# On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

Authors: Jing Cui; Litan Yan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.


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## Body

## 1. Introduction

The concept of almost periodicity is of great importance in probability for investigating stochastic processes [1–3]. The basic results on the almost periodic functions and their applications to deterministic differential equations may refer to [4, 5] and references therein. The concept of almost automorphy introduced initially by Bochner [6] is an important generalization of the classical almost periodicity. Since then, there has been an intense interest in studying several extensions of this concept such as asymptotic almost automorphy, p-almost automorphy, and Stepanov-like almost automorphy (see [5, 7–9] and references therein). Much of the motivation has come from mathematical physics, mathematical biology, and various fields of science and engineering [10–12].Besides, it should be pointed out that noise or stochastic perturbation is unavoidable and omnipresent in nature as well as that in man-made systems. Therefore, we must import the stochastic effects into the investigation of differential systems. In fact, the existence of almost periodic solutions for stochastic differential systems has been thoroughly investigated (see [13–17] and reference therein) while the existence of almost automorphic solutions for stochastic version has been in growing state. More precisely, in [18], the concept of square-mean almost automorphic process was introduced and investigated. Particularly, such a concept was utilized to study the existence and stability of square-mean almost automorphic mild solutions for a class of stochastic differential equations of the form
(1.1)dx(t)=Ax(t)+f(t,x(t))dt+g(t,x(t))dW(t),t∈R,
in a Hilbert space; Chang  et al. [19] extended the results in [18] to nonautonomous stochastic differential equations in Hilbert spaces; in [20], the square-mean pseudo almost automorphic process and its application to (1.2) were investigated; in [21], existence and exponential stability of almost automorphic mild solutions were considered to a class of stochastic differential equations with finite delay of the form
(1.2)dx(t)=Ax(t)+f(t,x(t),xt)dt+g(t,x(t),xt)dW(t),t∈R;
one can also see [22, 23] for the existence of square-mean almost automorphic mild solutions of stochastic differential equations.In this paper, we consider a general setting; that is, we make extensive use of intermediate space techniques to investigate the existence of square-mean almost automorphic mild solutions to the class of abstract nonautonomous neutral stochastic evolution equations of the form(1.3)d[x(t)+g(t,B1x(t))]=[A(t)x(t)+b(t,B2x(t))]dt+σ(t,B3x(t))dW(t),t∈R,
where A(t):𝒟(A(t))⊂L2(P,H)→L2(P,H) is a family of closed linear operators whose corresponding analytic semigroup is exponential dichotomy, Bi, i=1,2,3, are bounded operators, W(t) is a Q-Brownian motion defined on a probability space (Ω,ℱ,P) with a filtration ℱt=σ{(W(u)-W(v)):u,v≤t}, and g:L2(P,H)→L2(P,Hβ), b:L2(P,H)→L2(P,H), and σ:L2(P,H)→L2(P,ℒ20(K,H)) are jointly continuous functions to be specified later.The rest of this paper is organized as follows. In Section2 we present some basic notations and preliminary results. Section 3 is devoted to the study of existence of almost automorphic mild solutions for systems (1.3).


## 2. Preliminaries

For more details on this section, we refer to Da Prato et al. [24], Diagana [8], and Fu-Liu [18]. Throughout this paper, we assume that (H,∥·∥), (K,∥·∥K) are real separable Hilbert spaces and (Ω,ℱ,{ℱt}t≥0,P) is supposed to be a filtered complete probability space. Denote by L2(P,H) the Banach space of all H-valued random variables x such that E∥x∥2=∫Ω‍∥x∥2dP<+∞ endowed with the norm ∥x∥2=(𝔼∥x∥2)1/2. If K1,K2 are Banach spaces, we denote by ℒ(K1,K2) the Banach spaces of bounded linear operators from K1 to K2 equipped with natural operator norm; when K1=K2, this is simply denoted by ℒ(K1). Furthermore, ℒ20(K,H) denotes the space of all Q-Hilbert-Schmidt operators from K to H with the norm
(2.1)|ξ|L202:=tr⁡(ξQξ*)<∞,ξ∈L(K,H).Fort∈ℝ, A(t):𝒟(A(t))⊂L2(P,H)→L2(P,H) is a family of closed linear operators (not necessarily densely defined) satisfying the so-called Acquistapace-Terreni conditions (ATCs for short; see Lemma 2.3). If L is a linear operator on H, then the symbols 𝒟(L),ρ(L),σ(L),N(L),R(L) stand, respectively, for the domain, resolvent set, spectrum, kernel, and range of L. We also set R(λ,L):=(λI-L)-1 for all λ∈ρ(A) and Q=I-P for a projection P.Definition 2.1.
A family of bounded linear operators{U(t,s):t≥s,t,s∈ℝ} on L2(P,H) associated with A(t) is said to be an evolution family of operators if the following conditions hold:  (i)
U(t,s)U(s,r)=U(t,r) for all t,s,r∈ℝ, such that t≥s≥r;(ii)
U(t,t)=I, for t∈ℝ;(iii)
(t,s)→U(t,s)∈ℒ(L2(P,H)) is strongly continuous, for t>s;(iv)
U(·,s)∈C′((s,∞),ℒ(H)) and
(2.2)∂∂tU(t,s)=A(t)U(t,s).Definition 2.2 (see [8]).
One says that an evolution family{U(t,s):t≥s,t,s∈ℝ} is exponential dichotomy (or hyperbolic) if there are projections P(t), t∈ℝ, being uniformly bounded and strongly continuous in t and constants δ>0 and N≥1 such that  (1)
U(t,s)P(s)=P(t)U(t,s);(2)
the restrictionUQ(t,s):Q(s)H→Q(t)H of U(t,s) is invertible, and we set UQ(s,t):=UQ(t,s)-1;(3)
∥U(t,s)P(s)∥≤Ne-δ(t-s)∥ and ∥UQ(s,t)Q(t)∥≤Ne-δ(t-s), for t≥s, t,s∈ℝ.
IfU has an exponential dichotomy, then the operator family
(2.3)Γ(t,s):={U(t,s)P(s),t≥s,t,s∈R;-UQ(t,s)Q(s),t<s,t,s∈R
is called Green's function corresponding to U and P(·). If P(t)=I for t∈ℝ, then U is exponentially stable.
The following lemma holds by [25].Lemma 2.3.
IfA(t) satisfy the ATCs; that is, there exists a positive constant λ0 such that the operator A(t),t∈ℝ, satisfying
(2.4)Σθ∪{0}⊆ρ(A(t)-λ0),‖R(λ,A(t)-λ0)‖≤K1+|λ|,‖(A(t)-λ0)R(λ0,A(t)-λ0)[R(λ0,A(t))-R(λ0,A(s))]‖≤L|t-s|μ|λ|-ν
for t,s∈ℝ, λ∈Σθ:={λ∈ℂ-{0}:|argλ|≤θ} and constants θ∈(π/2,π), L,K≥0, μ,ν∈(0,1] with μ+ν>1, then there exists a unique evolution family {U(t,s):t≥s,t,s∈ℝ} on L2(P,H).Definition 2.4 (see [26]).
A linear operatorA:𝒟(A)⊂H→H (not necessarily densely defined) is said to be sectorial if the following hold.
There exist constantsζ∈ℝ, θ∈(π/2,π) and M>0 such that
(2.5)ρ(A)⊃Sθ,ζ:={λ∈C:λ≠ζ,|arg(λ-ζ)|<θ},‖R(λ,A)‖≤M|λ-ζ|,λ∈Sθ,ζ.
LetA be a sectorial operator on H and α∈(0,1). Define the real interpolation space
(2.6)HαA:={x∈H:‖x‖αA:=sup⁡‖rα(A-λ0)R(r,A-λ0)x‖<∞};
it is a Banach space endowed with the norm ∥·∥αA. Given a family of linear operators A(t), t∈ℝ, for α∈[0,1], we set Hαt:=HαA(t) with the corresponding norms.
The following estimates for the evolution familyU appeared in [8] are useful.Lemma 2.5.
Forx∈H, 0≤α≤1 and t>s, there exist some constants c(α), m(α) such that
(2.7)‖U(t,s)P(s)x‖αt≤c(α)e-(δ/2)(t-s)‖x‖,(2.8)‖U~Q(s,t)Q(t)x‖αs≤m(α)e-δ(t-s)‖x‖.Throughout the rest of this paper, we assume that the following conditions onA(·) and U hold:(C1)
ATCs are satisfied and the evolution familyU generated by A(·) has an exponential dichotomy with constants N,δ and dichotomy projections P(t) for t∈ℝ. Moreover, 0∈ρ(A(t)) for each t∈ℝ and the following holds:
(2.9)sup⁡t,s∈R‖A(s)A-1(t)‖L(H,Hα)≤c0(C2)
there exists0<α<β<1 with 2β>α+1 such that
(2.10)Hαt=Hα,Hβt=Hβ,for allt∈ℝ, with uniform equivalent norms. And there exist constants k(α), k1 such that
(2.11)E‖x‖2≤k1E‖x‖α2,x∈L2(P,Hα),E‖x‖α2≤k(α)E‖x‖β2,x∈L2(P,Hβ).Lemma 2.6 (see [8]).
Under the above assumptions, there exist constantsm(α,β),n(α,β)>0 such that
(2.12)‖A(s)U(t,s)P(s)x‖α≤n(α,β)(t-s)-αe-(δ/2)(t-s)‖x‖β,t>s,‖A(s)U~Q(t,s)Q(s)x‖α≤m(α,β)e-δ(s-t)‖x‖β,t≤s.We recall some basic definitions and results of square-mean almost automorphic processes (see [18, 19]).Let(𝔹,∥·∥𝔹) be a Banach space and L2(P,𝔹) its L2-space.Definition 2.7.
A stochastic processx:ℝ→L2(P,𝔹) is said to be stochastically continuous if
(2.13)lim⁡t→sE‖x(t)-x(s)‖B2=0.Definition 2.8 (see [18, 21]).
A stochastically continuous stochastic processx:ℝ→L2(P,𝔹) is said to be square-mean almost automorphic if for every sequence of real numbers {sn′}n∈ℕ there exists a subsequence {sn}n∈ℕ such that
(2.14)lim⁡m,n→∞E‖x(t+sn-sm)-x(t)‖B2=0.
This is equivalent to that there exists a stochastic processy:ℝ→L2(P,𝔹) such that, for each t∈ℝ,
(2.15)lim⁡n→∞E‖x(t+sn)-y(t)‖B2=0,lim⁡n→∞E‖y(t-sn)-x(t)‖B2=0.
Denote bySAA(ℝ,L2(P,𝔹)) the collection of all the square-mean almost automorphic processes x(t):ℝ→L2(P,𝔹). It is a Banach space equipped with the usual sup-norm
(2.16)‖x‖∞:=sup⁡t∈R(E‖x(t)‖B2)1/2.Lemma 2.9 (see [18]).
Ifx,y are square-mean almost automorphic processes, one has  (i)
x+y is square-mean almost automorphic;(ii)
λx is square-mean almost automorphic for every scalar λ;(iii)
there exists a positive constantM such that ∥x∥∞2≤M.Let(𝔹i,∥·∥𝔹i), i=1,2 be Banach spaces, and L2(P,𝔹i), i=1,2, their corresponding L2-spaces, respectively.Definition 2.10 (compare with [18, 21]).
A functionf:ℝ×L2(P,𝔹1)→L2(P,𝔹2), (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,𝔹1), if, for every sequence of real numbers {sn′}n∈ℕ, there exists a subsequence {sn}n∈ℕ such that
(2.17)lim⁡m,n→∞E‖f(t+sn-sm,x)-f(t,x)‖B22=0,
for each t∈ℝ and x∈L2(P,𝔹1).This is equivalent to that there exists a functionf~:ℝ×L2(P,𝔹1)→L2(P,𝔹2) such that, for each t∈ℝ and x∈L2(P,𝔹1),
(2.18)lim⁡n→∞E‖f(t+sn,x)-f~(t,x)‖B22=0,lim⁡n→∞E‖f~(t-sn,x)-f(t,x)‖B22=0.We need the following composition of square-mean almost automorphic processes.Lemma 2.11.
Suppose thatf:ℝ×L2(P,𝔹1)→L2(P,𝔹2) is square-mean almost automorphic in t, and assume that f(t,x) satisfies
(2.19)‖f(t,x)-f(t,y)‖B22≤κ(‖x-y‖B12),foranyx,y∈L2(P,B1),t∈R,
where κ is a concave nondecreasing function from ℝ+ to ℝ+ such that κ(0)=0, κ(u)>0 and ∫0+‍(du/κ(u))=+∞. Then for any square-mean almost automorphic process ϕ:ℝ→L2(P,𝔹1), the stochastic process F:ℝ→L2(P,𝔹2) given by F(s):=f(s,ϕ(s)) is square-mean almost automorphic.Proof.
Sincef and ϕ are square-mean almost automorphic processes, for every sequence of real numbers {sn′}n∈ℕ, there exist a subsequence {sn}n∈ℕ⊂{sn′}n∈ℕ and some functions f~, ϕ~ such that, for each t∈ℝ, x∈L2(P,𝔹1),
(2.20)lim⁡n→∞E‖f(t+sn,x)-f~(t,x)‖B22=0,lim⁡n→∞E‖ϕ(t+sn)-ϕ~(t)‖B12=0.
Let F~(t):=f(t,ϕ~(t)). Then we have
(2.21)E‖F(t+sn)-F~(t)‖B22≤2E‖f(t+sn,ϕ(t+sn))-f(t+sn,ϕ~(t))‖B22+2E‖f(t+sn,ϕ~(t))-f~(t,ϕ~(t))‖B22≤2Eκ(‖ϕ(t+sn)-ϕ~(t)‖B12)+2E‖f(t+sn,ϕ~(t))-f~(t,ϕ~(t))‖B22;
by Jensen's inequality, it follows that
(2.22)E‖F(t+sn)-F~(t)‖B22≤2κ(E‖ϕ(t+sn)-ϕ~(t)‖B12)+2E‖f(t+sn,ϕ~(t))-f~(t,ϕ~(t))‖B22;
noting that κ is concave and κ(0)=0, we deduce that
(2.23)lim⁡n→∞E∥F(t+sn)-F~(t)∥B22=0,foreacht∈R.
Similarly, we can prove that limn→∞E∥F~(t-sn)-F(t)∥𝔹22=0; this completes the proof.Lemma 2.12 (see [22]).
LetL∈ℒ(L2(P,𝔹1),L2(P,𝔹2)) and assume f∈SAA(ℝ,L2(P,𝔹1)). Then Lf∈SAA(ℝ,L2(P,𝔹2)).The consideration is mainly based on the following fixed point theorem of Krasnoselskii (see [27]).Lemma 2.13.
LetC be a closed, bounded, and convex subset of a Banach space X. Let A and B be operators, defined on C satisfying the conditions:(a)
Aϕ+Bφ∈C when ϕ,φ∈C;(b)
the operatorA is a contraction;(c)
the operatorB is continuous and B(C) is contained in a compact set.
Then the equationx=Ax+Bx has a solution in C.


## 3. Existence of Square-Mean Almost Automorphic Mild Solutions

Firstly, we present the definition of mild solution for system (1.3).Definition 3.1.
An continuous stochastic functionx:ℝ×L2(P,Hα) is called a mild solution of (1.3) provided that the function
(3.1)s⟶E‖A(s)U(t,s)P(s)g(s,B1x(s))‖2
is integrable on (-∞,t),
(3.2)s⟶E‖A(s)U(t,s)Q(s)g(s,B1x(s))‖2
is integrable on (t,∞) for each t∈ℝ, and x(t) satisfies the following stochastic integral equation:
(3.3)x(t)=-g(t,B1x(t))-∫-∞t‍A(s)U(t,s)P(s)g(s,B1x(s))ds+∫t∞‍A(s)U(t,s)Q(s)g(s,B1x(s))ds+∫-∞t‍U(t,s)P(s)b(s,B2x(s))ds-∫t∞‍U(t,s)Q(s)b(s,B2x(s))ds+∫-∞t‍U(t,s)P(s)σ(s,B3x(s))dW(s)-∫t∞‍U(t,s)Q(s)σ(s,B3x(s))dW(s).In order to obtain our main results, we need the following assumptions.(A1)
Bi:L2(P,Hα)→L2(P,H),i=1,2,3, are bounded linear operators, and we set
(3.4)η:={‖Bi‖L(L2(P,Hα),L2(P,H)),i=1,2,3}.(A2)
R(λ0,A(·))∈ℒ(SAA(ℝ,L2(P,Hα))). For any sequence of real numbers {sn′}n∈ℕ, there exists a subsequence {sn}n∈ℕ such that, for each ϵ>0, one can find N0∈ℕ such that
(3.5)‖A(s+sn-sm)U(t+sn-sm,s+sn-sm)P(s+sn-sm)-A(s)U(t,s)P(s)‖α≤ϵH(t-s)whenever n,m>N0, t,s∈ℝ, t>s, where H:[0,∞)→[0,∞) is integrable. (A3)
Let0≤α<1/2<β<1. g:ℝ×L2(P,H)→L2(P,Hβ) is square-mean almost automorphic in t∈ℝ, and there exists a small Kg>0 such that
(3.6)E‖g(t,x)-g(t,y)‖β2≤KgE‖x-y‖2,for all t∈ℝ and x,y∈L2(P,H). (A4)
b:ℝ×L2(P,H)→L2(P,H), σ:ℝ×L2(P,H)→L2(P,ℒ20(K,H)) are square-mean almost automorphic in t∈ℝ, and, for each t∈ℝ, ϕ,φ∈L2(P,H),
(3.7)‖b(t,ϕ)-b(t,φ)‖2∨‖σ(t,ϕ)-σ(t,φ)‖L202≤κ(‖ϕ-φ‖2),where κ(·):ℝ+→ℝ+ is a concave nondecreasing function such that κ(0)=0, κ(u)>0 for u>0 and ∫0+‍(du/κ(u))=+∞. (A5)
For anyϵ>0, there exist a constant ω>0 and nondecreasing continuous functions Λ:ℝ+→ℝ+ such that, for all t∈ℝ and x∈L2(P,H) with E∥x∥2>ω,
(3.8)E‖b(t,x)‖2∨E‖σ(t,x)‖L202≤ϵΛ(E‖x‖2).Remark 3.2.
Functions such asκ1(u)=u and
(3.9)κ2(u)={u⋅(log⁡1u),0≤u≤b<1,b⋅(log⁡1u)+κ2(b-)′(u-b),u≥b
satisfy assumption (A4); in particular, we see that the Lipschitz condition is a special case of the proposed assumptions.Throughout the rest of this paper, we denote by(3.10)‖x‖α,∞:=sup⁡t∈R(E‖x(t)‖α2)1/2
the sup-norm of the space SAA(R,L2(P,Hα)). Let Πi,i=1,…,6, be the operators defined by
(3.11)Π1x(t)=∫-∞t‍A(s)U(t,s)P(s)g(s,B1x(s))ds,Π2x(t)=∫t∞‍A(s)U(t,s)Q(s)g(s,B1x(s))ds,Π3x(t)=∫-∞t‍U(t,s)P(s)b(s,B2x(s))ds,Π4x(t)=∫t∞‍U(t,s)Q(s)b(s,B2x(s))ds,Π5x(t)=∫-∞t‍U(t,s)P(s)σ(s,B3x(s))dW(s),Π6x(t)=∫t∞‍U(t,s)Q(s)σ(s,B3x(s))dW(s).Lemma 3.3.
Under assumptions (A1)–(A3), the operators Πi, i=1,2, defined above map SAA(ℝ,L2(P,Hα)) into itself.Proof.
Letx∈SAA(ℝ,L2(P,Hα)). By Lemma 2.12, s→Bix(s)∈SAA(ℝ,L2(P,H)) as Bi∈ℒ(L2(P,Hα),L2(P,H)), i=1,2,3. And hence, G(s):=g(s,B1x(s))∈SAA(ℝ,L2(P,Hβ)) by Lemma 2.11. In particular, supt∈ℝE∥G(t)∥β2<∞. Let us show that Π1x∈SAA(ℝ,L2(P,Hα)). Indeed, since G(s)∈SAA(ℝ,L2(P,Hβ)), for every sequence of real numbers {sn′}n∈ℕ, there exists a subsequence {sn}n∈ℕ such that, for each t∈ℝ,
(3.12)lim⁡m,n→∞E‖G(t+sn-sm)-G(t)‖β2=0.
From (A2), for any ϵ>0, one can find N0∈ℕ such that
(3.13)‖A(s+sn-sm)U(t+sn-sm,s+sn-sm)P(s+sn-sm)-A(s)U(t,s)P(s)‖α≤ϵH(t-s),
whenever n,m>N0, t,s∈ℝ, t>s. Thus,
(3.14)E‖(Π1x)(t+sn-sm)-(Π1x)(t)‖α2=E‖∫-∞t+sn-sm‍A(s)U(t+sn-sm,s)P(s)G(s)ds-∫-∞t‍A(s)U(t,s)P(s)G(s)ds‖α2=E‖∫-∞t‍A(s+sn-sm)U(t+sn-sm,s+sn-sm)P(s+sn-sm)⋅G(s+sn-sm)ds-∫-∞t‍A(s)U(t,s)P(s)G(s)ds‖α2≤2E‖∫-∞t‍[A(s+sn-sm)U(t+sn-sm,s+sn-sm)⋅P(s+sn-sm)-A(s)U(t,s)P(s)]G(s)ds∫-∞t‍‖α2+2E‖∫-∞t‍A(s+sn-sm)U(t+sn-sm,s+sn-sm)⋅P(s+sn-sm)[G(s+sn-sm)-G(s)]ds∫-∞t‍‖α2=2(L1+L2).
Using (3.13) and condition (C1), one has
(3.15)L1≤E(∫-∞t‍‖(A(s+sn-sm)U(t+sn-sm,s+sn-sm)⋅P(s+sn-sm)-A(s)U(t,s)P(s))G(s)‖αds∫-∞t‍)2≤ϵ2∫-∞t‍H(t-s)ds∫-∞t‍H(t-s)E‖G(s)‖α2ds≤(∫-∞t‍H(t-s)ds)2k(α)sup⁡t∈RE‖G(t)‖β2≤ϵ2k(α)‖H‖L12sup⁡t∈RE‖G(t)‖β2.
For L2, we use Lemma 2.6 to get
(3.16)L2≤E(∫-∞t‍‖A(s+sn-sm)U(t+sn-sm,s+sn-sm)⋅P(s+sn-sm)[G(s+sn-sm)-G(s)]‖αds∫-∞t‍)2≤n2(α,β)E[∫-∞t‍(t-s)-αe-(δ/2)(t-s)‖G(s+sn-sm)-G(s)‖βds]2≤n2(α,β)Γ2(1-α)sup⁡t∈RE‖G(s+sn-sm)-G(s)‖β2.
Combing this estimates with (3.12), one obtains
(3.17)lim⁡m,n→∞E‖(Π1x)(t+sn-sm)-(Π1x)(t)‖α2=0,
which implies that Π1x∈SAA(ℝ,L2(P,Hα)). By a similar argument, we can show that Π2x∈SAA(ℝ,L2(P,Hα)).Lemma 3.4.
Under assumptions(A1),(A2), and (A4), the operators Πi, i=3,4, defined above map SAA(ℝ,L2(P,Hα)) into itself.Proof .
Letx∈SAA(ℝ,L2(P,Hα)). By Lemmas 2.11 and 2.12 it follows that Ξ(s):=b(s,B2x(s))∈SAA(ℝ,L2(P,H)). Particularly, supt∈ℝE∥Ξ(t)∥2<∞. We now show that Π3x∈SAA(ℝ,L2(P,Hα)). Since Ξ(s):=b(s,B2x(s))∈SAA(ℝ,L2(P,H)), for every sequence of real numbers {sn′}n∈ℕ, there exists a subsequence {sn}n∈ℕ such that, for each t∈ℝ,
(3.18)lim⁡m,n→∞E‖Ξ(t+sn-sm)-Ξ(t)‖2=0.
For anyϵ>0, by making changes of variables we have
(3.19)E‖(Π3x)(t+sn-sm)-(Π3x)(t)‖α2=E‖∫-∞t+sn-sm‍U(t+sn-sm,s)P(s)Ξ(s)ds-∫-∞t‍U(t,s)P(s)Ξ(s)ds‖α2=E‖∫-∞t‍U(t+sn-sm,s+sn-sm)P(s+sn-sm)Ξ(s+sn-sm)ds-∫-∞t‍U(t,s)P(s)Ξ(s)ds‖α2=E‖∫0∞‍U(t+sn-sm,t-s+sn-sm)P(t-s+sn-sm)Ξ(t-s+sn-sm)ds-∫-∞t‍U(t,t-s)P(t-s)Ξ(t-s)ds‖α2;
an elementary inequality shows that (3.20)E‖(Π3x)(t+sn-sm)-(Π3x)(t)‖α2≤3E‖∫0∞‍U(t+sn-sm,t-s+sn-sm)P(t-s+sn-sm)×[Ξ(t-s+sn-sm)-Ξ(t-s)]ds∫0∞‍‖α2+3E‖∫0ϵ‍[U(t+sn-sm,t-s+sn-sm)P(t-s+sn-sm)-U(t,t-s)P(t-s)]Ξ(t-s)ds∫0∞‍‖α2+3E‖∫ϵ∞‍[U(t+sn-sm,t-s+sn-sm)P(t-s+sn-sm)-U(t,t-s)P(t-s)]Ξ(t-s)ds∫0∞‍‖α2=3(I1+I2+I3).
Using Lemma 2.5, one has
(3.21)I1≤c2(α)E[∫0t‍s-αe-(δ/2)s‖Ξ(t-s+sn-sm)-Ξ(t-s)‖ds]2≤c2(α)(δ2)α-1Γ2(1-α)sup⁡t∈RE‖Ξ(t+sn-sm)-Ξ(t)‖2;
a straightforward computation yields
(3.22)I2≤E[∫0ϵ‍‖[U(t+sn-sm,t-s+sn-sm)P(t-s+sn-sm)-U(t,t-s)P(t-s)]Ξ(t-s)‖αds]2≤4ϵ2M2sup⁡t∈RE‖Ξ(t)‖2,
where M is a constant satisfying supt,s∥U(t,s)P(s)∥≤M.
ForI3, applying Proposition 4.4 in [4], we have
(3.23)I3≤E[∫ϵ∞‍‖[U(t+sn-sm,t-s+sn-sm)P(t-s+sn-sm)-U(t,t-s)P(t-s)]Ξ(t-s)∥αds∫ϵ∞‍]2≤ϵ2∫ϵ∞‍e-(δ/2)sds∫ϵ∞‍e-(δ/2)sE‖Ξ(t-s)‖2ds≤4ϵ2δ-2sup⁡t∈RE‖Ξ(t)‖2.
Combing this estimates with (3.18), we get
(3.24)lim⁡m,n→∞E‖Π3(t+sn-sm)-Π3(t)‖α2=0,
which implies that Π3x∈SAA(ℝ,L2(P,Hα)) whenever x∈SAA(ℝ,L2(P,Hα)). Similarly, we can verify that Π4x∈SAA(ℝ,L2(P,Hα)).Lemma 3.5.
Under assumptions(A1),(A2) and (A4), the operators Πi, i=5,6, defined above map SAA(ℝ,L2(P,Hα)) into itself.Proof.
Letx∈SAA(ℝ,L2(P,Hα)). Using Lemmas 2.11 and 2.12, we get Σ(s):=σ(s,B3x(s))∈SAA(ℝ,L2(P,ℒ20(K,H))). Particularly, supt∈ℝE∥Σ(t)∥ℒ202<∞. We now show that Π5x∈SAA(ℝ,L2(P,Hα)). Since Σ(s):=σ(s,B3x(s))∈SAA(ℝ,L2(P,H)), for every sequence of real numbers {sn′}n∈ℕ, there exists a subsequence {sn}n∈ℕ such that, for each t∈ℝ,
(3.25)lim⁡m,n→∞E‖Σ(t+sn-sm)-Σ(t)‖L202=0.
Note that the process W~(τ):=W(t+τ)-W(t) for each τ∈ℝ is also a Brownian motion and has the same distribution as W. For any ϵ>0, similar argument as above, we have
(3.26)E‖(Π5x)(t+sn-sm)-(Π5x)(t)‖α2=E‖∫-∞t‍U(t+sn-sm,s+sn-sm)P(s+sn-sm)Σ(s+sn-sm)dW~(s)-∫-∞t‍U(t,s)P(s)Σ(s)dW(s)‖α2=E‖∫0∞‍U(t+sn-sm,t-s+sn-sm)P(t-s+sn-sm)⋅Σ(t-s+sn-sm)dW(s)-∫0∞‍U(t,t-s)P(t-s)Σ(t-s)dW(s)‖α2≤3c2(α)∫0∞‍s-2αe-δsE‖Σ(t-s+sn-sm)-Σ(t-s)‖L202ds+12ϵM2sup⁡t∈RE‖Σ(t)‖L202+3ϵ2∫ϵ∞‍e-δsE‖Σ(t-s)‖L202ds≤3c2(α)δ2α-1Γ(2α-1)sup⁡t∈RE‖Σ(t+sn-sm)-Σ(t)‖L202+12ϵM2sup⁡t∈RE‖Σ(t)‖L202+3δ-1ϵ2sup⁡t∈RE‖Σ(t)‖L202.
From (3.25), we immediately get
(3.27)lim⁡m,n→∞E‖Π5(t+sn-sm)-Π5(t)‖α2=0,
which implies that Π5x∈SAA(ℝ,L2(P,Hα)). Similarly, we can show that Π6x∈SAA(ℝ,L2(P,Hα)) whenever x∈SAA(ℝ,L2(P,Hα)).
Consider the nonlinear operatorsQi, i=1,2, on SAA(R,L2(P,Hα)) defined by
(3.28)Q1x(t)=-g(t,B1x(t))-∫-∞t‍A(s)U(t,s)P(s)g(s,B1x(s))ds+∫t∞‍A(s)U(t,s)Q(s)g(s,B1x(s))ds,Q2x(t)=∫-∞t‍U(t,s)P(s)b(s,B2x(s))ds-∫t∞‍U(t,s)Q(s)b(s,B2x(s))ds+∫-∞t‍U(t,s)P(s)σ(s,B3x(s))dW(s)-∫t∞‍U(t,s)Q(s)σ(s,B3x(s))dW(s),
for each t∈ℝ. In view of Lemmas 3.3, 3.4, and 3.5, it follows that Qi, i=1,2, map SAA(ℝ,L2(P,Hα)) into itself. In what follows, we will prove that Qi, i=1,2, satisfy all the conditions in Lemma 2.13.Lemma 3.6.
Under assumptions (A1)–(A3), the operator Q1 defined above is a contraction provided that
(3.29)Θ:=3Kgη(k(α)+n2(α)Γ2(1-α)(2δ)2(1-α)+m2(α,β)δ-2)<1.Proof.
Letx,y∈SAA(ℝ,L2(P,Hα)). By using condition (C2) and assumptions (A1),(A3), we have
(3.30)E‖g(t,B1x(t))-g(t,B1y(t))‖α2≤k(α)E‖g(t,B1x(t))-g(t,B1y(t))‖β2≤k(α)KgE∥B1x(t)-B1y(t)∥2≤k(α)Kgη∥x-y∥α,∞2.
Now, using Lemma 2.6 together with Hölder's inequality, we obtain
(3.31)E‖∫-∞t‍A(s)U(t,s)P(s)[g(s,B1x(s))-g(s,B1y(s))]ds‖α2≤E[∫-∞t‍‖A(s)U(t,s)P(s)[g(s,B1x(s))-g(s,B1y(s))]‖αds]≤n2(α)∫-∞t‍(t-s)-αe-(δ/2)(t-s)ds×∫-∞t‍(t-s)-αe-(δ/2)(t-s)⋅E‖g(s,B1x(s))-g(s,B1y(s))‖β2ds≤n2(α)KgηΓ2(1-α)(2δ)2(1-α)‖x-y‖α,∞2.
Similarly,
(3.32)E‖∫t∞‍A(s)U(t,s)Q(s)[g(s,B1x(s))-g(s,B1y(s))]ds‖α2≤E[∫t∞‍‖A(s)U(t,s)Q(s)[g(s,B1x(s))-g(s,B1y(s))]‖αds]≤m2(α,β)∫t∞‍eδ(t-s)ds∫t∞‍eδ(t-s)E‖g(s,B1x(s))-g(s,B1y(s))‖β2ds≤m2(α,β)ηKgδ-2‖x-y‖α,∞2.
Thus,
(3.33)∥Q1x-Q1y∥α,∞2≤Θ∥x-y∥α,∞2.
The proof is completed.Lemma 3.7.
Under assumptions(A1) and (A4), the operator Q2 defined above is continuous; moreover, its image is contained in a compact set.Proof.
LetVr={x∈SAA(ℝ,L2(P,Hα)):∥x∥α,∞2≤r}, for some r>0. It is obvious that Vr is a closed bounded convex subset of L2(P,Hα). We begin with the continuity of Q2. Let {xn}∈SAA(ℝ,L2(P,Hα)) be a sequence with xn→x; that is, limn→∞∥xn-x∥α,∞2=0. Using the estimates in Lemma 2.5, we get
(3.34)E‖∫-∞t‍U(t,s)P(s)[b(s,B2xn(s))-b(s,B2x(s))]ds‖α2≤c2(α)E[∫-∞t‍(t-s)-αe-(δ/2)(t-s)‖b(s,B2xn(s))-b(s,B2x(s))‖ds]2,
by the continuity of b, B2, and Lebesgue’s dominated convergence theorem, it follows that
(3.35)E‖∫-∞t‍U(t,s)P(s)[b(s,B2xn(s))-b(s,B2x(s))]ds‖α2⟶0asn⟶∞.
Similarly, it is easy to show that
(3.36)E‖∫t∞‍U(t,s)Q(s)[b(s,B2xn(s))-b(s,B2x(s))]ds‖α2⟶0asn⟶∞.
Applying the isometry inequality, we obtain
(3.37)E‖∫-∞t‍U(t,s)P(s)[σ(s,B3xn(s))-σ(s,B3x(s))]ds‖α2≤c2(α)∫-∞t‍(t-s)-2αe-δ(t-s)E‖σ(s,B3xn(s))-σ(s,B3x(s))‖L202ds,
by the continuity of b, B2, and Lebesgue’s dominated convergence theorem yields
(3.38)E‖∫-∞t‍U(t,s)P(s)[σ(s,B3xn(s))-σ(s,B3x(s))]dW(s)‖α2⟶0asn⟶∞.
Similarly, it is easy to show that
(3.39)E‖∫t∞‍U(t,s)Q(s)[σ(s,B3xn(s))-σ(s,B3x(s))]dW(s)‖α2⟶0asn⟶∞.
Thus,
(3.40)E‖Q2x(t)-Q2xn(t)‖α2⟶0asn⟶∞,
which implies that ∥Q2x-Q2xn∥α,∞2→0 as n→∞.
Next, we show thatQ2(Vr) is contained in a compact set. In fact, by the Ascoli-Arzela theorem, it suffices to show that Q2 maps Vr into a equicontinuous family. Let x∈Vr be arbitrary and t1<t2.
An analogue argument as Lemma 4.8 in [13], we have
(3.41)E‖Π3x(t2)-Π3x(t1)‖α2≤2E‖∫-∞t1‍(U(t2,s)-U(t1,s))P(s)b(s,B2x(s))ds‖α2+2E‖∫t1t2‍U(t2,s)P(s)b(s,B2x(s))ds‖α2=2E‖∫-∞t1‍(∫t1t2‍∂U(τ,s)∂τdτ)P(s)b(s,B2x(s))ds‖α2+2E‖∫t1t2‍U(t2,s)P(s)b(s,B2x(s))ds‖α2≤2E[∫-∞t1‍∫t1t2‍‖A(τ)U(τ,s)P(s)b(s,B2x(s))‖αdτds]2+2E[∫t1t2‍‖U(t2,s)P(s)b(s,B2x(s))‖αds]2.
For the first term on the right-hand side of (3.41), we have
(3.42)2E[∫-∞t1‍∫t1t2‖A(τ)U(τ,s)P(s)b(s,B2x(s))‖α‍dτds]2=2E[∫-∞t1‍∫t1t2‍‖A(s)A-1(s)A(τ)U(τ,s)P(s)b(s,B2x(s))‖αdτds]2≤2c02E[∫-∞t1‍∫t1t2‍‖A(s)U(τ,s)P(s)b(s,B2x(s))‖αdτds]2≤2c02n2(α,β)E[∫-∞t1‍∫t1t2‍(τ-s)-αe-(δ/2)(τ-s)dτ‖b(s,B2x(s))‖ds]2,
and for the second term, we get
(3.43)2E[∫t1t2‖U(t2,s)P(s)b(s,B2x(s))‖α‍ds]2≤2c2(α)E[∫t1t2‍(t2-s)-αe-(δ/2)(t2-s)‖b(s,B2x(s))‖ds]2.
Combing these estimates with (3.41), it follows that there exists a positive constant k1(α,δ) such that
(3.44)E‖Π3x(t2)-Π3x(t1)‖α2≤k1(α,δ)(t2-t1)2sup⁡t∈RE‖b(s,B2x(s))‖2.
Similar computation can show that there exists a positive constant k2(α,δ) such that
(3.45)E‖Π4x(t2)-Π4x(t1)‖α2≤k2(α,δ)(t2-t1)2sup⁡t∈RE‖b(s,B2x(s))‖2.
As toΠ5x, we have
(3.46)E‖Π5x(t2)-Π5x(t1)‖α2≤2E‖∫-∞t1‍(U(t2,s)-U(t1,s))P(s)σ(s,B2x(s))dW(s)‖α2+2E‖∫t1t2‍U(t2,s)P(s)σ(s,B3x(s))dW(s)‖α2.
For the first term on the right-hand side of (3.46), we have
(3.47)2E‖∫-∞t1‍(U(t2,s)-U(t1,s))P(s)σ(s,B2x(s))dW(s)‖α2=2E‖∫-∞t1‍(∫t1t2‍∂U(τ,s)∂τdτ)P(s)σ(s,B2x(s))dW(s)‖α2=E‖∫t1t2‍A(τ)U(τ,t1)(∫-∞t1‍U(t1,s)P(s)σ(s,B2x(s))dW(s))dτ‖α2≤2E[∫t1t2‍‖A(τ)U(τ,s)U(s,t1)(∫-∞t1‍U(t1,s)P(s)σ(s,B2x(s))dW(s))‖αdτ]2;
using condition (C1), Hölder's inequality together with isometry inequality yields
(3.48)2E‖∫-∞t1‍(U(t2,s)-U(t1,s))P(s)σ(s,B3x(s))dW(s)‖α2≤2c02(t2-t1)∫t1t2‍E‖∫-∞t1‍A(s)U(τ,s)P(s)σ(s,B3x(s))dW(s)‖α2dτ≤2c02(t2-t1)n2(α,β)∫t1t2‍∫-∞t1‍(τ-s)-2αe-δ(τ-s)E∥σ(s,B3x(s))∥L202dsdτ≤k3(α,δ)(t2-t1)2sup⁡t∈RE‖σ(t,B3x(t))‖L202,
where k3(α,δ) is a positive constant depending on α,δ.
The second term is straightforward; we have(3.49)2E‖∫t1t2‍U(t2,s)P(s)σ(s,B3x(s))dW(s)‖α2≤c2(α)∫t1t2‍(t2-s)-αe-δ(t2-s)E‖σ(s,B3x(s))‖L202ds≤k4(α,δ)(t2-t1)sup⁡t∈RE‖σ(t,B3x(t))‖L202,
where k4(α,δ)>0.
Therefore,(3.50)E‖Π5x(t2)-Π5x(t1)‖α2≤(k3(α,δ)(t2-t1)2+k4(α,δ)(t2-t1))sup⁡t∈RE‖σ(t,B3x(t))‖L202.
Similarly, we can deduce that there exist some constantsk5(α,δ),k6(α,δ)>0 such that
(3.51)E‖Π6x(t2)-Π6x(t1)‖α2≤(k5(α,δ)(t2-t1)2+k6(α,δ)(t2-t1))sup⁡t∈RE‖σ(t,B3x(t))‖L202.
Since
(3.52)E‖Q2x(t2)-Q2x(t1)‖α2≤∑i=36‍E‖Π6x(t2)-Π6x(t1)‖α2,
combing the evaluations above, we conclude that the right-hand side of (3.52) tends to zero independent of x as t1→t2. This completes the proof.Theorem 3.8.
Assume that assumptions (A1)–(A5) are satisfied, and one further assumes that Θ<1. Then the system (1.3) has a square-mean almost automorphic mild solution which can be expressed as x=Q1x+Q2x.Proof.
Define an operatorQ on SAA(ℝ,L2(P,Hα)) by
(3.53)Qx=Q1x+Q2x,x∈SAA(R,L2(P,Hα)).
From Lemma 3.3 to Lemma 3.5, it is easy to see that Q maps SAA(ℝ,L2(P,Hα)) into itself. To complete the proof, it suffices to show that, for some closed bounded convex subset C of SAA(ℝ,L2(P,Hα)), we have
(3.54)Q1x+Q2y∈C,x,y∈C.
Let ϵ>0 be fixed. By (A5) it follows that there exist a positive constant ω and nondecreasing continuous functions Λ:ℝ+→ℝ+ such that, for all t∈ℝ and x∈L2(P,H) with E∥x∥2>ω,
(3.55)E‖b(t,x)‖2∨E‖σ(t,x)‖L202≤ϵΛ(E‖x‖2).
Thus, for all t∈ℝ, x∈SAA(ℝ,L2(P,Hα)),
(3.56)E‖b(t,B2x)‖2∨E‖σ(t,B3x)‖L202≤ϵΛ(ηE‖x‖α2)+ω-,
where ω-:=supt∈ℝ{E∥b(t,x)∥2,E∥σ(t,x)∥ℒ202:E∥x∥2≤ω}.
Letx,y∈SAA(ℝ,L2(P,Hα)). A standard computation involving assumptions (A1)–(A3), Lemma 2.5, and Hölder's inequality, we can deduce that
(3.57)E‖Q1x(t)+Q2y(t)‖α2≤7k(α)KgE‖B1x(t)‖2+7n2(α)Kg∫-∞t‍(t-s)-αe-(δ/2)(t-s)ds×∫-∞t‍(t-s)-αe-(δ/2)(t-s)E‖B1x(s)‖2ds+7m2(α,β)Kg∫t∞‍eδ(t-s)ds∫t∞‍eδ(t-s)dsE‖B1x(s)‖2ds+7c2(α)∫-∞t‍(t-s)-αe-(δ/2)(t-s)ds∫-∞t‍(t-s)-αe-(δ/2)(t-s)E‖b(s,B2x(s))‖2ds+7m2(α)∫t∞‍eδ(t-s)ds∫t∞‍eδ(t-s)E‖b(s,B2x(s))‖2ds+7c2(α)∫-∞t‍(t-s)-2αe-δ(t-s)E‖σ(s,B3x(s))‖L202ds+7m2(α)∫t∞‍e2δ(t-s)E‖σ(s,B3x(s))‖L202ds,
using (A5) and (3.56), we further derive that
(3.58)E‖Q1x(t)+Q2y(t)‖α2≤7Kgη(k(α)+n2(α)Γ2(1-α)(2δ)2(1-α)+m2(α,β)δ-2)‖x‖α,∞2+7[c2(α)(2δ)2(1-α)Γ2(1-α)+m2(α)δ-2+c2(α)δ2α-1Γ(1-2α)+m2(α)(2δ)-1(2δ)2(1-α)](ϵΛ(η‖x‖α,∞2)+ω-).
Note that, for ϵ,Kg sufficiently small, we can choose r->0 such that
(3.59)7Kgη(k(α)+n2(α)Γ2(1-α)(2δ)2(1-α)+m2(α,β)δ-2)r-+7[c2(α)(2δ)2(1-α)Γ2(1-α)+m2(α)δ-2+c2(α)δ2α-1Γ(1-2α)+m2(α)(2δ)-1(2δ)2(1-α)](ϵΛ(ηr-)+ω-)≤r-.
Let
(3.60)C={x∈SAA(R,L2(P,Hα)):‖x‖α,∞2≤r-}.
It is easy to see that C is a closed bounded convex subset of SAA(ℝ,L2(P,Hα)). Moreover, for all x,y∈C,
(3.61)E‖Q1x(t)+Q2y(t)‖α2≤r-.
Therefore, Q1x+Q2y∈C. By Lemmas 3.6 and 3.7 together with Krasnoselskii fixed point theorem we conclude that there exists a square-mean almost automorphic mild solution to (1.3). This completes the proof.Now, we give another main result by Banach fixed point theorem. We require the following assumptions.(A6)
b:R×L2(P,H)→L2(P,H), σ:R×L2(P,H)→L2(P,ℒ20(K,H)) are square-mean almost automorphic in t∈R and there exist some constants Kb,Kσ>0 such that for each t∈ℝ, ϕ,φ∈L2(P,H),
(3.62)∥b(t,ϕ)-b(t,φ)∥2≤Kb∥ϕ-φ∥2,∥σ(t,ϕ)-σ(t,φ)∥L202≤Kσ∥ϕ-φ∥2.Theorem 3.9.
Under assumptions (A1)–(A3) and (A6), (1.3) has a unique square-mean almost automorphic mild solution x(·)∈SAA(ℝ,L2(P,Hα)) provided that
(3.63)L:=7Kgη(k(α)+n2(α)Γ2(1-α)(2δ)2(1-α)+m2(α,β)δ-2)+7η[c2(α)(2δ)2(1-α)Γ2(1-α)+m2(α)δ-2+c2(α)δ2α-1Γ(1-2α)+m2(α)(2δ)-1]<1.Proof .
LetQ be the operator defined by (3.53). From Lemma 3.3 to Lemma 3.5, it is easy to see that Q maps SAA(ℝ,L2(P,Hα)) into itself. To complete the proof, it suffices to show that Q is a contractive map and has a unique fixed point. To this end, let x,y∈SAA(ℝ,L2(P,Hα)). By a similar argument as above we can deduce that
(3.64)E‖Qx(t)-Qy(t)‖α2≤7k(α)KgE‖B1x(t)-B1y(t)‖2+7n2(α)Kg∫-∞t‍(t-s)-αe(-δ/2)(t-s)ds×∫-∞t‍(t-s)-αe-(δ/2)(t-s)E‖B1x(s)-B1y(s)‖2ds+7m2(α,β)Kg∫t∞‍eδ(t-s)ds∫t∞‍eδ(t-s)E‖B1x(s)-B1y(s)‖2ds+7c2(α)∫-∞t‍(t-s)-αe-(δ/2)(t-s)ds∫-∞t‍(t-s)-αe-(δ/2)(t-s)×E‖b(s,B2x(s))-b(s,B2y(s))‖2ds+7m2(α)∫t∞‍eδ(t-s)ds∫t∞‍eδ(t-s)E‖b(s,B2x(s))-b(s,B2y(s))‖2ds+7c2(α)∫-∞t‍(t-s)-2αe-δ(t-s)E‖σ(s,B3x(s))-σ(s,B3y(s))‖L202ds+7m2(α)∫t∞‍e2δ(t-s)E‖σ(s,B3x(s))-σ(s,B3y(s))‖L202ds≤L‖x-y‖α,∞2,
which implies that
(3.65)‖Qx-Qy‖α,∞2≤L‖x-y‖α,∞2.
Hence, by the Banach fixed point principle, Q has a unique fixed point x(t) which is obviously the square-mean almost automorphic mild solution to (1.3). The proof is completed.Remark 3.10.
The results of Theorem3.9 can be applied to the existence of square-mean almost automorphic mild solutions to the example in [14].Remark 3.11.
Ifh≡0, A(t) is densely defined and the evolution family U(t,s) generated by A(t) is exponentially stable (that is, P=I), the existence of square-mean almost automorphic mild solutions has been studied in [23] by Banach fixed point theorem; if A(t)=A is the infinitesimal generator of an analytic semigroup of linear operators, the existence of square-mean almost automorphic mild solutions has been studied in [22] by Banach fixed point theorem. In other words, the results in [22, 23] have been generalized and improved.



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*Source: 870831-2012-07-05.xml*