On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

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.

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 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))𝑑𝑡+𝑔(𝑡,𝑥(𝑡))𝑑𝑊(𝑡),𝑡∈ℝ, (1.1) 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 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓𝑡,𝑥(𝑡),𝑥_𝑡𝑑𝑡+𝑔𝑡,𝑥(𝑡),𝑥_𝑡𝑑𝑊(𝑡),𝑡∈ℝ; (1.2) 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 𝑑𝑥(𝑡)+𝑔𝑡,𝐵₁𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑏𝑡,𝐵₂𝑥(𝑡)𝑑𝑡+𝜎𝑡,𝐵₃𝑥(𝑡)𝑑𝑊(𝑡),𝑡∈ℝ, (1.3) 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, 𝐵_𝑖, 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 ℱ_𝑡=𝜎{(𝑊(𝑢)−𝑊(𝑣))∶𝑢,𝑣≤𝑡}, and 𝑔∶𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻_𝛽), 𝑏∶𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻), and 𝜎∶𝐿²(𝑃,𝐻)→𝐿²(𝑃,ℒ⁰₂(𝐾,𝐻)) are jointly continuous functions to be specified later.

The rest of this paper is organized as follows. In Section 2 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,{ℱ_𝑡}_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, ℒ⁰₂(𝐾,𝐻) denotes the space of all 𝑄-Hilbert-Schmidt operators from 𝐾 to 𝐻 with the norm 𝜉²_ℒ02∶=tr𝜉𝑄𝜉^∗<∞,𝜉∈ℒ(𝐾,𝐻). (2.1)

For 𝑡∈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 𝜕𝑈()(𝑡)𝑈().𝜕𝑡𝑡,𝑠=𝐴𝑡,𝑠 (2.2)

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 restriction 𝑈_𝑄(𝑡,𝑠)∶𝑄(𝑠)𝐻→𝑄(𝑡)𝐻 of 𝑈(𝑡,𝑠) is invertible, and we set 𝑈_𝑄(𝑠,𝑡)∶=𝑈_𝑄(𝑡,𝑠)⁻¹; (3)‖𝑈(𝑡,𝑠)𝑃(𝑠)‖≤𝑁𝑒^{−𝛿(𝑡−𝑠)}‖ and ‖𝑈_𝑄(𝑠,𝑡)𝑄(𝑡)‖≤𝑁𝑒^{−𝛿(𝑡−𝑠)}, for 𝑡≥𝑠, 𝑡,𝑠∈ℝ.
If 𝑈 has an exponential dichotomy, then the operator family Γ()𝑈()𝑃(𝑠)𝑡,𝑠∶=𝑡,𝑠,𝑡≥𝑠,𝑡,𝑠∈ℝ;−𝑈_𝑄(𝑡,𝑠)𝑄(𝑠),𝑡<𝑠,𝑡,𝑠∈ℝ (2.3) 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. If 𝐴(𝑡) satisfy the ATCs; that is, there exists a positive constant 𝜆₀ such that the operator 𝐴(𝑡),𝑡∈ℝ, satisfying Σ_𝜃∪{0}⊆𝜌𝐴(𝑡)−𝜆₀,𝑅𝜆,𝐴(𝑡)−𝜆₀≤𝐾|𝜆|,𝐴(𝑡)1+−𝜆₀𝑅𝜆₀,𝐴(𝑡)−𝜆₀𝑅𝜆₀,𝐴(𝑡)−𝑅𝜆₀,𝐴(𝑠)≤𝐿|𝑡−𝑠|^𝜇|𝜆|^−𝜈 (2.4) for 𝑡,𝑠∈ℝ, 𝜆∈Σ_𝜃∶={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 operator 𝐴∶𝒟(𝐴)⊂𝐻→𝐻 (not necessarily densely defined) is said to be sectorial if the following hold.
There exist constants 𝜁∈ℝ, 𝜃∈(𝜋/2,𝜋) and 𝑀>0 such that 𝜌(𝐴)⊃𝑆_𝜃,𝜁(,‖𝑅(≤𝑀∶=𝜆∈ℂ∶𝜆≠𝜁,arg𝜆−𝜁)<𝜃𝜆,𝐴)‖𝜆−𝜁,𝜆∈𝑆_𝜃,𝜁. (2.5)
Let 𝐴 be a sectorial operator on 𝐻 and 𝛼∈(0,1). Define the real interpolation space 𝐻^𝐴_𝛼∶=𝑥∈𝐻∶‖𝑥‖^𝐴_𝛼∶=sup𝑟^𝛼𝐴−𝜆₀𝑅𝑟,𝐴−𝜆₀𝑥<∞; (2.6) it is a Banach space endowed with the norm ‖⋅‖^𝐴_𝛼. Given a family of linear operators 𝐴(𝑡), 𝑡∈ℝ, for 𝛼∈[0,1], we set 𝐻^𝑡_𝛼∶=𝐻_𝛼^𝐴(𝑡) with the corresponding norms.
The following estimates for the evolution family 𝑈 appeared in [8] are useful.

Lemma 2.5. For 𝑥∈𝐻, 0≤𝛼≤1 and 𝑡>𝑠, there exist some constants 𝑐(𝛼), 𝑚(𝛼) such that ‖𝑈(𝑡,𝑠)𝑃(𝑠)𝑥‖^𝑡_𝛼≤𝑐(𝛼)𝑒^{−(𝛿/2)(𝑡−𝑠)}‖𝑥‖,𝑈_𝑄(𝑠,𝑡)𝑄(𝑡)𝑥^𝑠_𝛼≤𝑚(𝛼)𝑒^{−𝛿(𝑡−𝑠)}‖𝑥‖. (2.7)(2.8)

Throughout the rest of this paper, we assume that the following conditions on 𝐴(⋅) and 𝑈 hold:(𝐶₁)ATCs are satisfied and the evolution family 𝑈 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: sup_{𝑡,𝑠∈ℝ}𝐴(𝑠)𝐴⁻¹(𝑡)_{ℒ(𝐻,𝐻𝛼)}≤𝑐₀ (2.9)(𝐶₂)there exists 0<𝛼<𝛽<1 with 2𝛽>𝛼+1 such that 𝐻^𝑡_𝛼=𝐻_𝛼,𝐻^𝑡_𝛽=𝐻_𝛽, (2.10)

for all 𝑡∈ℝ, with uniform equivalent norms. And there exist constants 𝑘(𝛼), 𝑘₁ such that 𝐸‖𝑥‖²≤𝑘₁𝐸‖𝑥‖²_𝛼,𝑥∈𝐿²𝑃,𝐻_𝛼,𝐸‖𝑥‖²_𝛼≤𝑘(𝛼)𝐸‖𝑥‖²_𝛽,𝑥∈𝐿²𝑃,𝐻_𝛽. (2.11)

Lemma 2.6 (see [8]). Under the above assumptions, there exist constants 𝑚(𝛼,𝛽),𝑛(𝛼,𝛽)>0 such that ‖𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝑥‖_𝛼≤𝑛𝛼,𝛽(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}‖𝑥‖_𝛽𝐴(𝑠)𝑈,𝑡>𝑠,_𝑄(𝑡,𝑠)𝑄(𝑠)𝑥_𝛼≤𝑚𝛼,𝛽𝑒^{−𝛿(𝑠−𝑡)}‖𝑥‖_𝛽,𝑡≤𝑠. (2.12)

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 process 𝑥∶ℝ→𝐿²(𝑃,𝔹) is said to be stochastically continuous if lim_𝑡→𝑠𝐸‖𝑥(𝑡)−𝑥(𝑠)‖²_𝔹=0. (2.13)

Definition 2.8 (see [18, 21]). A stochastically continuous stochastic process 𝑥∶ℝ→𝐿²(𝑃,𝔹) is said to be square-mean almost automorphic if for every sequence of real numbers {𝑠^′_𝑛}_𝑛∈ℕ there exists a subsequence {𝑠_𝑛}_𝑛∈ℕ such that lim_{𝑚,𝑛→∞}𝐸𝑥𝑡+𝑠_𝑛−𝑠_𝑚−𝑥(𝑡)²_𝔹=0. (2.14)
This is equivalent to that there exists a stochastic process 𝑦∶ℝ→𝐿²(𝑃,𝔹) such that, for each 𝑡∈ℝ, lim_𝑛→∞𝐸𝑥𝑡+𝑠_𝑛−𝑦(𝑡)²_𝔹=0,lim_𝑛→∞𝐸𝑦𝑡−𝑠_𝑛−𝑥(𝑡)²_𝔹=0. (2.15)
Denote by SAA(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 ‖𝑥‖_∞∶=sup_𝑡∈ℝ𝐸‖𝑥(𝑡)‖²_𝔹^1/2. (2.16)

Lemma 2.9 (see [18]). If 𝑥,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 constant 𝑀 such that ‖𝑥‖²_∞≤𝑀.

Let (𝔹_𝑖,‖⋅‖_𝔹𝑖), 𝑖=1,2 be Banach spaces, and 𝐿²(𝑃,𝔹_𝑖), 𝑖=1,2, their corresponding 𝐿²-spaces, respectively.

Definition 2.10 (compare with [18, 21]). A function 𝑓∶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 {𝑠^′_𝑛}_𝑛∈ℕ, there exists a subsequence {𝑠_𝑛}_𝑛∈ℕ such that lim_{𝑚,𝑛→∞}𝐸𝑓𝑡+𝑠_𝑛−𝑠_𝑚,𝑥−𝑓(𝑡,𝑥)²_𝔹2=0, (2.17) for each 𝑡∈ℝ and 𝑥∈𝐿²(𝑃,𝔹₁).

This is equivalent to that there exists a function 𝑓∶ℝ×𝐿²(𝑃,𝔹₁)→𝐿²(𝑃,𝔹₂) such that, for each 𝑡∈ℝ and 𝑥∈𝐿²(𝑃,𝔹₁), lim_𝑛→∞𝐸𝑓𝑡+𝑠_𝑛,𝑥−𝑓(𝑡,𝑥)²_𝔹2=0,lim_𝑛→∞𝐸𝑓𝑡−𝑠_𝑛,𝑥−𝑓(𝑡,𝑥)²_𝔹2=0. (2.18)

We need the following composition of square-mean almost automorphic processes.

Lemma 2.11. Suppose that 𝑓∶ℝ×𝐿²(𝑃,𝔹₁)→𝐿²(𝑃,𝔹₂) is square-mean almost automorphic in 𝑡, and assume that 𝑓(𝑡,𝑥) satisfies 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦²_𝔹2≤𝜅𝑥−𝑦²_𝔹1,forany𝑥,𝑦∈𝐿²𝑃,𝔹₁,𝑡∈ℝ, (2.19) 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. Since 𝑓 and 𝜙 are square-mean almost automorphic processes, for every sequence of real numbers {𝑠^′_𝑛}_𝑛∈ℕ, there exist a subsequence {𝑠_𝑛}_𝑛∈ℕ⊂{𝑠^′_𝑛}_𝑛∈ℕ and some functions 𝑓, 𝜙 such that, for each 𝑡∈ℝ, 𝑥∈𝐿²(𝑃,𝔹₁), lim_𝑛→∞𝐸𝑓𝑡+𝑠_𝑛,𝑥−𝑓(𝑡,𝑥)²_𝔹2=0,lim_𝑛→∞𝐸𝜙𝑡+𝑠_𝑛−𝜙(𝑡)²_𝔹1=0. (2.20) Let 𝐹(𝑡)∶=𝑓(𝑡,𝜙(𝑡)). Then we have 𝐸𝐹𝑡+𝑠_𝑛−𝐹(𝑡)²_𝔹2≤2𝐸𝑓𝑡+𝑠_𝑛,𝜙𝑡+𝑠_𝑛−𝑓𝑡+𝑠_𝑛,𝜙(𝑡)²_𝔹2+2𝐸𝑓𝑡+𝑠_𝑛,𝜙(𝑡)−𝑓𝑡,𝜙(𝑡)²_𝔹2≤2𝐸𝜅𝜙𝑡+𝑠_𝑛−𝜙(𝑡)²_𝔹1+2𝐸𝑓𝑡+𝑠_𝑛,𝜙(𝑡)−𝑓𝑡,𝜙(𝑡)²_𝔹2; (2.21) by Jensen's inequality, it follows that 𝐸𝐹𝑡+𝑠_𝑛−𝐹(𝑡)²_𝔹2≤2𝜅𝐸𝜙𝑡+𝑠_𝑛−𝜙(𝑡)²_𝔹1+2𝐸𝑓𝑡+𝑠_𝑛,𝜙(𝑡)−𝑓𝑡,𝜙(𝑡)²_𝔹2; (2.22) noting that 𝜅 is concave and 𝜅(0)=0, we deduce that lim_𝑛→∞𝐸‖𝐹𝑡+𝑠_𝑛−𝐹(𝑡)‖²_𝔹2=0,foreach𝑡∈ℝ. (2.23) Similarly, we can prove that lim_𝑛→∞𝐸‖𝐹(𝑡−𝑠_𝑛)−𝐹(𝑡)‖²_𝔹2=0; this completes the proof.

Lemma 2.12 (see [22]). Let 𝐿∈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. Let 𝐶 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 operator 𝐴 is a contraction;(c)the operator 𝐵 is continuous and 𝐵(𝐶) is contained in a compact set.
Then the equation 𝑥=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 function 𝑥∶ℝ×𝐿²(𝑃,𝐻_𝛼) is called a mild solution of (1.3) provided that the function 𝑠⟶𝐸𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)² (3.1) is integrable on (−∞,𝑡), 𝑠⟶𝐸𝐴(𝑠)𝑈(𝑡,𝑠)𝑄(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)² (3.2) is integrable on (𝑡,∞) for each 𝑡∈ℝ, and 𝑥(𝑡) satisfies the following stochastic integral equation: 𝑥(𝑡)=−𝑔𝑡,𝐵₁𝑥(𝑡)−^𝑡_−∞𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)+𝑑𝑠_∞𝑡𝐴(𝑠)𝑈(𝑡,𝑠)𝑄(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)𝑑𝑠+^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)−𝑑𝑠_∞𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠+^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)(𝑠)−𝑑𝑊_∞𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠). (3.3)

In order to obtain our main results, we need the following assumptions. (𝐴₁)𝐵_𝑖∶𝐿²(𝑃,𝐻_𝛼)→𝐿²(𝑃,𝐻),𝑖=1,2,3, are bounded linear operators, and we set 𝜂∶=𝐵_{𝑖ℒ(𝐿2(𝑃,𝐻𝛼),𝐿2(𝑃,𝐻))},𝑖=1,2,3. (3.4) (𝐴₂)𝑅(𝜆₀,𝐴(⋅))∈ℒ(SAA(ℝ,𝐿²(𝑃,𝐻_𝛼))). For any sequence of real numbers {𝑠^′_𝑛}_𝑛∈ℕ, there exists a subsequence {𝑠_𝑛}_𝑛∈ℕ such that, for each 𝜖>0, one can find 𝑁₀∈ℕ such that 𝐴𝑠+𝑠_𝑛−𝑠_𝑚𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑠+𝑠_𝑛−𝑠_𝑚−𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)_𝛼≤𝜖𝐻(𝑡−𝑠) (3.5)whenever 𝑛,𝑚>𝑁₀, 𝑡,𝑠∈ℝ, 𝑡>𝑠, where 𝐻∶[0,∞)→[0,∞) is integrable. (𝐴₃) Let 0≤𝛼<1/2<𝛽<1. 𝑔∶ℝ×𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻_𝛽) is square-mean almost automorphic in 𝑡∈ℝ, and there exists a small 𝐾_𝑔>0 such that 𝐸𝑔(𝑡,𝑥)−𝑔𝑡,𝑦²_𝛽≤𝐾_𝑔𝐸𝑥−𝑦², (3.6)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,ℒ⁰₂(𝐾,𝐻)) are square-mean almost automorphic in 𝑡∈ℝ, and, for each 𝑡∈ℝ, 𝜙,𝜑∈𝐿²(𝑃,𝐻), 𝑏𝑡,𝜙−𝑏𝑡,𝜑²∨𝜎𝑡,𝜙−𝜎𝑡,𝜑²_ℒ02≤𝜅𝜙−𝜑², (3.7)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 any 𝜖>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, 𝐸‖𝑏(𝑡,𝑥)‖²∨𝐸‖𝜎(𝑡,𝑥)‖²_ℒ02≤𝜖Λ𝐸‖𝑥‖². (3.8)

Remark 3.2. Functions such as 𝜅₁(𝑢)=𝑢 and 𝜅₂(𝑢)=1𝑢⋅log𝑢1,0≤𝑢≤𝑏<1,𝑏⋅log𝑢+𝜅₂(𝑏−)^′(𝑢−𝑏),𝑢≥𝑏 (3.9) 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 ‖𝑥‖_𝛼,∞∶=sup_𝑡∈ℝ𝐸‖𝑥(𝑡)‖²_𝛼^1/2 (3.10) the sup-norm of the space SAA(𝑅,𝐿²(𝑃,𝐻_𝛼)). Let Π_𝑖,𝑖=1,…,6, be the operators defined by Π₁𝑥(𝑡)=^𝑡_−∞𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)Π𝑑𝑠,₂𝑥(𝑡)=^∞_𝑡𝐴(𝑠)𝑈(𝑡,𝑠)𝑄(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)Π𝑑𝑠,₃𝑥(𝑡)=^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)Π𝑑𝑠,₄𝑥(𝑡)=^∞_𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)Π𝑑𝑠,₅𝑥(𝑡)=^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)(𝑠),Π𝑑𝑊₆𝑥(𝑡)=^∞_𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠). (3.11)

Lemma 3.3. Under assumptions (𝐴₁)–(𝐴₃), the operators Π_𝑖, 𝑖=1,2, defined above map SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) into itself.

Proof. Let 𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). By Lemma 2.12, 𝑠→𝐵_𝑖𝑥(𝑠)∈SAA(ℝ,𝐿²(𝑃,𝐻)) as 𝐵_𝑖∈ℒ(𝐿²(𝑃,𝐻_𝛼),𝐿²(𝑃,𝐻)), 𝑖=1,2,3. And hence, 𝐺(𝑠)∶=𝑔(𝑠,𝐵₁𝑥(𝑠))∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛽)) by Lemma 2.11. In particular, sup_𝑡∈ℝ𝐸‖𝐺(𝑡)‖²_𝛽<∞. Let us show that Π₁𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). Indeed, since 𝐺(𝑠)∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛽)), for every sequence of real numbers {𝑠^′_𝑛}_𝑛∈ℕ, there exists a subsequence {𝑠_𝑛}_𝑛∈ℕ such that, for each 𝑡∈ℝ, lim_{𝑚,𝑛→∞}𝐸𝐺𝑡+𝑠_𝑛−𝑠_𝑚−𝐺(𝑡)²_𝛽=0. (3.12) From (𝐴₂), for any 𝜖>0, one can find 𝑁₀∈ℕ such that 𝐴𝑠+𝑠_𝑛−𝑠_𝑚𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑠+𝑠_𝑛−𝑠_𝑚−𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)_𝛼≤𝜖𝐻(𝑡−𝑠), (3.13) whenever 𝑛,𝑚>𝑁₀, 𝑡,𝑠∈ℝ, 𝑡>𝑠. Thus, 𝐸Π₁𝑥𝑡+𝑠_𝑛−𝑠_𝑚−Π₁𝑥(𝑡)²_𝛼=𝐸^{𝑡+𝑠𝑛−𝑠𝑚}_−∞𝐴(𝑠)𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠𝑃(𝑠)𝐺(𝑠)𝑑𝑠−^𝑡_−∞𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝐺(𝑠)𝑑𝑠²_𝛼=𝐸^𝑡_−∞𝐴𝑠+𝑠_𝑛−𝑠_𝑚𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑠+𝑠_𝑛−𝑠_𝑚⋅𝐺𝑠+𝑠_𝑛−𝑠_𝑚𝑑𝑠−^𝑡_−∞𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝐺(𝑠)𝑑𝑠²_𝛼≤2𝐸^𝑡_−∞𝐴𝑠+𝑠_𝑛−𝑠_𝑚𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚⋅𝑃𝑠+𝑠_𝑛−𝑠_𝑚−𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝐺(𝑠)𝑑𝑠²_𝛼+2𝐸^𝑡_−∞𝐴𝑠+𝑠_𝑛−𝑠_𝑚𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚⋅𝑃𝑠+𝑠_𝑛−𝑠_𝑚𝐺𝑠+𝑠_𝑛−𝑠_𝑚−𝐺(𝑠)𝑑𝑠²_𝛼=2𝐿₁+𝐿₂. (3.14) Using (3.13) and condition (𝐶₁), one has 𝐿₁≤𝐸^𝑡_−∞𝐴𝑠+𝑠_𝑛−𝑠_𝑚𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚⋅𝑃𝑠+𝑠_𝑛−𝑠_𝑚−𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝐺(𝑠)_𝛼𝑑𝑠²≤𝜖^2𝑡_−∞𝐻(𝑡−𝑠)𝑑𝑠^𝑡_−∞𝐻(𝑡−𝑠)𝐸‖𝐺(𝑠)‖²_𝛼≤𝑑𝑠_𝑡−∞𝐻(𝑡−𝑠)𝑑𝑠²𝑘(𝛼)sup_𝑡∈ℝ𝐸‖𝐺(𝑡)‖²_𝛽≤𝜖²𝑘(𝛼)‖𝐻‖²_𝐿1sup_𝑡∈ℝ𝐸‖𝐺(𝑡)‖²_𝛽. (3.15) For 𝐿₂, we use Lemma 2.6 to get 𝐿₂≤𝐸^𝑡_−∞𝐴𝑠+𝑠_𝑛−𝑠_𝑚𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚⋅𝑃𝑠+𝑠_𝑛−𝑠_𝑚𝐺𝑠+𝑠_𝑛−𝑠_𝑚−𝐺(𝑠)_𝛼𝑑𝑠²≤𝑛²𝛼,𝛽𝐸^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}𝐺𝑠+𝑠_𝑛−𝑠_𝑚−𝐺(𝑠)_𝛽𝑑𝑠²≤𝑛²𝛼,𝛽Γ²(1−𝛼)sup_𝑡∈ℝ𝐸𝐺𝑠+𝑠_𝑛−𝑠_𝑚−𝐺(𝑠)²_𝛽. (3.16) Combing this estimates with (3.12), one obtains lim_{𝑚,𝑛→∞}𝐸Π₁𝑥𝑡+𝑠_𝑛−𝑠_𝑚−Π₁𝑥(𝑡)²_𝛼=0, (3.17) which implies that Π₁𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). By a similar argument, we can show that Π₂𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)).

Lemma 3.4. Under assumptions (𝐴₁),(𝐴₂), and (𝐴₄), the operators Π_𝑖, 𝑖=3,4, defined above map SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) into itself.

Proof . Let 𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). By Lemmas 2.11 and 2.12 it follows that Ξ(𝑠)∶=𝑏(𝑠,𝐵₂𝑥(𝑠))∈SAA(ℝ,𝐿²(𝑃,𝐻)). Particularly, sup_𝑡∈ℝ𝐸‖Ξ(𝑡)‖²<∞. We now show that Π₃𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). Since Ξ(𝑠)∶=𝑏(𝑠,𝐵₂𝑥(𝑠))∈SAA(ℝ,𝐿²(𝑃,𝐻)), for every sequence of real numbers {𝑠^′_𝑛}_𝑛∈ℕ, there exists a subsequence {𝑠_𝑛}_𝑛∈ℕ such that, for each 𝑡∈ℝ, lim_{𝑚,𝑛→∞}𝐸Ξ𝑡+𝑠_𝑛−𝑠_𝑚−Ξ(𝑡)²=0. (3.18)
For any 𝜖>0, by making changes of variables we have 𝐸Π₃𝑥𝑡+𝑠_𝑛−𝑠_𝑚−Π₃𝑥(𝑡)²_𝛼=𝐸^{𝑡+𝑠𝑛−𝑠𝑚}_−∞𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠𝑃(𝑠)Ξ(𝑠)𝑑𝑠−^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)Ξ(𝑠)𝑑𝑠²_𝛼=𝐸^𝑡_−∞𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑠+𝑠_𝑛−𝑠_𝑚Ξ𝑠+𝑠_𝑛−𝑠_𝑚−𝑑𝑠_𝑡−∞𝑈(𝑡,𝑠)𝑃(𝑠)Ξ(𝑠)𝑑𝑠²_𝛼=𝐸^∞₀𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑡−𝑠+𝑠_𝑛−𝑠_𝑚Ξ𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−𝑑𝑠_𝑡−∞𝑈(𝑡,𝑡−𝑠)𝑃(𝑡−𝑠)Ξ(𝑡−𝑠)𝑑𝑠²_𝛼; (3.19) an elementary inequality shows that 𝐸Π₃𝑥𝑡+𝑠_𝑛−𝑠_𝑚−Π₃𝑥(𝑡)²_𝛼≤3𝐸^∞₀𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑡−𝑠+𝑠_𝑛−𝑠_𝑚×Ξ𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−Ξ(𝑡−𝑠)𝑑𝑠²_𝛼+3𝐸^𝜖₀𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−𝑈(𝑡,𝑡−𝑠)𝑃(𝑡−𝑠)]Ξ(𝑡−𝑠)𝑑𝑠²_𝛼+3𝐸^∞_𝜖𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−𝑈(𝑡,𝑡−𝑠)𝑃(𝑡−𝑠)]Ξ(𝑡−𝑠)𝑑𝑠²_𝛼=3𝐼₁+𝐼₂+𝐼₃. (3.20) Using Lemma 2.5, one has 𝐼₁≤𝑐²(𝛼)𝐸^𝑡₀𝑠^−𝛼𝑒^{−(𝛿/2)𝑠}Ξ𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−Ξ(𝑡−𝑠)𝑑𝑠²≤𝑐²(𝛼)𝛿2^𝛼−1Γ²(1−𝛼)sup_𝑡∈ℝ𝐸Ξ𝑡+𝑠_𝑛−𝑠_𝑚−Ξ(𝑡)²; (3.21) a straightforward computation yields 𝐼₂≤𝐸^𝜖₀𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−𝑈(𝑡,𝑡−𝑠)𝑃(𝑡−𝑠)Ξ(𝑡−𝑠)_𝛼𝑑𝑠²≤4𝜖²𝑀²sup_𝑡∈ℝ𝐸‖Ξ(𝑡)‖², (3.22) where 𝑀 is a constant satisfying sup_𝑡,𝑠‖𝑈(𝑡,𝑠)𝑃(𝑠)‖≤𝑀.
For 𝐼₃, applying Proposition 4.4 in [4], we have 𝐼₃≤𝐸^∞_𝜖𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−𝑈(𝑡,𝑡−𝑠)𝑃(𝑡−𝑠)]Ξ(𝑡−𝑠)‖_𝛼𝑑𝑠²≤𝜖^2∞_𝜖𝑒^{−(𝛿/2)𝑠}𝑑𝑠^∞_𝜖𝑒^{−(𝛿/2)𝑠}𝐸‖Ξ(𝑡−𝑠)‖²𝑑𝑠≤4𝜖²𝛿⁻²sup_𝑡∈ℝ𝐸‖Ξ(𝑡)‖². (3.23) Combing this estimates with (3.18), we get lim_{𝑚,𝑛→∞}𝐸Π₃𝑡+𝑠_𝑛−𝑠_𝑚−Π₃(𝑡)²_𝛼=0, (3.24) which implies that Π₃𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) whenever 𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). Similarly, we can verify that Π₄𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)).

Lemma 3.5. Under assumptions (𝐴₁),(𝐴₂) and (𝐴₄), the operators Π_𝑖, 𝑖=5,6, defined above map SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) into itself.

Proof. Let 𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). Using Lemmas 2.11 and 2.12, we get Σ(𝑠)∶=𝜎(𝑠,𝐵₃𝑥(𝑠))∈SAA(ℝ,𝐿²(𝑃,ℒ⁰₂(𝐾,𝐻))). Particularly, sup_𝑡∈ℝ𝐸‖Σ(𝑡)‖²_ℒ02<∞. We now show that Π₅𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). Since Σ(𝑠)∶=𝜎(𝑠,𝐵₃𝑥(𝑠))∈SAA(ℝ,𝐿²(𝑃,𝐻)), for every sequence of real numbers {𝑠^′_𝑛}_𝑛∈ℕ, there exists a subsequence {𝑠_𝑛}_𝑛∈ℕ such that, for each 𝑡∈ℝ, lim_{𝑚,𝑛→∞}𝐸Σ𝑡+𝑠_𝑛−𝑠_𝑚−Σ(𝑡)²_ℒ02=0. (3.25) 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 𝐸Π₅𝑥𝑡+𝑠_𝑛−𝑠_𝑚−Π₅𝑥(𝑡)²_𝛼=𝐸^𝑡_−∞𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑠+𝑠_𝑛−𝑠_𝑚Σ𝑠+𝑠_𝑛−𝑠_𝑚𝑑𝑊(𝑠)−^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)Σ(𝑠)𝑑𝑊(𝑠)²_𝛼=𝐸^∞₀𝑈𝑡+𝑠_𝑛−𝑠_𝑚,𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑃𝑡−𝑠+𝑠_𝑛−𝑠_𝑚⋅Σ𝑡−𝑠+𝑠_𝑛−𝑠_𝑚𝑑𝑊(𝑠)−^∞₀𝑈(𝑡,𝑡−𝑠)𝑃(𝑡−𝑠)Σ(𝑡−𝑠)𝑑𝑊(𝑠)²_𝛼≤3𝑐²(𝛼)^∞₀𝑠^−2𝛼𝑒^−𝛿𝑠𝐸Σ𝑡−𝑠+𝑠_𝑛−𝑠_𝑚−Σ(𝑡−𝑠)²_ℒ02𝑑𝑠+12𝜖𝑀²sup_𝑡∈ℝ𝐸‖Σ(𝑡)‖²_ℒ02+3𝜖^2∞_𝜖𝑒^−𝛿𝑠𝐸‖Σ(𝑡−𝑠)‖²_ℒ02𝑑𝑠≤3𝑐²(𝛼)𝛿^2𝛼−1Γ(2𝛼−1)sup_𝑡∈ℝ𝐸Σ𝑡+𝑠_𝑛−𝑠_𝑚−Σ(𝑡)²_ℒ02+12𝜖𝑀²sup_𝑡∈ℝ𝐸‖Σ(𝑡)‖²_ℒ02+3𝛿⁻¹𝜖²sup_𝑡∈ℝ𝐸‖Σ(𝑡)‖²_ℒ02. (3.26) From (3.25), we immediately get lim_{𝑚,𝑛→∞}𝐸Π₅𝑡+𝑠_𝑛−𝑠_𝑚−Π₅(𝑡)²_𝛼=0, (3.27) which implies that Π₅𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). Similarly, we can show that Π₆𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) whenever 𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)).
Consider the nonlinear operators 𝑄_𝑖, 𝑖=1,2, on SAA(𝑅,𝐿²(𝑃,𝐻_𝛼)) defined by 𝑄₁𝑥(𝑡)=−𝑔𝑡,𝐵₁𝑥(𝑡)−^𝑡_−∞𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)+𝑑𝑠_∞𝑡𝐴(𝑠)𝑈(𝑡,𝑠)𝑄(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)𝑄𝑑𝑠,₂𝑥(𝑡)=^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠−^∞_𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)+𝑑𝑠_𝑡−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠)−^∞_𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠), (3.28) for each 𝑡∈ℝ. In view of Lemmas 3.3, 3.4, and 3.5, it follows that 𝑄_𝑖, 𝑖=1,2, map SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) into itself. In what follows, we will prove that 𝑄_𝑖, 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 Θ∶=3𝐾_𝑔𝜂𝑘(𝛼)+𝑛²(𝛼)Γ²()21−𝛼𝛿^2(1−𝛼)+𝑚²𝛼,𝛽𝛿⁻²<1. (3.29)

Proof. Let 𝑥,𝑦∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). By using condition (𝐶₂) and assumptions (𝐴₁),(𝐴₃), we have 𝐸𝑔(𝑡,𝐵₁𝑥(𝑡))−𝑔(𝑡,𝐵₁𝑦(𝑡))²_𝛼≤𝑘(𝛼)𝐸𝑔(𝑡,𝐵₁𝑥(𝑡))−𝑔(𝑡,𝐵₁𝑦(𝑡))²_𝛽≤𝑘(𝛼)𝐾_𝑔𝐸‖𝐵₁𝑥(𝑡)−𝐵₁𝑦(𝑡)‖²≤𝑘(𝛼)𝐾_𝑔𝜂‖𝑥−𝑦‖²_𝛼,∞. (3.30) Now, using Lemma 2.6 together with Hölder's inequality, we obtain 𝐸^𝑡_−∞𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)−𝑔𝑠,𝐵₁𝑦(𝑠)𝑑𝑠²_𝛼≤𝐸^𝑡_−∞𝐴(𝑠)𝑈(𝑡,𝑠)𝑃(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)−𝑔𝑠,𝐵₁𝑦(𝑠)_𝛼𝑑𝑠≤𝑛²(𝛼)^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}×𝑑𝑠_𝑡−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}⋅𝐸𝑔𝑠,𝐵₁𝑥(𝑠)−𝑔𝑠,𝐵₁𝑦(𝑠)²_𝛽𝑑𝑠≤𝑛²(𝛼)𝐾_𝑔𝜂Γ²()21−𝛼𝛿^2(1−𝛼)𝑥−𝑦²_𝛼,∞. (3.31) Similarly, 𝐸^∞_𝑡𝐴(𝑠)𝑈(𝑡,𝑠)𝑄(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)−𝑔𝑠,𝐵₁𝑦(𝑠)𝑑𝑠²_𝛼≤𝐸^∞_𝑡𝐴(𝑠)𝑈(𝑡,𝑠)𝑄(𝑠)𝑔𝑠,𝐵₁𝑥(𝑠)−𝑔𝑠,𝐵₁𝑦(𝑠)_𝛼𝑑𝑠≤𝑚²𝛼,𝛽^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝐸𝑔𝑠,𝐵₁𝑥(𝑠)−𝑔𝑠,𝐵₁𝑦(𝑠)²_𝛽𝑑𝑠≤𝑚²𝛼,𝛽𝜂𝐾_𝑔𝛿⁻²𝑥−𝑦²_𝛼,∞. (3.32) Thus, ‖𝑄₁𝑥−𝑄₁𝑦‖²_𝛼,∞≤Θ‖𝑥−𝑦‖²_𝛼,∞. (3.33) 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. Let 𝑉_𝑟={𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼))∶‖𝑥‖²_𝛼,∞≤𝑟}, for some 𝑟>0. It is obvious that 𝑉_𝑟 is a closed bounded convex subset of 𝐿²(𝑃,𝐻_𝛼). We begin with the continuity of 𝑄₂. Let {𝑥^𝑛}∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) be a sequence with 𝑥_𝑛→𝑥; that is, lim_𝑛→∞‖𝑥^𝑛−𝑥‖²_𝛼,∞=0. Using the estimates in Lemma 2.5, we get 𝐸^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥^𝑛(𝑠)−𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠²_𝛼≤𝑐²(𝛼)𝐸^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}𝑏𝑠,𝐵₂𝑥^𝑛(𝑠)−𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠², (3.34) by the continuity of 𝑏, 𝐵₂, and Lebesgue’s dominated convergence theorem, it follows that 𝐸^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥^𝑛(𝑠)−𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠²_𝛼⟶0as𝑛⟶∞. (3.35) Similarly, it is easy to show that 𝐸^∞_𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝑏𝑠,𝐵₂𝑥^𝑛(𝑠)−𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠²_𝛼⟶0as𝑛⟶∞. (3.36) Applying the isometry inequality, we obtain 𝐸^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝜎𝑠,𝐵₃𝑥^𝑛(𝑠)−𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑠²_𝛼≤𝑐²(𝛼)^𝑡_−∞(𝑡−𝑠)^−2𝛼𝑒^{−𝛿(𝑡−𝑠)}𝐸𝜎𝑠,𝐵₃𝑥^𝑛(𝑠)−𝜎𝑠,𝐵₃𝑥(𝑠)²_ℒ02𝑑𝑠, (3.37) by the continuity of 𝑏, 𝐵₂, and Lebesgue’s dominated convergence theorem yields 𝐸^𝑡_−∞𝑈(𝑡,𝑠)𝑃(𝑠)𝜎𝑠,𝐵₃𝑥^𝑛(𝑠)−𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼⟶0as𝑛⟶∞. (3.38) Similarly, it is easy to show that 𝐸^∞_𝑡𝑈(𝑡,𝑠)𝑄(𝑠)𝜎𝑠,𝐵₃𝑥^𝑛(𝑠)−𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼⟶0as𝑛⟶∞. (3.39) Thus, 𝐸𝑄₂𝑥(𝑡)−𝑄₂𝑥^𝑛(𝑡)²_𝛼⟶0as𝑛⟶∞, (3.40) which implies that ‖𝑄₂𝑥−𝑄₂𝑥^𝑛‖²_𝛼,∞→0 as 𝑛→∞.
Next, we show that 𝑄₂(𝑉_𝑟) is contained in a compact set. In fact, by the Ascoli-Arzela theorem, it suffices to show that 𝑄₂ maps 𝑉_𝑟 into a equicontinuous family. Let 𝑥∈𝑉_𝑟 be arbitrary and 𝑡₁<𝑡₂.
An analogue argument as Lemma 4.8 in [13], we have 𝐸Π₃𝑥𝑡₂−Π₃𝑥𝑡₁²_𝛼≤2𝐸^𝑡1_−∞𝑈𝑡₂,𝑠−𝑈𝑡₁,𝑠𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠²_𝛼+2𝐸^𝑡2_𝑡1𝑈𝑡₂,𝑠𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠²_𝛼=2𝐸^𝑡1_−∞^𝑡2_𝑡1𝜕𝑈(𝜏,𝑠)𝑃(𝑠)𝑏𝜕𝜏𝑑𝜏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠²_𝛼+2𝐸^𝑡2_𝑡1𝑈𝑡₂,𝑠𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠²_𝛼≤2𝐸^𝑡1_−∞^𝑡2_𝑡1𝐴(𝜏)𝑈(𝜏,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)_𝛼𝑑𝜏𝑑𝑠²+2𝐸^𝑡2_𝑡1𝑈𝑡₂,𝑠𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)_𝛼𝑑𝑠². (3.41) For the first term on the right-hand side of (3.41), we have 2𝐸^𝑡1_−∞^𝑡2_𝑡1𝐴(𝜏)𝑈(𝜏,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)_𝛼𝑑𝜏𝑑𝑠²=2𝐸^𝑡1_−∞^𝑡2_𝑡1𝐴(𝑠)𝐴⁻¹(𝑠)𝐴(𝜏)𝑈(𝜏,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)_𝛼𝑑𝜏𝑑𝑠²≤2𝑐²₀𝐸^𝑡1_−∞^𝑡2_𝑡1𝐴(𝑠)𝑈(𝜏,𝑠)𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)_𝛼𝑑𝜏𝑑𝑠²≤2𝑐²₀𝑛²𝛼,𝛽𝐸^𝑡1_−∞^𝑡2_𝑡1(𝜏−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝜏−𝑠)}𝑑𝜏𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠², (3.42) and for the second term, we get 2𝐸^𝑡2_𝑡1𝑈𝑡₂,𝑠𝑃(𝑠)𝑏𝑠,𝐵₂𝑥(𝑠)_𝛼𝑑𝑠²≤2𝑐²(𝛼)𝐸^𝑡2_𝑡1𝑡₂−𝑠^−𝛼𝑒^{−(𝛿/2)(𝑡2−𝑠)}𝑏𝑠,𝐵₂𝑥(𝑠)𝑑𝑠². (3.43) Combing these estimates with (3.41), it follows that there exists a positive constant 𝑘₁(𝛼,𝛿) such that 𝐸Π₃𝑥𝑡₂−Π₃𝑥𝑡₁²_𝛼≤𝑘₁(𝛼,𝛿)𝑡₂−𝑡₁²sup_𝑡∈ℝ𝐸𝑏𝑠,𝐵₂𝑥(𝑠)². (3.44) Similar computation can show that there exists a positive constant 𝑘₂(𝛼,𝛿) such that 𝐸Π₄𝑥𝑡₂−Π₄𝑥𝑡₁²_𝛼≤𝑘₂(𝛼,𝛿)𝑡₂−𝑡₁²sup_𝑡∈ℝ𝐸𝑏𝑠,𝐵₂𝑥(𝑠)². (3.45)
As to Π₅𝑥, we have 𝐸Π₅𝑥𝑡₂−Π₅𝑥𝑡₁²_𝛼≤2𝐸^𝑡1_−∞𝑈𝑡₂,𝑠−𝑈𝑡₁,𝑠𝑃(𝑠)𝜎𝑠,𝐵₂𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼+2𝐸^𝑡2_𝑡1𝑈𝑡₂,𝑠𝑃(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼. (3.46)
For the first term on the right-hand side of (3.46), we have 2𝐸^𝑡1_−∞𝑈𝑡₂,𝑠−𝑈𝑡₁,𝑠𝑃(𝑠)𝜎𝑠,𝐵₂𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼=2𝐸^𝑡1_−∞^𝑡2_𝑡1𝜕𝑈(𝜏,𝑠)𝑃(𝑠)𝜎𝜕𝜏𝑑𝜏𝑠,𝐵₂𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼=𝐸^𝑡2_𝑡1𝐴(𝜏)𝑈𝜏,𝑡₁^𝑡1_−∞𝑈𝑡₁,𝑠𝑃(𝑠)𝜎𝑠,𝐵₂𝑥(𝑠)𝑑𝑊(𝑠)𝑑𝜏²_𝛼≤2𝐸^𝑡2_𝑡1𝐴(𝜏)𝑈(𝜏,𝑠)𝑈𝑠,𝑡₁^𝑡1_−∞𝑈𝑡₁,𝑠𝑃(𝑠)𝜎𝑠,𝐵₂𝑥(𝑠)𝑑𝑊(𝑠)_𝛼𝑑𝜏²; (3.47) using condition (𝐶₁), Hölder's inequality together with isometry inequality yields 2𝐸^𝑡1_−∞𝑈𝑡₂,𝑠−𝑈𝑡₁,𝑠𝑃(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼≤2𝑐²₀𝑡₂−𝑡₁^𝑡2_𝑡1𝐸^𝑡1_−∞𝐴(𝑠)𝑈(𝜏,𝑠)𝑃(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼𝑑𝜏≤2𝑐²₀𝑡₂−𝑡₁𝑛²𝛼,𝛽^𝑡2_𝑡1^𝑡1_−∞(𝜏−𝑠)^−2𝛼𝑒^{−𝛿(𝜏−𝑠)}𝐸‖𝜎𝑠,𝐵₃𝑥(𝑠)‖²_ℒ02𝑑𝑠𝑑𝜏≤𝑘₃(𝛼,𝛿)𝑡₂−𝑡₁²sup_𝑡∈ℝ𝐸𝜎𝑡,𝐵₃𝑥(𝑡)²_ℒ02, (3.48) where 𝑘₃(𝛼,𝛿) is a positive constant depending on 𝛼,𝛿.
The second term is straightforward; we have 2𝐸^𝑡2_𝑡1𝑈𝑡₂,𝑠𝑃(𝑠)𝜎𝑠,𝐵₃𝑥(𝑠)𝑑𝑊(𝑠)²_𝛼≤𝑐²(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^−𝛼𝑒^{−𝛿(𝑡2−𝑠)}𝐸𝜎𝑠,𝐵₃𝑥(𝑠)²_ℒ02𝑑𝑠≤𝑘₄(𝛼,𝛿)𝑡₂−𝑡₁sup_𝑡∈ℝ𝐸𝜎𝑡,𝐵₃𝑥(𝑡)²_ℒ02, (3.49) where 𝑘₄(𝛼,𝛿)>0.
Therefore, 𝐸Π₅𝑥𝑡₂−Π₅𝑥𝑡₁²_𝛼≤𝑘₃(𝛼,𝛿)𝑡₂−𝑡₁²+𝑘₄(𝛼,𝛿)𝑡₂−𝑡₁sup_𝑡∈ℝ𝐸𝜎𝑡,𝐵₃𝑥(𝑡)²_ℒ02. (3.50)
Similarly, we can deduce that there exist some constants 𝑘₅(𝛼,𝛿),𝑘₆(𝛼,𝛿)>0 such that 𝐸Π₆𝑥𝑡₂−Π₆𝑥𝑡₁²_𝛼≤𝑘₅(𝛼,𝛿)𝑡₂−𝑡₁²+𝑘₆(𝛼,𝛿)𝑡₂−𝑡₁sup_𝑡∈ℝ𝐸𝜎𝑡,𝐵₃𝑥(𝑡)²_ℒ02. (3.51) Since 𝐸𝑄₂𝑥𝑡₂−𝑄₂𝑥𝑡₁²_𝛼≤⁶_𝑖=3𝐸Π₆𝑥𝑡₂−Π₆𝑥𝑡₁²_𝛼, (3.52) 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 operator 𝑄 on SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) by 𝑄𝑥=𝑄₁𝑥+𝑄₂𝑥,𝑥∈SAAℝ,𝐿²𝑃,𝐻_𝛼. (3.53) From Lemma 3.3 to Lemma 3.5, it is easy to see that 𝑄 maps SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)) into itself. To complete the proof, it suffices to show that, for some closed bounded convex subset 𝐶 of SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)), we have 𝑄₁𝑥+𝑄₂𝑦∈𝐶,𝑥,𝑦∈𝐶. (3.54) 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, 𝐸‖𝑏(𝑡,𝑥)‖²∨𝐸‖𝜎(𝑡,𝑥)‖²_ℒ02≤𝜖Λ𝐸‖𝑥‖². (3.55) Thus, for all 𝑡∈ℝ, 𝑥∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)), 𝐸𝑏𝑡,𝐵₂𝑥²∨𝐸𝜎𝑡,𝐵₃𝑥²_ℒ02≤𝜖Λ𝜂𝐸‖𝑥‖²_𝛼+𝜔, (3.56) where 𝜔∶=sup_𝑡∈ℝ{𝐸‖𝑏(𝑡,𝑥)‖²,𝐸‖𝜎(𝑡,𝑥)‖²_ℒ02∶𝐸‖𝑥‖²≤𝜔}.
Let 𝑥,𝑦∈SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). A standard computation involving assumptions (𝐴₁)–(𝐴₃), Lemma 2.5, and Hölder's inequality, we can deduce that 𝐸𝑄₁𝑥(𝑡)+𝑄₂𝑦(𝑡)²_𝛼≤7𝑘(𝛼)𝐾_𝑔𝐸𝐵₁𝑥(𝑡)²+7𝑛²(𝛼)𝐾_𝑔^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}×𝑑𝑠_𝑡−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}𝐸𝐵₁𝑥(𝑠)²𝑑𝑠+7𝑚²𝛼,𝛽𝐾_𝑔^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠𝐸𝐵₁𝑥(𝑠)²𝑑𝑠+7𝑐²(𝛼)^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}𝑑𝑠^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}𝐸𝑏𝑠,𝐵₂𝑥(𝑠)²𝑑𝑠+7𝑚²(𝛼)^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝐸𝑏𝑠,𝐵₂𝑥(𝑠)²𝑑𝑠+7𝑐²(𝛼)^𝑡_−∞(𝑡−𝑠)^−2𝛼𝑒^{−𝛿(𝑡−𝑠)}𝐸𝜎𝑠,𝐵₃𝑥(𝑠)²_ℒ02𝑑𝑠+7𝑚²(𝛼)^∞_𝑡𝑒^{2𝛿(𝑡−𝑠)}𝐸𝜎𝑠,𝐵₃𝑥(𝑠)²_ℒ02𝑑𝑠, (3.57) using (𝐴₅) and (3.56), we further derive that 𝐸𝑄₁𝑥(𝑡)+𝑄₂𝑦(𝑡)²_𝛼≤7𝐾_𝑔𝜂𝑘(𝛼)+𝑛²(𝛼)Γ²()21−𝛼𝛿^2(1−𝛼)+𝑚²𝛼,𝛽𝛿⁻²‖𝑥‖²_𝛼,∞+7𝑐²(𝛼)2𝛿^2(1−𝛼)Γ²(1−𝛼)+𝑚²(𝛼)𝛿⁻²+𝑐²(𝛼)𝛿^2𝛼−1Γ(1−2𝛼)+𝑚²(𝛼)(2𝛿)⁻¹𝜖Λ𝜂‖𝑥‖²_𝛼,∞+𝜔. (3.58) Note that, for 𝜖,𝐾_𝑔 sufficiently small, we can choose 𝑟>0 such that 7𝐾_𝑔𝜂𝑘(𝛼)+𝑛²(𝛼)Γ²()21−𝛼𝛿^2(1−𝛼)+𝑚²𝛼,𝛽𝛿⁻²𝑟+7𝑐²(𝛼)2𝛿^2(1−𝛼)Γ²(1−𝛼)+𝑚²(𝛼)𝛿⁻²+𝑐²(𝛼)𝛿^2𝛼−1Γ(1−2𝛼)+𝑚²(𝛼)(2𝛿)⁻¹𝜖Λ𝜂𝑟+𝜔≤𝑟. (3.59) Let 𝐶=𝑥∈SAAℝ,𝐿²𝑃,𝐻_𝛼∶‖𝑥‖²_𝛼,∞≤𝑟. (3.60) It is easy to see that 𝐶 is a closed bounded convex subset of SAA(ℝ,𝐿²(𝑃,𝐻_𝛼)). Moreover, for all 𝑥,𝑦∈𝐶, 𝐸𝑄₁𝑥(𝑡)+𝑄₂𝑦(𝑡)²_𝛼≤𝑟. (3.61) 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,ℒ⁰₂(𝐾,𝐻)) are square-mean almost automorphic in 𝑡∈𝑅 and there exist some constants 𝐾_𝑏,𝐾_𝜎>0 such that for each 𝑡∈ℝ, 𝜙,𝜑∈𝐿²(𝑃,𝐻), ‖𝑏𝑡,𝜙−𝑏𝑡,𝜑‖²≤𝐾_𝑏‖𝜙−𝜑‖²,‖𝜎𝑡,𝜙−𝜎𝑡,𝜑‖²_ℒ02≤𝐾_𝜎‖𝜙−𝜑‖². (3.62)

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,𝐻_𝛼)) provided that 𝐿∶=7𝐾_𝑔𝜂𝑘(𝛼)+𝑛²(𝛼)Γ²()21−𝛼𝛿^2(1−𝛼)+𝑚²𝛼,𝛽𝛿⁻²+7𝜂𝑐²(𝛼)2𝛿^2(1−𝛼)Γ²(1−𝛼)+𝑚²(𝛼)𝛿⁻²+𝑐²(𝛼)𝛿^2𝛼−1Γ(1−2𝛼)+𝑚²(𝛼)(2𝛿)⁻¹<1. (3.63)

Proof . Let 𝑄 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,𝐻_𝛼)) 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,𝐻_𝛼)). By a similar argument as above we can deduce that 𝐸𝑄𝑥(𝑡)−𝑄𝑦(𝑡)²_𝛼≤7𝑘(𝛼)𝐾_𝑔𝐸𝐵₁𝑥(𝑡)−𝐵₁𝑦(𝑡)²+7𝑛²(𝛼)𝐾_𝑔^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{(−𝛿/2)(𝑡−𝑠)}×𝑑𝑠_𝑡−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}𝐸𝐵₁𝑥(𝑠)−𝐵₁𝑦(𝑠)²𝑑𝑠+7𝑚²𝛼,𝛽𝐾_𝑔^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝐸𝐵₁𝑥(𝑠)−𝐵₁𝑦(𝑠)²𝑑𝑠+7𝑐²(𝛼)^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}𝑑𝑠^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−(𝛿/2)(𝑡−𝑠)}×𝐸𝑏𝑠,𝐵₂𝑥(𝑠)−𝑏𝑠,𝐵₂𝑦(𝑠)²𝑑𝑠+7𝑚²(𝛼)^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠^∞_𝑡𝑒^{𝛿(𝑡−𝑠)}𝐸𝑏𝑠,𝐵₂𝑥(𝑠)−𝑏𝑠,𝐵₂𝑦(𝑠)²𝑑𝑠+7𝑐²(𝛼)^𝑡_−∞(𝑡−𝑠)^−2𝛼𝑒^{−𝛿(𝑡−𝑠)}𝐸𝜎𝑠,𝐵₃𝑥(𝑠)−𝜎𝑠,𝐵₃𝑦(𝑠)²_ℒ02𝑑𝑠+7𝑚²(𝛼)^∞_𝑡𝑒^{2𝛿(𝑡−𝑠)}𝐸𝜎𝑠,𝐵₃𝑥(𝑠)−𝜎𝑠,𝐵₃𝑦(𝑠)²_ℒ02𝑑𝑠≤𝐿𝑥−𝑦²_𝛼,∞, (3.64) which implies that 𝑄𝑥−𝑄𝑦²_𝛼,∞≤𝐿𝑥−𝑦²_𝛼,∞. (3.65) 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 Theorem 3.9 can be applied to the existence of square-mean almost automorphic mild solutions to the example in [14].

Remark 3.11. If ℎ≡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.

Acknowledgment

The Project is sponsored by the NSFC (11171062), the Innovation Program of Shanghai Municipal Education Commission (12ZZ063), the Natural Science Foundation of Anhui Province (1208085MA11), and the NSF of Anhui Educational Committee (KJ2011A139).