Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integrodifferential Equations in Hilbert Spaces

Abstract

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

1. Introduction

The almost periodic type solutions to stochastic differential equations are among the most attractive topics in mathematical analysis due to their extensive applications in areas such as physics, economics, mathematical biology, and engineering. And the concept of almost automorphic functions, which was initially introduced in the literature by Bochner [1], is an important generalization of the almost periodic functions. From then on, the almost automorphic functions and the almost automorphic solutions for differential systems have been investigated by many mathematicians [2–13].

Integrodifferential equations arose naturally in mechanics, electromagnetic theory, heat flow, nuclear reactor dynamics, and population dynamics [14]. The papers [5, 14–20] are concerned with the existence of almost periodic type solutions to stochastic functional integrodifferential equations. Ding et al. [21] investigated the existence of pseudo almost periodic solutions for an equation arising in the study of heat conduction in materials with memory, which could be transformed into the following abstract integrodifferential equation: 𝑢^′(𝑡)=𝐴𝑢(𝑡)+^𝑡₀𝐵(𝑡−𝑠)𝑢(𝑠)𝑑𝑠+𝑓(𝑡,𝑢(𝑡)),𝑡≥0. (1) Diagana et al. [5] established the existence and uniqueness of asymptotically almost automorphic mild solution to an abstract partial neutral integrodifferential equation with unbounded delay 𝑑𝐷𝑑𝑡𝑡,𝑢_𝑡=𝐴𝐷𝑡,𝑢_𝑡+^𝑡₀𝐵(𝑡−𝑠)𝐷𝑠,𝑢_𝑠𝑑𝑠+𝑔𝑡,𝑢_𝑡[),𝑢,𝑡∈𝜎,𝜎+𝑎_𝜎=𝜑, (2) where 𝐷(𝑡,𝜑)=𝜑(0)+𝑓(𝑡,𝜑).

Furthermore, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. This paper is mainly focused on the existence and uniqueness of square-mean almost automorphic mild solutions to the following stochastic functional integrodifferential equations in the abstract form: 𝑑𝑁(𝑡,𝑥(𝑡))=𝐴𝑁(𝑡,𝑥(𝑡))𝑑𝑡+^𝑡₀𝐵()𝑁((𝑠𝛾𝑡−𝑠𝑠,𝑥))𝑑𝑠𝑑𝑡+ℎ𝑡,𝑥₂(𝑡)𝑑𝑡+𝑓𝑡,𝑥𝛾₃(𝑡)𝑑𝑊(𝑡),𝑡≥0, (3) where 𝐴∶𝐷(𝐴)⊆𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻), 𝐵(𝑡)∶𝐷(𝐵(𝑡))⊆𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻), and 𝑡≥0 are linear, closed, and densely defined operators on 𝐿²(𝑃,𝐻) and 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 𝑁(𝑡,𝑥(𝑡))=𝑥(𝑡)−𝑎(𝑡,𝑥(𝛾₁(𝑡))), 𝑎, ℎ, 𝑓, and 𝛾_𝑖(𝑖=1,2,3) are appropriate functions to be specified later.

The Lipschitz condition is a very important condition in the field of the existence and uniqueness of solutions for differential equations. In [15], Chang et al. established a new composition theorem for square-mean almost automorphic functions under conditions which were different from Lipschitz conditions in the literature. And they apply this new composition theorem to investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation. It can be proved that the conditions in [15] (see assumption (H3)(ii)) are weaker than Lipschitz condition. In other words, a function with Lipschitz condition is satisfied (H3)(ii). However, there exists such a function which satisfies (H3)(ii) but does not satisfy Lipschitz condition (see Remark 11).

In this paper, firstly, upon using Lipschitz condition and other some appropriate assumptions, some sufficient conditions for the existence and uniqueness of square-mean almost automorphic mild solution to (3) are given. Secondly, by virtue of new composition theorem in [15] together with Schauder’s fixed point theorem, we investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation in a real separable Hilbert space, which is different from Lipschitz condition in the literature. Finally, we discuss the existence and uniqueness of an almost automorphic mild solution to a concrete integrodifferential equation, which is an illustration to demonstrate our main analyses.

2. Preliminaries

Throughout this 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‖²𝑑𝑃<∞. (4) For 𝑥∈𝐿²(𝑃,𝐻), let ‖𝑥‖₂=_Ω‖𝑥‖²𝑑𝑃^1/2. (5) Then it is routine to check that 𝐿²(𝑃,𝐻) is a Hilbert space equipped with the norm ‖⋅‖₂. The notations 𝐶(𝑅;𝐿²(𝑃,𝐻)) and 𝐵𝐶(𝑅;𝐿²(𝑃,𝐻)) stand for the collection of all continuous stochastic processes from 𝑅 into 𝐿²(𝑃,𝐻) and the space of all bounded continuous stochastic processes 𝑥∶𝑅→𝐿²(𝑃,𝐻), respectively. It is then easy to check that 𝐵𝐶(𝑅;𝐿²(𝑃,𝐻)) is a Banach space when it is endowed with the norm ‖𝑥‖_{𝐵𝐶(𝑅;𝐿}²_(𝑃,𝐻))∶=sup_𝑡∈𝑅‖𝑥(𝑡)‖₂. Let 𝐿(𝐾,𝐻) denote the space of all linear bounded operators from 𝐾 into 𝐻, which are equipped with the usual operator norm ‖⋅‖_{𝐿(𝐾,𝐻)}; in particular, it is simply denoted by ℒ(𝐻) when 𝐾=𝐻. In addition, 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}.

Definition 1 (see [18]). A family of bounded linear operators 𝑇(𝑡)∶𝑡≥0 from 𝐿²(𝑃,𝐻) into 𝐿²(𝑃,𝐻) is a resolvent operator family for the problem 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)𝑑𝑡+^𝑡₀𝐵()𝑥(𝑠)𝑥(0)𝑡−𝑠𝑑𝑠𝑑𝑡,𝑡≥0,=𝑥₀∈𝐿²(𝑃,𝐻),𝑡≥0, (6) if the following conditions are satisfied:(a)𝑇(0)=𝐼 (the identity operator on 𝐿²(𝑃,𝐻)) and the function 𝑇(𝑡)𝑥 is continuous on [0,+∞) for every 𝑥∈𝐿²(𝑃,𝐻);(b)𝑇(𝑡)𝐷(𝐴)⊆𝐷(𝐴) for all 𝑡≥0 and for 𝑥∈𝐷(𝐴), 𝐴𝑇(𝑡)𝑥 is continuous on [0,+∞), and 𝑇(𝑡)𝑥 is continuously differentiable on [0,+∞);(c)for 𝑥∈𝐷(𝐴), the following resolvent equations hold: 𝑑𝑇(𝑡)(𝑡)𝑑𝑡𝑥=𝐴𝑇𝑥+^𝑡₀𝐵()𝑇(𝑠)𝑑𝑡−𝑠𝑥𝑑𝑠,𝑡≥0,𝑇(𝑡)(𝑡)𝑑𝑡𝑥=𝑇𝐴𝑥+^𝑡₀𝑇(𝑡−𝑠)𝐴𝐵(𝑠)𝑥𝑑𝑠,𝑡≥0. (7)

For more details on semigroup theory and resolvent operators, we refer the reader to [16, 18, 22].

Definition 2 (see [17]). A stochastic process 𝑥∶𝑅→𝐿²(𝑃,𝐻) is said to be stochastically continuous if lim_𝑡→𝑠𝐸‖𝑥(𝑡)−𝑥(𝑠)‖²=0. (8)

Definition 3 (see [17]). 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 exists a subsequence {𝑠_𝑛}_𝑛∈𝑁 and a stochastic process 𝑦∶𝑅→𝐿²(𝑃,𝐻) such that lim_𝑛→∞𝐸𝑥𝑡+𝑠_𝑛−𝑦(𝑡)²=0,lim_𝑛→∞𝐸𝑦𝑡−𝑠_𝑛−𝑥(𝑡)²=0 (9) 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)).

Definition 4. 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 if 𝑓(𝑡,𝑥) is square-mean almost automorphic in 𝑡∈𝑅 uniformly for all 𝑥∈𝐾, where 𝐾 is any bounded subset of 𝐿²(𝑃,𝐻). That is to say, for every sequence of real numbers {𝑠^′_𝑛}_𝑛∈𝑁, there exist a subsequence {𝑠_𝑛}_𝑛∈𝑁 and a function 𝑓∶𝑅×𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻) such that lim_𝑛→∞𝐸𝑓𝑡+𝑠_𝑛,𝑥−𝑓(𝑡,𝑥)²=0,lim_𝑛→∞𝐸𝑓𝑡−𝑠_𝑛,𝑥−𝑓(𝑡,𝑥)²=0 (10) for each 𝑡∈𝑅 and each 𝑥∈𝐾.

Lemma 5 (see [17]). (𝐴𝐴(𝑅;𝐿²(𝑃,𝐻)),‖⋅‖_∞) is a Banach space when it is equipped with the norm ‖⋅‖_∞∶=sup_𝑡∈𝑅‖𝑥(𝑡)‖²=sup_𝑡∈𝑅𝐸‖𝑥(𝑡)‖^21/2 (11) for 𝑥∈𝐴𝐴(𝑅;𝐿²(𝑃,𝐻)).

Lemma 6 (see [17]). If 𝑥, 𝑥₁, and 𝑥₂ are all square-mean almost automorphic stochastic processes, then the following hold:(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).

Theorem 7 (see [17]). 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: 𝐸𝑓(𝑡,𝑥)−𝑓𝑡,𝑦²≤ME𝑥−𝑦² (12) for all 𝑥,𝑦∈𝐿²(𝑃,𝐻) and for each 𝑡∈𝑅, where 𝑀>0 is independent of 𝑡. Then for any square-mean almost automorphic process 𝑥∶𝑅→𝐿²(𝑃,𝐻), the stochastic process 𝐹∶𝑅→𝐿²(𝑃,𝐻) given by 𝐹(⋅)=𝑓(⋅,𝑥(⋅)) is square-mean almost automorphic.

Theorem 8 (see [15]). Let 𝑓∶𝑅×𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻), (𝑡,𝑥)→𝑓(𝑡,𝑥) be square-mean almost automorphic, and assume that 𝑓(𝑡,⋅) is uniformly continuous on each bounded subset 𝐾⊂𝐿²(𝑃,𝐻) uniformly for 𝑡∈𝑅; that is, for all 𝜀>0, there exists 𝛿>0 such that 𝑥,𝑦∈𝐾 and 𝐸‖𝑥−𝑦‖²<𝛿 imply that 𝐸‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖²<𝜀 for all 𝑡∈𝑅. Then for any square-mean almost automorphic process 𝑥∶𝑅→𝐿²(𝑃,𝐻), the stochastic process 𝐹∶𝑅→𝐿²(𝑃,𝐻) given by 𝐹(⋅)∶=𝑓(⋅,𝑥(⋅)) is square-mean almost automorphic.

3. Some Lemmas

In this section, we apply Theorems 7 and 8 to investigate the existence of square-mean almost automorphic mild solutions to problem (3).

Definition 9. A stochastically continuous stochastic process 𝑥∶[0,∞)→𝐿²(𝑃,𝐻) is called a mild solution of the system (3) if 𝑥(0)=𝑥₀ and 𝑥(𝑡) satisfies 𝑥(𝑡)=𝑇(𝑡)𝑥₀−𝑎0,𝑥𝛾₁(0)+𝑎𝑡,𝑥𝛾₁(𝑡)+^𝑡₀𝑇(𝑡−𝑠)ℎ𝑠,𝑥𝛾₂(𝑠)+𝑑𝑠_𝑡0𝑇(𝑡−𝑠)𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠) (13) for all 𝑡≥0.

Remark 10. It is easy to see that if 𝑇(𝑡) is exponentially stable, then we obtain that the stochastic process 𝑥 is a mild solution to problem (3) if and only if 𝑥 satisfies the stochastic integral equation 𝑥(𝑡)=𝑎𝑡,𝑥𝛾₁(𝑡)+^𝑡_−∞𝑇(𝑡−𝑠)ℎ𝑠,𝑥𝛾₂(𝑠)+𝑑𝑠_𝑡−∞𝑇(𝑡−𝑠)𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠) (14) for all 𝑡≥0.

Let us list the following assumptions.(H1)There exists a resolvent operator 𝑇(⋅) of (3), and 𝑇(⋅) is exponentially stable; that is, ‖𝑇(𝑡)‖≤𝑀𝑒^−𝛿𝑡 (15) for all 𝑡≥0 and some constants 𝑀,𝛿>0. Moreover, 𝑇(𝑡) is compact for 𝑡≥0.(H2)The functions 𝑎,ℎ,𝑓∈𝐴𝐴(𝑅⁺×𝐿²(𝑃,𝐻),𝐿²(𝑃,𝐻)) and there exist positive numbers 𝑙_𝑎, 𝑙_ℎ, 𝑙_𝑓 such that 𝐸𝑎(𝑡,𝑥)−𝑎𝑡,𝑦²≤𝑙_𝑎𝐸𝑥−𝑦²,𝐸ℎ(𝑡,𝑥)−ℎ𝑡,𝑦²≤𝑙_ℎ𝐸𝑥−𝑦²,𝐸𝑓(𝑡,𝑥)−𝑓𝑡,𝑦²≤𝑙_𝑓𝐸𝑥−𝑦² (16) for all 𝑡≥0 and each 𝑥,𝑦∈𝐿²(𝑃,𝐻).(H3)𝛾_𝑖(⋅)∈𝐶(𝑅⁺,𝑅⁺),  𝑖=1,2,3.(H4)The functions 𝑎,ℎ,𝑓∶𝑅⁺×𝐿²(𝑃,𝐻)→𝐿²(𝑃,𝐻) satisfy the following conditions.(i)𝑎, ℎ, 𝑓 are square-mean almost automorphic and 𝑎(𝑡,⋅), 𝑓(𝑡,⋅), 𝑔(𝑡,⋅) are uniformly continuous in every uniformly bounded subset 𝐾⊂𝐿²(𝑃,𝐻) for 𝑡≥0.(ii)There exist integrable functions 𝑚_𝑎,𝑚_ℎ,𝑚_𝑓∶𝑅→[0,∞) and continuous nondecreasing functions 𝑊_𝑎,𝑊_ℎ,𝑊_𝑓∶[0,∞)→(0,∞) such that 𝐸𝑎𝑡,𝜑²≤𝑚_𝑎(𝑡)𝑊_𝑎𝐸𝜑²,𝐸𝑓𝑡,𝜑²≤𝑚_ℎ(𝑡)𝑊_ℎ𝐸𝜑²,𝐸𝑔𝑡,𝜑²≤𝑚_𝑓(𝑡)𝑊_𝑓𝐸𝜑² (17) for all (𝑡,𝜑)∈𝑅⁺×𝐾.(iii)Let 𝑥_𝑛⊂𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) be uniformly bounded in 𝑅⁺ and uniformly convergent in each compact subset of 𝑅⁺. Then 𝑎(⋅,𝑥_𝑛(⋅)), ℎ(⋅,𝑥_𝑛(⋅)), and 𝑓(⋅,𝑥_𝑛(⋅)) are relatively compact in 𝐵𝐶(𝑅⁺;𝐿²(𝑃,𝐻)).

Remark 11. Obviously, when the continuous function 𝑓 satisfies Lipschitz condition like (H2) and 𝑓(𝑡,0)=0, then function 𝑓 satisfies (H4)(ii). However, the function 𝑓(𝑡,𝑥)=√𝑥[sin𝑡+sin√2𝑡] satisfies (H4)(ii) but does not satisfy (H2).

In the proof of the existence theorem, we need the following technical lemmas.

Lemma 12. Assume that conditions (H1)–(H3) hold. Let 𝐿 be the operator defined by, for each 𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)), (𝐿𝑥)(𝑡)∶=𝑎𝑡,𝑥𝛾₁(𝑡)+^𝑡_−∞𝑇(𝑡−𝑠)ℎ𝑠,𝑥𝛾₂(𝑠)+𝑑𝑠_𝑡−∞𝑇(𝑡−𝑠)𝑓𝑠,𝑥𝛾₃(𝑠)(𝑠)𝛾𝑑𝑊∶=𝑎𝑡,𝑥₁(𝑡)+Φ𝑥(𝑡)+Ψ𝑥(𝑡). (18) Then 𝐿 maps 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) into itself.

Proof. Firstly, let 𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)); then 𝑠→𝑥(𝛾₁(𝑠)) is in 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) as 𝛾₁(⋅)∈𝐶(𝑅⁺,𝑅⁺). Hence, by (H2) and Theorem 7, one can easily see that 𝑠→𝑎(𝑠,𝑥(𝛾₁(𝑠))) belongs to 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)).
Similarly, 𝑠→ℎ(𝑠,𝑥(𝛾₂(𝑠))) and 𝑠→𝑓(𝑠,𝑥(𝛾₃(𝑠))) belong to 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) whenever 𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)).
Secondly, we show that Φ is square-mean almost automorphic.
Let {𝑠^′_𝑛}_𝑛∈𝑁 be an arbitrary sequence of real numbers. Since 𝐻(⋅)=ℎ(⋅,𝑥(𝛾₂(⋅)))∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)), there exists a subsequence {𝑠_𝑛}_𝑛∈𝑁 of {𝑠^′_𝑛}_𝑛∈𝑁 such that, for a certain stochastic process 𝐻, lim_𝑛→∞𝐸𝐻𝑡+𝑠_𝑛−𝐻(𝑡)²=0,lim_𝑛→∞𝐸𝐻𝑡−𝑠_𝑛−𝐻(𝑡)²=0 (19) hold for each 𝑡≥0. Moreover, if we let Φ𝑥(𝑡)=∫^𝑡_−∞𝑇(𝑡−𝑠)𝐻(𝑠)𝑑𝑠, by using Cauchy-Schwarz inequality, we have 𝐸Φ𝑥(𝑡+𝑠_𝑛)−Φ𝑥(𝑡)²=𝐸^{𝑡+𝑠𝑛}_−∞𝑇𝑡+𝑠_𝑛−𝑠𝐻(𝑠)𝑑𝑠−^𝑡_−∞𝑇(𝑡−𝑠)𝐻(𝑠)𝑑𝑠²≤𝐸^𝑡_−∞𝑇(𝑡−𝑠)𝐻𝑠+𝑠_𝑛−𝐻(𝑠)𝑑𝑠²≤𝑀²𝐸^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝐻𝑠+𝑠_𝑛−𝐻(𝑠)𝑑𝑠²≤𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}×𝑑𝑠_𝑡−∞𝑒^{−𝛿(𝑡−𝑠)}𝐸𝐻𝑠+𝑠_𝑛−𝐻(𝑠)²𝑑𝑠≤𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²sup_𝑡≥0𝐸𝐻𝑡+𝑠_𝑛−𝐻(𝑡)²≤𝑀²𝛿²sup_𝑡≥0𝐸𝐻(𝑡+𝑠_𝑛)−𝐻(𝑡)². (20)
Thus, by (19), we immediately obtain that lim_𝑛→∞𝐸Φ𝑥𝑡+𝑠_𝑛−Φ𝑥(𝑡)²=0 (21) for each 𝑡≥0. And we can show in a similar way that lim_𝑛→∞𝐸Φ𝑥𝑡−𝑠_𝑛−Φ𝑥(𝑡)²=0 (22) for each 𝑡≥0. Thus we conclude that Φ𝑥(𝑡)∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)).
Thirdly, we show that Ψ is square-mean almost automorphic.
Since 𝐹(⋅)=𝑓(⋅,𝑥(𝛾₃(⋅)))∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)), there exists a subsequence {𝑠_𝑛}_𝑛∈𝑁 of {𝑠^′_𝑛}_𝑛∈𝑁 such that, for a certain stochastic process 𝐹, lim_𝑛→∞𝐸𝐹𝑡+𝑠_𝑛−𝐹(𝑡)²=0,lim_𝑛→∞𝐸‖𝐹𝑡−𝑠_𝑛−𝐹(𝑡)²=0 (23) hold for each 𝑡≥0.
This is more complicated than the previous case because of the involvement of the Brownian motion 𝑊. To overcome such a difficulty, we make extensive use of the properties of 𝑊 defined by 𝑊(𝜎)∶=𝑊𝜎+𝑠_𝑛−𝑊𝑠_𝑛 (24) for each 𝜎. Note that 𝑊 is a Brownian motion and has the same distribution as 𝑊. Moreover, if we let Ψ𝑥(𝑡)=∫^𝑡_−∞𝑇(𝑡−𝑠)𝐹(𝑠)𝑑𝑠, then, by making a change of variable 𝜎=𝑠−𝑠_𝑛, we get 𝐸Ψ𝑥(𝑡+𝑠_𝑛)−Ψ𝑥(𝑡)²=𝐸^{𝑡+𝑠𝑛}_−∞𝑇𝑡+𝑠_𝑛𝐹(𝑠)(𝑠)−−𝑠𝑑𝑊_𝑡−∞𝑇(𝑡−𝑠)𝐹(𝑠)𝑑𝑊(𝑠)²=𝐸^𝑡_−∞𝑇(𝑡−𝜎)𝐹(𝜎+𝑠_𝑛)−𝐹(𝜎)𝑑𝑊(𝜎)². (25) Then, using an estimate on the Ito integral established, we obtain 𝐸Ψ𝑥𝑡+𝑠_𝑛−Ψ𝑥(𝑡)²≤𝐸^𝑡_−∞𝑇(𝑡−𝜎)𝐹𝜎+𝑠_𝑛−𝐹(𝜎)²𝑑𝜎≤𝑀^2𝑡_−∞𝑒^{−2𝛿(𝑡−𝜎)}𝐸𝐹𝜎+𝑠_𝑛−𝐹(𝜎)²≤𝑀𝑑𝜎₂2𝛿sup_𝑡≥0𝐸𝐹𝑡+𝑠_𝑛−𝐹(𝑡)². (26) Thus, by (23), we immediately obtain that lim_𝑛→∞𝐸Ψ𝑥𝑡+𝑠_𝑛−Ψ𝑥(𝑡)²=0 (27) for each 𝑡≥0. And we can show in a similar way that lim_𝑛→∞𝐸Ψ𝑥𝑡−𝑠_𝑛−Ψ𝑥(𝑡)²=0 (28) for each 𝑡≥0. Thus we conclude that Ψ𝑥(𝑡)∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)).
According to Lemma 6, we can easily obtain that 𝐿 maps 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) into itself.

Lemma 13. Assume that conditions (H1), (H3), and (H4)(i) are satisfied. Then 𝐿 is continuous and maps 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) into itself, where 𝐿 is defined by Lemma 12.

Proof.
Step 1. We prove that 𝐿 maps 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) into itself.
Firstly, let 𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)); then 𝑠→𝑥(𝛾₁(𝑠)) is in 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) as 𝛾₁(⋅)∈𝐶(𝑅⁺,𝑅⁺). Hence, by (H4)(i) and Theorem 8, one can easily see that 𝑠→𝑎(𝑠,𝑥(𝛾₁(𝑠))) belongs to 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)).
Similarly, 𝑠→ℎ(𝑠,𝑥(𝛾₂(𝑠))) and 𝑠→𝑓(𝑠,𝑥(𝛾₃(𝑠))) belong to 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) whenever 𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)).
Secondly, from Lemma 12, we can prove that Φ and Ψ are square-mean almost automorphic.
According to Lemma 6, we can easily obtain that 𝐿 is continuous and maps 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) into itself.
Step 2. We prove that 𝐿 is continuous on 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)).
Let {𝑥_𝑛}⊂(𝑅⁺;𝐿²(𝑃,𝐻)) be a sequence which converges to some 𝑥∈(𝑅⁺;𝐿²(𝑃,𝐻)) with respect to ‖⋅‖_∞; that is, ‖𝑥_𝑛−𝑥‖_∞→0 as 𝑛→∞. There exists a bounded subset 𝐾⊂𝐿²(𝑃,𝐻) such that 𝑥_𝑛(𝑡), 𝑥(𝑡)⊂𝐾 for 𝑡≥0, 𝑛=1,2,…. By (H3), (H4)(i), and Theorem 8, for any 𝜀>0, there exist 𝛿>0 and 𝑁>0 such that 𝐸‖𝑥_𝑛(𝛾_𝑖(𝑡))−𝑥(𝛾_𝑖(𝑡))‖²<𝛿(𝑖=1,2,3), which imply that 𝐸𝑎𝑡,𝑥_𝑛𝛾₁(𝑡)−𝑎𝑡,𝑥𝛾₁(𝑡)²<𝜀9,𝐸ℎ𝑡,𝑥_𝑛𝛾₂(𝑡)−ℎ𝑡,𝑥𝛾₂(𝑡)²<𝛿²𝜀9𝑀²,𝐸𝑓𝑡,𝑥_𝑛𝛾₃(𝑡)−𝑓𝑡,𝑥𝛾₃(𝑡)²<2𝛿𝜀9𝑀² (29) for all 𝑛>𝑁 and 𝑡≥0, where 𝛿,𝑀 are given in (H1). Then 𝐸𝐿𝑥_𝑛(𝑡)−(𝐿𝑥)(𝑡)²≤3𝐸𝑎𝑡,𝑥_𝑛𝛾₁(𝑡)−𝑎𝑡,𝑥𝛾₁(𝑡)²+3𝐸^𝑡_−∞𝑇(𝑡−𝑠)ℎ𝑡,𝑥_𝑛𝛾₂(𝑡)−ℎ𝑡,𝑥𝛾₂(𝑡)𝑑𝑠²+3𝐸^𝑡_−∞𝑇()×𝑓𝑡−𝑠𝑡,𝑥_𝑛𝛾₃(𝑡)−𝑓𝑡,𝑥𝛾₃(𝑡)𝑑𝑊(𝑠)²≤𝜀3+3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}×𝑑𝑠_𝑡−∞𝑒^{−𝛿(𝑡−𝑠)}𝐸ℎ𝑡,𝑥_𝑛𝛾₂(𝑡)−ℎ𝑡,𝑥𝛾₂(𝑡)²𝑑𝑠+3𝑀^2𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}𝐸𝑓𝑡,𝑥_𝑛𝛾₃(𝑡)−𝑓𝑡,𝑥𝛾₃(𝑡)²≤𝜀𝑑𝑠3+3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝛿²𝜀9𝑀²𝑑𝑠+3𝑀^2𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}2𝛿𝜀9𝑀²≤𝜀𝑑𝑠3+𝛿²𝜀3^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²+2𝛿𝜀3^𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}𝑑𝑠=𝜀 (30) for all 𝑛>𝑁 and all 𝑡≥0. This implies that 𝐿 is continuous. The proof is completed.

4. Existence of Square-Mean Almost Automorphic Solutions

Theorem 14. If the assumptions (H1)–(H3) are satisfied, then the system (3) has a unique square-mean almost periodic automorphic mild solution, whenever Θ<1/3, where Θ=𝑙_𝑎+𝑀²𝛿²𝑙_ℎ+𝑀²𝑙2𝛿^𝑓. (31)

Proof. Let 𝐿∶𝐴𝐴(𝑅⁺,𝐿²(𝑃,𝐻))→𝐶(𝑅⁺,𝐿²(𝑃,𝐻)) be the operator defined by 𝐿𝑥(𝑡)∶=𝑎𝑡,𝑥𝛾₁(𝑡)+^𝑡_−∞𝑇(𝑡−𝑠)ℎ𝑠,𝑥𝛾₂(𝑠)+𝑑𝑠_𝑡−∞𝑇(𝑡−𝑠)𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠). (32)
From the proof of Lemma 12, we see that 𝐻(⋅),𝐹(⋅)∈𝐴𝐴(𝑅⁺,𝐿²(𝑃,𝐻)). Therefore, 𝐿𝑥(𝑡)∈𝐴𝐴(𝑅⁺,𝐿²(𝑃,𝐻)); thus 𝐿𝑥(⋅) maps 𝐴𝐴(𝑅⁺,𝐿²(𝑃,𝐻)) into itself.
To complete the proof, it suffices to prove that 𝐿𝑥(⋅) is a contraction. Since (𝑎+𝑏+𝑐)²≤3𝑎²+3𝑏²+3𝑐², for 𝑥(𝑡),𝑦(𝑡)∈𝐴𝐴(𝑅⁺,𝐿²(𝑃,𝐻)), we obtain 𝐸𝐿𝑥(𝑡)−𝐿𝑦(𝑡)²≤3𝐸𝑎(𝑡,𝑥(𝛾₁(𝑡)))−𝑎(𝑡,𝑦(𝛾₁(𝑡)))²+3𝐸^𝑡_−∞𝑇(𝑡−𝑠)ℎ(𝑠,𝑥(𝛾₂(𝑠)))−ℎ(𝑠,𝑦(𝛾₂(𝑠)))𝑑𝑠²+3𝐸^𝑡_−∞𝑇(𝑡−𝑠)𝑓𝑠,𝑥𝛾₃(𝑠𝛾)−𝑓𝑠,𝑦₃(𝑠)𝑑𝑊(𝑠)². (33)
Evaluating the three terms of the right-hand side, we have 3𝐸𝑎(𝑡,𝑥(𝛾₁(𝑡)))−𝑎(𝑡,𝑦(𝛾₁(𝑡)))²≤3𝑙_𝑎sup_𝑠≥0𝐸𝑥𝛾₂(𝑠)−𝑦𝛾₂(𝑠)²≤3𝑙_𝑎𝑥−𝑦²_∞,3𝐸^𝑡_−∞𝑇(𝑡−𝑠)ℎ(𝑠,𝑥(𝛾₂(𝑠)))−ℎ(𝑠,𝑦(𝛾₂(𝑠)))𝑑𝑠²≤3𝑀^2𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}×𝑑𝑠_𝑡−∞𝑒^{−𝛿(𝑡−𝑠)}𝐸ℎ𝑠,𝑥𝛾₂(𝑠)−ℎ𝑠,𝑦𝛾₂(𝑠)²𝑑𝑠≤3𝑀²𝑙_ℎ^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}×𝑑𝑠_𝑡−∞𝑒^{−𝛿(𝑡−𝑠)}𝐸𝑥𝛾₂(𝑠)−𝑦𝛾₂(𝑠)²𝑑𝑠≤3𝑀²𝑙_ℎ^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²sup_𝑠≥0𝐸𝑥𝛾₂(𝑠)−𝑦𝛾₂(𝑠)²≤3𝑀²𝛿²𝑙_ℎ𝑥−𝑦²_∞,3𝐸^𝑡_−∞𝑇(𝑡−𝑠)𝑓(𝑠,𝑥(𝛾₃(𝑠)))−𝑓(𝑠,𝑦(𝛾₃(𝑠)))𝑑𝑊(𝑠)²≤3𝑀^2𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}𝐸𝑓𝑠,𝑥𝛾₃(𝑠)−𝑓𝑠,𝑦𝛾₃(𝑠)²𝑑𝑠≤3𝑀²𝑙_𝑓^𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}𝑑𝑠sup_𝑠≥0𝐸𝑥𝛾₃(𝑠)−𝑦𝛾₃(𝑠)²≤3𝑀²𝑙2𝛿^𝑓𝑥−𝑦²_∞. (34)
Thus, we obtain that 𝐸𝐿𝑥(𝑡)−𝐿𝑦(𝑡)²≤3𝑙_𝑎+3𝑀²𝛿²𝑙_ℎ+3𝑀²𝑙2𝛿^𝑓𝑥−𝑦²_∞=3Θ𝑥−𝑦²_∞ (35) for each 𝑡≥0.
Consequently, if Θ<1/3, then (3) has a unique fixed point, which is the unique square-mean almost periodic automorphic mild solution to (3), such that 𝐿𝑥(𝑡)=𝑥(𝑡).

Theorem 15. Assume that conditions (H1), (H3), and (H4) are satisfied; then problem (3) admits at least one square-mean almost automorphic mild solution on 𝑅 provided that 𝐿_ℎ=sup_𝑡≥0^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑚_ℎ(𝑠)𝐿𝑑𝑠<∞,_𝑓=sup_𝑡≥0^𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}𝑚_𝑓(𝑠)𝑚𝑑𝑠<∞,_𝑎(𝑡)liminf_𝑟→∞𝑊_𝑎(𝑟)𝑟+𝑀²𝐿_ℎ𝛿liminf_𝑟→∞𝑊_ℎ(𝑟)𝑟+𝑀²𝐿_𝑓liminf_𝑟→∞𝑊_𝑓(𝑟)𝑟<13. (36)(37)

Proof. For the sake of convenience, we break the proof into several steps.
Step 1. 𝐿 is continuous.
From the proof of the previous lemmas it is clear that the nonlinear operator 𝐿 is well defined and continuous. Moreover, by Lemma 13, we infer that 𝐿𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) whenever 𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)); that is, 𝐿 maps 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) into itself.
Step 2. 𝐿 maps bounded sets into bounded sets.
Let 𝐵_𝑟={𝑥∈𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻))∶‖𝑥‖_∞≤𝑟} for each 𝑟>0. We prove that there exists a number 𝑟 such that 𝐿(𝐵_𝑟)⊆𝐵_𝑟.
Clearly, for each positive number 𝑟, 𝐵_𝑟 is a bounded closed convex set in 𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)). We claim that there exists a positive number 𝑟 such that 𝐿(𝐵_𝑟)⊆𝐵_𝑟. If it is not true, then for every 𝑟>0 there exist 𝑥_𝑟(⋅)∈𝐵_𝑟 and 𝑡_𝑟≥0 such that 𝐸‖(𝐿𝑥_𝑟)(𝑡_𝑟)‖²>𝑟. However, on the other hand, we have 𝑟<𝐸(𝐿𝑥_𝑟)(𝑡_𝑟)²≤3𝐸𝑎𝑡_𝑟,𝑥𝛾₁𝑡_𝑟²+3𝐸^𝑡𝑟_−∞𝑇𝑡_𝑟−𝑠ℎ𝑠,𝑥𝛾₂(𝑠)𝑑𝑠²+3𝐸^𝑡𝑟_−∞𝑇𝑡_𝑟−𝑠𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠)²≤3𝑚_𝑎(𝑡)𝑊_𝑎(𝑟)+3𝑀^2𝑡𝑟_−∞𝑒^{−𝛿(𝑡𝑟−𝑠)}×𝑑𝑠_{𝑡𝑟−∞}𝑒^{−𝛿(𝑡𝑟−𝑠)}𝐸ℎ𝑠,𝑥𝛾₂(𝑠)²𝑑𝑠+3𝑀^2𝑡𝑟_−∞𝑒^{−2𝛿(𝑡𝑟−𝑠)}𝐸𝑓𝑠,𝑥𝛾₃(𝑠)²𝑑𝑠≤3𝑚_𝑎(𝑡)𝑊_𝑎(𝑟)+3𝑀²𝛿^𝑡𝑟_−∞𝑒^{−𝛿(𝑡𝑟−𝑠)}𝑚_ℎ(𝑠)𝑊_ℎ(𝑟)𝑑𝑠+3𝑀^2𝑡𝑟_−∞𝑒^{−2𝛿(𝑡𝑟−𝑠)}𝑚_𝑓(𝑠)𝑊_𝑓(𝑟)𝑑𝑠≤3𝑚_𝑎(𝑡)𝑊_𝑎(𝑟)+3𝑀²𝐿_ℎ𝛿𝑊_ℎ(𝑟)+3𝑀²𝐿_𝑓𝑊_𝑓(𝑟). (38) Dividing both sides by 𝑟 and taking the lower limit as 𝑟→∞, we obtain 1≤3𝑚_𝑎(𝑡)liminf_𝑟→∞𝑊_𝑎(𝑟)𝑟+3𝑀²𝐿_ℎ𝛿liminf_𝑟→∞𝑊_ℎ(𝑟)𝑟+3𝑀²𝐿_𝑓liminf_𝑟→∞𝑊_𝑓(𝑟)𝑟, (39) which contradicts condition (37). Thus, for some positive number 𝑟, 𝐿(𝐵_𝑟)⊆𝐵_𝑟.
Step 3. 𝐿 maps bounded sets into equicontinuous sets.
Next we prove that the operator 𝐿 is completely continuous on 𝐵_𝑟. It suffices to prove that the following statements are true.(i)𝑉(𝑡)={(𝐿𝑥)(𝑡)∶𝑥∈𝐵_𝑟} is relatively compact in 𝐿²(𝑃,𝐻) for each 𝑡≥0.(ii){𝐿𝑥∶𝑥∈𝐵_𝑟}⊂𝐴𝐴(𝑅⁺;𝐿²(𝑃,𝐻)) is a family of equicontinuous functions.
Firstly, we show that (i) holds. Let 0<𝜀<1 be given. For each 𝑡≥0 and ∈𝐵_𝑟, we define 𝐿_𝜀𝑥(𝑡)=𝑎𝑡−𝜀,𝑥𝛾₁()(𝜀)×𝑡−𝜀+𝑇_{𝑡−𝜀−∞}𝑇(𝑡−𝜀−𝑠)ℎ𝑠,𝑥𝛾₂(𝑠)+𝑑𝑠_{𝑡−𝜀−∞}𝑇(𝑡−𝜀−𝑠)𝑓𝑠,𝑥𝛾₃(𝑠)(𝑠𝛾𝑑𝑊)=𝑎𝑡−𝜀,𝑥₁()(𝜀)+(.𝑡−𝜀+𝑇)[(Φ𝑥)(𝑡−𝜀Ψ𝑥)(𝑡−𝜀)] (40)
Since 𝑇(𝑡)(𝑡≥0) is compact and (H4)(iii), then the set 𝑉_𝜀(𝑡)={(𝐿_𝜀𝑥)(𝑡)∶𝑥∈𝐵_𝑟} is relatively compact in 𝐿²(𝑃,𝐻) for each 𝑡≥0. Moreover, for every 𝑥∈𝐵_𝑟, we have 𝐸(𝐿𝑥)(𝑡)−𝐿_𝜀𝑥(𝑡)²≤3𝐸𝑎𝑡,𝑥𝛾₁(𝑡)−𝑎𝑡−𝜀,𝑥𝛾₁(𝑡−𝜀)²+3𝐸^𝑡_𝑡−𝜀𝑇(𝑡−𝑠)ℎ𝑠,𝑥𝛾₂(𝑠)𝑑𝑠²+3𝐸^𝑡_𝑡−𝜀𝑇(𝑡−𝑠)𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠)²≤3𝐸𝑎𝑡,𝑥𝛾₁(𝑡)−𝑎𝑡−𝜀,𝑥𝛾₁(𝑡−𝜀)+3𝑀^2𝑡_𝑡−𝜀𝑒^{−𝛿(𝑡−𝑠)}𝑑𝑠²sup_𝑡≥0𝐸‖ℎ(𝑡)‖²+3𝑀^2𝑡_𝑡−𝜀𝑒^{−2𝛿(𝑡−𝑠)}𝑑𝑠sup_𝑡≥0𝐸𝑓(𝑡)². (41)
Observe that 𝑎(⋅,𝑥(𝛾₁(⋅))) is continuous; therefore, letting 𝜀→0, there are relatively compact sets 𝑉_𝜀(𝑡) arbitrarily close to 𝑉(𝑡) and hence 𝑉(𝑡) is also relatively compact in 𝐿²(𝑃,𝐻) for each 𝑡≥0 and 𝐿 is completely continuous on 𝐵_𝑟.
We now show that (ii) holds. Let 𝑡₁,𝑡₂≥0,𝑡₁<𝑡₂,𝑥∈𝐵_𝑟. Then 𝐸(𝐿𝑥)𝑡₂−(𝐿𝑥)𝑡₁²≤𝐸𝑎𝑡₂,𝑥𝛾₁𝑡₂−𝑎𝑡₁,𝑥𝛾₁𝑡₁+^𝑡2_−∞𝑇𝑡₂−𝑠ℎ𝑠,𝑥𝛾₂(𝑠)−𝑑𝑠_𝑡1−∞𝑇𝑡₁−𝑠ℎ𝑠,𝑥𝛾₂(𝑠)+𝑑𝑠_𝑡2−∞𝑇𝑡₂−𝑠𝑓𝑠,𝑥𝛾₃(𝑠)(𝑠)−𝑑𝑊_𝑡1−∞𝑇𝑡₁−𝑠𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠)²≤4𝐸𝑎𝑡₂,𝑥𝛾₁𝑡₂−𝑎𝑡₁,𝑥𝛾₁𝑡₁²+4𝐸^𝑡2_𝑡1𝑇𝑡₂−𝑡₁𝑇𝑡₁−𝑠ℎ𝑠,𝑥𝛾₂(𝑠)𝑑𝑠²+4𝐸^𝑡2_𝑡1𝑇𝑡₂−𝑡₁𝑇𝑡₁−𝑠𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠)²+4𝐸𝑇𝑡₂−𝑡₁×−𝐼_𝑡1−∞𝑇𝑡₁−𝑠ℎ𝑠,𝑥𝛾₂(𝑠−)𝑑𝑠_𝑡1−∞𝑇𝑡₁−𝑠𝑓𝑠,𝑥𝛾₃(𝑠)𝑑𝑊(𝑠)²≤4𝐸𝑎𝑡₂,𝑥𝛾₁𝑡₂−𝑎𝑡₁,𝑥𝛾₁𝑡₁²+4𝑀^2𝑡2_𝑡1𝑒^{−𝛿(𝑡2−𝑠)𝑑𝑠2}sup_𝑡≥0‖ℎ(𝑡)‖²+4𝑀^2𝑡2_𝑡1𝑒^{−2𝛿(𝑡2−𝑠)𝑑𝑠}sup_𝑡≥0𝑓(𝑡)²+4sup_{(𝑡)𝑦∈𝑉}𝐸𝑇𝑡₂−𝑡₁−𝐼𝑦². (42) The right-hand side tends to 0 independently of 𝑥∈𝐵_𝑟 as 𝑡₁→𝑡₂, which implies that the set {𝐿𝑥∶𝑥∈𝐵_𝑟} is right equicontinuous at 𝑡. By a similar procedure we can show that {𝐿𝑥∶𝑥∈𝐵_𝑟} is left equicontinuous at 𝑡. Thus, the set {𝐿𝑥∶𝑥∈𝐵_𝑟} is equicontinuous; that is, 𝐿 maps 𝐵_𝑟 into a family of equicontinuous functions.
As a consequence of Steps 1–3 together with Schauder’s fixed point theorem, we deduce that 𝐿 has a fixed point in 𝐵_𝑟 which is a square-mean almost periodic mild solution to (3). The proof is completed.

5. Example

Consider the neutral stochastic partial functional integrodifferential equations of the form 𝑑𝑁(𝑡,𝑥(𝑡))=𝐴𝑁(𝑡,𝑥(𝑡))𝑑𝑡+^𝑡₀𝐵()𝑁((𝑠𝛾𝑡−𝑠𝑠,𝑥))𝑑𝑠𝑑𝑡+ℎ𝑡,𝑥₂(𝑡)𝑑𝑡+𝑓𝑡,𝑥𝛾₃(𝑡)𝑑𝑊(𝑡),𝑡≥0, (43) for (𝑡,𝜉)∈[0,∞)×[0,𝜋], where 𝑊(𝑡) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω,ℱ,𝑃,ℱ_𝑡), ℱ_𝑡=𝜎{𝑊(𝑢)−𝑊(𝑣);𝑢,𝑣≤𝑡}. Here 𝑁(𝑡,𝑥(𝑡))=𝑥(𝑡)−𝑎(𝑡,𝑥(𝛾₁(𝑡))) and 𝛾₁(𝑡)=𝛾₂(𝑡)=𝛾₃(𝑡)=𝜆𝑡+𝜆₀, the constants 𝜆>0, 𝜆₀>0.

For that, let Ω⊂𝑅² be an open subset whose boundary 𝜕Ω is sufficiently regular. Let 𝑋=𝐻¹₀(Ω)×𝐿²(Ω) and consider the linear operator 𝐴 whose domain is given by 𝐷(𝐴)=(𝐻²(Ω)⋂𝐻¹₀(Ω))×𝐻¹₀(Ω) and 𝐴𝑥𝑦=𝑦𝛼(0)𝑥^′′−𝛽(0)𝑦, (44) where 𝛼(⋅), 𝛽(⋅) are real-valued functions of class 𝐶² on [0,∞) such that 𝛼(0)>0, and 𝛽(0)>0.

From Chen [23], we see that 𝐴 is the infinitesimal generator of a uniformly exponentially stable 𝐶₀-semigroup (𝑇(𝑡))_𝑡≥0 on 𝐻¹₀(Ω)×𝐿²(Ω). In what follows, we will assume that 𝑀,𝛿>0 are positive constants such that ‖𝑇(𝑡)‖≤𝑀𝑒^−2𝛿𝑡 for all 𝑡>0.

Let 𝐵(𝑡)=𝐴𝐹(𝑡), where 𝐹∶𝐻¹₀(Ω)×𝐿²(Ω)→𝐻¹₀(Ω)×𝐿²(Ω) is the operator family defined by 𝐹=𝐹_𝑖𝑗=00−𝛽^′(𝑡)(0)𝛼+𝛽^′(𝑡)𝛼(0)𝛼^′(𝑡)𝛼(0), (45) and assume that max𝐹₂₂(𝑡),𝐹₂₁(𝑡≤𝛿)𝑀𝑒^−2𝛿𝑡𝐹,𝑡≥0,max^′₂₂(𝑡),𝐹^′₂₁(𝑡≤𝛿)²𝑀²𝑒^−2𝛿𝑡,𝑡≥0. (46) From the results in Grimmer [22], we see that the abstract integrodifferential system, 𝑥^′(𝑡)=𝐴𝑥(𝑡)+^𝑡₀𝐴𝐹(𝑡−𝑠)𝑥(𝑠)𝑑𝑠, (47) has an associated uniformly exponentially stable resolvent of operators {𝑇(𝑡)}_𝑡≥0 with ‖𝑇(𝑡)‖≤𝑀𝑒^−𝛿𝑡 for 𝑡≥0.

Let 𝑎𝑡,𝜑(𝜉)=𝛽𝜑(𝜉)sin𝑡+sin√,ℎ(𝜉)(𝜉√2𝑡𝑡,𝜑=𝛽𝜑)sin𝑡+sin,𝑓(𝜉)(𝜉√3𝑡𝑡,𝜑=𝛽𝜑)sin𝑡+sin5𝑡. (48) Thus, one has 𝐸𝑎(𝑡,𝑥)−𝑎𝑡,𝑦²≤4𝛽²𝑥−𝑦²,𝐸ℎ(𝑡,𝑥)−ℎ𝑡,𝑦²≤4𝛽²𝑥−𝑦²,𝐸𝑓(𝑡,𝑥)−𝑓𝑡,𝑦²≤4𝛽²𝑥−𝑦². (49) It follows that 𝑙_𝑎=𝑙_ℎ=𝑙_𝑓=4𝛽². Therefore, according to Theorem 14, (43) has a unique square-mean almost periodic mild solution provided that 𝛽<𝛿/6(2𝛿²+2𝑀²+𝑀²𝛿). Let 𝑎𝑡,𝜑(𝜉)=𝛽sin𝜑(𝜉)sin𝑡+sin√,ℎ(𝜉)𝜑(𝜉√2𝑡𝑡,𝜑=𝛽sin)sin𝑡+sin,𝑓(𝜉)𝜑(𝜉)√3𝑡𝑡,𝜑=𝛽sinsin𝑡+sin5𝑡. (50) Thus, one has 𝐸𝑎𝑡,𝜑²≤𝛽²sin𝑡+sin√2𝑡²𝐸𝜑(𝜉)²,𝐸ℎ𝑡,𝜑²≤𝛽²sin𝑡+sin√3𝑡²𝐸𝜑(𝜉)²,𝐸𝑓𝑡,𝜑²≤𝛽²sin𝑡+sin√5𝑡²𝐸𝜑(𝜉)². (51) It follows that 𝑚_𝑎(𝑡)=𝛽²sin𝑡+sin√2𝑡²,𝑚_ℎ(𝑡)=𝛽²sin𝑡+sin√3𝑡²,𝑚_𝑓(𝑡)=𝛽²sin𝑡+sin√5𝑡² (52) and 𝑊_𝑎(𝑟)=𝑊_ℎ(𝑟)=𝑊_𝑓(𝑟)=𝑟.

Obviously, 𝐿_ℎ=sup_𝑡≥0^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}𝑚_ℎ(𝑠)𝑑𝑠=𝛽²sup_𝑡≥0^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}sin𝑡+sin√3𝑡²𝐿𝑑𝑠<∞,_𝑓=sup_𝑡≥0^𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}𝑚_𝑓(𝑠)𝑑𝑠=𝛽²sup_𝑡≥0^𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}sin𝑡+sin√5𝑡²𝑑𝑠<∞. (53) Let 𝐿_𝑎=sup_𝑡≥0sin𝑡+sin√2𝑡²,𝐿_ℎ=sup_𝑡≥0^𝑡_−∞𝑒^{−𝛿(𝑡−𝑠)}sin𝑡+sin√3𝑡²𝐿𝑑𝑠,_𝑓=sup_𝑡≥0^𝑡_−∞𝑒^{−2𝛿(𝑡−𝑠)}sin𝑡+sin√5𝑡²𝑑𝑠. (54) Therefore, according to Theorem 15, (43) has a square-mean almost periodic mild solution provided that 𝛽<𝛿3𝛿𝐿_𝑎+𝑀²𝐿_ℎ+𝛿𝑀²𝐿_𝑓. (55)

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11161051), the Scientific Research Foundation of the Higher Education Institutions of Guangxi Province of China (no. 201202ZD080) and the Scientific Research Foundation of Guangxi Education Office (no. 201012MS183).