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Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations
Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations Yong-Kui Chang a,b,∗ , Zhi-Han Zhao a , G.M. N’Guérékata c , Ruyun Ma b a
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, PR China
b
Department of Mathematics, Northwest Normal University, Lanzhou 730070, PR China
c
Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA
article
info
Article history: Received 21 July 2010 Accepted 10 September 2010 Keywords: Stepanov-like almost automorphy Square-mean almost automorphy Stochastic differential equations
abstract In this paper, we introduce a new concept of Stepanov-like almost automorphy (or S 2 -almost automorphy) for stochastic processes. We use the results obtained to investigate the existence and uniqueness of a Stepanov-like almost automorphic mild solution to a class of nonlinear stochastic differential equations in a real separable Hilbert space. Our main results extend some known ones in the sense of square-mean almost automorphy. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction This paper is mainly concerned with the existence and uniqueness of Stepanov-like almost automorphic mild solutions to the class of nonlinear stochastic differential equations in the abstract form dx(t ) = Ax(t )dt + f (t , x(t ))dt + g (t , x(t ))dW (t ),
t ∈ R,
(1.1)
where A is the infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 on L2 (P, H), and W (t ) is a two-sided standard onedimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u)− W (v); u, v ≤ t }. Here f and g are appropriate functions specified later. The concept of almost automorphy is an important generalization of the classical almost periodicity. It was introduced by Bochner [1,2], for more details about this topics we refer the reader to [3,4]. In recent years, the existence of almost periodic and almost automorphic solutions on different kinds of deterministic differential equations has been considerably investigated in lots of publications [5–16] because of its significance and applications in physics, mechanics and mathematical biology. Recently, the existence of almost periodic or pseudo almost periodic solutions to some stochastic differential equations has been considered in many publications such as [17–23] and references therein. In a very recent paper [24], the authors 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 the following stochastic differential equations dx(t ) = Ax(t )dt + f (t )dt + g (t )dW (t ),
t ∈ R,
dx(t ) = Ax(t )dt + f (t , x(t ))dt + g (t , x(t ))dW (t ),
t ∈ R,
in a Hilbert space L (P, H), where A is an infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 , and W (t ) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u)− W (v); u, v ≤ t }. 2
∗
Corresponding author at: Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, PR China. E-mail addresses: [email protected] (Y.-K. Chang), [email protected] (Z.-H. Zhao), Gaston.N’[email protected] (G.M. N’Guérékata), [email protected] (R. Ma). 1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.09.007
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Motivated by the works [14,19,24], the main purpose of this paper is to introduce the notion of Stepanov-like almost automorphy for stochastic processes and apply this new concept to investigate the existence and uniqueness of Stepanovlike almost automorphic solutions to the problem (1.1). The obtained result can be seen as a contribution to this emerging field. The rest of this paper is organized as follows. In Section 2, we introduce the notion of Stepanov-like almost automorphic processes and study some of their basic properties. In Section 3, we prove the existence and uniqueness of Stepanov-like almost automorphic mild solutions to some linear and nonlinear stochastic differential equations, respectively. 2. Preliminaries In this section, we introduce some basic definitions, notations, lemmas and technical results which will be used in the sequel. For more details on this section, we refer the reader to [24,25]. Throughout the paper, we assume that (H, ‖ · ‖, ⟨·, ·⟩) and (K, ‖ · ‖K , ⟨·, ·⟩K ) are two real separable Hilbert spaces. Let (Ω , F , P) be a complete probability space. The notation L2 (P, H) stands for the space of all H-valued random variables x such that E ‖x ‖2 =
∫ Ω
‖x‖2 dP < ∞.
For x ∈ L2 (P, H), let
∫ ‖x ‖2 =
Ω
12 ‖x ‖2 d P .
Then it is routine to check that L2 (P, H) is a Hilbert space equipped with the norm ‖ · ‖2 . We let C (R, L2 (P, H)) (respectively, C (R × L2 (P, H), L2 (P, H))) denote the collection of continuous stochastic processes from R into L2 (P, H) (respectively, the collection of continuous stochastic processes from R × L2 (P, H) into L2 (P, H)). In addition, W (t ) is a two-sided standard onedimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u)− W (v); u, v ≤ t }. Definition 2.1 ([24]). A stochastic process x : R → L2 (P, H) is said to be stochastically continuous if lim E ‖x(t ) − x(s)‖2 = 0. t →s
Definition 2.2 ([24]). A stochastically continuous stochastic process x : R → L2 (P, H) is said to be square-mean almost automorphic if for every sequence of real numbers there exist a subsequence {sn }n∈N and a stochastic process y : R → L2 (P, H) such that lim E ‖x(t + sn ) − y(t )‖2 = 0
n→∞
and
lim E ‖y(t − sn ) − x(t )‖2 = 0
n→∞
hold for each t ∈ R. The collection of all square-mean almost automorphic stochastic processes x : R → L2 (P, H) is denoted by AA(R; L2 (P, H)). Lemma 2.1 ([24]). (AA(R; L2 (P, H)), ‖ · ‖∞ ) is a Banach space when it is equipped with the norm 1
‖x‖∞ := sup ‖x(t )‖2 = sup(E ‖x(t )‖2 ) 2 , t ∈R
t ∈R
for x ∈ AA(R; L2 (P, H)). Definition 2.3 ([24]). A function f : R × L2 (P, H) → L2 (P, H), (t , x) → f (t , x), which is jointly continuous, is said to be square-mean almost automorphic in t ∈ R for each x ∈ L2 (P, H) if for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a stochastic process f : f : R × L2 (P, H) → L2 (P, H) such that lim E ‖f (t + sn , x) − f (t , x)‖2 = 0
n→∞
and
lim E ‖ f (t − sn , x) − f (t , x)‖2 = 0
n→∞
for each t ∈ R and each x ∈ L2 (P, H). Definition 2.4 ([19]). The Bochner transform xb (t , s), t ∈ R, s ∈ [0, 1], of a stochastic process x : R → L2 (P, H) is defined by xb (t , s) := x(t + s). Remark 2.1 ([19]). A stochastic process y(t , s), t ∈ R, s ∈ [0, 1], is the Bochner transform of a certain stochastic process x, y(t , s) = xb (t , s),
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if and only if y(t + τ , s − τ ) = y(s, t ) for all t ∈ R, s ∈ [0, 1] and τ ∈ [s − 1, s]. Definition 2.5 ([14]). The Bochner transform f b (t , s, u), t ∈ R, s ∈ [0, 1], u ∈ L2 (P, H), of a function f : R × L2 (P, H) → L2 (P, H) is defined by f b (t , s, u) := f (t + s, u) for each u ∈ L2 (P, H). Definition 2.6 ([19]). The space BS 2 (L2 (P, H)) of all Stepanov bounded stochastic processes consists of all measurable stochastic processes x : R → L2 (P, H) such that xb = L∞ (R; L2 (0, 1; L2 (P, H))). This is a Banach space with the norm
‖x‖S 2 = ‖x ‖L∞ (R;L2 ) = sup t ∈R
E ‖x(t + s)‖ ds 2
E ‖x(τ )‖ dτ 2
= sup
21
0
t +1
∫ t ∈R
1
∫
b
12
.
t
Definition 2.7. A stochastic process x ∈ BS 2 (L2 (P, H)) is called Stepanov-like almost automorphic (or S 2 -almost automorphic) if xb ∈ AA(R; L2 (0, 1; L2 (P, H))). In other words, a stochastic process x ∈ L2loc (R; L2 (P, H)) is said to be Stepanov-like almost automorphic if its Bochner transform xb : R → L2 (0, 1; L2 (P, H)) is square-mean almost automorphic in the sense that for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a stochastic process y ∈ L2loc (R; L2 (P, H)) such that t +1
∫
t +1
∫
E ‖x(s + sn ) − y(s)‖ ds → 0 2
E ‖y(s − sn ) − x(s)‖2 ds → 0
and
t
t
as n → ∞ pointwise on R. The collection of all such functions will be denoted by AS 2 (R; L2 (P, H)). Remark 2.2. It is clear that, if x : R → L2 (P, H) is a square-mean almost automorphic stochastic process, then x is S 2 -almost automorphic, that is, AA(R; L2 (P, H)) ⊂ AS 2 (R; L2 (P, H)). Lemma 2.2. Let (xn (t ))n∈N be a sequence of S 2 -almost automorphic stochastic processes such that t +1
∫
E ‖xn (s) − x(s)‖2 ds → 0 for each t ∈ R,
(2.1)
t
as n → ∞, then x ∈ AS 2 (R; L2 (P, H)). Proof. For any i ∈ N fixed, since xi (t ) ∈ AS 2 (R; L2 (P, H)), then for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a stochastic process yi ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖xi (s + sn ) − yi (s)‖2 ds → 0
t +1
∫
E ‖yi (s − sn ) − xi (s)‖2 ds → 0
and
t
(2.2)
t
as n → ∞ pointwise on R. On the other hand, from (2.1), we easily deduce that (xn (t ))n∈N is a Cauchy sequence with respect to ‖ · ‖S 2 . We observe that, for each t ∈ R, the sequence (yi (t )) is also a Cauchy sequence in L2loc (R; L2 (P, H)). Indeed, if we write yi (t ) − yj (t ) = yi (t ) − xi (t + sn ) + xi (t + sn ) − xj (t + sn ) + xj (t + sn ) − yj (t ), then for a sufficiently large n, we get t +1
∫
E ‖yi (s) − yj (s)‖ ds ≤ 3 2
t
t +1
∫
E ‖yi (s) − xi (s + sn )‖ ds + 3 2
t
t +1
∫
E ‖xi (s + sn ) − xj (s + sn )‖2 ds t
t +1
∫
E ‖xj (s + sn ) − yj (s)‖2 ds.
+3 t
By (2.1) and (2.2), the sequence of (yi (t )) is Cauchy.
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
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Using the completeness of L2loc (R; L2 (P, H)), we denote by y(t ) the pointwise limit of (yi (t )). Now let us prove that x(t ) ∈ AS 2 (R; L2 (P, H)). Note that the inequality below holds for any index i and any t ∈ R t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds ≤ 3
t +1
∫
E ‖x(s + sn ) − xi (s + sn )‖2 ds + 3
E ‖xi (s + sn ) − yi (s)‖2 ds t
t
t
t +1
∫
t +1
∫
E ‖yi (s) − y(s)‖ ds. 2
+3 t
So, from (2.1) and the fact that y(t ) is the pointwise limit of (yi (t )), for any sufficiently small ε > 0 there exists a sufficiently large i, such that for each t ∈ R t +1
∫
E ‖x(s + sn ) − xi (s + sn )‖2 ds < t
ε 9
∫
,
t +1
E ‖yi (s) − y(s)‖2 ds <
t
ε 9
.
Now for this sufficiently large i, from (2.2), there exists a sufficient N such that n > N we have t +1
∫
E ‖xi (s + sn ) − yi (s)‖2 ds < t
ε 9
.
Thus t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds < ε
for n > N ,
t
which implies t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds → 0 t
as n → ∞ pointwise on R. We can use the same step to prove that t +1
∫
E ‖y(s − sn ) − x(s)‖2 ds → 0 t
as n → ∞ pointwise on R. That is, x(t ) ∈ AS 2 (R; L2 (P, H)). The proof is finished.
Lemma 2.3. Let (xn (t ))n∈N be a sequence of square-mean almost automorphic stochastic processes such that lim E ‖xn (t ) − x(t )‖2 = 0 for each t ∈ R.
n→∞
Then x ∈ AA(R; L2 (P, H)). Proof. It is similar to the proof of Lemma 2.2, so we omit the details.
Remark 2.3. From the proof of [24, Theorem 2.4.], we can easily see that the conclusion of Lemma 2.3 is true. Theorem 2.1. AS 2 (R; L2 (P, H)) is a Banach space when it is equipped with the norm t +1
∫
E ‖x(s)‖ ds 2
‖x‖S 2 = sup t ∈R
21
for x ∈ AS 2 (R; L2 (P, H)).
t
Proof. We can easily obtain this result by using the inclusion AS 2 (R; L2 (P, H)) ⊂ BS 2 (L2 (P, H)) combined with the fact that (BS 2 (L2 (P, H)), ‖ · ‖S 2 ) is a Banach space and Lemma 2.2. Definition 2.8. A function f : R × L2 (P, H) → L2 (P, H), (t , x) → f (t , x) with f (·, x) ∈ L2loc (R; L2 (P, H)) for each x ∈ L2 (P, H), is said to be S 2 -almost automorphic in t ∈ R uniformly in x ∈ L2 (P, H) if t → f (t , x) is S 2 -almost automorphic for each x ∈ L2 (P, H). That means, for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a function f : R × L2 (P, H) → L2 (P, H) with f (·, x) ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖f (s + sn , x) − f (s, x)‖2 ds → 0 t
t +1
∫
E ‖ f (s − sn , x) − f (s, x)‖2 ds → 0
and t
as n → ∞ pointwise on R and for each x ∈ L2 (P, H). We denoted by AS 2 (R × L2 (P, H); L2 (P, H)) the set of all such functions.
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Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
Theorem 2.2. Let f : R × L2 (P, H) → L2 (P, H), (t , x) → f (t , x) be S 2 -almost automorphic in t ∈ R uniformly in x ∈ L2 (P, H), and assume that f satisfies the Lipschitz condition in the following sense:
˜ ‖x − y ‖2 E ‖f (t , x) − f (t , y)‖2 ≤ ME ˜ > 0 is independent of t. Then for any S 2 -almost automorphic process for all x, y ∈ L2 (P, H) and for each t ∈ R, where M x : R → L2 (P, H), the stochastic process F : R → L2 (P, H) given by F (t ) = f (t , x(t )) is S 2 -almost automorphic. Proof. Let {s′n }n∈N be an arbitrary sequence of real numbers. Since x ∈ AS 2 (R; L2 (P, H)), there exist a subsequence {sn }n∈N of {s′n }n∈N and a stochastic process y ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds → 0
t +1
∫
E ‖y(s − sn ) − x(s)‖2 ds → 0
and
(2.3)
t
t
as n → ∞ pointwise on R. On the other hand, Since (t , x) → f (t , x) is S 2 -almost automorphic in t ∈ R uniformly in x ∈ L2 (P, H), we can extract a subsequence {sn }n∈N of {s′n }n∈N (for convenience, we also denote it by {sn }n∈N ) and a function f (·, x) ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖f (s + sn , x) − f (s, x)‖2 ds → 0
t +1
∫
E ‖ f (s − sn , x) − f (s, x)‖2 ds → 0
and
(2.4)
t
t
as n → ∞ pointwise on R and for each x ∈ L2 (P, H). Now, let us consider the function F : R → L2 (P, H) defined by F (t ) := f (t , y(t )), t ∈ R. Note that F (t + sn ) − F (t ) = f (t + sn , x(t + sn )) − f (t + sn , y(t )) + f (t + sn , y(t )) − f (t , y(t )). So, we obtain E ‖F ( t + s n ) − F (t )‖2 ≤ 2E ‖f (t + sn , x(t + sn )) − f (t + sn , y(t ))‖2 + 2E ‖f (t + sn , y(t )) − f (t , y(t ))‖2
˜ ‖x(t + sn ) − y(t )‖2 + 2E ‖f (t + sn , y(t )) − ≤ 2ME f (t , y(t ))‖2 . We can deduce from (2.3) and (2.4) that t +1
∫
E ‖F (s + sn ) − F (s)‖2 ds → 0 t
as n → ∞ pointwise on R. Similarly, using the same argument as above, we can prove that t +1
∫
E ‖ F (s − sn ) − F (s)‖2 ds → 0 t
as n → ∞ pointwise on R, which proves that F (t ) is S 2 -almost automorphic.
Definition 2.9. An Ft -progressively measurable stochastic process {x(t )}t ∈R is called a mild solution of problem (1.1) on R if it satisfies the corresponding stochastic integral equation x(t ) = T (t − a)x(a) +
t
∫
T (t − s)f (s, x(s))ds + a
t
∫
T (t − s)g (s, x(s))dW (s) a
for all t ≥ a and for each a ∈ R. 3. Main results In this section, we investigate the existence of a S 2 -almost automorphic solution for the problem (1.1). To study the existence of S 2 -almost automorphic mild solutions to Eq. (1.1), we first consider the existence of S 2 -almost automorphic mild solutions to the linear stochastic differential equation dx(t ) = Ax(t )dt + f (t )dt + g (t )dW (t ),
t ∈ R,
(3.1)
where A is the infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 on L2 (P, H), f , g ∈ AS 2 (R; L2 (P, H)) ∩ C (R, L2 (P, H)), and W (t ) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u) − W (v); u, v ≤ t }.
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
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Definition 3.1. An Ft -progressively measurable stochastic process {x(t )}t ∈R is called a mild solution of problem (3.1) on R if it satisfies the corresponding stochastic integral equation x(t ) = T (t − a)x(a) +
t
∫
t
∫
T (t − s)f (s)ds +
T (t − s)g (s)dW (s) a
a
for all t ≥ a and for each a ∈ R. We first list the following basic assumption: (H1) The operator A is the infinitesimal generator of an exponentially stable C0 -semigroup {T (t )}t ≥0 on L2 (P, H); that is, there exist M > 0, δ > 0 such that ‖T (t )‖ ≤ Me−δ t , for all t ≥ 0. Theorem 3.1. Under previous assumptions, if we assume that (H1) holds, then the problem (3.1) has a unique mild solution x ∈ AS 2 (R; L2 (P, H)). Proof. Let us first prove uniqueness. Assume that x : R → L2 (P, H) is a bounded stochastic process and satisfies the homogeneous equation dx(t ) = Ax(t )dt ,
t ∈ R.
(3.2) −δ(t −s)
Then x(t ) = T (t − s)x(s), for any t ≥ s. Thus ‖x(t )‖ ≤ MK e with ‖x(s)‖ ≤ K for s ∈ R almost surely. Take a sequence of real numbers {sn }n∈N such that sn → −∞ as n → ∞. For any t ∈ R fixed, one can find a subsequence {snk }k∈N ⊂ {sn }n∈N such that snk < t for all k = 1, 2, . . . . By letting k → ∞, we get x(t ) = 0 almost surely. Now, if x1 , x2 : R → L2 (P, H) are bounded solutions to Eq. (3.1), then x = x1 − x2 is a bounded solution to Eq. (3.2). In view of the above, x = x1 − x2 = 0 almost surely, that is, x1 = x2 almost surely. Now let us investigate the existence. Consider for each n = 1, 2, . . . , the integrals xn (t ) =
∫
n
T (σ )f (t − σ )dσ
and
yn (t ) =
n−1
∫
n
T (σ )g (t − σ )dW (σ ) n−1
for each t ∈ R. First, by using Hölder’s inequality, we get t +1
∫
E ‖xn (s)‖ ds = 2
∫
t
t +1
∫ E
t +1
∫
t
∫
n n−1
n
‖T (σ )‖2 E ‖f (s − σ )‖2 dσ ds
≤ n−1
t
≤ M2
t +1
∫
∫
n
∫
n
e−2δσ
n −1
2
M2 2δ
t +1
∫
E ‖f (s − σ )‖2 ds dσ t
≤ M 2 ‖f ‖2S 2 ≤
e−2δσ E ‖f (s − σ )‖2 dσ ds
n−1
t
≤ M2
2 T (σ )f (s − σ )dσ ds
∫
n
e−2δσ dσ
n −1
‖f ‖2S 2 e−2δn (e2δ − 1).
Since M2δ ‖f ‖2S 2 (e2δ − 1) n=1 e−2δ n < ∞, we deduce from the well-known Weierstrass test that the series convergent in the sense of the norm ‖ · ‖S 2 uniformly on R. Now let
Φ (t ) :=
∑∞
∞ −
xn (t ) for each t ∈ R.
n=1
Observe that
Φ (t ) =
∫
t
T (t − s)f (s)ds −∞
Clearly, Φ (t ) ∈ C (R, L2 (P, H)).
for each t ∈ R.
∑∞
n=1
xn (t ) is
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Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
Now, by using an estimate on the Ito integral established in [26], we obtain that t +1
∫
E ‖yn (s)‖2 ds =
t +1
∫ t
t
t +1
∫
n −1
‖T (σ )‖2 E ‖g (s − σ )‖2 dσ ds n−1
t
t +1
∫
∫
∫
n
e−2δσ E ‖g (s − σ )‖2 dσ ds
n−1
t
≤ M2
T (σ )g (s − σ )dW (σ ) ds
n
∫
≤ ≤ M2
2
n
∫
E
n
e−2δσ
n −1
≤ M 2 ‖g ‖2S 2 ≤ 2
M2 2δ
t +1
∫
E ‖g (s − σ )‖2 ds dσ t
∫
n
e−2δσ dσ
n−1
‖g ‖2S 2 e−2δn (e2δ − 1).
Since M2δ ‖g ‖2S 2 (e2δ − 1) n=1 e−2δ n < ∞, we deduce from the Weierstrass test that the series the sense of the norm ‖ · ‖S 2 uniformly on R. Furthermore,
∑∞
Ψ (t ) :=
∫
t
T (t − s)g (s)dW (s) =
−∞
∞ −
yn (t ),
∑∞
n =1
yn (t ) is convergent in
t ∈ R,
n =1
and clearly Ψ (t ) ∈ C (R, L2 (P, H)). Now let us show that each xn , yn ∈ AS 2 (R; L2 (P, H)). First, we prove that xn ∈ AS 2 (R; L2 (P, H)). Indeed, let {s′m }m∈N be a sequence of real numbers. Since f ∈ AS 2 (R; L2 (P, H)), there exist a subsequence {sm }m∈N of {s′m }m∈N and a stochastic process f ∈ L2loc (R; L2 (P, H)) t +1
∫
t +1
∫
E ‖f ( s + s m ) − f (s)‖2 ds → 0
E ‖ f (s − sm ) − f (s)‖2 ds → 0
and
t
t
as m → ∞ pointwise on R. Moreover, if we let xn (t ) = t +1
∫
E ‖xn (s + sm ) − xn (s)‖2 ds =
t +1
∫
t
t
∫ = t
T (σ ) f (t − σ )dσ , we have
T (σ )f (s + sm − σ )dσ − n −1
∫ t +1 E t +1
∫
n−1
n
∫
E
n
∫
n −1
T (σ ) f (s − σ )dσ ds
2
n n −1
2
n
T (σ )[f (s + sm − σ ) − f (s − σ )]dσ ds
n
∫
‖T (σ )‖2 E ‖f (s + sm − σ ) − f (s − σ )‖2 dσ ds
≤ n−1
t
≤ M2
t +1
∫
∫
≤ M2
e−2δσ E ‖f (s + sm − σ ) − f (s − σ )‖2 dσ ds
n −1
t n
∫
n
e−2δσ
n −1
t +1
∫
E ‖f (s + sm − σ ) − f (s − σ )‖2 ds dσ . t
Obviously, the last inequality goes to 0 as m → ∞ pointwise on R. Similarly we can prove that t +1
∫
E ‖ xn (s − sm ) − xn (s)‖2 ds → 0 t
as m → ∞ pointwise on R. Thus we conclude that each xn ∈ AS 2 (R; L2 (P, H)) and consequently their uniform limit Φ (t ) ∈ AS 2 (R; L2 (P, H)), by using Lemma 2.2. Next, we show that each yn ∈ AS 2 (R; L2 (P, H)). Since g ∈ AS 2 (R; L2 (P, H)), then for every sequence of real numbers {s′m }m∈N there exist a subsequence {sm }m∈N ⊂ {s′m }m∈N and a stochastic process g ∈ L2loc (R; L2 (P, H)) such that t +1
∫ t
E ‖g (s + sm ) − g (s)‖2 ds → 0
∫ and t
t +1
E ‖ g (s − sm ) − g (s)‖2 ds → 0
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
1137
n
as m → ∞ pointwise on R. Moreover, if we let yn (t ) =
T (σ ) g (t − σ )dW (σ ), by using the Ito integral, we get n −1 2 ∫ n ∫ t +1 ∫ t +1 ∫ n 2 T (σ ) g (s − σ )dW (σ ) E T (σ )g (s + sm − σ )dW (σ ) − E ‖yn (s + sm ) − yn (s)‖ ds = ds n−1 t n−1 t 2 ∫ t +1 ∫ n ds E T (σ )[ g ( s + s − σ ) − g ( s − σ )] dW (σ ) = m t n−1 ∫ t +1 ∫ n ‖T (σ )‖2 E ‖g (s + sm − σ ) − g (s − σ )‖2 dσ ds ≤ n −1
t
≤ M2
t +1
∫
n −1
t
≤ M2
∫
n
∫
n
e−2δσ
e−2δσ E ‖g (s + sm − σ ) − g (s − σ )‖2 dσ ds t +1
∫
n−1
E ‖g (s + sm − σ ) − g (s − σ )‖2 ds dσ .
t
Obviously, the last inequality goes to 0 as m → ∞ pointwise on R. Arguing in a similar way, we infer that t +1
∫
E ‖ yn (s − sm ) − yn (s)‖2 ds → 0 t
as m → ∞ pointwise on R. Thus we conclude that each yn ∈ AS 2 (R; L2 (P, H)) and consequently their uniform limit Ψ (t ) ∈ AS 2 (R; L2 (P, H)), by using Lemma 2.2. Define x(t ) =
∫
t
T (t − s)f (s)ds +
∫
−∞
t
T (t − s)g (s)dW (s),
t ∈ R.
−∞
Obviously to Eq. (3.1). Let us prove that x(t ) is a mild solution of Eq. (3.1). Indeed, if we let a x is a bounded solution a x(a) = −∞ T (a − s)f (s)ds + −∞ T (a − s)g (s)dW (s), then T (t − a)x(a) =
∫
a
T (t − s)f (s)ds + −∞
∫
a
T (t − s)g (s)dW (s). −∞
But for t ≥ a, t
∫
∫
T (t − s)g (s)dW (s) =
t
T (t − s)g (s)dW (s) − −∞
a
= x(t ) −
a
∫
T (t − s)g (s)dW (s) −∞
∫
t
T (t − s)f (s)ds + −∞
∫
a
T (t − s)f (s)ds − T (t − a)x(a) −∞
= x(t ) − T (t − a)x(a) −
t
∫
T (t − s)f (s)ds. a
It follows that x(t ) = T (t − a)x(a) +
∫
t
T (t − s)f (s)ds +
a
t
∫
T (t − s)g (s)dW (s). a
In view of the above, it follows that x is the only bounded S 2 -almost automorphic mild solution to Eq. (3.1). The proof is now complete. In order to investigate the solution to (1.1), we need the following assumptions: (H2) The function f ∈ AS 2 (R × L2 (P, H); L2 (P, H)) ∩ C (R × L2 (P, H), L2 (P, H)) and there exists a constant Lf > 0 such that E ‖f (t , x) − f (t , y)‖2 ≤ Lf E ‖x − y‖2 for all t ∈ R and each x, y ∈ L2 (P, H). (H3) The function g ∈ AS 2 (R × L2 (P, H); L2 (P, H)) ∩ C (R × L2 (P, H), L2 (P, H)) and there exists a positive number Lg such that E ‖g (t , x) − g (t , y)‖2 ≤ Lg E ‖x − y‖2 for all t ∈ R and each x, y ∈ L2 (P, H).
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Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
Theorem 3.2. Assume the conditions (H1)–(H3) are satisfied, then the problem (1.1) has a unique S 2 -almost automorphic mild solution on R provided that
[ L0 =
2
M2
2
δ2
M Lf +
δ
] Lg
< 1.
(3.3)
Proof. Let Λ : AS 2 (R; L2 (P, H)) → AS 2 (R; L2 (P, H)) be the operator defined by t
∫
Λx(t ) =
T (t − s)f (s, x(s))ds +
∫
t
T (t − s)g (s, x(s))dW (s),
t ∈ R.
−∞
−∞
From previous assumptions and the properties of {T (t )}t ≥0 one can easily see that Λx is well-defined and continuous. By Theorem 2.2, we infer that both F (·) = f (·, x(·)) and G(·) = g (·, x(·)) ∈ AS 2 (R; L2 (P, H)). Then by using the proof of Theorem 3.1 with the above Lemma 2.2, we have that Λx ∈ AS 2 (R; L2 (P, H)) whenever x ∈ AS 2 (R; L2 (P, H)). Thus Λ maps AS 2 (R; L2 (P, H)) into itself. Now we prove that Λ is a contraction mapping on AS 2 (R; L2 (P, H)). Indeed, for each t ∈ R, x, y ∈ AS 2 (R; L2 (P, H)), we see that t +1
∫
E ‖(Λx)(s) − (Λy)(s)‖ ds = 2
t +1
∫
t
t
2
t +1
E
t
∫ +2 t
2
2
s
∫
≤2
−∞
∫ t +1 E
−∞
∫ t +1
∫
t +1
t +1
s
∫
≤
≤
δ 2
t +1
∫
δ [
E ‖f (σ , x(σ )) − f (σ , y(σ ))‖ dσ 2
ds
−∞
e−δ(s−σ ) E ‖f (σ , x(σ )) − f (σ , y(σ ))‖2 dσ ds s
∫
e−2δ(s−σ ) E ‖g (σ , x(σ )) − g (σ , y(σ ))‖2 dσ ds
−∞ t +1
∫
∫
s
∞
∫
e−δ(s−σ ) E ‖x(σ ) − y(σ )‖2 dσ ds + 2M 2 Lg
−∞
t
M 2 Lf
e
−δ(s−σ )
−∞
t
M 2 Lf
s
‖T (s − σ )‖2 E ‖g (σ , x(σ )) − g (σ , y(σ ))‖2 dσ ds
t
2
dσ
∫
−∞
∫
+ 2M 2 ≤
−δ(s−σ )
s
∫
t
δ
s
e
∫
M2
2 T (s − σ )[g (σ , x(σ )) − g (σ , y(σ ))]dW (σ ) ds
−∞
+2 2
T (s − σ )[f (σ , x(σ )) − f (σ , y(σ ))]dσ ds
s
t
≤
T (s − σ )[f (σ , x(σ )) − f (σ , y(σ ))]dσ −∞
T (s − σ )[g (σ , x(σ )) − g (σ , y(σ ))]dW (σ ) ds
−∞
≤ 2M
s
s
∫ + ∫
∫ E
e−δξ
2
M2
δ
δ
M 2 Lf + 2
∫
∫
t −ξ +1
E ‖x(s) − y(s)‖2 dsdξ + 2M 2 Lg
∞
∫ 0
s
e−2δ(s−σ ) E ‖x(σ ) − y(σ )‖2 dσ ds
−∞
t
t −ξ
0
t +1
∫
e−2δξ
∫
t −ξ +1
E ‖x(s) − y(s)‖2 dsdξ t −ξ
]
Lg ‖x − y‖2S 2 .
Hence
‖Λx − Λy‖S 2 = sup t ∈R
t +1
∫
E ‖(Λx)(s) − (Λy)(s)‖2 ds
21 ≤
L0 ‖x − y‖S 2 ,
t
which implies that Λ is a contraction by (3.3). So by the Banach contraction principle, we draw a conclusion that there exists a unique fixed point x(·) for Λ in AS 2(R; L2 (P, H)), such that Λx = x. Moreover, using the same proof as in Theorem 3.1, we t t can see that x(t ) = T (t − a)x(a) + a T (t − s)f (s, x(s))ds + a T (t − s)g (s, x(s))dW (s) is a mild solution of Eq. (1.1) and x(·) ∈ AS 2 (R; L2 (P, H)). The proof is complete.
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
1139
Acknowledgements The first author’s work was supported by NNSF of China (10901075), the Key Project of Chinese Ministry of Education, China Postdoctoral Science Foundation funded project (20100471645), the Scientific Research Fund of Gansu Provincial Education Department (0804-08), and Qing Lan Talent Engineering Funds (QL-05-16A) by Lanzhou Jiaotong University. The fourth author’s work was supported by NNSF of China (11061030). References [1] S. Bochner, A new approach to almost automorphicity, Proc. Natl. Acad. Sci. USA 48 (1962) 2039–2043. [2] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA 52 (1964) 907–910. [3] G.M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Space, Kluwer Academic Plenum Publishers, New York, London, Moscow, 2001. [4] G.M. N’Guérékata, Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005. [5] E. Hernández, H.R. Henríquez, Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl. 221 (1998) 499–522. [6] H.R. Henríquez, C.H. Vasquez, Almost periodic solutions of abstract retarded functional differential equations with unbounded delay, Acta Appl. Math. 57 (1999) 105–132. [7] S. Abbas, D. Bahuguna, Almost periodic solutions of neutral functional differential equations, Comput. Math. Appl. 55 (2008) 2593–2601. [8] Z.H. Zhao, Y.K. Chang, W.S. Li, Asymptotically almost periodic, almost periodic and pseudo almost periodic mild solutions for neutral differential equations, Nonlinear Anal. RWA 11 (2010) 3037–3044. [9] Z.H. Zhao, Y.K. Chang, J.J. Nieto, Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in Banach spaces, Nonlinear Anal. TMA 72 (2010) 1886–1894. [10] G.M. N’Guérékata, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum 69 (2004) 80–86. [11] T. Diagana, G.M. N’Guérékata, Almost automorphic solutions to semilinear evolution equations, Funct. Differ. Equ. 13 (2006) 195–206. [12] T. Diagana, G.M. N’Guérékata, Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett. 20 (2007) 462–466. [13] T. Diagana, G.M. N’Guérékata, Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal. 86 (2007) 723–733. [14] G.M. N’Guérékata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal. 68 (2008) 2658–2667. [15] H. Lee, H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Anal. 69 (2008) 2158–2166. [16] Y.K. Chang, Z.H. Zhao, J.J. Nieto, Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces, Rev. Mat. Complut. (2010), doi:10.1007/s13163-010-0047-2. [17] P. Bezandry, T. Diagana, Existence of almost periodic solutions to some stochastic differential equations, Appl. Anal. 86 (2007) 819–827. [18] P. Bezandry, Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, Statist. Probab. Lett. 78 (2008) 2844–2849. [19] P. Bezandry, T. Diagana, Existence of S 2 -almost periodic solutions to a class of nonautonomous stochastic evolution equations, Electron. J. Qual. Theory Differ. Equ. 35 (2008) 1–19. [20] A.Ya. Dorogovtsev, O.A. Ortega, On the existence of periodic solutions of a stochastic equation in a Hilbert space, Visnik Kiiv. Univ. Ser. Mat. Mekh. 30 (1988) 21–30. 115. [21] G. Da Prato, C. Tudor, Periodic and almost periodic solutions for semilinear stochastic evolution equations, Stoch. Anal. Appl. 13 (1995) 13–33. [22] C. Tudor, Almost periodic solutions of affine stochastic evolutions equations, Stoch. Stoch. Rep. 38 (1992) 251–266. [23] C.A. Tudor, M. Tudor, Pseudo almost periodic solutions of some stochastic differential equations, Math. Rep. (Bucur.) 1 (1999) 305–314. [24] M.M. Fu, Z.X. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc. (in press), arXiv:1001.3049v1 [math.DS]. [25] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [26] A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl. 90 (1982) 12–44.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations Yong-Kui Chang a,b,∗ , Zhi-Han Zhao a , G.M. N’Guérékata c , Ruyun Ma b a
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, PR China
b
Department of Mathematics, Northwest Normal University, Lanzhou 730070, PR China
c
Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA
article
info
Article history: Received 21 July 2010 Accepted 10 September 2010 Keywords: Stepanov-like almost automorphy Square-mean almost automorphy Stochastic differential equations
abstract In this paper, we introduce a new concept of Stepanov-like almost automorphy (or S 2 -almost automorphy) for stochastic processes. We use the results obtained to investigate the existence and uniqueness of a Stepanov-like almost automorphic mild solution to a class of nonlinear stochastic differential equations in a real separable Hilbert space. Our main results extend some known ones in the sense of square-mean almost automorphy. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction This paper is mainly concerned with the existence and uniqueness of Stepanov-like almost automorphic mild solutions to the class of nonlinear stochastic differential equations in the abstract form dx(t ) = Ax(t )dt + f (t , x(t ))dt + g (t , x(t ))dW (t ),
t ∈ R,
(1.1)
where A is the infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 on L2 (P, H), and W (t ) is a two-sided standard onedimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u)− W (v); u, v ≤ t }. Here f and g are appropriate functions specified later. The concept of almost automorphy is an important generalization of the classical almost periodicity. It was introduced by Bochner [1,2], for more details about this topics we refer the reader to [3,4]. In recent years, the existence of almost periodic and almost automorphic solutions on different kinds of deterministic differential equations has been considerably investigated in lots of publications [5–16] because of its significance and applications in physics, mechanics and mathematical biology. Recently, the existence of almost periodic or pseudo almost periodic solutions to some stochastic differential equations has been considered in many publications such as [17–23] and references therein. In a very recent paper [24], the authors 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 the following stochastic differential equations dx(t ) = Ax(t )dt + f (t )dt + g (t )dW (t ),
t ∈ R,
dx(t ) = Ax(t )dt + f (t , x(t ))dt + g (t , x(t ))dW (t ),
t ∈ R,
in a Hilbert space L (P, H), where A is an infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 , and W (t ) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u)− W (v); u, v ≤ t }. 2
∗
Corresponding author at: Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, PR China. E-mail addresses: [email protected] (Y.-K. Chang), [email protected] (Z.-H. Zhao), Gaston.N’[email protected] (G.M. N’Guérékata), [email protected] (R. Ma). 1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.09.007
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
1131
Motivated by the works [14,19,24], the main purpose of this paper is to introduce the notion of Stepanov-like almost automorphy for stochastic processes and apply this new concept to investigate the existence and uniqueness of Stepanovlike almost automorphic solutions to the problem (1.1). The obtained result can be seen as a contribution to this emerging field. The rest of this paper is organized as follows. In Section 2, we introduce the notion of Stepanov-like almost automorphic processes and study some of their basic properties. In Section 3, we prove the existence and uniqueness of Stepanov-like almost automorphic mild solutions to some linear and nonlinear stochastic differential equations, respectively. 2. Preliminaries In this section, we introduce some basic definitions, notations, lemmas and technical results which will be used in the sequel. For more details on this section, we refer the reader to [24,25]. Throughout the paper, we assume that (H, ‖ · ‖, ⟨·, ·⟩) and (K, ‖ · ‖K , ⟨·, ·⟩K ) are two real separable Hilbert spaces. Let (Ω , F , P) be a complete probability space. The notation L2 (P, H) stands for the space of all H-valued random variables x such that E ‖x ‖2 =
∫ Ω
‖x‖2 dP < ∞.
For x ∈ L2 (P, H), let
∫ ‖x ‖2 =
Ω
12 ‖x ‖2 d P .
Then it is routine to check that L2 (P, H) is a Hilbert space equipped with the norm ‖ · ‖2 . We let C (R, L2 (P, H)) (respectively, C (R × L2 (P, H), L2 (P, H))) denote the collection of continuous stochastic processes from R into L2 (P, H) (respectively, the collection of continuous stochastic processes from R × L2 (P, H) into L2 (P, H)). In addition, W (t ) is a two-sided standard onedimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u)− W (v); u, v ≤ t }. Definition 2.1 ([24]). A stochastic process x : R → L2 (P, H) is said to be stochastically continuous if lim E ‖x(t ) − x(s)‖2 = 0. t →s
Definition 2.2 ([24]). A stochastically continuous stochastic process x : R → L2 (P, H) is said to be square-mean almost automorphic if for every sequence of real numbers there exist a subsequence {sn }n∈N and a stochastic process y : R → L2 (P, H) such that lim E ‖x(t + sn ) − y(t )‖2 = 0
n→∞
and
lim E ‖y(t − sn ) − x(t )‖2 = 0
n→∞
hold for each t ∈ R. The collection of all square-mean almost automorphic stochastic processes x : R → L2 (P, H) is denoted by AA(R; L2 (P, H)). Lemma 2.1 ([24]). (AA(R; L2 (P, H)), ‖ · ‖∞ ) is a Banach space when it is equipped with the norm 1
‖x‖∞ := sup ‖x(t )‖2 = sup(E ‖x(t )‖2 ) 2 , t ∈R
t ∈R
for x ∈ AA(R; L2 (P, H)). Definition 2.3 ([24]). A function f : R × L2 (P, H) → L2 (P, H), (t , x) → f (t , x), which is jointly continuous, is said to be square-mean almost automorphic in t ∈ R for each x ∈ L2 (P, H) if for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a stochastic process f : f : R × L2 (P, H) → L2 (P, H) such that lim E ‖f (t + sn , x) − f (t , x)‖2 = 0
n→∞
and
lim E ‖ f (t − sn , x) − f (t , x)‖2 = 0
n→∞
for each t ∈ R and each x ∈ L2 (P, H). Definition 2.4 ([19]). The Bochner transform xb (t , s), t ∈ R, s ∈ [0, 1], of a stochastic process x : R → L2 (P, H) is defined by xb (t , s) := x(t + s). Remark 2.1 ([19]). A stochastic process y(t , s), t ∈ R, s ∈ [0, 1], is the Bochner transform of a certain stochastic process x, y(t , s) = xb (t , s),
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Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
if and only if y(t + τ , s − τ ) = y(s, t ) for all t ∈ R, s ∈ [0, 1] and τ ∈ [s − 1, s]. Definition 2.5 ([14]). The Bochner transform f b (t , s, u), t ∈ R, s ∈ [0, 1], u ∈ L2 (P, H), of a function f : R × L2 (P, H) → L2 (P, H) is defined by f b (t , s, u) := f (t + s, u) for each u ∈ L2 (P, H). Definition 2.6 ([19]). The space BS 2 (L2 (P, H)) of all Stepanov bounded stochastic processes consists of all measurable stochastic processes x : R → L2 (P, H) such that xb = L∞ (R; L2 (0, 1; L2 (P, H))). This is a Banach space with the norm
‖x‖S 2 = ‖x ‖L∞ (R;L2 ) = sup t ∈R
E ‖x(t + s)‖ ds 2
E ‖x(τ )‖ dτ 2
= sup
21
0
t +1
∫ t ∈R
1
∫
b
12
.
t
Definition 2.7. A stochastic process x ∈ BS 2 (L2 (P, H)) is called Stepanov-like almost automorphic (or S 2 -almost automorphic) if xb ∈ AA(R; L2 (0, 1; L2 (P, H))). In other words, a stochastic process x ∈ L2loc (R; L2 (P, H)) is said to be Stepanov-like almost automorphic if its Bochner transform xb : R → L2 (0, 1; L2 (P, H)) is square-mean almost automorphic in the sense that for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a stochastic process y ∈ L2loc (R; L2 (P, H)) such that t +1
∫
t +1
∫
E ‖x(s + sn ) − y(s)‖ ds → 0 2
E ‖y(s − sn ) − x(s)‖2 ds → 0
and
t
t
as n → ∞ pointwise on R. The collection of all such functions will be denoted by AS 2 (R; L2 (P, H)). Remark 2.2. It is clear that, if x : R → L2 (P, H) is a square-mean almost automorphic stochastic process, then x is S 2 -almost automorphic, that is, AA(R; L2 (P, H)) ⊂ AS 2 (R; L2 (P, H)). Lemma 2.2. Let (xn (t ))n∈N be a sequence of S 2 -almost automorphic stochastic processes such that t +1
∫
E ‖xn (s) − x(s)‖2 ds → 0 for each t ∈ R,
(2.1)
t
as n → ∞, then x ∈ AS 2 (R; L2 (P, H)). Proof. For any i ∈ N fixed, since xi (t ) ∈ AS 2 (R; L2 (P, H)), then for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a stochastic process yi ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖xi (s + sn ) − yi (s)‖2 ds → 0
t +1
∫
E ‖yi (s − sn ) − xi (s)‖2 ds → 0
and
t
(2.2)
t
as n → ∞ pointwise on R. On the other hand, from (2.1), we easily deduce that (xn (t ))n∈N is a Cauchy sequence with respect to ‖ · ‖S 2 . We observe that, for each t ∈ R, the sequence (yi (t )) is also a Cauchy sequence in L2loc (R; L2 (P, H)). Indeed, if we write yi (t ) − yj (t ) = yi (t ) − xi (t + sn ) + xi (t + sn ) − xj (t + sn ) + xj (t + sn ) − yj (t ), then for a sufficiently large n, we get t +1
∫
E ‖yi (s) − yj (s)‖ ds ≤ 3 2
t
t +1
∫
E ‖yi (s) − xi (s + sn )‖ ds + 3 2
t
t +1
∫
E ‖xi (s + sn ) − xj (s + sn )‖2 ds t
t +1
∫
E ‖xj (s + sn ) − yj (s)‖2 ds.
+3 t
By (2.1) and (2.2), the sequence of (yi (t )) is Cauchy.
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
1133
Using the completeness of L2loc (R; L2 (P, H)), we denote by y(t ) the pointwise limit of (yi (t )). Now let us prove that x(t ) ∈ AS 2 (R; L2 (P, H)). Note that the inequality below holds for any index i and any t ∈ R t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds ≤ 3
t +1
∫
E ‖x(s + sn ) − xi (s + sn )‖2 ds + 3
E ‖xi (s + sn ) − yi (s)‖2 ds t
t
t
t +1
∫
t +1
∫
E ‖yi (s) − y(s)‖ ds. 2
+3 t
So, from (2.1) and the fact that y(t ) is the pointwise limit of (yi (t )), for any sufficiently small ε > 0 there exists a sufficiently large i, such that for each t ∈ R t +1
∫
E ‖x(s + sn ) − xi (s + sn )‖2 ds < t
ε 9
∫
,
t +1
E ‖yi (s) − y(s)‖2 ds <
t
ε 9
.
Now for this sufficiently large i, from (2.2), there exists a sufficient N such that n > N we have t +1
∫
E ‖xi (s + sn ) − yi (s)‖2 ds < t
ε 9
.
Thus t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds < ε
for n > N ,
t
which implies t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds → 0 t
as n → ∞ pointwise on R. We can use the same step to prove that t +1
∫
E ‖y(s − sn ) − x(s)‖2 ds → 0 t
as n → ∞ pointwise on R. That is, x(t ) ∈ AS 2 (R; L2 (P, H)). The proof is finished.
Lemma 2.3. Let (xn (t ))n∈N be a sequence of square-mean almost automorphic stochastic processes such that lim E ‖xn (t ) − x(t )‖2 = 0 for each t ∈ R.
n→∞
Then x ∈ AA(R; L2 (P, H)). Proof. It is similar to the proof of Lemma 2.2, so we omit the details.
Remark 2.3. From the proof of [24, Theorem 2.4.], we can easily see that the conclusion of Lemma 2.3 is true. Theorem 2.1. AS 2 (R; L2 (P, H)) is a Banach space when it is equipped with the norm t +1
∫
E ‖x(s)‖ ds 2
‖x‖S 2 = sup t ∈R
21
for x ∈ AS 2 (R; L2 (P, H)).
t
Proof. We can easily obtain this result by using the inclusion AS 2 (R; L2 (P, H)) ⊂ BS 2 (L2 (P, H)) combined with the fact that (BS 2 (L2 (P, H)), ‖ · ‖S 2 ) is a Banach space and Lemma 2.2. Definition 2.8. A function f : R × L2 (P, H) → L2 (P, H), (t , x) → f (t , x) with f (·, x) ∈ L2loc (R; L2 (P, H)) for each x ∈ L2 (P, H), is said to be S 2 -almost automorphic in t ∈ R uniformly in x ∈ L2 (P, H) if t → f (t , x) is S 2 -almost automorphic for each x ∈ L2 (P, H). That means, for every sequence of real numbers {s′n }n∈N , there exist a subsequence {sn }n∈N and a function f : R × L2 (P, H) → L2 (P, H) with f (·, x) ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖f (s + sn , x) − f (s, x)‖2 ds → 0 t
t +1
∫
E ‖ f (s − sn , x) − f (s, x)‖2 ds → 0
and t
as n → ∞ pointwise on R and for each x ∈ L2 (P, H). We denoted by AS 2 (R × L2 (P, H); L2 (P, H)) the set of all such functions.
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Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
Theorem 2.2. Let f : R × L2 (P, H) → L2 (P, H), (t , x) → f (t , x) be S 2 -almost automorphic in t ∈ R uniformly in x ∈ L2 (P, H), and assume that f satisfies the Lipschitz condition in the following sense:
˜ ‖x − y ‖2 E ‖f (t , x) − f (t , y)‖2 ≤ ME ˜ > 0 is independent of t. Then for any S 2 -almost automorphic process for all x, y ∈ L2 (P, H) and for each t ∈ R, where M x : R → L2 (P, H), the stochastic process F : R → L2 (P, H) given by F (t ) = f (t , x(t )) is S 2 -almost automorphic. Proof. Let {s′n }n∈N be an arbitrary sequence of real numbers. Since x ∈ AS 2 (R; L2 (P, H)), there exist a subsequence {sn }n∈N of {s′n }n∈N and a stochastic process y ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖x(s + sn ) − y(s)‖2 ds → 0
t +1
∫
E ‖y(s − sn ) − x(s)‖2 ds → 0
and
(2.3)
t
t
as n → ∞ pointwise on R. On the other hand, Since (t , x) → f (t , x) is S 2 -almost automorphic in t ∈ R uniformly in x ∈ L2 (P, H), we can extract a subsequence {sn }n∈N of {s′n }n∈N (for convenience, we also denote it by {sn }n∈N ) and a function f (·, x) ∈ L2loc (R; L2 (P, H)) such that t +1
∫
E ‖f (s + sn , x) − f (s, x)‖2 ds → 0
t +1
∫
E ‖ f (s − sn , x) − f (s, x)‖2 ds → 0
and
(2.4)
t
t
as n → ∞ pointwise on R and for each x ∈ L2 (P, H). Now, let us consider the function F : R → L2 (P, H) defined by F (t ) := f (t , y(t )), t ∈ R. Note that F (t + sn ) − F (t ) = f (t + sn , x(t + sn )) − f (t + sn , y(t )) + f (t + sn , y(t )) − f (t , y(t )). So, we obtain E ‖F ( t + s n ) − F (t )‖2 ≤ 2E ‖f (t + sn , x(t + sn )) − f (t + sn , y(t ))‖2 + 2E ‖f (t + sn , y(t )) − f (t , y(t ))‖2
˜ ‖x(t + sn ) − y(t )‖2 + 2E ‖f (t + sn , y(t )) − ≤ 2ME f (t , y(t ))‖2 . We can deduce from (2.3) and (2.4) that t +1
∫
E ‖F (s + sn ) − F (s)‖2 ds → 0 t
as n → ∞ pointwise on R. Similarly, using the same argument as above, we can prove that t +1
∫
E ‖ F (s − sn ) − F (s)‖2 ds → 0 t
as n → ∞ pointwise on R, which proves that F (t ) is S 2 -almost automorphic.
Definition 2.9. An Ft -progressively measurable stochastic process {x(t )}t ∈R is called a mild solution of problem (1.1) on R if it satisfies the corresponding stochastic integral equation x(t ) = T (t − a)x(a) +
t
∫
T (t − s)f (s, x(s))ds + a
t
∫
T (t − s)g (s, x(s))dW (s) a
for all t ≥ a and for each a ∈ R. 3. Main results In this section, we investigate the existence of a S 2 -almost automorphic solution for the problem (1.1). To study the existence of S 2 -almost automorphic mild solutions to Eq. (1.1), we first consider the existence of S 2 -almost automorphic mild solutions to the linear stochastic differential equation dx(t ) = Ax(t )dt + f (t )dt + g (t )dW (t ),
t ∈ R,
(3.1)
where A is the infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 on L2 (P, H), f , g ∈ AS 2 (R; L2 (P, H)) ∩ C (R, L2 (P, H)), and W (t ) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω , F , P, Ft ), where Ft = σ {W (u) − W (v); u, v ≤ t }.
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
1135
Definition 3.1. An Ft -progressively measurable stochastic process {x(t )}t ∈R is called a mild solution of problem (3.1) on R if it satisfies the corresponding stochastic integral equation x(t ) = T (t − a)x(a) +
t
∫
t
∫
T (t − s)f (s)ds +
T (t − s)g (s)dW (s) a
a
for all t ≥ a and for each a ∈ R. We first list the following basic assumption: (H1) The operator A is the infinitesimal generator of an exponentially stable C0 -semigroup {T (t )}t ≥0 on L2 (P, H); that is, there exist M > 0, δ > 0 such that ‖T (t )‖ ≤ Me−δ t , for all t ≥ 0. Theorem 3.1. Under previous assumptions, if we assume that (H1) holds, then the problem (3.1) has a unique mild solution x ∈ AS 2 (R; L2 (P, H)). Proof. Let us first prove uniqueness. Assume that x : R → L2 (P, H) is a bounded stochastic process and satisfies the homogeneous equation dx(t ) = Ax(t )dt ,
t ∈ R.
(3.2) −δ(t −s)
Then x(t ) = T (t − s)x(s), for any t ≥ s. Thus ‖x(t )‖ ≤ MK e with ‖x(s)‖ ≤ K for s ∈ R almost surely. Take a sequence of real numbers {sn }n∈N such that sn → −∞ as n → ∞. For any t ∈ R fixed, one can find a subsequence {snk }k∈N ⊂ {sn }n∈N such that snk < t for all k = 1, 2, . . . . By letting k → ∞, we get x(t ) = 0 almost surely. Now, if x1 , x2 : R → L2 (P, H) are bounded solutions to Eq. (3.1), then x = x1 − x2 is a bounded solution to Eq. (3.2). In view of the above, x = x1 − x2 = 0 almost surely, that is, x1 = x2 almost surely. Now let us investigate the existence. Consider for each n = 1, 2, . . . , the integrals xn (t ) =
∫
n
T (σ )f (t − σ )dσ
and
yn (t ) =
n−1
∫
n
T (σ )g (t − σ )dW (σ ) n−1
for each t ∈ R. First, by using Hölder’s inequality, we get t +1
∫
E ‖xn (s)‖ ds = 2
∫
t
t +1
∫ E
t +1
∫
t
∫
n n−1
n
‖T (σ )‖2 E ‖f (s − σ )‖2 dσ ds
≤ n−1
t
≤ M2
t +1
∫
∫
n
∫
n
e−2δσ
n −1
2
M2 2δ
t +1
∫
E ‖f (s − σ )‖2 ds dσ t
≤ M 2 ‖f ‖2S 2 ≤
e−2δσ E ‖f (s − σ )‖2 dσ ds
n−1
t
≤ M2
2 T (σ )f (s − σ )dσ ds
∫
n
e−2δσ dσ
n −1
‖f ‖2S 2 e−2δn (e2δ − 1).
Since M2δ ‖f ‖2S 2 (e2δ − 1) n=1 e−2δ n < ∞, we deduce from the well-known Weierstrass test that the series convergent in the sense of the norm ‖ · ‖S 2 uniformly on R. Now let
Φ (t ) :=
∑∞
∞ −
xn (t ) for each t ∈ R.
n=1
Observe that
Φ (t ) =
∫
t
T (t − s)f (s)ds −∞
Clearly, Φ (t ) ∈ C (R, L2 (P, H)).
for each t ∈ R.
∑∞
n=1
xn (t ) is
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Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
Now, by using an estimate on the Ito integral established in [26], we obtain that t +1
∫
E ‖yn (s)‖2 ds =
t +1
∫ t
t
t +1
∫
n −1
‖T (σ )‖2 E ‖g (s − σ )‖2 dσ ds n−1
t
t +1
∫
∫
∫
n
e−2δσ E ‖g (s − σ )‖2 dσ ds
n−1
t
≤ M2
T (σ )g (s − σ )dW (σ ) ds
n
∫
≤ ≤ M2
2
n
∫
E
n
e−2δσ
n −1
≤ M 2 ‖g ‖2S 2 ≤ 2
M2 2δ
t +1
∫
E ‖g (s − σ )‖2 ds dσ t
∫
n
e−2δσ dσ
n−1
‖g ‖2S 2 e−2δn (e2δ − 1).
Since M2δ ‖g ‖2S 2 (e2δ − 1) n=1 e−2δ n < ∞, we deduce from the Weierstrass test that the series the sense of the norm ‖ · ‖S 2 uniformly on R. Furthermore,
∑∞
Ψ (t ) :=
∫
t
T (t − s)g (s)dW (s) =
−∞
∞ −
yn (t ),
∑∞
n =1
yn (t ) is convergent in
t ∈ R,
n =1
and clearly Ψ (t ) ∈ C (R, L2 (P, H)). Now let us show that each xn , yn ∈ AS 2 (R; L2 (P, H)). First, we prove that xn ∈ AS 2 (R; L2 (P, H)). Indeed, let {s′m }m∈N be a sequence of real numbers. Since f ∈ AS 2 (R; L2 (P, H)), there exist a subsequence {sm }m∈N of {s′m }m∈N and a stochastic process f ∈ L2loc (R; L2 (P, H)) t +1
∫
t +1
∫
E ‖f ( s + s m ) − f (s)‖2 ds → 0
E ‖ f (s − sm ) − f (s)‖2 ds → 0
and
t
t
as m → ∞ pointwise on R. Moreover, if we let xn (t ) = t +1
∫
E ‖xn (s + sm ) − xn (s)‖2 ds =
t +1
∫
t
t
∫ = t
T (σ ) f (t − σ )dσ , we have
T (σ )f (s + sm − σ )dσ − n −1
∫ t +1 E t +1
∫
n−1
n
∫
E
n
∫
n −1
T (σ ) f (s − σ )dσ ds
2
n n −1
2
n
T (σ )[f (s + sm − σ ) − f (s − σ )]dσ ds
n
∫
‖T (σ )‖2 E ‖f (s + sm − σ ) − f (s − σ )‖2 dσ ds
≤ n−1
t
≤ M2
t +1
∫
∫
≤ M2
e−2δσ E ‖f (s + sm − σ ) − f (s − σ )‖2 dσ ds
n −1
t n
∫
n
e−2δσ
n −1
t +1
∫
E ‖f (s + sm − σ ) − f (s − σ )‖2 ds dσ . t
Obviously, the last inequality goes to 0 as m → ∞ pointwise on R. Similarly we can prove that t +1
∫
E ‖ xn (s − sm ) − xn (s)‖2 ds → 0 t
as m → ∞ pointwise on R. Thus we conclude that each xn ∈ AS 2 (R; L2 (P, H)) and consequently their uniform limit Φ (t ) ∈ AS 2 (R; L2 (P, H)), by using Lemma 2.2. Next, we show that each yn ∈ AS 2 (R; L2 (P, H)). Since g ∈ AS 2 (R; L2 (P, H)), then for every sequence of real numbers {s′m }m∈N there exist a subsequence {sm }m∈N ⊂ {s′m }m∈N and a stochastic process g ∈ L2loc (R; L2 (P, H)) such that t +1
∫ t
E ‖g (s + sm ) − g (s)‖2 ds → 0
∫ and t
t +1
E ‖ g (s − sm ) − g (s)‖2 ds → 0
Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
1137
n
as m → ∞ pointwise on R. Moreover, if we let yn (t ) =
T (σ ) g (t − σ )dW (σ ), by using the Ito integral, we get n −1 2 ∫ n ∫ t +1 ∫ t +1 ∫ n 2 T (σ ) g (s − σ )dW (σ ) E T (σ )g (s + sm − σ )dW (σ ) − E ‖yn (s + sm ) − yn (s)‖ ds = ds n−1 t n−1 t 2 ∫ t +1 ∫ n ds E T (σ )[ g ( s + s − σ ) − g ( s − σ )] dW (σ ) = m t n−1 ∫ t +1 ∫ n ‖T (σ )‖2 E ‖g (s + sm − σ ) − g (s − σ )‖2 dσ ds ≤ n −1
t
≤ M2
t +1
∫
n −1
t
≤ M2
∫
n
∫
n
e−2δσ
e−2δσ E ‖g (s + sm − σ ) − g (s − σ )‖2 dσ ds t +1
∫
n−1
E ‖g (s + sm − σ ) − g (s − σ )‖2 ds dσ .
t
Obviously, the last inequality goes to 0 as m → ∞ pointwise on R. Arguing in a similar way, we infer that t +1
∫
E ‖ yn (s − sm ) − yn (s)‖2 ds → 0 t
as m → ∞ pointwise on R. Thus we conclude that each yn ∈ AS 2 (R; L2 (P, H)) and consequently their uniform limit Ψ (t ) ∈ AS 2 (R; L2 (P, H)), by using Lemma 2.2. Define x(t ) =
∫
t
T (t − s)f (s)ds +
∫
−∞
t
T (t − s)g (s)dW (s),
t ∈ R.
−∞
Obviously to Eq. (3.1). Let us prove that x(t ) is a mild solution of Eq. (3.1). Indeed, if we let a x is a bounded solution a x(a) = −∞ T (a − s)f (s)ds + −∞ T (a − s)g (s)dW (s), then T (t − a)x(a) =
∫
a
T (t − s)f (s)ds + −∞
∫
a
T (t − s)g (s)dW (s). −∞
But for t ≥ a, t
∫
∫
T (t − s)g (s)dW (s) =
t
T (t − s)g (s)dW (s) − −∞
a
= x(t ) −
a
∫
T (t − s)g (s)dW (s) −∞
∫
t
T (t − s)f (s)ds + −∞
∫
a
T (t − s)f (s)ds − T (t − a)x(a) −∞
= x(t ) − T (t − a)x(a) −
t
∫
T (t − s)f (s)ds. a
It follows that x(t ) = T (t − a)x(a) +
∫
t
T (t − s)f (s)ds +
a
t
∫
T (t − s)g (s)dW (s). a
In view of the above, it follows that x is the only bounded S 2 -almost automorphic mild solution to Eq. (3.1). The proof is now complete. In order to investigate the solution to (1.1), we need the following assumptions: (H2) The function f ∈ AS 2 (R × L2 (P, H); L2 (P, H)) ∩ C (R × L2 (P, H), L2 (P, H)) and there exists a constant Lf > 0 such that E ‖f (t , x) − f (t , y)‖2 ≤ Lf E ‖x − y‖2 for all t ∈ R and each x, y ∈ L2 (P, H). (H3) The function g ∈ AS 2 (R × L2 (P, H); L2 (P, H)) ∩ C (R × L2 (P, H), L2 (P, H)) and there exists a positive number Lg such that E ‖g (t , x) − g (t , y)‖2 ≤ Lg E ‖x − y‖2 for all t ∈ R and each x, y ∈ L2 (P, H).
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Y.-K. Chang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 1130–1139
Theorem 3.2. Assume the conditions (H1)–(H3) are satisfied, then the problem (1.1) has a unique S 2 -almost automorphic mild solution on R provided that
[ L0 =
2
M2
2
δ2
M Lf +
δ
] Lg
< 1.
(3.3)
Proof. Let Λ : AS 2 (R; L2 (P, H)) → AS 2 (R; L2 (P, H)) be the operator defined by t
∫
Λx(t ) =
T (t − s)f (s, x(s))ds +
∫
t
T (t − s)g (s, x(s))dW (s),
t ∈ R.
−∞
−∞
From previous assumptions and the properties of {T (t )}t ≥0 one can easily see that Λx is well-defined and continuous. By Theorem 2.2, we infer that both F (·) = f (·, x(·)) and G(·) = g (·, x(·)) ∈ AS 2 (R; L2 (P, H)). Then by using the proof of Theorem 3.1 with the above Lemma 2.2, we have that Λx ∈ AS 2 (R; L2 (P, H)) whenever x ∈ AS 2 (R; L2 (P, H)). Thus Λ maps AS 2 (R; L2 (P, H)) into itself. Now we prove that Λ is a contraction mapping on AS 2 (R; L2 (P, H)). Indeed, for each t ∈ R, x, y ∈ AS 2 (R; L2 (P, H)), we see that t +1
∫
E ‖(Λx)(s) − (Λy)(s)‖ ds = 2
t +1
∫
t
t
2
t +1
E
t
∫ +2 t
2
2
s
∫
≤2
−∞
∫ t +1 E
−∞
∫ t +1
∫
t +1
t +1
s
∫
≤
≤
δ 2
t +1
∫
δ [
E ‖f (σ , x(σ )) − f (σ , y(σ ))‖ dσ 2
ds
−∞
e−δ(s−σ ) E ‖f (σ , x(σ )) − f (σ , y(σ ))‖2 dσ ds s
∫
e−2δ(s−σ ) E ‖g (σ , x(σ )) − g (σ , y(σ ))‖2 dσ ds
−∞ t +1
∫
∫
s
∞
∫
e−δ(s−σ ) E ‖x(σ ) − y(σ )‖2 dσ ds + 2M 2 Lg
−∞
t
M 2 Lf
e
−δ(s−σ )
−∞
t
M 2 Lf
s
‖T (s − σ )‖2 E ‖g (σ , x(σ )) − g (σ , y(σ ))‖2 dσ ds
t
2
dσ
∫
−∞
∫
+ 2M 2 ≤
−δ(s−σ )
s
∫
t
δ
s
e
∫
M2
2 T (s − σ )[g (σ , x(σ )) − g (σ , y(σ ))]dW (σ ) ds
−∞
+2 2
T (s − σ )[f (σ , x(σ )) − f (σ , y(σ ))]dσ ds
s
t
≤
T (s − σ )[f (σ , x(σ )) − f (σ , y(σ ))]dσ −∞
T (s − σ )[g (σ , x(σ )) − g (σ , y(σ ))]dW (σ ) ds
−∞
≤ 2M
s
s
∫ + ∫
∫ E
e−δξ
2
M2
δ
δ
M 2 Lf + 2
∫
∫
t −ξ +1
E ‖x(s) − y(s)‖2 dsdξ + 2M 2 Lg
∞
∫ 0
s
e−2δ(s−σ ) E ‖x(σ ) − y(σ )‖2 dσ ds
−∞
t
t −ξ
0
t +1
∫
e−2δξ
∫
t −ξ +1
E ‖x(s) − y(s)‖2 dsdξ t −ξ
]
Lg ‖x − y‖2S 2 .
Hence
‖Λx − Λy‖S 2 = sup t ∈R
t +1
∫
E ‖(Λx)(s) − (Λy)(s)‖2 ds
21 ≤
L0 ‖x − y‖S 2 ,
t
which implies that Λ is a contraction by (3.3). So by the Banach contraction principle, we draw a conclusion that there exists a unique fixed point x(·) for Λ in AS 2(R; L2 (P, H)), such that Λx = x. Moreover, using the same proof as in Theorem 3.1, we t t can see that x(t ) = T (t − a)x(a) + a T (t − s)f (s, x(s))ds + a T (t − s)g (s, x(s))dW (s) is a mild solution of Eq. (1.1) and x(·) ∈ AS 2 (R; L2 (P, H)). The proof is complete.
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