppose that F is
Lipschitz in the following sense:
E(cid:3)F (t,Y)−(t,Z)(cid:3)2 ≤ ME(cid:3)Y −Z(cid:3)2
B2 B1
for all Y,Z ∈ L2(P;B1) and for each t ∈ R, where M > 0. Then for any square-mean almost
periodic process Φ : R → L2(P;B1), the stochastic process t → F (t,Φ(t)) is square-mean
almost periodic.
Definition 2.3. A continuous stochastic process X : R → L2(P;B) is said to be square mean
almost automorphic if for every sequence of real numbers {s(cid:6) } there exist a subsequence
n n∈N
{s } and a stochastic process Y :R → L2(P;B) such that
n n∈N
lim E(cid:3)X(t+s )−Y (t)(cid:3)2 = 0 and lim E(cid:3)Y (t−s )−X(t)(cid:3)2 =0
n→∞ n B n→∞ n B
hold for each t ∈ R.
Thecollectionofallsquare-meanalmostautomorphicstochasticprocessesX :R → L2(P;B)
(cid:6) (cid:7)
2
is denoted by AA R;L (P;B) .
(cid:6) (cid:7) (cid:6) (cid:7)
Lemma 2.4. [17]AA R;L2(P;B) ⊂ CUB R;L2(P;B) equipped with the norm (cid:3)·(cid:3) is
B,∞
a Banach space.
Definition 2.4. Afunction F : R×L2(P;B1)→ L2(P;B2),(t,Y)→ F (t,Y),whichisjointly
continuous, is said to be square-mean almost automorphic in t ∈ R for each Y ∈ L2(P;B1) if
for every sequence of real numbers {s(cid:6) } there exist a subsequence {s } and a stochastic
n n∈N n n∈N
process F˜ : R×L2(P;B1) → L2(P;B2) such that
(cid:5) (cid:5) (cid:5) (cid:5)
(cid:5) (cid:5)2 (cid:5) (cid:5)2
lim E(cid:5)F (t+s ,Y)−F(cid:8)(t,Y)(cid:5) = 0 and lim E(cid:5)F(cid:8)(t−s ,Y)−F (t,Y)(cid:5) = 0
n n
n→∞ B2 n→∞ B2
Almost periodic and almost automorphic solutions 95
Lemma 2.5. [17] Let F : R × L2(P;B1) → L2(P;B2),(t,Y) → F (t,Y) be square-mean
almost automorphic in t ∈ R uniformly in Y ∈ L2(P;B1). Suppose that F is Lipschitz in the
following sense:
E(cid:3)F (t,Y)−(t,Z)(cid:3)2 ≤ ME(cid:3)Y −Z(cid:3)2
B2 B1
for all Y,Z ∈ L2(P;B1) and for each t ∈ R, where M > 0. Then for any square-mean almost
automorphic process Φ : R → L2(P;B1), the stochastic process t → F (t,Φ(t)) is square mean
almost automorphic.
For convenience, we recall from [19] the mild solution to (1) as follows.
Definition 2.5. A stochastic process x(t) : [δ, δ+a) → L2(P;H), a > 0, is a mild solution of
the neutral stochastic system (1) on [δ,δ +a), if the function s → AT (t−s)f(s,x(s−r)) is
intergrable on [δ, t) for every δ < t <δ+a, and
(cid:9)
t
x(t) = T (t)[x(δ)+f(δ,Bx(δ))]−f(t,Bx(t))− AT (t−s)f (s,Bx(s))ds
(cid:9) (cid:9) δ
t t
+ T (t−s)g(s,Cx(s))ds+ T (t−s)σ(s,Dx(s))dW (s), t ∈ [δ, δ+a)(.2)
δ δ
3 Existence results
Throughout the rest of the work, (G,(cid:3)·(cid:3) ) denotes an arbitrary real separable Hilbert spaces,
G
which is continuously embedded into the ground real separable Hilbert spaces H.
To prove our results we always assume that the following conditions are satisfied.
(H1) The operator A is the infinitesimal generator of an exponentially stable semigroup
(T(t))t≥0 such that there exist constants M > 0 and δ > 0 with
(cid:3)T (t)(cid:3) ≤ Me−δt, ∀t ≥ 0.
L(L2(P;G))
(cid:6) (cid:7)
Furthermore,thefunctions → AT (s)definedfrom(0,∞)intoL L2(P;G) isstrongly(Lebesgue)
measurable and there exist a function γ : [0,∞) → [0,∞) and a constant ω > 0 with ρ :=
(cid:4)
∞e−ωsγ(s)ds < ∞ such that
0
(cid:3)AT (s)(cid:3) ≤ e−ωsγ(s), s > 0.
L(L2(P;G))
(H2) The functions f,g(t,x) :R×L2(P;H) → L2(P;G) is a square-mean almost periodic
in t ∈ R uniformly in x ∈ Ω ( Ω ⊂ L2(P;H) being a compact subspace ). Moreover, f,g is
96 Yan Zhao
Lipschitz in the following sense: there exist L , L for which
f g
E(cid:3)f (t,x)−f(t,y)(cid:3)2 ≤ L E(cid:3)x−y(cid:3)2 ,
G f H
E(cid:3)g(t,x)−g(t,y)(cid:3)2 ≤ L E(cid:3)x−y(cid:3)2 ,
G g H
for all stochastic processes x,y ∈ L2(P;H) and t ∈ R.
(H3) The functions σ(t,x) : R×L2(P;H) → L2(P;L2(K0;G)) is a square-mean almost
periodic in t ∈R uniformly in x ∈ Ω ( Ω⊂ L2(P;H) being a compact subspace ). Moreover, σ
is Lipschitz in the following sense: there exists L for which
σ
E(cid:3)σ(t,x)−σ(t,y)(cid:3)2 ≤ L E(cid:3)x−y(cid:3)2
L2(K0;G) σ H
for all stochastic processes x,y ∈ L2(P;H) and t ∈ R. (cid:10)
(cid:6) (cid:7)
(H4)B,C andDbelongtoL L2(P;G);L2(P;H) withmax (cid:3)B(cid:3) ,(cid:3)C(cid:3)
(cid:11) L(L2(P;G);L2(P;H)) L(L2(P;G);L2(P;H))
(cid:3)D(cid:3) = h.
L(L2(P;G);L2(P;H))
(H5) The functions f,g(t,x) : R ×L2(P;H) → L2(P;G) is a square-mean almost auto-
morphic in t ∈ R uniformly in x ∈ L2(P;H). Moreover, f,g is Lipschitz in the following sense:
there exist L , L for which
f g
E(cid:3)f (t,x)−f(t,y)(cid:3)2 ≤ L E(cid:3)x−y(cid:3)2 ,
G f H
E(cid:3)g(t,x)−g(t,y)(cid:3)2 ≤ L E(cid:3)x−y(cid:3)2 ,
G g H
for all stochastic processes x,y ∈ L2(P;H) and t ∈ R.
(H6) The functions σ(t,x) : R×L2(P;H) → L2(P;L2(K0;G)) is a square-mean almost
automorphic in t ∈ R uniformly in x ∈ L2(P;H) . Moreover, σ is Lipschitz in the following
sense: there exists L for which
σ
E(cid:3)σ(t,x)−σ(t,y)(cid:3)2 ≤ L E(cid:3)x−y(cid:3)2
L2(K0;G) σ H
for all stochastic processes x,y ∈ L2(P;H) and t ∈ R.
The proof of our main results require the following technical lemmas.
Lemma 3.1. Suppose that assumptions (H1) − (H2) and (H4) hold. Define the nonlinear
(cid:6) (cid:7)
operator Λ1 by: for each φ∈ AP R;L2(P;G) ,
(cid:9)
t
(Λ1φ)(t):= AT (t−s)f (s,Bφ(s))ds.
−∞
(cid:6) (cid:7)
2
Then Λ1 maps AP R;L (P;G) into itself.
Almost periodic and almost automorphic solutions 97
(cid:6) (cid:7)
Proof. First, let us check that Λ is well defined. Indeed, if φ ∈ AP R;L2(P;G) , then
(cid:6) (cid:7) (cid:6) (cid:7)
Bφ(s) is in AP R;L2(P;G) as B ∈ L L2(P;G);L2(P;H) . And hence, by Lemma (2.3),
(cid:6) (cid:7) (cid:6) (cid:7)
the function f(t,Bφ(t)) ∈ AP R;L2(P;G) when φ ∈ AP R;L2(P;G) . In particular,
(cid:3)f (t,Bφ(t))(cid:3) < ∞. So, we have from the above discussion and (H1)
G,∞
(cid:5)(cid:9) (cid:5)
(cid:5) t (cid:5)2
E(cid:3)(Λφ)(t)(cid:3)2 = E(cid:5)(cid:5) AT (t−s)f (s,Bφ(s))ds(cid:5)(cid:5)
G
(cid:10)(cid:9) −∞ (cid:11)(cid:10)(cid:9) G (cid:11)
t t
≤ e−ω(t−s)γ(t−s)ds e−ω(t−s)γ(t−s)E(cid:3)f (s,Bφ(s))(cid:3)2ds
G
−∞ −∞
≤ ρ2(cid:3)f (t,Bφ(t))(cid:3)2 < ∞.
G,∞
(cid:6) (cid:7)
Now, Λ1(cid:6)is well defin(cid:7)ed. By Lemma (2.3), the function f(t,Bφ(t)) ∈ AP R;L2(P;G) when
φ∈ AP R;L2(P;G) . Therefore, for each ε > 0 there exists l(ε) > 0 such that any interval of
length l(ε) contains at least τ for which
E(cid:3)f (t+τ,Bφ(t+τ))−f (t,Bφ(t))(cid:3)2 < ε
G ρ2
for each t ∈ R.
We also have that
(cid:5)(cid:9) (cid:5)
(cid:5) t (cid:5)2
E(cid:3)(Λ1φ)(t+τ)−(Λ1φ)(t)(cid:3)2 = E(cid:5)(cid:5) AT (t−s)[f(s+τ,Bφ(s+τ))−f (s,Bφ(s))]ds(cid:5)(cid:5)
G
(cid:10)(cid:9) −∞ (cid:11) G
t
≤ e−w(t−s)γ(t−s)ds
−∞
(cid:10)(cid:9) (cid:11)
t
× e−w(t−s)γ(t−s)E(cid:3)f (s+τ,Bφ(s+τ))−f (s,Bφ(s))(cid:3)2ds
G
−∞
≤ ρ2(cid:3)f(s+τ,Bφ(s+τ))−f (s,Bφ(s))(cid:3)2
G,∞
< ε
In view of the above, E(cid:3)(Λ1φ)(t+τ)−(Λ1φ)(t)(cid:3)2 < ε for each t ∈ R, that is, (Λ1φ)t is
G
square-mean almost periodic.
Lemma 3.2. Suppose that assumptions (H1) − (H2) and (H4) hold. Define the nonlinear
(cid:6) (cid:7)
operator Λ2 by: for each φ∈ AP R;L2(P;G) ,
(cid:9)
t
(Λ2φ)(t):= T (t−s)g(s,Cφ(s))ds.
−∞
(cid:6) (cid:7)
2
Then Λ2 maps AP R;L (P;G) into itself.
Proof. The proof is similar to that of Lemma (3.1) with appropriate adaptations. So we omit
it.
98 Yan Zhao
Lemma 3.3. Suppose that assumptions (H1),(H3) and (H4) hold. Define the nonlinear oper-
(cid:6) (cid:7)
ator Λ3 by: for each φ ∈ AP R;L2(P;G) ,
(cid:9)
t
(Λ3φ)(t):= T (t−s)σ(s,Dx(s))dW (s).
−∞
(cid:6) (cid:7)
2
Then Λ3 maps AP R;L (P;G) into itself.
Proof. The proof is similar to that of Theorem 3.2 in [9] with appropriate adaptations. So we
omit it.
Lemma 3.4. Suppo(cid:6)se that assum(cid:7)ptions (H4)−(H5) hold. Define the nonlinear operator Λ4
by: for each φ ∈ AA R;L2(P;G) ,
(cid:9)
t
(Λ4φ)(t):= AT (t−s)f (s,Bφ(s))ds.
−∞
(cid:6) (cid:7)
2
Then Λ4 maps AA R;L (P;G) into itself.
(cid:6) (cid:7)
Proof. First, let(cid:6)us check th(cid:7)at Λ4 is we(cid:6)ll defined. Indeed, (cid:7)if φ ∈ AA R;L2(P;G) , then
Bφ(s) is in AA R;L2(P;G) as B ∈ L L2(P;G);L2(P;H) . And hence, by Lemma (2.5),
(cid:6) (cid:7) (cid:6) (cid:7)
the function f(t,Bφ(t)) ∈ AA R;L2(P;G) when φ ∈ AA R;L2(P;G) . In particular,
(cid:3)f (t,Bφ(t))(cid:3) < ∞. So, we have from the above discussion and (H1)
G,∞
(cid:5)(cid:9) (cid:5)
(cid:5) t (cid:5)2
E(cid:3)(Λφ)(t)(cid:3)2 = E(cid:5)(cid:5) AT (t−s)f (s,Bφ(s))ds(cid:5)(cid:5)
G
−∞
(cid:10)(cid:9) (cid:11)(cid:10)(cid:9) G (cid:11)
t t
≤ e−ω(t−s)γ(t−s)ds e−ω(t−s)γ(t−s)E(cid:3)f (s,Bφ(s))(cid:3)2ds
G
−∞ −∞
≤ ρ2(cid:3)f (t,Bφ(t))(cid:3)2 < ∞.
G,∞
(cid:6) (cid:7)
Now, Λ4 is well defined. To prove that (Λ4φ)∈ AA R;L2(P;G) , one has to prove that
(cid:9)
t
t → Λ4φ(t)= lim AT (t−s)ϕ(s)ds
r→−∞
r
is square-mean almost automorphic, where ϕ(s) = f (s,Bx(s)) for each s ∈ R.
(cid:6)
Now let s be an arbitrary sequence of real numbers, Since ϕ is square-mean almost auto-
n
morphic, there exists a subsequence (s ) of s(cid:6) such that lim E(cid:3)ϕ(t+s )−h(t)(cid:3)2 = 0 for
n n n→∞ n G
each t ∈R and lim E(cid:3)ϕ(t)−h(t−s )(cid:3)2 = 0 for each t ∈ R.
n→∞ n G
Now
(cid:5) (cid:5)
(cid:5)(cid:4) (cid:4) (cid:5)2
E(cid:5) t+snAT (t+s −s)ϕ(s)ds− t AT (t−s)h(s)ds(cid:5)
−∞(cid:5) n −∞ (cid:5)
(cid:5)(cid:4) (cid:4) (cid:5)2
= E(cid:5) t AT (t−s)ϕ(s+s )ds− t AT (t−s)h(s)ds(cid:5)
(cid:5) −∞ n −∞ (cid:5)
(cid:5)(cid:4) (cid:5)2
≤ E(cid:5) t AT (t−s)[ϕ(s+s )−h(s)]ds(cid:5)
(cid:4) −∞ n
≤ ρ t e−ω(t−s)γ(t−s)E(cid:3)ϕ(s+s )−h(s)(cid:3)2ds.
−∞ n
Almost periodic and almost automorphic solutions 99
(cid:5) (cid:5)
(cid:5)(cid:4) (cid:4) (cid:5)2
So,byLebesguesdominatedconvergencetheorem,E(cid:5) t+snAT (t+s −s)ϕ(s)ds− t AT (t−s)ϕ(s)ds(cid:5) =
−∞ n −∞
0 as n → ∞ for each t ∈ R. (cid:5) (cid:5)
(cid:5)(cid:4) (cid:4) (cid:5)2
Similarly,itcanbeshownthatE(cid:5) t AT (t−s)ϕ(s)ds− t−snAT (t−s t−s)h(s)ds(cid:5) =
−∞ −∞ n
0 as n → ∞ for each t ∈ R.
This shows that Λ4φ(t) is a square-mean almost automorphic function.
Lemma 3.5. Suppose that assumptions (H1) and (H4) − (H5) hold. Define the nonlinear
(cid:6) (cid:7)
operator Λ5 by: for each φ∈ AA R;L2(P;G) ,
(cid:9)
t
(Λ5φ)(t):= T (t−s)g(s,Cφ(s))ds.
−∞
(cid:6) (cid:7)
2
Then Λ5 maps AA R;L (P;G) into itself.
Proof. The proof is similar to that of Lemma 3.4 with appropriate adaptations. So we omit
it.
Lemma 3.6. Suppose that assumptions (H1), (H4) and (H6) hold. Define the nonlinear op-
(cid:6) (cid:7)
erator Λ6 by: for each φ ∈ AA R;L2(P;G) ,
(cid:9)
t
(Λ6φ)(t):= T (t−s)σ(s,Dx(s))dW (s).
−∞
(cid:6) (cid:7)
2
Then Λ6 maps AA R;L (P;G) into itself.
Proof. The proof is similar to that of Theorem 3.1 in [20] with appropriate adaptations. So we
omit it.
Theorem 3.1. Assume the conditions (H1)−(H4) hold. Then (1) has a unique square-mean
almost period mild solution on R provided that
(cid:12) (cid:13)
1 1
2 2 2 2 2 2 2
4 h L +ρ R L +h M L +h M L < 1. (3)
f f δ2 g δ2 σ
(cid:6) (cid:7) (cid:6) (cid:7)
Proof. Let Γ: AP R;L2(P;G) → AP R;L2(P;G) be the map defined by
(cid:9)
t
Γx(t) = −f(t,Bx(t))− AT (t−s)f (s,Bx(s))ds
−∞
(cid:9) (cid:9)
t t
+ T (t−s)g(s,Cx(s))ds+ T (t−s)σ(s,Dx(s))dW (s), t ∈ R.(4)
−∞ −∞
From Lemma2.3, Lemma3.1, Lemma3.2and Lemma3.3, it follows that thenonlinear operator
(cid:6) (cid:7) (cid:6) (cid:7)
Γ is well defined, that is, it maps AP R;L2(P;G) → AP R;L2(P;G) . In the following, we
prove that Γ is a contraction.
100 Yan Zhao
Clearly,
(cid:5)(cid:9) (cid:5)
(cid:5) t (cid:5)
(cid:3)(Γx)(t)−(Γy)(t)(cid:3) ≤ (cid:3)f(t,Bx(t))−f (t,By(t))(cid:3) +(cid:5)(cid:5) AT (t−s)[f(s,Bx(s))−f (s,By(s))]ds(cid:5)(cid:5)
G G
(cid:5)(cid:9) −∞ (cid:5) G
(cid:5) t (cid:5)
+(cid:5)(cid:5) T (t−s)[g(s,Cx(s))−g(s,Cy(s))]ds(cid:5)(cid:5)
(cid:5)(cid:9)−∞ G (cid:5)
(cid:5) t (cid:5)
+(cid:5)(cid:5) T (t−s)[σ(s,Dx(s))−σ(s,Dy(s))]dW (s)(cid:5)(cid:5) .
−∞ G
Since (a+b+c+d)2 ≤ 4a2+4b2+4c2+4d2, we have
E(cid:3)(Γx)(t)−(Γy)(t)(cid:3)2 ≤ 4E(cid:3)f (t,Bx(t))−f(t,By(t))(cid:3)2
G (cid:5)(cid:9) G (cid:5)
(cid:5) t (cid:5)2
+4E(cid:5)(cid:5) AT (t−s)[f(s,Bx(s))−f (s,By(s))]ds(cid:5)(cid:5)
(cid:5)(cid:9)−∞ (cid:5) G
(cid:5) t (cid:5)2
+4E(cid:5)(cid:5) T (t−s)[g(s,Cx(s))−g(s,Cy(s))]ds(cid:5)(cid:5)
(cid:5)(cid:9)−∞ G (cid:5)
(cid:5) t (cid:5)2
+4E(cid:5)(cid:5) T (t−s)[σ(s,Dx(s))−σ(s,Dy(s))]dW (s)(cid:5)(cid:5) .
−∞ G
We first evaluate the first term of the right-hand side as follows:
E(cid:3)f(t,Bx(t))−f (t,By(t))(cid:3)2 ≤ h2L (cid:3)x−y(cid:3)2 .
f G,∞
As to the second term, it follows from (H1) that
(cid:5) (cid:5)
(cid:5)(cid:4) (cid:5)2
E(cid:5) t AT (t−s)[f (s,Bx(s))−f (s,By(s))]ds(cid:5)
−(cid:14)∞ (cid:15)
(cid:4) 2
≤ E t e−ω(t−s)γ(t−s)(cid:3)f (s,Bx(s))−f(s,By(s))(cid:3)ds
(cid:2) −∞ (cid:3)
(cid:4) 2
≤ t e−ω(t−s)γ(t−s)ds supE(cid:3)f(s,Bx(s))−f(s,By(s))(cid:3)2
−∞
t∈R
≤ ρ2h2L (cid:3)x−y(cid:3)2 .
f G,∞
As to the third term, we obtain that
(cid:5) (cid:5)
(cid:5)(cid:4) (cid:5)2
E(cid:5) t T (t−s)[g(s,Cx(s))−g(s,Cy(s))]ds(cid:5)
−∞(cid:5) (cid:5)
(cid:5)(cid:4) (cid:5)2
≤ M2E(cid:5) t e−δ(t−s)[g(s,Cx(s))−g(s,Cy(s))]ds(cid:5)
−∞
≤ h2 1 M2L (cid:3)x−y(cid:3)2 .
δ2 g G,∞
As far as the last term is concerned, we use an estimate on the Itoˆ integral established
in [21, Proposition 1.9] and obtain:
(cid:5) (cid:5)
(cid:5)(cid:4) (cid:5)2
E(cid:5) t T (t−s)[σ(s,Dx(s))−σ(s,Dy(s))]dW (s)(cid:5)
−∞ (cid:2) (cid:3)
(cid:4) (cid:4)
≤ M2E t e−δ(t−s)ds t e−δ(t−s)(cid:3)σ(s,Dx(s))−σ(s,Dy(s))(cid:3)2ds
−∞ −∞
≤ h2 1 M2L (cid:3)x−y(cid:3)2 .
δ2 σ G,∞