Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations

Abstract

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

1. Introduction

Stochastic differential equations in both finite and infinite dimensions, which are important from the viewpoint of applications since they incorporate natural randomness into the mathematical description of the phenomena and hence provide a more accurate description of it, have received considerable attention. Based on this viewpoint, there has been an increasing interest in extending certain classical deterministic results to stochastic differential equations in recent years. As a good case in point, the existence of almost periodic or pseudo-almost periodic solutions to stochastic evolution equations has been extensively considered in many publications; see [1–8] and the references therein.

Integrodifferential equations are used to describe lots of phenomena arising naturally from many fields such as fluid dynamics, number reactor dynamics, population dynamics, electromagnetic theory, and biological models, most of which cannot be described by classical differential equations, and hence they have attracted more and more attention in recent years; see [1, 9–12] for more details.

Recently, Keck and McKibben [9, 10] proposed a general abstract model for semilinear functional stochastic integrodifferential equations and studied the existence and uniqueness of mild solutions to these equations. Based on their works, the existence and uniqueness of square-mean almost periodic solutions to some functional integrodifferential stochastic evolution equations was carefully investigated in [1] for the autonomous case and in our forthcoming paper for the nonautonomous case. In a very recent paper, as a natural generalization of the notion of square-mean almost periodicity, a new concept of square-mean almost automorphic stochastic process was introduced by Fu and Liu [13], and the existence results of square-mean almost automorphic mild solutions to some linear and semilinear autonomous stochastic differential equations were formulated, while paper [14] investigated the same issue for nonautonomous stochastic differential equations. Under some suitable assumptions, the authors established in a forthcoming paper the existence and uniqueness of square-mean almost automorphic solutions to a class of autonomous functional integrodifferential stochastic evolution equations.

In this paper, we are concerned with a class of semilinear nonautonomous functional stochastic integrodifferential equations in a real separable Hilbert space in the abstract form: 𝑋^′(𝑡)=𝐴(𝑡)𝑋(𝑡)+^𝑡_−∞𝐶()𝐺((𝑠(𝑠)+𝑡−𝑠𝑠,𝑋))𝑑𝑊_𝑡−∞𝐵(𝑡−𝑠)𝐹₂(𝑠,𝑋(𝑠))𝑑𝑠+𝐹₁(𝑡,𝑋(𝑡)),𝑡∈ℝ, (1) where 𝐴(𝑡)∶𝐷(𝐴(𝑡))⊂ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ) is a family of densely defined closed (possibly unbounded) linear operator satisfying the so-called “Acquistapace-Terreni” conditions introduced in [15], 𝐵 and 𝐶 are convolution-type kernels in ℒ¹(0,∞) and ℒ²(0,∞), respectively, satisfying Assumption  3.2 in [16], 𝑊(𝑡) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω,ℱ,𝐏,ℱ_𝑡), where ℱ_𝑡=𝜎{𝑊(𝑢)−𝑊(𝑣);𝑢,𝑣≤𝑡}. Here 𝐹₁,𝐹₂,𝐺∶ℝ×ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ) are jointly continuous functions satisfying some additional conditions to be specified later in Section 3.

Motivated by the aforementioned works [1, 13, 14], we investigate in this paper the existence and uniqueness of square-mean almost automorphic solutions to nonautonomous equation (1). The main tools employed here are Banach contraction principle and an estimate on the Ito integral. The obtained results can be seen as a contribution to this emerging field.

The paper is organized as follows. In Section 2, we review some basic definitions and preliminary facts on square-mean almost automorphic processes which will be used throughout this paper. Section 3 is devoted to establish the existence, uniqueness, and the asymptotical stability of square-mean almost automorphic mild solution to (1). As an illustration of our abstract result, square-mean almost automorphic solution to a concrete nonautonomous integrodifferential stochastic evolution equation is investigated in Section 4.

2. Preliminaries

To begin this paper, we recall some primary definitions, notations, lemmas, and technical results which will be used in the sequel. For more details on almost automorphy and stochastic differential equations, the readers are referred to [13, 17–23] and the references therein.

Throughout this paper, we assume that (ℍ,‖⋅‖) is a real separable Hilbert space, (Ω,ℱ,𝐏) is a probability space, and ℒ²(𝐏;ℍ) stands for the space of all ℍ-valued random variables 𝑋 such that 𝐄‖𝑋‖²=_Ω‖𝑋‖²𝑑𝐏<∞. (2) For 𝑋∈ℒ²(𝐏;ℍ), let ‖𝑋‖₂∶=_Ω‖𝑋‖²𝑑𝐏^1/2. (3) It is routine to check that ℒ²(𝐏;ℍ) is a Banach space equipped with the norm ‖⋅‖₂.

It is well-known that Brownian motion plays a key role in the construction of stochastic integrals. Throughout this paper, 𝑊(𝑡) denotes a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω,ℱ,𝐏,ℱ_𝑡), where ℱ_𝑡=𝜎{𝑊(𝑢)−𝑊(𝑣);𝑢,𝑣≤𝑡}.

Definition 1. A standard one-dimensional Brownian motion is a continuous, adapted real-valued stochastic process (𝑊(𝑡),𝑡≥0) such that(i)𝑊(0)=0 a.s.;(ii)𝑊(𝑡)−𝑊(𝑠) is independent of ℱ_𝑠 for all 0≤𝑠<𝑡;(iii)𝑊(𝑡)−𝑊(𝑠) is 𝑁(0,𝑡−𝑠) distributed for all 0≤𝑠≤𝑡.

The following definitions and lemmas concerning square-mean almost automorphic functions can be found in [13, 14].

Definition 2. A stochastic process 𝑋∶ℝ→ℒ²(𝐏;ℍ) is said to be stochastically continuous if lim_𝑡→𝑠𝐄‖𝑋(𝑡)−𝑋(𝑠)‖²=0. (4)

Definition 3. 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 {𝑠^′_𝑛} there exists a subsequence {𝑠_𝑛} and a stochastic process 𝑌∶ℝ→ℒ²(𝐏;ℍ) such that lim_𝑛→∞𝐄𝑋𝑡+𝑠_𝑛−𝑌(𝑡)²=0,lim_𝑛→∞𝐄𝑋𝑡+𝑠_𝑛−𝑌(𝑡)²=0 (5) hold for each 𝑡∈ℝ. The collection of all square-mean almost automorphic stochastic processes is denoted by 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)).

Lemma 4. If 𝑋, 𝑋₁, and 𝑋₂ are all square-mean almost automorphic stochastic processes, then the following statements hold true:(i)𝑋₁+𝑋₂ is square-mean almost automorphic;(ii)𝜆𝑋 is square-mean almost automorphic for every scalar 𝜆;(iii)There exists a constant 𝑀>0 such that sup_𝑡∈ℝ‖𝑋(𝑡)‖₂≤𝑀. That is, 𝑋 is bounded in ℒ²(𝐏;ℍ).

Lemma 5. 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)) is a Banach space when it is equipped with the norm ‖𝑋‖_∞∶=sup_𝑡∈ℝ‖𝑋(𝑡)‖₂=sup_𝑡∈ℝ𝐄‖𝑋(𝑡)‖^21/2 (6) for 𝑋∈𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)).

Definition 6. A jointly continuous function 𝐹∶ℝ×ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ), (𝑡,𝑋)↦𝐹(𝑡,𝑋) 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} there exists a subsequence {𝑠_𝑛} and a stochastic process 𝐺∶ℝ×ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ) such that lim_𝑛→∞𝐄𝐹𝑡+𝑠_𝑛,𝑋−𝐺(𝑡,𝑋)²=0,lim_𝑛→∞𝐄𝐺𝑡−𝑠_𝑛,𝑋−𝐹(𝑡,𝑋)²=0 (7) hold for each 𝑡∈ℝ and each 𝑋∈ℒ²(𝐏;ℍ).

Lemma 7. Let 𝑓∶ℝ×ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ), (𝑡,𝑋)↦𝑓(𝑡,𝑋) be square-mean almost automorphic in 𝑡∈ℝ for each 𝑋∈ℒ²(𝐏;ℍ), and assume that 𝑓 satisfies a Lipschitz condition in the following sense: 𝐄𝑓𝑡,𝜑−𝑓𝑡,𝜓²≤𝐿𝐄𝜑−𝜓² (8) for all 𝜑,𝜓∈ℒ²(𝐏;ℍ) and each 𝑡∈ℝ, where 𝐿>0 is independent of 𝑡. Then for any square-mean almost automorphic stochastic process 𝑋∶ℝ→ℒ²(𝐏;ℍ), the stochastic process 𝐹∶ℝ→ℒ²(𝐏;ℍ) given by 𝐹(𝑡)∶=𝑓(𝑡,𝑋(𝑡)) is square-mean almost automorphic.

The Acquistapace-Terreni conditions (ATCs, for short) play an important role in the study of nonautonomous stochastic differential equations. We state it below for the readers’ convenience.

ATCs. There exist constants 𝜆₀≥0,𝜃∈(𝜋/2,𝜋),𝐿,𝐾≥0, and 𝛼,𝛽∈(0,1] with 𝛼+𝛽>1 such that Σ_𝜃∪{0}⊂𝜌𝐴(𝑡)−𝜆₀,𝑅𝜆,𝐴(𝑡)−𝜆₀≤𝐾|𝜆|,𝐴(𝑡)1+−𝜆₀𝑅𝜆,𝐴(𝑡)−𝜆₀×𝑅𝜆₀,𝐴(𝑡)−𝑅𝜆₀,𝐴(𝑠)≤𝐿|𝑡−𝑠|^𝛼|𝜆|^𝛽 (9) for 𝑡,𝑠∈ℝ,𝜆∈Σ_𝜃∶={𝜆∈ℂ−{0}∶|arg𝜆|≤𝜃}.

The following lemma can be found in [15, 24, 25].

Lemma 8. Suppose that the ATCs are satisfied, and then there exists a unique evolution family {𝑈(𝑡,𝑠)}_{−∞<𝑠≤𝑡<∞} on ℒ²(𝐏;ℍ), which governs the linear part of (1).

3. Main Results

In this section, we investigate the existence and uniqueness of square-mean almost automorphic solution to the nonautonomous stochastic integrodifferential evolution equation (1). The following assumptions are imposed on (1) which will be assumed throughout the manuscript.(H1) The operator 𝐴(𝑡)∶𝐷(𝐴(𝑡))⊂ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ) is a family of densely defined closed linear operators satisfying the ATCs, and the generated evolution family 𝑈(𝑡,𝑠) is uniformly exponentially stable; that is, there exist constants 𝑀≥1 and 𝛿>0 such that ‖𝑈(𝑡,𝑠)‖≤𝑀𝑒^{−𝛿(𝑡−𝑠)},∀𝑡≥𝑠. (10)(H2) The evolution family {𝑈(𝑡,𝑠),𝑡≥𝑠} generated by 𝐴(𝑡) satisfies the following condition: from every sequence of real numbers {𝑠^′_𝑛}_𝑛∈ℕ, we can extract a subsequence {𝑠_𝑛}_𝑛∈ℕ such that, for any 𝜀>0, there exists an 𝑁∈ℕ such that 𝑛>𝑁 implies that 𝑈𝑡+𝑠_𝑛,𝑠+𝑠_𝑛−𝑈(𝑡,𝑠)≤𝜀𝑒^{−𝛿(𝑡−𝑠)},𝑈𝑡−𝑠_𝑛,𝑠−𝑠_𝑛−𝑈(𝑡,𝑠)≤𝜀𝑒^{−𝛿(𝑡−𝑠)} (11) for all 𝑡≥𝑠, where 𝛿>0 is the constant required in (H1).(H3) The functions 𝐹_𝑖∶ℝ×ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ),(𝑡,𝑋)↦𝐹_𝑖(𝑡,𝑋)(𝑖=1,2), and 𝐺∶ℝ×ℒ²(𝐏;ℍ)→ℒ²(𝐏;ℍ),(𝑡,𝑋)↦𝐺(𝑡,𝑋) are square-mean almost automorphic in 𝑡∈ℝ for each 𝑋∈ℒ²(𝐏;ℍ). Moreover, 𝐹₁, 𝐹₂, and 𝐺 are Lipschitz in 𝑋 uniformly for 𝑡 in the following sense: there exist constants 𝐾_𝑖>0(𝑖=1,2,3) such that 𝐄𝐹_𝑖(𝑡,𝑋)−𝐹_𝑖(𝑡,𝑌)²≤𝐾_𝑖𝐄‖𝑋−𝑌‖²𝐄‖𝐺()(,𝑖=1,2,𝑡,𝑋−𝐺𝑡,𝑌)‖²≤𝐾₃𝐄‖𝑋−𝑌‖² (12) for all stochastic processes 𝑋,𝑌∈ℒ²(𝐏;ℍ) and 𝑡∈ℝ.

Definition 9. An ℱ_𝑡 progressively measurable process (𝑋(𝑡))_𝑡∈ℝ is called a mild solution of (1) if it satisfies the corresponding stochastic integral equation: 𝑋(𝑡)()𝑋(𝑎)+=𝑈𝑡,𝑎_𝑡𝑎𝑈(𝑡,𝜎)^𝜎_𝑎𝐶()𝐺((𝑠(𝑠)+𝜎−𝑠𝑠,𝑋))𝑑𝑊𝑑𝜎_𝑡𝑎𝑈(𝑡,𝜎)^𝜎_𝑎𝐵(𝜎−𝑠)𝐹₂((𝑠+𝑠,𝑋))𝑑𝑠𝑑𝜎_𝑡𝑎𝑈(𝑡,𝑠)𝐹₁(𝑠,𝑋(𝑠))𝑑𝑠 (13) for all 𝑡≥𝑎 and each 𝑎∈ℝ.

Now we are in a position to show the existence and uniqueness of square-mean almost automorphic solution to (1).

Theorem 10. Assume that conditions (H1)–(H3) are satisfied, then (1) has a unique square-mean almost automorphic mild solution 𝑋(⋅)∈𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)) which can be explicitly expressed as 𝑋(𝑡)=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶()𝐺((𝑠(𝑠)+𝜎−𝑠𝑠,𝑋))𝑑𝑊𝑑𝜎_𝑡−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝐹₂((𝑠+𝑠,𝑋))𝑑𝑠𝑑𝜎_𝑡−∞𝑈(𝑡,𝑠)𝐹₁(𝑠,𝑋(𝑠))𝑑𝑠 (14) provided that 𝑀Θ∶=3²𝛿²𝐾₁+𝐾₂⋅‖𝐵‖²_ℒ1()0,∞+𝐾₃⋅‖𝐶‖²_ℒ2()0,∞<1. (15)

Proof. First of all, it is not difficult to verify that the stochastic process 𝑋(𝑡)=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶()𝐺((𝑠(𝑠)+𝜎−𝑠𝑠,𝑋))𝑑𝑊𝑑𝜎_𝑡−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝐹₂((𝑠+𝑠,𝑋))𝑑𝑠𝑑𝜎_𝑡−∞𝑈(𝑡,𝑠)𝐹₁(𝑠,𝑋(𝑠))𝑑𝑠 (16) is well defined and satisfies 𝑋(𝑡)()𝑋(𝑎)+=𝑈𝑡,𝑎_𝑡𝑎𝑈(𝑡,𝜎)^𝜎_𝑎𝐶()𝐺((𝑠(𝑠)+𝜎−𝑠𝑠,𝑋))𝑑𝑊𝑑𝜎_𝑡𝑎𝑈(𝑡,𝜎)^𝜎_𝑎𝐵(𝜎−𝑠)𝐹₂((𝑠+𝑠,𝑋))𝑑𝑠𝑑𝜎_𝑡𝑎𝑈(𝑡,𝑠)𝐹₁(𝑠,𝑋(𝑠))𝑑𝑠 (17) for all 𝑡≥𝑎 and each 𝑎∈ℝ, and hence it is a mild solution of the original (1).
To seek the square-mean almost automorphic mild solution to (1), let us consider the nonlinear operator 𝒮 acting on the Banach space 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)) given by (𝒮𝑋)(𝑡)∶=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶()𝐺((𝑠(𝑠)+𝜎−𝑠𝑠,𝑋))𝑑𝑊𝑑𝜎_𝑡−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝐹₂((𝑠+𝑠,𝑋))𝑑𝑠𝑑𝜎_𝑡−∞𝑈(𝑡,𝑠)𝐹₁(𝑠,𝑋(𝑠))𝑑𝑠. (18) If we can show that the operator 𝒮 maps 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)) into itself and it is a contraction mapping, then, by Banach contraction principle, we can conclude that there is a unique square-mean almost automorphic mild solution to (1).
Now define three nonlinear operators acting on the Banach space 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)) as follows: 𝒮₁𝑋(𝑡)∶=^𝑡_−∞𝑈(𝑡,𝑠)𝐹₁((𝑠𝒮𝑠,𝑋))𝑑𝑠,₂𝑋(𝑡)∶=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝐹₂((𝑠𝒮𝑠,𝑋))𝑑𝑠𝑑𝜎,₃𝑋(𝑡)∶=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶(𝜎−𝑠)𝐺(𝑠,𝑋(𝑠))𝑑𝑊(𝑠)𝑑𝜎. (19) We show first that 𝒮₁𝑋 is square-mean almost automorphic whenever 𝑋 is. Indeed, assuming that 𝑋 is square-mean almost automorphic, then Lemma 7 implies that 𝑓₁(⋅)∶=𝐹₁(⋅,𝑋(⋅)) is also square-mean almost automorphic. Hence, for any sequence of real numbers {𝑠^′_𝑛}, there exists a subsequence {𝑠_𝑛} of {𝑠^′_𝑛} and a stochastic process 𝑓₁∶ℝ→ℒ²(𝐏;ℍ) such that lim_𝑛→∞𝐄𝑓₁𝑡+𝑠_𝑛−𝑓₁(𝑡)²=0,lim_𝑛→∞𝐄𝑓₁𝑡−𝑠_𝑛−𝑓₁(𝑡)²=0 (20) hold for each 𝑡∈ℝ. By assumption (H2), for any 𝜀>0, there exists 𝑁₁=𝑁₁(𝜀)∈ℕ such that 𝑛>𝑁₁ implies that 𝑈𝑡+𝑠_𝑛,𝑠+𝑠_𝑛−𝑈(𝑡,𝑠)≤𝜀𝑒^{−𝛿(𝑡−𝑠)}𝐄𝑓∀𝑡≥𝑠,₁𝑡+𝑠_𝑛−𝑓₁(𝑡)²𝐄𝑓<𝜀,₁𝑡−𝑠_𝑛−𝑓₁(𝑡)²<𝜀∀𝑡∈ℝ. (21) Now, define functions on ℝ as follows: 𝑢(𝑡)∶=^𝑡_−∞𝑈(𝑡,𝑠)𝑓₁(𝑠)𝑢(𝑡)𝑑𝑠∶=^𝑡_−∞𝑈(𝑡,𝑠)𝑓₁(𝑠)𝑑𝑠, (22) and then the above observation together with (3.1) and Cauchy-Schwarz inequality implies that for any 𝜀>0 and for the aforementioned 𝑁₁=𝑁₁(𝜀)∈ℕ if 𝑛>𝑁₁ it yields that 𝐄𝑢𝑡+𝑠_𝑛−𝑢(𝑡)²=𝐄^{𝑡+𝑠𝑛}_−∞𝑈𝑡+𝑠_𝑛,𝑠𝑓₁(𝑠)𝑑𝑠−^𝑡_−∞𝑈(𝑡,𝑠)𝑓₁(𝑠)𝑑𝑠²=𝐄^𝑡_−∞𝑈𝑡+𝑠_𝑛,𝑠+𝑠_𝑛𝑓₁𝑠+𝑠_𝑛−𝑑𝑠_𝑡−∞𝑈(𝑡,𝑠)𝑓₁(𝑠)𝑑𝑠²≤2𝐄^𝑡_−∞𝑈𝑡+𝑠_𝑛,𝑠+𝑠_𝑛−𝑈(𝑡,𝑠)𝑓₁𝑠+𝑠_𝑛𝑑𝑠²+2𝐄^𝑡_−∞𝑈(𝑡,𝑠)𝑓₁𝑠+𝑠_𝑛−𝑓₁(𝑠)𝑑𝑠²≤2𝜀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑓₁𝑠+𝑠_𝑛𝑑𝑠²+2𝑀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑓₁𝑠+𝑠_𝑛−𝑓₁(𝑠)𝑑𝑠²≤2𝜀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝐄𝑓₁𝑠+𝑠_𝑛²𝑑𝑠+2𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}×𝑑𝑠_𝑡−∞𝑒^{−𝛿(𝑡−𝑠)}𝐄𝑓₁𝑠+𝑠_𝑛−𝑓₁(𝑠)²𝑑𝑠≤2𝜀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²sup_𝑡∈ℝ𝐄𝑓₁𝑡+𝑠_𝑛²+2𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²sup_𝑡∈ℝ𝐄𝑓₁𝑡+𝑠_𝑛−𝑓₁(𝑡)²≤2𝑀₁𝛿²⋅𝜀²+2𝑀²𝛿²⋅𝜀, (23) where 𝑀₁∶=sup_𝑡∈ℝ𝐄‖𝑓₁(𝑡)‖²<+∞. Hence, lim_𝑛→∞𝐄𝑢𝑡+𝑠_𝑛−𝑢(𝑡)²=0 (24) for each 𝑡∈ℝ. And we can show in a similar way that lim_𝑛→∞𝐄𝑢𝑡−𝑠_𝑛−𝑢(𝑡)²=0 (25) for each 𝑡∈ℝ. Thus, we conclude that 𝑢=𝒮₁𝑋∈𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)).
In an analogous way, assuming that 𝑋 is square-mean almost automorphic and using Lemma 7, one can easily see that 𝑓₂(⋅)∶=𝐹₂(⋅,𝑋(⋅)) is also square-mean almost automorphic. Let {𝑠^′_𝑛} be an arbitrary sequence of real numbers, and then there exists a subsequence {𝑠_𝑛} of {𝑠^′_𝑛} and a stochastic process 𝑓₂∶ℝ→ℒ²(𝐏;ℍ) such that lim_𝑛→∞𝐄𝑓₂𝑡+𝑠_𝑛−𝑓₂(𝑡)²=0,lim_𝑛→∞𝐄𝑓₂𝑡−𝑠_𝑛−𝑓₂(𝑡)²=0 (26) hold for each 𝑡∈ℝ. Now define functions on ℝ as follows: 𝑣(𝑡)∶=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂(𝑠)𝑣(𝑡)𝑑𝑠𝑑𝜎∶=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂(𝑠)𝑑𝑠𝑑𝜎, (27) and then, by making change of variables 𝜏=𝜎−𝑠_𝑛 and 𝑟=𝑠−𝑠_𝑛, we obtain that 𝐄𝑣𝑡+𝑠_𝑛−𝑣(𝑡)²=𝐄^{𝑡+𝑠𝑛}_−∞𝑈𝑡+𝑠_𝑛,𝜎^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂(𝑠)𝑑𝑠𝑑𝜎−^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂(𝑠)𝑑𝑠𝑑𝜎²=𝐄^𝑡_−∞𝑈𝑡+𝑠_𝑛,𝜎+𝑠_𝑛×^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂𝑠+𝑠_𝑛−𝑑𝑠𝑑𝜎_𝑡−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂(𝑠)𝑑𝑠𝑑𝜎²≤2𝐄^𝑡_−∞𝑈𝑡+𝑠_𝑛,𝜎+𝑠_𝑛−𝑈(𝑡,𝜎)×^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂𝑠+𝑠_𝑛𝑑𝑠𝑑𝜎²+2𝐄^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐵()×𝑓𝜎−𝑠₂𝑠+𝑠_𝑛−𝑓₂(𝑠)𝑑𝑠𝑑𝜎²≤2𝜀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂𝑠+𝑠_𝑛𝑑𝑠𝑑𝜎²+2𝑀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐵()×𝑓𝜎−𝑠₂𝑠+𝑠_𝑛−𝑓₂(𝑠)𝑑𝑠𝑑𝜎². (28) Let us evaluate the first term of the right-handed side by using Cauchy-Schwarz inequality: 𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)𝜎}_−∞𝐵(𝜎−𝑠)𝑓₂𝑠+𝑠_𝑛𝑑𝑠𝑑𝜎²≤^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎×^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}𝐄^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂𝑠+𝑠_𝑛𝑑𝑠²≤𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)𝜎}_−∞‖𝐵(×𝜎−𝑠)‖𝑑𝑠_𝜎−∞‖𝐵(𝜎−𝑠)‖𝐄𝑓₂𝑠+𝑠_𝑛²≤𝑑𝑠𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎²⋅₀^+∞‖𝐵(𝑢)‖𝑑𝑢²⋅sup_𝑠∈ℝ𝐄𝑓₂𝑠+𝑠_𝑛²≤‖𝐵‖²_ℒ1(0,∞)𝛿²⋅sup_𝑡∈ℝ𝐄𝑓₂(𝑡)². (29) As to the second term, in a similar manner as above, we have the following observation: 𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)𝜎}_−∞𝐵(𝜎−𝑠)𝑓₂𝑠+𝑠_𝑛−𝑓₂(𝑠)𝑑𝑠𝑑𝜎²≤𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐵()×𝑓𝜎−𝑠₂𝑠+𝑠_𝑛−𝑓₂(𝑠)𝑑𝑠²≤𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝐄^𝜎_−∞𝐵()×𝑓𝜎−𝑠₂𝑠+𝑠_𝑛−𝑓₂(𝑠)𝑑𝑠²≤𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)𝜎}_−∞‖𝐵(×𝜎−𝑠)‖𝑑𝑠_𝜎−∞‖𝐵(𝑓𝜎−𝑠)‖×𝐄₂𝑠+𝑠_𝑛−𝑓₂(𝑠)²≤𝑑𝑠𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎²⋅₀^+∞‖𝐵(𝑢)‖𝑑𝑢²⋅sup_𝑡∈ℝ𝐄𝑓₂𝑡+𝑠_𝑛−𝑓₂(𝑡)²≤1𝛿²⋅‖𝐵‖²_ℒ1(0,∞)⋅sup_𝑡∈ℝ𝐄𝑓₂𝑡+𝑠_𝑛−𝑓₂(𝑡)²≤‖𝐵‖²_ℒ1(0,∞)𝛿²⋅𝜀. (30) Based on the above argument, for arbitrary 𝜀>0, thanks to the boundedness and square-mean almost automorphy of 𝑓₂, there exists certain 𝑁₂=𝑁₂(𝜀)∈ℕ such that 𝑛>𝑁₂ implies that 𝐄𝑣𝑡+𝑠_𝑛−𝑣(𝑡)²≤2𝜀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐵(𝜎−𝑠)𝑓₂𝑠+𝑠_𝑛𝑑𝑠𝑑𝜎²+2𝑀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐵()×𝑓𝜎−𝑠₂𝑠+𝑠_𝑛−𝑓₂(𝑠)𝑑𝑠𝑑𝜎²≤2‖𝐵‖²_ℒ1(0,∞)⋅sup_𝑡∈ℝ𝐄𝑓₂(𝑡)²𝛿²⋅𝜀²+2𝑀²⋅‖𝐵‖²_ℒ1(0,∞)𝛿²⋅𝜀, (31) where we use the fact that, for arbitrary 𝜀>0, there exists 𝑁₂=𝑁₂(𝜀)∈ℕ such that, for all 𝑛>𝑁₂, 𝐄𝑓₂𝑡+𝑠_𝑛−𝑓₂(𝑡)²<𝜀 (32) holds for all 𝑡∈ℝ. This indicates that lim_𝑛→∞𝐄𝑣𝑡+𝑠_𝑛−𝑣(𝑡)²=0 (33) for each 𝑡∈ℝ. In an analogous way, we can show that lim_𝑛→∞𝐄𝑣𝑡−𝑠_𝑛−𝑣(𝑡)²=0 (34) for each 𝑡∈ℝ. Combining (33) with (34), we obtain that 𝑣=𝒮₂𝑋 is square-mean almost automorphic.
Assuming that 𝑋∈𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)), then similar argument as above ensures that 𝑔(⋅)∶=𝐺(⋅,𝑋(⋅))∈𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)). As a consequence, for every sequence of real numbers {𝑠^′_𝑛} there exist a subsequence {𝑠_𝑛}⊂{𝑠^′_𝑛} and a stochastic process 𝑔 such that lim_𝑛→∞𝐄𝑔𝑡+𝑠_𝑛−𝑔(𝑡)²=0,lim_𝑛→∞𝐄𝑔𝑡−𝑠_𝑛−𝑔(𝑡)²=0 (35) hold for each 𝑡∈ℝ. Hence, for arbitrary 𝜀>0, there exists certain 𝑁₃=𝑁₃(𝜀)∈ℕ such that, for all 𝑛>𝑁₃, there holds 𝐄𝑔𝑡+𝑠_𝑛−𝑔(𝑡)²<𝜀 (36) for all 𝑡∈ℝ.
The next step aims to prove the square-mean almost automorphy of 𝒮₃𝑋. This is more complicated because the involvement of the Brownian motion 𝑊. Consider the Brownian motion 𝑊 defined by 𝑊(𝑠)=𝑊𝑠+𝑠_𝑛−𝑊𝑠_𝑛 (37) for each 𝑠∈ℝ, which has the same distribution as 𝑊. Define two functions on ℝ as below: 𝜔(𝑡)∶=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶()𝑔(𝑠)(𝑠)𝜔(𝑡)𝜎−𝑠𝑑𝑊𝑑𝜎,∶=^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶(𝜎−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)𝑑𝜎. (38) By making change of variables 𝜏=𝜎−𝑠_𝑛 and 𝑟=𝑠−𝑠_𝑛, and then using Cauchy-Schwarz inequality, we obtain that 𝐄𝜔𝑡+𝑠_𝑛−𝜔(𝑡)²=𝐄^{𝑡+𝑠𝑛}_−∞𝑈𝑡+𝑠_𝑛,𝜎^𝜎_−∞𝐶(𝜎−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)𝑑𝜎−^𝑡_−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶(𝜎−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)𝑑𝜎²=𝐄^𝑡_−∞𝑈𝑡+𝑠_𝑛,𝜎+𝑠_𝑛×^𝜎_−∞𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛𝑑𝑊𝑠+𝑠_𝑛−𝑑𝜎_𝑡−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶(𝜎−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)𝑑𝜎²=𝐄^𝑡_−∞𝑈𝑡+𝑠_𝑛,𝜎+𝑠_𝑛×^𝜎_−∞𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛𝑑𝑊(𝑠)−𝑑𝜎_𝑡−∞𝑈(𝑡,𝜎)^𝜎_−∞𝐶(𝜎−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)𝑑𝜎²≤2𝐄^𝑡_−∞𝑈𝑡+𝑠_𝑛,𝜎+𝑠_𝑛(×−𝑈𝑡,𝜎)_𝜎−∞𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛𝑑𝑊(𝑠)𝑑𝜎²+2𝐄^𝑡_−∞𝑈()×𝑡,𝜎_𝜎−∞𝐶()×𝑔𝜎−𝑠𝑠+𝑠_𝑛−𝑔(𝑠)𝑑𝑊(𝑠)𝑑𝜎²≤2𝜀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛𝑑𝑊(𝑠)𝑑𝜎²+2𝑀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐶()×𝑔𝜎−𝑠𝑠+𝑠_𝑛−𝑔(𝑠)𝑑𝑊(𝑠)𝑑𝜎²≤2𝜀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝐄^𝜎_−∞𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛𝑑𝑊(𝑠)²𝑑𝜎+2𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝐄^𝜎_−∞𝐶()×𝑔𝜎−𝑠𝑠+𝑠_𝑛−𝑔(𝑠)𝑑𝑊(𝑠)²𝑑𝜎. (39) For the above-mentioned 𝜀>0, by using an estimate on Ito integral established in [26], it follows that once 𝑛>𝑁₃, then the first term of the above inequality can be evaluated as ^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝐄^𝜎_−∞𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛𝑑𝑊(𝑠)²≤𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐄𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛²≤𝑑𝑠𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞‖𝐶(𝜎−𝑠)‖²𝐄𝑔𝑠+𝑠_𝑛²≤𝑑𝑠𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)𝜎}_−∞‖𝐶(𝜎−𝑠)‖²𝑑𝑠𝑑𝜎⋅sup_𝑠∈ℝ𝐄𝑔𝑠+𝑠_𝑛²≤^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎²⋅^∞₀‖𝐶(𝑢)‖²𝑑𝑢⋅sup_𝑡∈ℝ𝐄𝑔(𝑡)²≤‖𝐶‖²_ℒ2(0,∞)𝛿²⋅sup_𝑡∈ℝ𝐄𝑔(𝑡)², (40) and the second term can be evaluated analogously as ^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}𝐄^𝜎_−∞𝐶()×𝑔𝜎−𝑠𝑠+𝑠_𝑛−𝑔(𝑠)𝑑𝑊(𝑠)²≤𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)𝜎}_−∞𝐄𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛−𝑔(𝑠)²≤𝑑𝑠𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)𝜎}_−∞‖𝐶(𝜎−𝑠)‖²𝐄×𝑔𝑠+𝑠_𝑛−𝑔(𝑠)²≤𝑑𝑠𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞‖𝐶(𝜎−𝑠)‖²𝑑𝑠𝑑𝜎⋅sup_𝑡∈ℝ𝑔𝑠+𝑠_𝑛−𝑔(𝑠)²≤^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎²⋅^∞₀‖𝐶(𝑢)‖²𝑑𝑢⋅sup_𝑡∈ℝ𝑔𝑠+𝑠_𝑛−𝑔(𝑠)²≤‖𝐶‖²_ℒ2(0,∞)𝛿²⋅𝜀. (41) The above argument yields that 𝐄𝜔𝑡+𝑠_𝑛−𝜔(𝑡)²≤2𝜀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝐄^𝜎_−∞𝐶(𝜎−𝑠)𝑔𝑠+𝑠_𝑛𝑑𝑊(𝑠)²𝑑𝜎+2𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝐄^𝜎_−∞𝐶()×𝑔𝜎−𝑠𝑠+𝑠_𝑛−𝑔(𝑠)𝑑𝑊(𝑠)²≤2‖𝐶‖𝑑𝜎_2ℒ2(0,∞)⋅sup_𝑡∈ℝ𝐄𝑔(𝑡)²𝛿²⋅𝜀²+2𝑀²⋅‖𝐶‖²_ℒ2(0,∞)𝛿²⋅𝜀, (42) which implies that lim_𝑛→∞𝐄𝜔𝑡+𝑠_𝑛−𝜔(𝑡)²=0 (43) for each 𝑡∈ℝ. Analogously, we can show that lim_𝑛→∞𝐄𝜔𝑡−𝑠_𝑛−𝜔(𝑡)²=0 (44) for each 𝑡∈ℝ. Combining (43) with (44), we obtain that 𝜔=𝒮₃𝑋 is square-mean almost automorphic.
In view of the above arguments, it follows from (18) that the nonlinear operator 𝒮=𝒮₁+𝒮₂+𝒮₃ maps 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)) into itself. To complete the proof, it suffices to show that 𝒮 is a contraction mapping on 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)). Indeed, for each 𝑋,𝑌∈𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)), thanks to the fact that (𝑎+𝑏+𝑐)²≤3𝑎²+3𝑏²+3𝑐², we have the following observation: 𝐄‖(𝒮𝑋)(𝑡)−(𝒮𝑌)(𝑡)‖²≤3𝐄^𝑡_−∞𝑈(𝑡,𝑠)𝐹₁(𝑠,𝑋(𝑠))−𝐹₁(𝑠,𝑌(𝑠))𝑑𝑠²+3𝐄^𝑡_−∞𝑈(𝑡,𝜎)×^𝜎_−∞𝐵()×𝐹𝜎−𝑠₂(𝑠,𝑋(𝑠))−𝐹₂(𝑠,𝑌(𝑠))𝑑𝑠𝑑𝜎²+3𝐄^𝑡_−∞𝑈()×𝑡,𝜎_𝜎−∞𝐶()×[𝐺((𝑠((𝑠(𝑠)𝜎−𝑠𝑠,𝑋))−𝐺𝑠,𝑌))]𝑑𝑊𝑑𝜎²≤3𝐄^𝑡_−∞‖𝑈(𝑡,𝑠)‖𝐹₁(𝑠,𝑋(𝑠))−𝐹₁(𝑠,𝑌(𝑠))𝑑𝑠²+3𝐄^𝑡_−∞‖𝑈(×𝑡,𝜎)‖_𝜎−∞𝐵()×𝐹𝜎−𝑠₂(𝑠,𝑋(𝑠))−𝐹₂(𝑠,𝑌(𝑠))𝑑𝑠𝑑𝜎²+3𝐄^𝑡_−∞‖𝑈(×𝑡,𝜎)‖_𝜎−∞𝐶(𝐺((𝑠((𝑠(𝑠𝜎−𝑠)[𝑠,𝑋))−𝐺𝑠,𝑌))]𝑑𝑊)𝑑𝜎². (45) Now, we evaluate the first term of the right-hand side with the help of Cauchy-Schwarz inequality as follows: 3𝐄^𝑡_−∞‖𝑈(𝑡,𝑠)‖𝐹₁(𝑠,𝑋(𝑠))−𝐹₁(𝑠,𝑌(𝑠))𝑑𝑠²≤3𝑀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝐹₁(𝑠,𝑋(𝑠))−𝐹₁(𝑠,𝑌(𝑠))𝑑𝑠²≤3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}×𝑑𝑠_𝑡−∞𝑒^{−𝛿(𝑡−𝑠)}𝐄𝐹₁(𝑠,𝑋(𝑠))−𝐹₁(𝑠,𝑌(𝑠))²𝑑𝑠≤3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²×sup_𝑠∈ℝ𝐄𝐹₁(𝑠,𝑋(𝑠))−𝐹₁(𝑠,𝑌(𝑠))²≤3𝐾₁𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²sup_𝑡∈ℝ𝐄‖𝑋(𝑡)−𝑌(𝑡)‖²=3𝐾₁𝛿²⋅𝑀²⋅sup_𝑡∈ℝ𝐄‖𝑋(𝑡)−𝑌(𝑡)‖². (46) As to the second term, we can proceed in the same manner as above and obtain 3𝐄^𝑡_−∞‖𝑈(×𝑡,𝜎)‖_𝜎−∞𝐵(𝜎−𝑠)𝐹₂(𝑠,𝑋(𝑠))−𝐹₂(𝑠,𝑌(𝑠))𝑑𝑠𝑑𝜎²≤3𝐾₂𝛿²⋅𝑀²⋅‖𝐵‖²_ℒ1(0,∞)⋅sup_𝑡∈ℝ𝐄‖𝑋(𝑡)−𝑌(𝑡)‖². (47) As far as the last term of the right-hand side is concerned, we use again the estimate on the Ito integral to obtain 3𝐄^𝑡_−∞‖𝑈(×𝑡,𝜎)‖_𝜎−∞𝐶()×[𝐺((𝑠((𝑠(𝑠𝜎−𝑠𝑠,𝑋))−𝐺𝑠,𝑌))]𝑑𝑊)𝑑𝜎²≤3𝑀²𝐄^𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞𝐶()×[𝐺((𝑠((𝑠𝜎−𝑠𝑠,𝑋))−𝐺𝑠,𝑌))]𝑑𝑊𝑑𝜎²≤3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×𝐄^𝜎_−∞𝐶()×[𝐺((𝑠((𝑠(𝑠𝜎−𝑠𝑠,𝑋))−𝐺𝑠,𝑌))]𝑑𝑊)²𝑑𝜎≤3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}×𝑑𝜎_𝑡−∞𝑒^{−𝛿(𝑡−𝜎)}×^𝜎_−∞‖𝐶(𝜎−𝑠)‖²×𝐄‖𝐺(𝑠,𝑋(𝑠))−𝐺(𝑠,𝑌(𝑠))‖²𝑑𝑠𝑑𝜎≤3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎²⋅‖𝐶‖²_ℒ2(0,∞)⋅sup_𝑠∈ℝ𝐄‖𝐺(𝑠,𝑋(𝑠))−𝐺(𝑠,𝑌(𝑠))‖²≤3𝐾₃𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝜎)}𝑑𝜎²⋅‖𝐶‖²_ℒ2(0,∞)⋅sup_𝑡∈ℝ𝐄‖𝑋(𝑡)−𝑌(𝑡)‖²=3𝐾₃𝛿²⋅𝑀²⋅‖𝐶‖²_ℒ2(0,∞)⋅sup_𝑡∈ℝ𝐄‖𝑋(𝑡)−𝑌(𝑡)‖². (48) Thus, by combining the three inequalities together, we obtain that, for each 𝑡∈ℝ, 𝐄‖(𝒮𝑋)(𝑡)−(𝒮𝑌)(𝑡)‖²≤3𝑀²𝛿²𝐾₁+𝐾₂⋅‖𝐵‖²_ℒ1()0,∞+𝐾₃⋅‖𝐶‖²_ℒ2()0,∞⋅sup_𝑡∈ℝ𝐄‖𝑋(𝑡)−𝑌(𝑡)‖². (49) That is, ‖(𝒮𝑋)(𝑡)−(𝒮𝑌)(𝑡)‖²₂≤Θsup_𝑡∈ℝ‖𝑋(𝑡)−𝑌(𝑡)‖²₂, (50) where Θ∶=3(𝑀²/𝛿²)[𝐾₁+𝐾₂⋅‖𝐵‖²_ℒ1(0,∞)+𝐾₃⋅‖𝐶‖²_ℒ2(0,∞)]. Notice that sup_𝑡∈ℝ‖𝑋(𝑡)−𝑌(𝑡)‖²₂≤sup_𝑡∈ℝ‖𝑋(𝑡)−𝑌(𝑡)‖₂². (51) As a result, (50) together with (51) gives that, for each 𝑡∈ℝ, ‖(𝒮𝑋)(𝑡)−(𝒮𝑌)(𝑡)‖₂≤√Θsup_𝑡∈ℝ‖𝑋(𝑡)−𝑌(𝑡)‖₂=√Θ‖𝑋−𝑌‖_∞. (52) It follows that ‖𝒮𝑋−𝒮𝑌‖_∞=sup_𝑡∈ℝ‖(𝒮𝑋)(𝑡)−(𝒮𝑌)(𝑡)‖₂≤√Θ‖𝑋−𝑌‖_∞, (53) which implies that 𝒮 is a contraction mapping by the assumption (15) imposed on Θ. Therefore, by the Banach contraction principle, we conclude that there exists a unique fixed point 𝑋 for 𝒮 in 𝐴𝐴(ℝ;ℒ²(𝐏;ℍ)), which is the unique square-mean almost automorphic mild solution to the functional integrodifferential semilinear stochastic evolution equation (1) as we have claimed. The proof is complete.

Remark 11. If the functions 𝐹₁,𝐹₂,𝐺 in (1) are square-mean almost periodic in 𝑡, then the unique square-mean almost automorphic solution obtained in Theorem 10 is actually square-mean almost periodic; see paper [27].

Now we are in a position to show the asymptotically stable property of the unique square-mean almost automorphic solution to (1). Recall that the unique square-mean almost automorphic solution 𝑋^∗(𝑡) of (1) is said to be stable in square-mean sense if, for arbitrary 𝜖>0, there exists 𝛿>0 such that 𝐄𝑋(𝑡)−𝑋^∗(𝑡)²<𝜖,𝑡≥0 (54) whenever 𝐄‖𝑋(0)−𝑋^∗(0)‖²<𝛿, where 𝑋(𝑡) stands for a solution of (1) with initial value 𝑋(0). The solution 𝑋^∗(𝑡) is said to be asymptotically stable in square-mean sense if it is stable in square-mean sense and lim_𝑡→∞𝐄𝑋(𝑡)−𝑋^∗(𝑡)²=0. (55)

The following Gronwall-type inequality is proved to be useful in our asymptotic stability analysis.

Lemma 12. Let 𝑢(𝑡),𝑏(𝑡) be nonnegative continuous functions for 𝑡≥𝑎, and 𝛼,𝛾 be some positive constants. If 𝑢(𝑡)≤𝛼𝑒^{−𝛽(𝑡−𝑎)}+^𝑡_𝑎𝑒^{−𝛽(𝑡−𝑠)}𝑏(𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥𝑎, (56) then 𝑢(𝑡)≤𝛼exp−𝛽(𝑡−𝑎)+^𝑡_𝑎𝑏(𝑠)𝑑𝑠. (57)

Theorem 13. Let all the assumptions in Theorem 10 hold and assume that 𝑀²𝛿²𝐾₁+𝐾₂⋅‖𝐵‖²_ℒ1()0,∞+𝐾₃⋅‖𝐶‖²_ℒ2()0,∞<14. (58) Then the unique square-mean almost automorphic mild solution 𝑋^∗(𝑡) of (1) is asymptotically stable in square-mean sense.

Proof. Let 𝑋(𝑡) be any mild solution of (1) with initial value 𝑋(0). Then, on account of (H1)-(H2) and the assumptions imposed on 𝐵 and 𝐶, along the same line as in [9] we could show that, for any 𝑡≥0, 𝐄𝑋(𝑡)−𝑋^∗(𝑡)²=𝐄𝑈(𝑡,0)𝑋(0)−𝑋^∗(0)+^𝑡₀𝑈()×𝐹𝑡,𝑠₁(𝑠,𝑋(𝑠))−𝐹₁𝑠,𝑋^∗(𝑠+)𝑑𝑠_𝑡0𝑈()×𝑡,𝜎_𝜎0𝐵()×𝐹𝜎−𝑠₂(𝑠,𝑋(𝑠))−𝐹₂𝑠,𝑋^∗(𝑠)+𝑑𝑠𝑑𝜎_𝑡0𝑈()×𝑡,𝜎_𝜎0𝐶()×𝐺((𝑠𝜎−𝑠𝑠,𝑋))−𝐺𝑠,𝑋^∗(𝑠)𝑑𝑊(𝑠)𝑑𝜎²≤4𝐄𝑈(𝑡,0)𝑋(0)−𝑋^∗(0)²+4𝐄^𝑡₀𝑈(𝑡,𝑠)𝐹₁(𝑠,𝑋(𝑠))−𝐹₁𝑠,𝑋^∗(𝑠)²+4𝐄^𝑡₀𝑈()×𝑡,𝜎_𝜎0𝐵()×𝐹𝜎−𝑠₂(𝑠,𝑋(𝑠))−𝐹₂𝑠,𝑋^∗(𝑠)𝑑𝑠𝑑𝜎²+4𝐄^𝑡₀𝑈(𝑡,𝜎)×^𝜎₀𝐶()×𝐺((𝑠𝜎−𝑠𝑠,𝑋))−𝐺𝑠,𝑋^∗(𝑠)𝑑𝑊(𝑠)𝑑𝜎²≤4𝑀²𝑒^−𝛿𝑡𝐄𝑋(0)−𝑋^∗(0)²+𝜅^𝑡₀𝑒^{−𝛿(𝑡−𝑠)}𝐄𝑋(𝑠)−𝑋^∗(𝑠)²𝑑𝑠, (59) where 𝜅∶=(4𝑀²(𝐾₁+𝐾₂⋅‖𝐵‖²_ℒ1(0,∞)+𝐾₃⋅‖𝐶‖²_ℒ2(0,∞)))/𝛿.
Define 𝑌(𝑡)∶=𝐄‖𝑋(𝑡)−𝑋^∗(𝑡)‖², and it yields that 𝑌(𝑡)≤4𝑀²𝑌(0)𝑒^−𝛿𝑡+𝜅^𝑡₀𝑒^{−𝛿(𝑡−𝑠)}𝑌(𝑠)𝑑𝑠. (60) Hence, it follows from Lemma 12 that 𝑌(𝑡)≤4𝑀²𝑌(0)exp{(−𝛿+𝜅)𝑡}. (61) Straightforwardly, we obtain that 𝑌(𝑡) converges to 0 exponentially fast if −𝛿+𝜅<0, which is equivalent to our condition (58). Thus, we come to the conclusion that the unique square-mean almost automorphic mild solution 𝑋^∗(𝑡) of (1) is asymptotically stable in square-mean sense. The proof is completed.

4. Applications

To illustrate the applications of our abstract results, let us consider the following nonautonomous functional integrodifferential stochastic partial differential equation: 𝜕𝑋=𝜕𝜕𝑡²𝑋𝜕𝑥²+^𝑡_−∞𝐶()𝐺(((𝑠)+𝑡−𝑠𝑠,𝑋𝑠,𝑥))𝑑𝑊_𝑡−∞𝐵(𝑡−𝑠)𝐹₂(𝑠,𝑋(𝑠,𝑥))𝑑𝑠+𝐹₁(𝑡,𝑋(𝑡,𝑥)) (62) for 𝑡∈ℝ and 𝑥∈Ω, where Ω⊂ℝ^𝑛 is a bounded subset whose boundary 𝜕Ω is both of class 𝐶² and locally on one side of Ω. Suppose further that (62) satisfies the following boundary conditions: ^𝑛_𝑖,𝑗=1𝑛_𝑖(𝑥)𝑎_𝑖𝑗()()𝑡,𝑥𝑑𝑋𝑡,𝑥𝑑𝑥_𝑖=0,𝑡∈ℝ,𝑥∈𝜕Ω, (63) where 𝑛(𝑥)=(𝑛₁(𝑥),𝑛₂(𝑥),…,𝑛_𝑛(𝑥)) is the outer unit normal vector. A family of operators 𝐴(𝑡,𝑥) defined by 𝜕²𝑋/𝜕𝑥²=𝐴(𝑡,𝑥)𝑋(𝑡,𝑥) is formally assigned to be 𝐴(𝑡,𝑥)=^𝑛_𝑖,𝑗=1𝜕𝜕𝑥_𝑖𝑎_𝑖𝑗()𝜕𝑡,𝑥𝜕𝑥_𝑗+𝑐(𝑡,𝑥),𝑡∈ℝ,𝑥∈Ω, (64) where 𝑎_𝑖𝑗(𝑖,𝑗=1,2,…,𝑛) and 𝑐 satisfy the following conditions.(H4) The coefficients 𝑎_𝑖𝑗(𝑖,𝑗=1,2,…,𝑛) are symmetric, that is, 𝑎_𝑖𝑗=𝑎_𝑗𝑖 for all 𝑖,𝑗=1,2,…,𝑛. In addition, 𝑎_𝑖𝑗∈𝐶^𝜇_𝑏ℝ;ℒ²𝐏;𝐶Ω∩𝐶_𝑏ℝ;ℒ²𝐏;𝐶¹Ω∩𝐴𝐴ℝ;ℒ²𝐏;ℒ²(Ω) (65) for all 𝑖,𝑗=1,2,…,𝑛 and 𝑐∈𝐶^𝜇_𝑏ℝ;ℒ²𝐏;ℒ²(Ω)∩𝐶_𝑏ℝ;ℒ²𝐏;𝐶Ω∩𝐴𝐴ℝ;ℒ²𝐏;ℒ¹(Ω) (66) for some 𝜇∈(1/2,1], where Ω means the closure of Ω.(H5) There exists 𝛿₀>0 such that ^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝑡,𝑥)𝜂_𝑖𝜂_𝑗≥𝛿₀𝜂² (67) for all (𝑡,𝑥)∈ℝ×Ω and 𝜂∈ℝ^𝑛.

Now, let ℍ=ℒ²(Ω) and let ℋ²(Ω) be the Sobolev space of order 2 on Ω. For each 𝑡∈ℝ, define an operator 𝐴(𝑡) on ℒ²(𝐏;ℍ) by 𝐴(𝑡)𝑋=𝐴(𝑡,𝑥)𝑋∀𝑋∈𝒟(𝐴(𝑡)), (68) where 𝒟(𝐴(𝑡))=𝑋∈ℒ²𝐏,ℋ²(Ω)∶_{𝑛𝑖,𝑗=1}𝑛_𝑖(⋅)𝑎_𝑖𝑗()()𝑡,⋅𝑑𝑋𝑡,⋅𝑑𝑥_𝑖=0on𝜕Ω. (69) Under assumptions (H4)-(H5), the existence of an evolution family 𝑈(𝑡,𝑠) satisfying (H1) is guaranteed; see, for example, [28].

And thus, as an immediate consequence of Theorem 10, it yields the following.

Theorem 14. Under assumptions (H2), (H3), (H4), and (H5), the nonautonomous integrodifferential stochastic evolution equation (62)-(63) has a unique mild solution which is square-mean almost automorphic provided that (15) holds. If, in addition, (58) is valid, then the unique almost automorphic solution is asymptotically stable in square-mean sense.

Acknowledgments

This paper is jointly supported by the National Natural Science Foundation of China under Grant nos. 11201266 and 11171191 and the Tianyuan Youth Foundation of National Natural Science Foundation of China under Grant nos. 11026150 and 11026098.