Electronic Journal of Differential Equations, Vol. 2010(2010), No. 124, pp. 1–25.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SQUARE-MEAN ALMOST PERIODIC MILD
SOLUTIONS TO SOME NONAUTONOMOUS STOCHASTIC
SECOND-ORDER DIFFERENTIAL EQUATIONS
PAUL H. BEZANDRY, TOKA DIAGANA
Abstract. In this paper we use the well-known Schauder fixed point principle
to obtain the existence of square-mean almost periodic solutions to some classes
of nonautonomous second order stochastic differential equations on a Hilbert
space.
1. Introduction
Let B be a Banach space. In Goldstein and N’Guérékata [30], the existence of
almost automorphic solutions to the evolution
u0 (t) = Au(t) + F (t, u(t)),
t∈R
where A : D(A) ⊂ B → B is a closed linear operator on a Banach space B which
generates an exponentially stable C0 -semigroup T = (T (t))t≥0 and the function
F : R × B → B is given by F (t, u) = P (t)Q(u) with P , Q being some appropriate continuous functions satisfying some additional conditions, was established.
The main tools used in [30] are fractional powers of operators and the fixed-point
theorem of Schauder.
Recently Diagana [20] generalized the results of [30] to the nonautonomous case
by obtaining the existence of almost automorphic mild solutions to
u0 (t) = A(t)u(t) + f (t, u(t)),
t∈R
(1.1)
where A(t) for t ∈ R is a family of closed linear operators with domains D(A(t))
satisfying Acquistapace-Terreni conditions, and the function f : R × B 7→ B is
almost automorphic in t ∈ R uniformly in the second variable. For that, Diagana
utilized similar techniques as in [30], dichotomy tools, and the Schauder fixed point
theorem.
Let H be a Hilbert space. Motivated by the above mentioned papers, the present
paper is aimed at utilizing Schauder fixed point theorem to study the existence of
p-th mean almost periodic solutions to the nonautonomous stochastic differential
equations
dX(t) = A(t)X(t) dt + F1 (t, X(t)) dt + F2 (t, X(t)) dW(t),
2000 Mathematics Subject Classification. 34K14, 60H10, 35B15, 34F05.
Key words and phrases. Stochastic differential equation; Wiener process.
c
2010
Texas State University - San Marcos.
Submitted May 4, 2010. Published August 30, 2010.
1
t ∈ R,
(1.2)
2
P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
where (A(t))t∈R is a family of densely defined closed linear operators satisfying
Acquistapace and Terreni conditions, the functions F1 : R × Lp (Ω, H) → Lp (Ω, H)
and F2 : R×Lp (Ω, H) → Lp (Ω, L02 ) are jointly continuous satisfying some additional
conditions, and W is a Wiener process.
Then, we utilize our main results to study the existence of square-mean almost
periodic solutions to the second order stochastic differential equations
dX 0 (ω, t) + a(t) dX(ω, t)
h
i
= − b(t) AX(ω, t) + f1 (t, X(ω, t)) dt
(1.3)
+f2 (t, X(ω, t)) dW(ω, t),
for all ω ∈ Ω and t ∈ R, where A : D(A) ⊂ H → H is a self-adjoint linear operator
whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < · · · < λn → ∞
with each eigenvalue having a finite multiplicity γj equals to the multiplicity of
the corresponding eigenspace, the functions a, b : R → (0, ∞) are almost periodic
functions, and the function fi (i = 1, 2) : R × L2 (Ω, H) → L2 (Ω, H) are jointly continuous functions satisfying some additional conditions and W is a one dimensional
Brownian motion.
It should be mentioned the existence of almost periodic to (1.2) in the case
when A(t) is periodic, that is, A(t + T ) = A(t) for each t ∈ R for some T > 0
was established by Da Prato and Tudor in [17]. In the paper by Bezandry and
Diagana [9], upon assuming that the operators A(t) satisfy Acquistapace-Terreni
conditions and that Fi (i = 1, 2, 3,) satisfy Lipschitz conditions, the Banach fixed
point principle was utilized to obtain the existence of a square-mean almost periodic
solutions to (1.2). In this paper is goes back to utilizing Schauder fixed theorem
to establish the existence of p-th mean almost periodic solutions to (1.2). Next, we
make extensive use of those abstract results to deal with the existence of squaremean almost periodic solutions to the second-order stochastic differential equations
formulated in (1.3).
2. Preliminaries
In this section, A : D(A) ⊂ H → H stands for a self-adjoint linear operator
whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < · · · < λn → ∞
with each eigenvalue having a finite multiplicity γj equals to the multiplicity of the
corresponding eigenspace.
Let {ekj } be a (complete) orthonormal sequence of eigenvectors associated with
the eigenvalues {λj }j≥1 . Clearly, for each
∞
n
o
X
u ∈ D(A) := x ∈ H :
λ2j kEj xk2 < ∞ ,
j=1
Ax =
∞
X
j=1
γj
λj
X
hx, ekj iekj =
∞
X
λ j Ej x
j=1
k=1
Pγj
where Ej x = k=1
hx, ekj iekj .
Note that {Ej }j≥1 is a sequence of orthogonal projections on H. Moreover, each
x ∈ H can written as follows:
∞
X
x=
Ej x.
j=1
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3
It should also be mentioned that the operator −A is the infinitesimal generator
of an analytic semigroup {T (t)}t≥0 , which is explicitly expressed in terms of those
orthogonal projections Ej by, for all x ∈ H,
T (t)x =
∞
X
e−λj t Ej x.
j=1
r
In addition, the fractional powers A (r ≥ 0) of A exist and are given by
∞
n
o
X
2
D(Ar ) = x ∈ H :
λ2r
kE
xk
<
∞
j
j
j=1
and
Ar x =
∞
X
λ2r
j Ej x,
∀x ∈ D(Ar ).
j=1
Let B, k · k be a Banach space. If L is a linear operator on the Banach space B,
then D(L), ρ(L), σ(L), N (L), N (L), and R(L) stand respectively for the domain,
resolvent, spectrum, null space, and the range of L. also, we set R(λ, L) := (λI −
L)−1 for all λ ∈ ρ(L). If P is a projection, we then set Q = I − P . If B1 , B2
are Banach spaces, then the space B(B1 , B2 ) denotes the collection of all bounded
linear operators from B1 into B2 equipped with its natural topology. This is simply
denoted by B(B1 ) when B1 = B2 .
2.1. Evolution Families. Let B be a Banach space equipped with the norm k · k.
The family of closed linear operators A(t) for t ∈ R on B with domain D(A(t))
(possibly not densely defined) is said to
satisfy Acquistapace-Terreni conditions if:
π
there exist constants ω ≥ 0, θ ∈ 2 , π , K, L ≥ 0 and µ, ν ∈ (0, 1] with µ + ν > 1
such that
K
Sθ ∪ {0} ⊂ ρ A(t) − ω 3 λ, kR λ, A(t) − ω k ≤
(2.1)
1 + |λ|
and
i
h k A(t) − ω R λ, A(t) − ω R ω, A(t) − R ω, A(s) k ≤ L|t − s|µ |λ|−ν (2.2)
for t, s ∈ R, λ ∈ Sθ := λ ∈ C \ {0} : | arg λ| ≤ θ .
It should mentioned that the conditions (2.1) and (2.2) were introduced in the
literature by Acquistapace and Terreni in [2, 3] for ω = 0. Among other things,
it ensures that there exists a unique evolution family U = U (t, s) on B associated
with A(t) satisfying
(a)
(b)
(c)
(d)
U (t, s)U (s, r) = U (t, r);
U (t, t) = I for t ≥ s ≥ r in R;
(t, s) 7→ U (t, s) ∈ B(B) is continuous for t > s;
U (·, s) ∈ C 1 ((s, ∞), B(B)), ∂U
∂t (t, s) = A(t)U (t, s) and
kA(t)k U (t, s)k ≤ K (t − s)−k
(2.3)
for 0 < t − s ≤ 1, k = 0, 1; and
(e) ∂s+ U (t, s)x = −U (t, s)A(s)x for t > s and x ∈ D(A(s)) with A(s)x ∈
D(A(s)).
4
P. H. BEZANDRY, T. DIAGANA
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It should also be mentioned that the above-mentioned properties were mainly
established in [1, Theorem 2.3] and [50, Theorem 2.1], see also [3, 49]. In that case
we say that A(·) generates the evolution family U (·, ·).
One says that an evolution family U has an exponential dichotomy (or is hyperbolic) if there are projections P (t) (t ∈ R) that are uniformly bounded and strongly
continuous in t and constants δ > 0 and N ≥ 1 such that
(f) U (t, s)P (s) = P (t)U (t, s);
(g) the restriction UQ (t, s) : Q(s)B → Q(t)B of U (t, s) is invertible (we then
eQ (s, t) := UQ (t, s)−1 ); and
set U
eQ (s, t)Q(t)k ≤ N e−δ(t−s) for t ≥ s and
(h) kU (t, s)P (s)k ≤ N e−δ(t−s) and kU
t, s ∈ R.
This setting requires some estimates related to U (t, s). For that, we introduce the
interpolation spaces for A(t). We refer the reader to the following excellent books
[29], and [38] for proofs and further information on theses interpolation spaces.
Let A be a sectorial operator on B (for that, in (2.1)-(2.2), replace A(t) with A)
and let α ∈ (0, 1). Define the real interpolation space
n
o
A
α
BA
α := x ∈ B : kxkα := sup kr (A − ω)R(r, A − ω)xk < ∞ ,
r>0
which, by the way, is a Banach space when endowed with the norm k · kA
α . For
convenience we further write
BA
0 := B,
kxkA
0 := kxk,
BA
1 := D(A)
and
kxkA
1 := k(ω − A)xk.
Moreover, let B̂A := D(A) of B. In particular, we have the following continuous
embedding
α
A
A
D(A) ,→ BA
(2.4)
β ,→ D((ω − A) ) ,→ Bα ,→ B̂ ,→ B,
for all 0 < α < β < 1, where the fractional powers are defined in the usual way.
In general, D(A) is not dense in the spaces BA
α and B. However, we have the
following continuous injection
k·kA
α
BA
β → D(A)
(2.5)
for 0 < α < β < 1.
Given the family of linear operators A(t) for t ∈ R, satisfying (2.1)-(2.2), we set
Btα := BA(t)
,
α
B̂t := B̂A(t)
for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms. Then the embedding in
(2.4) holds with constants independent of t ∈ R. These interpolation spaces are of
class Jα [38, Definition 1.1.1 ] and hence there is a constant c(α) such that
kyktα ≤ c(α)kyk1−α kA(t)ykα ,
y ∈ D(A(t)).
(2.6)
We have the following fundamental estimates for the evolution family U (t, s).
Proposition 2.1. [7] Suppose the evolution family U = U (t, s) has exponential
dichotomy. For x ∈ B, 0 ≤ α ≤ 1 and t > s, the following hold:
(i) There is a constant c(α), such that
δ
kU (t, s)P (s)xktα ≤ c(α)e− 2 (t−s) (t − s)−α kxk.
(2.7)
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5
(ii) There is a constant m(α), such that
eQ (s, t)Q(t)xksα ≤ m(α)e−δ(t−s) kxk.
kU
(2.8)
We need the following technical lemma.
Lemma 2.2 ([20, 21, Diagana]). For each x ∈ B, suppose that the family of operators A(t) (t ∈ R) satisfy Acquistapce-Terreni conditions, assumption (H.2) holds,
and that there exist real numbers µ, α, β such that 0 ≤ µ < α < β < 1 with
2α > µ + 1. Then there is a constant r(µ, α) > 0 such that
δ
kA(t)U (t, s)xkα ≤ r(µ, α)e− 4 (t−s) (t − s)−α kxk.
(2.9)
for all t > s.
Proof. Let x ∈ B. First of all, note that kA(t)U (t, s)kB(B,Bα ) ≤ K(t − s)−(1−α) for
all t, s such that 0 < t − s ≤ 1 and α ∈ [0, 1]. Letting t − s ≥ 1 and using (H2) and
the above-mentioned approximate, we obtain
kA(t)U (t, s)xkα = kA(t)U (t, t − 1)U (t − 1, s)xkα
≤ kA(t)U (t, t − 1)kB(B,Bα ) kU (t − 1, s)xk
≤ M Keδ e−δ(t−s) kxk
= K1 e−δ(t−s) kxk
δ
3δ
= K1 e− 4 (t−s) (t − s)α (t − s)−α e− 4 (t−s) kxk.
3δ
Now since e− 4 (t−s) (t − s)α → 0 as t → ∞ it follows that there exists c4 (α) > 0
such that
δ
kA(t)U (t, s)xkα ≤ c4 (α)(t − s)−α e− 4 (t−s) kxk.
Now, let 0 < t − s ≤ 1. Using (2.7) and the fact 2α > µ + 1, we obtain
t+s
t+s
)U (
, s)xkα
2
2
t+s
t+s
)kB(B,Bα ) kU (
, s)xk
≤ kA(t)U (t,
2
2
t+s
t+s
≤ k1 kA(t)U (t,
)kB(B,Bα ) kU (
, s)xkµ
2
2
t − s −µ δ
t − s α−1
c(µ)
e− 4 (t−s) kxk
≤ k1 K
2
2
δ
≤ c5 (α, µ)(t − s)α−1−µ e− 4 (t−s) kxk
kA(t)U (t, s)xkα = kA(t)U (t,
δ
≤ c5 (α, µ)(t − s)−α e− 4 (t−s) kxk.
Therefore there exists r(α, µ) > 0 such that
δ
kA(t)U (t, s)xkα ≤ r(α, µ)(t − s)−α e− 4 (t−s) kxk
for all t, s ∈ R with t ≥ s.
It should be mentioned that if U (t, s) is exponentially stable, then P (t) = I and
Q(t) = I − P (t) = 0 for all t ∈ R. In that case, (2.7) still holds and be rewritten as
follows: for all x ∈ B,
δ
kU (t, s)xktα ≤ c(α)e− 2 (t−s) (t − s)−α kxk.
(2.10)
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P. H. BEZANDRY, T. DIAGANA
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2.2. Wiener process and P -th mean almost periodic stochastic processes.
For details of this subsection, we refer the reader to Bezandry and Diagana [9],
Corduneanu [14], and the references therein. Throughout this paper, H and K will
denote real separable Hilbert spaces with respective norms k · k and k · kK . Let
(Ω, F, P) be a complete probability space. We denote by L2 (K, H) the space of
all Hilbert-Schmidt operators acting between K and H equipped with the HilbertSchmidt norm k · k2 .
For a symmetric nonnegative operator Q ∈ L2 (K, H) with finite trace we assume
that {W(t), t ∈ R} is a Q-Wiener process defined on (Ω, F, P) and with values
in K. Recall that W can obtained as follows: let {Wi (t), t ∈ R}, i = 1, 2, be
independent K-valued Q-Wiener processes, then
(
W1 (t)
if t ≥ 0
W(t) =
W2 (−t) if t ≤ 0
is Q-Wiener process with R as time parameter. We let Ft = σ{W(s), s ≤ t}.
Let p ≥ 2. The collection of all strongly measurable, p-th integrable H-valued
random variables, denoted by Lp (Ω, H), is a Banach space equipped with norm
kXkLp (Ω,H) = (EkXkp )1/p ,
where the expectation E is defined by
Z
E[g] =
g(ω)dP(ω) .
Ω
1
2
Let K0 = Q K and
L02
= L2 (K0 , H) with respect to the norm
1
kΦk2L0 = kΦ Q 2 k22 = Tr(ΦQΦ∗ ) .
2
Definition 2.3. A stochastic process X : R → Lp (Ω; B) is said to be continuous
whenever
lim EkX(t) − X(s)kp = 0.
t→s
Definition 2.4. A stochastic process X : R → Lp (Ω; B) is said to be stochastically
bounded whenever
n
o
lim sup P kX(t)k > N = 0.
N →∞ t∈R
Definition 2.5. A continuous stochastic process X : R → Lp (Ω; B) is said to be
p-th 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
sup EkX(t + τ ) − X(t)kp < ε .
(2.11)
t∈R
A continuous stochastic process X, which is 2-nd mean almost periodic will be
called square-mean almost periodic.
Like for classical almost periodic functions, the number τ will be called an εtranslation of X.
The collection of all p-th mean almost periodic stochastic processes X : R →
Lp (Ω; B) will be denoted by AP (R; Lp (Ω; B)).
The next lemma provides with some properties of p-th mean almost periodic
processes.
Lemma 2.6. If X belongs to AP (R; Lp (Ω; B)), then
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SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
7
(i) the mapping t → EkX(t)kp is uniformly continuous;
(ii) there exists a constant M > 0 such that EkX(t)kp ≤ M , for each t ∈ R;
(iii) X is stochastically bounded.
Lemma 2.7. AP (R; Lp (Ω; B)) ⊂ BU C(R; Lp (Ω; B)) is a closed subspace.
In view of Lemma 2.7, it follows that the space AP (R; Lp (Ω; B)) of p-th mean
almost periodic processes equipped
with the sup norm k · k∞ is a Banach space.
Let B1 , k · k1 and B2 , k · k2 be Banach spaces and let Lp (Ω; B1 ) and Lp (Ω; B2 )
be their corresponding Lp -spaces, respectively.
Definition 2.8. A function F : R × Lp (Ω; B1 ) → Lp (Ω; B2 )), (t, Y ) 7→ F (t, Y ),
which is jointly continuous, is said to be p-th mean almost periodic in t ∈ R
uniformly in Y ∈ K where K ⊂ Lp (Ω; B1 ) is a compact if for any ε > 0, there
exists lε (K) > 0 such that any interval of length lε (K) contains at least a number
τ for which
sup EkF (t + τ, Y ) − F (t, Y )kp2 < ε
t∈R
for each stochastic process Y : R → K.
We have the following composition result.
Theorem 2.9. Let F : R × Lp (Ω; B1 ) → Lp (Ω; B2 ), (t, Y ) 7→ F (t, Y ) be a p-th
mean almost periodic process in t ∈ R uniformly in Y ∈ K, where K ⊂ Lp (Ω; B1 )
is any compact subset. Suppose that F (t, ·) is uniformly continuous on bounded
subsets K 0 ⊂ Lp (Ω; B1 ) in the following sense: for all ε > 0 there exists δε > 0 such
that X, Y ∈ K 0 and EkX − Y kp1 < δε , then
EkF (t, Y ) − F (t, Z)kp2 < ε,
∀t ∈ R.
Then for any p-th mean almost periodic process Φ : R → Lp (Ω; B1 ), the stochastic
process t 7→ F (t, Φ(t)) is p-th mean almost periodic.
3. Main Results
In this section, we study the existence of p-th mean almost periodic solutions
to the class of nonautonomous stochastic differential equations of type (1.2) where
(A(t))t∈R is a family of closed linear operators on Lp (Ω; H) satisfying (2.1)-(2.2),
and the functions F1 : R × Lp (Ω, H) → Lp (Ω, H), F2 : R × Lp (Ω, H) → Lp (Ω, L02 )
are p-th mean almost periodic in t ∈ R uniformly in the second variable, and
W is Q-Wiener process taking its values in K with the real number line as time
parameter.
Our method for investigating the existence and uniqueness of a p-th mean almost
periodic solution to (1.2) consists of making extensive use of ideas and techniques
utilized in [30], [21], and the Schauder fixed-point theorem.
To study the existence of p-th mean almost periodic solutions to (1.2), we suppose
that the following assumptions hold:
(H1) The injection Hα ,→ H is compact.
(H2) The family of operators A(t) satisfy Acquistapace-Terreni conditions and
the evolution family U (t, s) associated with A(t) is exponentially stable;
that is, there exist constant M , δ > 0 such that
kU (t, s)k ≤ M e−δ(t−s)
for all t ≥ s.
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P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
(H3) Let µ, α, β be real numbers such that 0 ≤ µ < α < β < 1 with 2α > µ + 1.
Moreover, Htα = Hα and Htβ = Hβ for all t ∈ R, with uniform equivalent
norms.
(H4) R(ζ, A(·)) ∈ AP (R, Lp (Ω; H)).
(H5) The function F1 : R × Lp (Ω, H) → Lp (Ω, H) is p-th mean almost periodic
in the first variable uniformly in the second variable. Furthermore, X →
F1 (t, X) is uniformly continuous on any bounded subset O of Lp (Ω, H) for
each t ∈ R. Finally,
sup EkF1 (t, X)kp ≤ M1 kXk∞
t∈R
+
where M1 : R → R+ is a continuous function satisfying
M1 (r)
= 0.
r
(H6) The function F2 : R × Lp (Ω, H) → Lp (Ω, L02 ) is p-th mean almost periodic
in the first variable uniformly in the second variable. Furthermore, X →
F2 (t, X) is uniformly continuous on any bounded subset O0 of Lp (Ω, H) for
each t ∈ R. Finally,
sup EkF2 (t, X)kp ≤ M2 kXk∞
lim
r→∞
t∈R
+
where M2 : R → R+ is a continuous function satisfying lim M2 (r)/r =
r→∞
0.
In this section, Γ1 and Γ2 stand respectively for the nonlinear integral operators
defined by
Z t
(Γ1 X)(t) :=
U (t, s)F1 (s, X(s)) ds,
−∞
Z
t
(Γ2 X)(t) :=
U (t, s)F2 (s, X(s)) dW(s) .
−∞
In addition to the above-mentioned assumptions, we assume that α ∈ 0, 12 −
p > 2 and α ∈ 0, 12 if p = 2.
1
p
if
Lemma 3.1. Under assumptions (H2)–(H6), the mappings Γi : BC(R, Lp (Ω, H))
→ BC(R, Lp (Ω, Hα )) (i = 1, 2) are well defined and continuous.
Proof. We first show that Γi BC R, Lp (Ω, H) ⊂ BC R, Lp (Ω, Hα ) (i = 1, 2).
Let us start with Γ1 X. Using (2.10) it follows that for all X ∈ BC(R, Lp (Ω, H)),
EkΓ1 X(t)kpα
hZ t
ip
δ
≤E
c(α)(t − s)−α e− 2 (t−s) kF1 (s, X(s))k ds
−∞
≤ c(α)p
Z
t
−∞
p
δ
(t − s)− p−1 α e− 2 (t−s) ds
p−1 Z
t
δ
e− 2 (t−s) EkF1 (s, X(s))kp ds
−∞
p
α 2 p−1
2 1− p−1
p
≤ c(α)p Γ 1 −
α
M1 kXk∞
p−1
δ
δ
p−1 2 p(1−α)
p
≤ c(α)p Γ 1 −
α
M1 kXk∞ ,
p−1
δ
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SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
9
and hence
kΓ1 Xkpα,∞ := sup EkΓ1 X(t)kpα ≤ l(α, δ, p)M1 kXk∞ ,
t∈R
p−1 2 p(1−α)
p
where l(α, δ, p) = c(α)p Γ 1 − p−1
α
.
δ
As to Γ2 X, we proceed into two steps. For p > 2, we need the following estimates.
Lemma 3.2. Let p > 2, 0 < α < 1, α + p1 < ξ < 1/2, and Ψ : Ω × R → L02 be an
(Ft )-adapted measurable stochastic process such that
sup EkΨ(t)kpL0 < ∞ .
2
t∈R
Then
Rt
(i) Ek −∞ (t − s)−ξ U (t, s)Ψ(s) dW(s)kp ≤ s(Γ, ξ, δ, p) supt∈R EkΨ(t)kpL0 ;
2
Rt
(ii) Ek −∞ U (t, s)Ψ(s) dW(s)kpα ≤ k(Γ, α, ξ, δ, p) supt∈R EkΨ(t)kpL0
2
where s(Γ, ξ, δ, p) and k(Γ, α, ξ, δ, p) are positive constants with Γ a classical Gamma
function.
Proof. (i) A direct application of a Proposition due to De Prato and Zabczyk [18]
and Holder’s inequality allows us to write
Z t
Ek
(t − σ)−ξ U (t, σ)Ψ(σ) dW(σ)kp
−∞
hZ
≤ Cp E
t
(t − σ)−2ξ kU (t, σ)Ψ(σ)k2 dσ
ip/2
−∞
t
hZ
≤ Cp N E
p
≤ Cp N p
Z
−∞
t
(t − σ)−2ξ e−2δ(t−σ) kΨ(σ)k2L0 dσ
ip/2
2
(t − σ)−2ξ e−2δ(t−σ) dσ
p−1 Z
−∞
t
−∞
e−2δ(t−σ) EkΨ(σ)kpL0 dσ
2
p−2
1
2pξ
2pξ
2
)(2δ) p−2 −1
sup EkΨ(t)kpL0
≤ Cp N p Γ(1 −
2
p−2
2δ t∈R
p
≤ s(Γ, ξ, δ, p) sup EkΨ(t)kL0 .
2
t∈R
To prove (ii), we use the factorization method of the stochastic convolution integral.
Z
t
U (t, s)Ψ(s) dW(s) =
−∞
where
Z
sin πξ
(Rξ SΨ )(t)
π
a.s.
t
(Rξ SΨ )(t) =
(t − s)ξ−1 U (t, s)SΨ (s) ds
−∞
with
Z
s
SΨ (s) =
(s − σ)−ξ U (s, σ)Ψ(σ) dW(σ) ,
−∞
and ξ satisfying α + p1 < ξ < 1/2. We can now evaluate
Z t
Ek
U (t, s)Ψ(s) dW(s)kpα
−∞
(3.1)
10
P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
Z
ip
sin(πξ) p h t
E
(t − s)−ξ kU (t, s)SΨ (s)kα ds
≤
π
−∞
Z
ip
p h t
sin(πξ)
p
≤ M (α)
E
(t − s)ξ−α−1 e−δ(t−s) kSΨ (s)kα ds
π
−∞
Z t
p−1
p
sin(πξ)
p
(t − s) p−1 (ξ−α−1) e−δ(t−s) ds
×
≤ M (α)p π
−∞
Z t
×
e−δ(t−s) EkSΨ (s)kp ds
−∞
≤ r(Γ, α, ξ, δ, p) sup EkSΨ (s)kp .
s∈R
On the other hand, it follows from part (i) that
EkSΨ (t)kp ≤ s(Γ, ξ, δ, p) sup EkΨ(t)kpL0 .
(3.2)
2
t∈R
Thus,
Z
t
Ek
U (t, s)Ψ(s) dW(s)kpα
−∞
≤ r(Γ, α, ξ, δ, p)s(Γ, ξ, δ, p) sup EkΨ(t)kpL0
≤
t∈R
k(Γ, α, ξ, δ, p) sup EkΨ(t)kpL0
2
t∈R
2
.
We now use the estimates obtained in Lemma 3.2 (ii) to obtain
EkΓ2 X(t)kpα ≤ k(α, ξ, δ, p) sup EkF2 (s, X(s))kpL0
2
t∈R
≤ k(α, ξ, δ, p)M2 kXk∞ ,
and hence
kΓ2 Xkpα,∞ ≤ k(α, ξ, δ, p)M2 kXk∞ ,
where k(α, ξ, δ, p) is a positive constant. For p = 2, we have
Z t
EkΓ2 X(t)k2α = Ek
U (t, s)F2 (s, X(s)) dW(s)k2α
−∞
Z
t
(t − s)−2α e−δ(t−s) EkF2 (s, X(s))k2L0
2
1−2α
2
≤ c(α) Γ 1 − 2α δ
M2 kXk∞ ,
2
≤ c(α)
−∞
and hence
kΓ2 Xk2α,∞ ≤ s(α, δ)M2 kXk∞ ,
where s(α, δ) = c(α)2 Γ 1 − 2α δ 1−2α .
For the continuity, let X n ∈ AP (R; Lp (Ω, H)) be a sequence which converges to
some X ∈ AP (R; Lp (Ω, H)); that is, kX n − Xk∞ → 0 as n → ∞. It follows from
the estimates in Proposition 2.1 that
Z t
Ek
U (t, s)[F1 (s, X n (s)) − F1 (s, X(s))] dskpα
−∞
EJDE-2010/124
SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
hZ
≤E
t
δ
c(α)(t − s)−α e− 2 (t−s) kF1 (s, X n (s)) − F1 (s, X(s))k ds
11
ip
.
−∞
Now, using the continuity of F1 and the Lebesgue Dominated Convergence Theorem
we obtain that
Z t
Ek
U (t, s)[F1 (s, X n (s)) − F1 (s, X(s))] dskpα → 0 as n → ∞ .
−∞
Therefore,
kΓ1 X n − Γ1 Xk∞,α → 0 as n → ∞.
For the term containing the Wiener process W, we use the estimates in Lemma 3.2
to obtain
Z t
Ek
U (t, s)[F2 (s, X n (s)) − F2 (s, X(s))] dW(s)kpα
−∞
≤ k(α, ξ, δ, p) sup EkF2 (t, X n (t)) − F2 (t, X(t))kp
t∈R
for p > 2 and
Z t
Ek
U (t, s)[F2 (s, X n (s)) − F2 (s, X(s))] dW(s)k2α
−∞
≤ n(α)2
Z
t
(t − s)−2α e−δ(t−s) EkF2 (s, X(s)n ) − F2 (s, X(s))k2 ds
−∞
for p = 2.
Now, using the continuity of G and the Lebesgue Dominated Convergence Theorem we obtain that
Z t
Ek
U (t, s)[F2 (s, X n (s)) − F2 (s, X(s))] dW(s)kpα → 0 as n → ∞ .
−∞
Therefore,
kΓ2 X n − Γ2 Xk∞,α → 0
as n → ∞.
Lemma 3.3. Under assumptions (H2)–(H6), the integral operator Γi (i = 1, 2)
maps AP R, Lp (Ω, H) into itself.
Proof. Let us first show that Γ1 X(·) is p-th mean almost periodic et let f1 (t) =
F1 (t, X(t)). Indeed, assuming that X is p-th mean almost periodic and using
assumption (H5), Theorem 2.9, and [39, Proposition 4.4], given ε > 0, one can find
lε > 0 such that any interval of length lε contains at least τ with the property that
δ
kU (t + τ, s + τ ) − U (t, s)k ≤ εe− 2 (t−s)
for all t − s ≥ ε, and
Ekf1 (σ + τ ) − f1 (σ)kp < η
for each σ ∈ R, where η(ε) → 0 as ε → 0. Moreover, it follows from Lemma 2.6 (ii)
that there exists a positive constant K1 such that
sup Ekf1 (σ)kp ≤ K1 .
σ∈R
Now, using assumption (H2) and Holder’s inequality, we obtain
EkΓ1 X(t + τ ) − Γ1 X(t)kp
12
P. H. BEZANDRY, T. DIAGANA
hZ
≤ 3p−1 E
EJDE-2010/124
∞
kU (t + τ, t + τ − s)kkf1 (t + τ − s) − f1 (t − s)k ds
ip
0
hZ ∞
ip
p−1
+3
E
kU (t + τ, t + τ − s) − U (t, t − s)kkf1 (t − s)k ds
ε
hZ ε
ip
+ 3p−1 E
kU (t + τ, t + τ − s) − U (t, t − s)kkf1 (t − s)k ds
0
hZ ∞
ip
p−1
p
≤3 M E
e−δs kf1 (t + τ − s) − f1 (t − s)k ds
0
ip
hZ ∞ δ
hZ ε
ip
p−1 p
+3 ε E
e− 2 s kf1 (t − s)k ds + 3p−1 M p E
2e−δs kf1 (t − s)k ds
ε
0
Z ∞
p−1 Z ∞
e−δs ds
≤ 3p−1 M p
e−δs Ekf1 (t + τ − s) − f1 (t − s)kp ds
0
0
Z ∞
p−1 Z ∞ δps
e−δs ds
e− 2 Ekf1 (t − s)kp ds
+ 3p−1 εp
0
ε
Z ε
p−1 Z ε δps
+ 6p−1 M p
e−δs ds
e− 2 Ekf1 (t − s)kp ds
0
0
Z ∞
p
p−1
p
−δs
≤3 M
e
ds sup Ekf1 (t + τ − s) − f1 (t − s)kp
s∈R
0
Z ∞
p
+ 3p−1 εp
e−δs ds sup Ekf1 (t − s)kp
s∈R
ε
Z ε
p
+ 6p−1 M p
e−δs ds sup Ekf1 (t − s)kp
s∈R
0
1
1
≤ 3p−1 M p p η + 3p−1 M p K1 p εp + 6p−1 M p εp K1 εp .
δ
δ
As for Γ2 X(·), we split the proof in two cases: p > 2 and p = 2. To this end,
we let f2 (t) = F2 (t, X(t)). Let us start with the case where p > 2. Assuming that
X is p-th mean almost periodic and using assumption (H6), Theorem 2.9, and [39,
Proposition 4.4], given ε > 0, one can find lε > 0 such that any interval of length
lε contains at least τ with the property that
δ
kU (t + τ, s + τ ) − U (t, s)k ≤ εe− 2 (t−s)
for all t − s ≥ ε, and
Ekf2 (σ + τ ) − f2 (σ)kp < η
for each σ ∈ R, where η(ε) → 0 as ε → 0.
Moreover, it follows from Lemma 2.6 (ii) that there exists a positive constant
K2 such that
sup Ekf2 (σ)kp ≤ K2 .
σ∈R
Now
Ekf2 (t + τ ) − f2 (t)kp
Z ∞
h
i
≤ 3p−1 E
U (t + τ, t + τ − s) f2 (t + τ − s) − f2 (t − s) dW(s)kp
0
Z ∞ h
i
+ 3p−1 E
U (t + τ, t + τ − s) − U (t, t − s) f2 (t − s) dW(s)kp
ε
EJDE-2010/124
SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
Z
+ 3p−1 E
ε
h
13
i
U (t + τ, t + τ − s) − U (t, t − s) f2 (t − s) dW(s)kp .
0
We then have
EkΓ2 X(t + τ ) − Γ2 X(t)kp
hZ ∞
ip/2
≤ 3p−1 Cp E
kU (t + τ, t + τ − s)k2 kf2 (t + τ − s) − f2 (t − s)k2L0 ds
2
0
hZ ∞
ip/2
+ 3p−1 Cp E
kU (t + τ, t + τ − s) − U (t, t − s)k2 kf2 (t − s)k2L0 ds
2
ε
hZ ε
ip/2
+ 3p−1 Cp E
kU (t + τ, t + τ − s) − U (t, t − s)k2 kf2 (t − s)k2L0 ds
2
0
ip/2
hZ ∞
≤ 3p−1 Cp M p E
e−2δs kf2 (t + τ − s) − f2 (t − s)k2L0 ds
2
0
hZ ∞
ip/2
+ 3p−1 Cp εp E
e−δs kf2 (t − s)k2L0 ds
2
ε
hZ ε
ip/2
+ 3p−1 2p/2 Cp E
e−2δs kf2 (t − s)k2L0 ds
2
0
≤ 3p−1 Cp M p
∞
Z
pδs
e− p−2 ds
0
+ 3p−1 Cp εp
∞
Z
p−2
2
e−
pδs
e− 2(p−2) ds
p−2
Z
2
ε
∞
ε
Z
pδs
4
pδs
e− p−2 ds
∞
Ekf2 (t − s)kpL0 ds
e−
pδs
2
pδs
∞
p−2
Z
2
0
Ekf2 (t − s)kpL0 ds
2
0
e− p−2 ds
2
ε
p−2
Z
2
0
+ 3p−1 Cp εp K2
e−
ε
+ 3p−1 2p/2 Cp M p
Z
kf2 (t + τ − s) − f2 (t − s)kpL0 ds
2
0
∞
Z
≤ 3p−1 Cp M p η
pδs
2
e−
pδs
2
ds
0
Z
∞
pδs
− 2(p−2)
e
ds
p−2
Z
2
e−
pδs
4
ds
ε
ε
+ 3p−1 2p/2 Cp M p K2
∞
Z
ε
pδs
− p−2
e
ds
p−2
Z
2
0
ε
e−
pδs
2
ds
0
p − 2 p−2 2 ≤ 3 Cp M η
pδ
pδ
2(p − 2) p−2
4
2
+ 3p−1 Cp εp K2
+ 3p−1 2p/2 Cp M p K2 εp .
pδ
pδ
p−1
p
As to the case p = 2, we proceed in the same way an using isometry inequality to
obtain
EkΓ2 X(t + τ ) − Γ2 X(t)k2
Z ∞
≤ 3 M2
e−2δs ds sup Ekf2 (σ + τ ) − f − 2(σ)k2L0
2
σ∈R
0
Z ∞
Z ε
e−δ s ds sup Ekf2 (σ)k2L0 + 6M 2
e−2δs ds sup Ekf2 (σ)k2L0
+ 3ε2
ε
σ∈R
h M2
i
K2
≤3 η
+ε
+ 2εK2 .
2δ
δ
2
0
σ∈R
2
14
P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
Hence, Γ2 X(·) is p-th mean almost periodic.
Let γ ∈ (0, 1] and let
o
n
BC γ R, Lp (Ω, Hα ) = X ∈ BC R, Lp (Ω, Hα ) : kXkα,γ < ∞ ,
where
h
h
kXkα,γ = sup EkX(t)kpα
i1/p
+γ
t∈R
sup
t,s∈R,s6=t
EkX(t) − X(s)kpα
t − s|γ
i1/p
.
Clearly, the space BC γ R, Lp (Ω, Hα ) equipped with the norm k · kα,γ is a Banach
space, which is in fact the Banach space of all bounded continuous Holder functions
from R to Lp (Ω, Hα ) whose Holder exponent is γ.
Lemma 3.4. Under assumptions (H1)–(H6),
the mapping Γ1 defined previously
maps bounded sets of BC R, Lp (Ω, H) into bounded sets of BC γ (R, Lp (Ω, Hα ))
for some 0 < γ < 1.
Proof. Let X ∈ BC(R, Lp (Ω, H)) and let f1 (t) = F1 (t, X(t)) for each t ∈ R. Proceeding as before, we have
EkΓ1 X(t)kpα ≤ cEkΓ1 X(t)kpβ ≤ c · l(β, δ, p)M1 kXk∞ .
Let t1 < t2 . Clearly, we have
Ek(Γ1 X)(t2 ) − (Γ1 X)(t1 )kpα
Z t2
Z
≤ 2p−1 Ek
U (t2 , s)f1 (s) dskpα + 2p−1 Ek
t1
Z t2
t1
[U (t2 , s) − U (t1 , s)]f1 (s) dskpα
−∞
Z t1
Z
t2
∂U (τ, s) dτ f1 (s) dskpα
∂τ
t1
−∞
t1
Z t2
Z t1 Z t2
= 2p−1 Ek
U (t2 , s)f1 (s) dskpα + 2p−1 Ek
A(τ )U (τ, s)f1 (s) dτ dskpα
= 2p−1 Ek
U (t2 , s)f1 (s) dskpα + 2p−1 Ek
−∞
t1
t1
= N1 + N2 .
Clearly,
nZ
N1 ≤ E
t2
op
kU (t2 , s)f1 (s)kα ds
t1
nZ
≤ c(α)p E
t2
δ
(t2 − s)−α e− 2 (t2 −s) kf1 (s)k ds
op
t1
p
M1 kXk
Z
≤ c(α)
≤ c(α)p
M1 kXk
Z
t2
p
− p−1
α − δ2 (t2 −s)
(t2 − s)
t1
t2
e
p−1 Z
t1
p
− p−1
α
(t2 − s)
p−1 t2 − t1
t1
−(p−1)
p
α
(t2 − t1 )p(1−α) .
p−1
Similarly, using estimates in Lemma 2.2
n Z t1 Z t2
op
N2 ≤ E
kA(τ )U (τ, s)f1 (s)kα dτ ds
≤ c(α)p M1 kXk 1 −
−∞
t1
t2
δ
e− 2 (t2 −s) ds
EJDE-2010/124
SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
t1
nZ
≤ r(µ, α)p E
Z
−∞
t1
t2 Z t1
hZ
p
≤ r(µ, α) E
×
t1
op
δ
(τ − s)−α e− 4 (τ −s) kf1 (s)k dτ ds
p
δ
(τ − s)− p−1 α e− 4 (τ −s) ds
p−1
p
−∞
t1
Z
t2
15
− δ4 (τ −s)
e
kf1 (s)kp ds
1/p
dτ
ip
−∞
≤ r(µ, α)p
×
hZ
t1
Z
−∞
t1
t2
Z
t1
≤ r(µ, α)p
Z
δ
e− 4 (t1 −s) Ekf1 (s)kp ds
p
δ
(τ − s)− p−1 α e− 4 (τ −s) ds
−∞
t1
δ
e− 4 (t1 −s) Ekf1 (s)kp ds
p−1
p
dτ
ip
−∞
×
hZ
t2
−α
(τ − t1 )
Z
t1
− δ4 (τ −s)
e
ds
p−1
p
dτ
ip
−∞
t1
≤ r(µ, α)p
Z
t1
− δ4 (t1 −s)
e
Ekf1 (s)kp ds
−∞
×
hZ
t2
(τ − t1 )−α
Z
∞
δ
e− 4 r dr
p−1
p
dτ
ip
τ −t1
t1
2 p
≤ r(µ, α) M1 kXk
(1 − β)−p (t2 − t1 )p(1−α) .
p
p
For γ = 1 − α, one has
pγ
Ek(Γ1 X)(t2 ) − (Γ1 X)(t1 )kpα ≤ s(α, β, δ)M1 kXk t2 − t1 where s(α, β, δ) is a positive constant.
the
Lemma 3.5. Let α, β ∈ 0, 21 with α < β. Under assumptions (H1)-(H6),
mapping Γ2 defined previously maps bounded sets of BC R, Lp (Ω, H) into bounded
sets of BC γ (R, Lp (Ω, Hα )) for some 0 < γ < 1.
Proof. Let X ∈ BC(R, Lp (Ω, H)) and let f2 (t) = F2 (t, X(t)) for each t ∈ R. We
break down the computations in two cases: p > 2 and p = 2.
For p > 2, we have
EkΓ2 X(t)kpα ≤ cEkΓ2 X(t)kpβ ≤ c · k(β, ξ, δ, p)M2 kXk∞ .
Let t1 < t2 . Clearly,
Ek(Γ2 X)(t2 ) − (Γ2 X)(t1 )kpα
Z t2
≤ 2p−1 Ek
U (t2 , s)f2 (s) dW(s)kpα
t1
+ 2p−1 Ek
Z
t1
−∞
= N10 + N20 .
[U (t2 , s) − U (t1 , s)]f2 (s) dW(s)kpα
16
P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
We use the factorization method (3.1) to obtain
Z t2
sin(πξ) p
0
Ek
(t2 − s)ξ−1 U (t2 , s)Sf2 (s) dskpα
N1 =
π
t1
Z
ip
sin(πξ) p h t2
E
≤
(t2 − s)ξ−1 kU (t2 , s)Sf2 (s)kα ds
π
t1
Z
ip
δ
sin(πξ) p h t2
≤ M (α)p E
(t2 − s)ξ−1 (t2 − s)α e− 2 (t2 −s) kSf2 (s)k ds
π
t1
Z t2
p−1
p
sin(πξ)
p
(t2 − s)− p−1 α ds
≤ M (α)p π
t1
Z t2
δ
×
(t2 − s)−p(1−ξ) e−p 2 (t2 −s) EkSf2 (s)kp ds
t1
Z
p sin(πξ) p
≤ M (α)
×
Z
t2
π
t2
p
(t2 − s)− p−1 α ds
p−1
×
t1
δ
(t2 − s)−p(1−ξ) e−p 2 (t2 −s) ds sup EkSf2 (t)kp
t∈R
t1
≤ s(ξ, δ, Γ, p) 1 −
−(p−1)
p
α
M2 kXk∞ (t2 − t1 )p(1−α)
p−1
where s(ξ, δ, Γ, p) is a positive constant. Similarly,
Z t1 h Z t2
i
∂
U (τ, s) dτ f2 (s) dW(s)kpα
N20 = Ek
−∞
t1 ∂τ
Z t1 h Z t2
i
A(τ )U (τ, s) dτ f2 (s) dW(s)kpα .
= Ek
−∞
t1
Now, using the representation (3.1) together with a stochastic version of the Fubini
theorem with the help of Lemma 2.2 gives us
Z t2 Z t1
sin(πξ) p
Ek
N20 = A(τ )U (τ, t1 )
(t1 − s)ξ−1 U (t1 , s)Sf2 (s) ds dτ kpα
π
t1
−∞
Z
Z
ip
sin(πξ) p h t2 t1
E
(t1 − s)ξ−1 kA(τ )U (τ, s)Sf2 (s)kα ds dτ
≤
π
t1
−∞
Z
Z
ip
sin(πξ) p h t2 t1
δ
E
≤ r(µ, α)
(t1 − s)ξ−1 (τ − s)−α e 4 (τ −s) kSf2 (s)k ds dτ
π
t1
−∞
where ξ satisfies β +
that
1
p
< ξ < 1/2. Since τ > t1 , it follows from Holder’s inequality
N20
Z
Z t1
ip
sin(πξ p h t2
δ
E
≤ r(µ, α)
(τ − t1 )−α
(t1 − s)ξ−1 e− 4 (τ −s) kSf2 (s)k ds dτ
π
t1
−∞
Z
p
sin(πξ p h t2
E
≤ r(µ, α)
(τ − t1 )−α dτ
π
t1
EJDE-2010/124
×
Z
t1
SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
δ
(t1 − s)ξ−1 e− 4 (t1 −s) kSf2 (s)k ds
17
p i
−∞
Z t1
p−1
sin(πξ p
p
δ
p(1−α)
≤ r(µ, α)
(t2 − t1 )
(t1 − s) p−1 (ξ−α−1) e 4 (t1 −s) ds
π
−∞
Z t1
δ
×
e− 4 (t1 −s) ds sup EkSf2 (s)kp
s∈R
−∞
−p
≤ r(ξ, β, δ, Γ, p)(1 − α)
M2 kXk∞ (t2 − t1 )p(1−α) .
For γ = 1 − α, one has
i1/p
h
Ek(Γ2 X)(t2 ) − (Γ2 X)(t1 )kpα
h
i1/p
≤ r(ξ, β, δ, Γ, p)(1 − α)−1 M2 kXk∞
(t2 − t1 )γ .
As for p = 2, we have
EkΓ2 X(t)k2α ≤ cEkΓ2 X(t)k2β ≤ c · s(β, δ)M2 kXk∞ .
For t1 < t2 , let us start with the first term. By Ito isometry identity, we have
n Z t2
N10 ≤ c(α)2
(t2 − s)−2α e−δ(t2 −s) Ekf2 (s)k2L0 ds
2
t1
t2
Z
(t2 − s)−2α ds sup Ekf2 (s)k2L0
2
s∈R
t1
−1
1−2α
≤ c(α)(1 − 2α) M2 kXk∞ (t2 − t1 )
.
≤ c(α)2
Similarly, using the estimates in Lemma 2.2 we have
Z t1 h Z t2
i
∂
0
U (τ, s) dτ f2 (s) dW(s)k2α
N2 = Ek
−∞
t1 ∂τ
Z t1 h Z t2
i
A(τ )U (τ, s) dτ f2 (s) dW(s)k2α
= Ek
−∞
t2
Z
t1
nZ
A(τ )U (τ, t1 )
= Ek
hZ
≤E
t1
o
U (t1 , s)f2 (s) dW(s) dτ k2α
−∞
t1
t2
Z
t1
k
A(τ )U (τ, s)f2 (s) dW(s)k2α dτ
i2
−∞
t1
Z
≤ r(µ, α)2 (t2 − t1 )
t2
nZ
−∞
t1
Z
2
≤ r(µ, α) (t2 − t1 )
t1
t2
o
δ
(τ − s)−2α e− 2 (τ −s) Ekf2 (s)k2L0 ds dτ
2
(τ − t1 )−2α dτ
t1
2
−1
≤ r(µ, α) (1 − 2α)
Z
t1
−∞
2(1−α)
M2 kXk∞ (t2 − t1 )
δ
e− 2 (t1 −s) Ekf2 (s)k2L0 ds
2
.
For γ = 12 − α, one has
h
i1/2
h
i1/2
Ek(Γ2 X)(t2 )−(Γ2 X)(t1 )k2α
≤ r(ξ, β, δ)(1−2β)−1/2 M2 kXk∞
(t2 −t1 )γ .
Therefore, for each X ∈ BC(R, Lp (Ω, H)) such that EkX(t)kp ≤ R for all t ∈
R, then Γi X(t) belongs to BC γ (R, Lp (Ω, Hα )) with EkΓi X(t)kp ≤ R0 where R0
depends on R.
18
P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
Lemma 3.6. The integral operators Γi map bounded sets of AP (Ω, Lp (Ω, H)) into
bounded sets of BC γ (R, Lp (Ω, Hα )) ∩ AP (R, Lp (Ω, H)) for 0 < γ < α , i = 1, 2.
The proof of the above lemma follows the same lines as that of Lemma 3.4, and
hence it is omitted. Similarly, the next lemma is a consequence of [30, Proposition
3.3]. Note in this context that X = Lp (Ω, H) and Y = Lp (Ω, Hα ).
Lemma 3.7. For 0 < γ < α, BC γ (R, Lp (Ω, Hα )) is compactly contained in
BC(R, Lp (Ω, H)); that is, the canonical injection
id : BC γ (R, Lp (Ω, Hα )) ,→ BC(R, Lp (Ω, H))
is compact, which yields
id : BC γ (R, Lp (Ω, Hα )) ∩ AP (R, Lp (Ω, H)) → AP (R, Lp (Ω, H))
is also compact.
The next theorem is the main result of Section 3 and is a nondeterministic
counterpart of the main result in Diagana [21].
Theorem 3.8. Suppose assumptions (H1)–(H6) hold, then the nonautonomous
differential equation Equation (1.2) has at least one p-th mean almost periodic solution.
Proof. Let us recall that in view of Lemmas 3.7 and 3.3, we have
k Γ1 + Γ2 Xkα,∞ ≤ d(β, δ) M1 kXk∞ + M2 kXk∞
and
Ek Γ1 + Γ2 X(t2 ) − Γ1 + Γ2 X(t1 )kpα
γ
≤ s(α, β, δ) M1 kXk∞ + M2 kXk∞ t2 − t1 for all X ∈ BC(R, Lp (Ω, Hα )), t1 , t2 ∈ R with t1 6= t2 , where d(β, δ) and s(α, β, δ)
are positive constants. Consequently, X ∈ BC(R, Lp (Ω, H)) and kXk∞ < R
yield (Γ1 + Γ2 )X ∈ BC γ (R, Lp (Ω, Hα )) and k Γ1 + Γ2 Xkpα,∞ < R1 where R1 =
c(α, β, δ) M1 (R) + M2 (R) . since M(R)/R → 0 as R → ∞, and since EkXkp ≤
cEkXkpα for all X ∈ Lp (Ω, Hα ), it follows that exists an r > 0 such that for all
R ≥ r, the following hold
Γ1 + Γ2 BAP (R,Lp (Ω,H)) (0, R) ⊂ BBC γ (R,Lp (Ω,Hα )) ∩ BAP (R,Lp (Ω,H)) (0, R) .
In view of the above, it follows that Γ1 + Γ2 : D → D is continuous and compact,
where D is the ball in AP (R, Lp (Ω, H))
of radius R with R ≥ r. Using the Schauder
fixed point it follows that Γ1 + Γ2 has a fixed point, which is obviously a p-th
mean almost periodic mild solution to (1.2).
4. Square-mean almost periodic solutions to some second order
stochastic differential equations
In this section we study and obtain under some reasonable assumptions, the existence of square-mean almost periodic solutions to some classes of nonautonomous
second-order stochastic differential equations of type (1.3) on a Hilbert space H
using Schauder’s fixed-point theorem.
For that, the main idea consists of rewriting (1.3) as a nonautonomous first-order
differential equation on H × H involving the family of 2×2-operator matrices L(t).
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SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
Indeed, setting Z :=
19
X
, Equation (1.3) can be rewritten in the Hilbert space
dX(t)
H × H in the form
dZ(ω, t) = [L(t)Z(ω, t) + F1 (t, Z(ω, t))] dt + F2 (t, Z(ω, t))dW(ω, t),
(4.1)
where t ∈ R, L(t) is the family of 2 × 2-operator matrices defined on H = H × H
by
0
IH
L(t) =
(4.2)
−b(t)A −a(t)IH
whose domain D = D(L(t)) is constant in t ∈ R and is given by D(L(t)) = D(A) ×
H. Moreover, the semilinear term Fi (i = 1, 2) appearing in (4.1) is defined on
R × Hα for some α ∈ (0, 1) by
0
Fi (t, Z) =
,
fi (t, X)
where Hα = H̃α× H withH̃α is the real interpolation space between B and D(A)
given by H̃α := H, D(A)
.
α,∞
First of all, note that for 0 < α < β < 1, then
L2 (Ω, Hβ ) ,→ L2 (Ω, Hα ) ,→ L2 (Ω; H)
are continuously embedded and hence therefore exist constants k1 > 0, k(α) > 0
such that
EkZk2 ≤ k1 EkZk2α
EkZk2α ≤ k(α)EkZk2β
for each Z ∈ L2 (Ω, Hα ),
for each Z ∈ L2 (Ω, Hβ ).
To study the existence of square-mean solutions of (4.1), in addition to (H1) we
adopt the following assumptions.
(H7) Let fi (i = 1, 2) : R × L2 (Ω; H) → L2 (Ω; H) be square-mean almost periodic.
Furthermore, X 7→ fi (t, X) is uniformly continuous on any bounded subset
K of L2 (Ω; H) for each t ∈ R. Finally,
sup Ekfi (t, X)k2 ≤ Mi kXk∞
t∈R
+
where Mi : R → R+ is continuous function satisfying
Mi (r)
= 0.
r→∞
r
Under the above assumptions, it will be shown that the linear operator matrices
L(t) satisfy the well-known Acquistapace-Terreni conditions, which does guarantee
the existence of an evolution family U(t, s) associated with it. Moreover, it will be
shown that U(t, s) is exponentially stable under those assumptions.
lim
4.1. Square-Mean Almost Periodic Solutions. To analyze (4.1), our strategy
consists in studying the existence of square-mean almost periodic solutions to the
corresponding class of stochastic differential equations of the form
dZ(t) = [L(t)Z(t) + F1 (t, Z(t))]dt + F2 (t, Z(t))dW(t)
2
(4.3)
2
for all t ∈ R, where the operators L(t) : D(L(t)) ⊂ L (Ω, H) → L (Ω, H) satisfy
Acquistapace-Terreni conditions, Fi (i = 1, 2) as before, and W is a one-dimensional
Brownian motion.
20
P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
Note that each Z ∈ L2 (Ω, H) can be written in terms of the sequence of orthogonal projections En as
γn
∞ X
∞
X
X
X=
hX, ekn iekn =
En X.
n=1 k=1
n=1
Moreover, for each X ∈ D(A),
AX =
∞
X
λj
j=1
γj
X
hX, ekj iekj =
∞
X
λj Ej X.
j=1
k=1
X
∈ D(L) = D(A) × L2 (Ω, H), we obtain
Y
0
IL2 (Ω,H)
X
L(t)Z =
−b(t)A −a(t)IL2 (Ω,H)
Y
P∞
Y
En Y P
n=1
P
=
=
∞
∞
−b(t)AX − a(t)Y
−b(t) n=1 λn En X − a(t) n=1 En Y
∞ X
0
1
En 0
X
=
−b(t)λn −a(t)
0 En
Y
Therefore, for all Z :=
=
n=1
∞
X
An (t)Pn Z,
n=1
where
Pn :=
and
En
0
0
,
En
n ≥ 1,
0
1
,
−b(t)λn −a(t)
Now, the characteristic equation for An (t) is
An (t) :=
n ≥ 1.
λ2 + a(t)λ + λn b(t) = 0
(4.4)
2
with discriminant ∆n (t) = a (t) − 4λn b(t) for all t ∈ R. We assume that there
exists δ0 , γ0 > 0 such that
inf a(t) > 2δ0 > 0,
t∈R
inf b(t) > γ0 > 0.
t∈R
(4.5)
From (4.5) it easily follows that all the roots of (4.4) are nonzero (with nonzero
real parts) given by
p
p
−a(t) − ∆n (t)
−a(t) + ∆n (t)
n
n
, λ2 (t) =
;
λ1 (t) =
2
2
that is,
n
o
σ(An (t)) = λn1 (t), λn2 (t) .
In view of the above, it is easy to see that there exist γ0 ≥ 0 and θ ∈ π2 , π such
that
Sθ ∪ {0} ⊂ ρ (L(t) − γ0 I)
for each t ∈ R where
n
o
Sθ = z ∈ C \ {0} : arg z ≤ θ .
EJDE-2010/124
SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
21
On the other hand, one can show without difficulty that An (t) = Kn−1 (t)Jn (t)Kn (t),
where
n
λ1 (t)
0
1
1
Jn (t) =
,
K
(t)
=
n
0
λn2 (t)
λn1 (t) λn2 (t)
and
Kn−1 (t)
1
= n
λ1 (t) − λn2 (t)
−λn2 (t) 1
.
λn1 (t) −1
For λ ∈ Sθ and Z ∈ L2 (Ω, H), one has
R(λ, L)Z =
=
∞
X
(λ − An (t))−1 Pn Z
n=1
∞
X
Kn (t)(λ − Jn (t)Pn )−1 Kn−1 (t)Pn Z.
n=1
Hence,
EkR(λ, L)Zk2 ≤
≤
∞
X
n=1
∞
X
kKn (t)Pn (λ − Jn (t)Pn )−1 Kn−1 (t)Pn k2B(H) EkPn Zk2
kKn (t)Pn k2B(H) k(λ − Jn (t)Pn )−1 k2B(H) kKn−1 (t)Pn k2B(H) EkPn Zk2 .
n=1
Moreover, for Z :=
Z1
Z2
∈ L2 (Ω, H), we obtain
EkKn (t)Pn Zk2 = EkEn Z1 + En Z2 k2 + Ekλn1 En Z1 + λn2 En Z2 k2
2 ≤ 3 1 + λ1n (t) EkZk2 .
Thus, there exists C1 > 0 such that
EkKn (t)Pn Zk2 ≤ C1 λ1n (t)EkZk2 for all n ≥ 1.
Z1
Similarly, for Z :=
∈ L2 (Ω, H), one can show that there is C2 > 0 such that
Z2
C2
EkKn−1 (t)Pn Zk2 ≤ 1 EkZk2
λn (t)
for all n ≥ 1.
Now, for Z ∈ L2 (Ω, H), we have
−1
Ek(λ − Jn (t)Pn )
Zk = E
2
≤
1
λ−λ1n (t)
0
0
1
λ−λ2n
!
Z1 2
Z2
1
1
EkZ1 k2 +
EkZ2 k2 .
|λ − λ1n (t)|2
|λ − λ2n (t)|2
Let λ0 > 0. Define the function
ηt (λ) :=
1 + |λ|
.
|λ − λ2n (t)|
It is clear that ηt is continuous and bounded on the closed set
Σ := {λ ∈ C : |λ| ≤ λ0 , | arg λ| ≤ θ}.
22
P. H. BEZANDRY, T. DIAGANA
EJDE-2010/124
On the other hand, it is clear that η is bounded for |λ| > λ0 . Thus η is bounded
on Sθ . If we take
1 + |λ|
N = sup
: λ ∈ Sθ , n ≥ 1, j = 1, 2, .
j
|λ − λn (t)|
Therefore,
N
Ek(λ − Jn (t)Pn )−1 Zk2 ≤
EkZk2 , λ ∈ Sθ .
1 + |λ|
Consequently,
K
kR(λ, L(t))k ≤
1 + |λ|
for all λ ∈ Sθ .
First of all, note that the domain D = D(L(t)) is independent of t. Now note
that the operator L(t) is invertible with
−a(t)b−1 (t)A−1 −b−1 (t)A−1
−1
L(t) =
, t ∈ R.
IH
0
Hence, for t, s, r ∈ R, computing L(t) − L(s) L(r)−1 and assuming that there exist
La , Lb ≥ 0 and µ ∈ (0, 1] such that
a(t) − a(s) ≤ La t − sµ , b(t) − b(s) ≤ Lb t − sµ ,
(4.6)
it easily follows that there exists C > 0 such that
2µ
Ek(L(t) − L(s))L(r)−1 Zk2 ≤ C t − s EkZk2 .
In summary, the family of operators L(t) t∈R satisfy Acquistpace-Terreni conditions. Consequently, there exists an evolution family U (t, s) associated with it. Let
us now check that U (t, s) has exponential dichotomy. First of all note that For
every t ∈ R, the family of linear operators L(t) generate an analytic semigroup
(eτ L(t) )τ ≥0 on L2 (Ω, H) given by
eτ L(t) Z =
∞
X
Kl (t)−1 Pl eτ Jl Pl Kl (t)Pl Z, Z ∈ L2 (Ω, H).
l=1
On the other hand,
Ekeτ L(t) Zk2 =
∞
X
kKl (t)−1 Pl k2B(H) keτ Jl Pl k2B(H) kKl (t)Pl k2B(H) EkPl Zk2 ,
l=1
with for each Z =
Z1
,
Z2
τ Jl
Eke
ρl1 τ
e El
Pl Zk = 0
0
2
!
l
eρ2 τ El
l
Z1 2
Z2
l
≤ Ekeρ1 τ El Z1 k2 + Ekeρ2 τ El Z2 k2
≤ e−2δ0 τ EkZk2 .
Therefore,
keτ L(t) k ≤ Ce−δ0 τ ,
Using the continuity of a, b and the equality
τ ≥ 0.
R(λ, L(t)) − R(λ, L(s)) = R(λ, L(t))(L(t) − L(s))R(λ, L(s)),
(4.7)
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SQUARE-MEAN ALMOST PERIODIC SOLUTIONS
23
it follows that the mapping J 3 t 7→ R(λ, L(t)) is strongly continuous for λ ∈
Sω where J ⊂ R is an arbitrary compact interval. Therefore, L(t) satisfies the
assumptions of [42, Corollary 2.3], and thus the evolution family (U (t, s))t≥s is
exponentially stable.
It remains to verify that R(γ0 , L(·)) ∈ AP (R, B(L2 (Ω; H))). For that we need to
show that L−1 (·) ∈ AP (R, B(L2 (Ω, H))). Since t → a(t), t → b(t), and t → b(t)−1
are almost periodic it follows that t → d(t) = − a(t)
b(t) is almost periodic, too. So for
all ε > 0 there exists l(ε) > 0 such that every interval of length l(ε) contains a τ
such that
1 ε
1
ε
√ , d(t + τ ) − d(t) <
√
<
−
b(t + τ ) b(t)
kA−1 k 2
kA−1 k 2
for all t ∈ R. Clearly,
kL−1 (t + τ ) − L−1 (t)k ≤ 2 1/2 −1
1
1 2 kA kB(H)
+ d(t + τ ) − d(t)
−
b(t + τ ) b(t)
<ε
and hence t → L−1 (t) is almost periodic with respect to L2 (Ω, H)-operator topology. Therefore, R(γ0 , L(·)) ∈ AP (R, B(L2 (Ω; H))).
To study the existence of square-mean almost periodic solutions of (4.3), we use
the general results obtained in Section 3.
Definition 4.1. A continuous random function, Z : R → L2 (Ω; H) is said to be a
bounded solution of (4.3) on R provided that
Z t
Z t
Z(t) =
U (t, s)F1 (s, Z(s)) ds +
U (t, s)P (s) F2 (s, Z(s)) dW(s)
s
s
for each t ≥ s and for all t, s ∈ R.
Remark 4.2. Note that it follows from (H7) that Fi (i = 1, 2) : R × L2 (Ω; H) →
L2 (Ω; H) is square-mean almost periodic. Furthermore, Z 7→ Fi (t, Z) is uniformly
continuous on any bounded subset K of L2 (Ω; H) for each t ∈ R. Finally,
sup EkFi (t, Z)k2 ≤ Mi kZk∞
t∈R
where Mi : R+ → R+ is continuous function satisfying
lim
r→∞
Mi (r)
= 0.
r
Theorem 4.3. Suppose assumptions (H1), (H3), (H7) hold, then the nonautonomous differential equation (4.3) has at least one square-mean almost periodic
solution.
In view of Remark 4.2, the proof of the above theorem follows along the same
lines as that of Theorem 3.8 and hence it is omitted.
Acknowledgments. The authors would like thanks the anonymous referee for the
careful reading of the manuscript and insightful comments.
24
P. H. BEZANDRY, T. DIAGANA
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Paul H. Bezandry
Department of Mathematics, Howard University, Washington, DC 20059, USA
E-mail address: pbezandry@howard.edu
Toka Diagana
Department of Mathematics, Howard University, Washington, DC 20059, USA
E-mail address: tdiagana@howard.edu