Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

Abstract

The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.

1. Introduction

This paper is mainly concerned with the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the following stochastic fractional partial integrodifferential equation in the form 𝑑𝑥(𝑡)−𝑓(𝑡,𝑥(𝑡))=^𝑡₀(𝑡−𝑠)^𝛼−2Γ()𝐴𝑥(𝑠)((𝑠((𝑡(𝑡)𝑥(0)𝛼−1−𝑓𝑠,𝑥))𝑑𝑠𝑑𝑡+𝑔𝑡,𝑥))𝑑𝑊,𝑡≥0,=𝑢₀, (1) where 1<𝛼<2, 𝐴∶𝐷(𝐴)⊂𝐿²(ℙ,ℍ)→𝐿²(ℙ,ℍ) is a linear densely defined operator of sectorial type on a Hilbert space 𝐿²(ℙ,ℍ), 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}, and 𝑢₀ is an ℱ₀-adapted, ℍ-valued random variable independent of the Wiener process 𝑊. Here 𝑓 and 𝑔 are appropriate functions to be specified later. The convolution integral in (1) is understood in the Riemann-Liouville fractional integral (see, e.g., [1, 2]). We notice that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proven to be valuable in various fields of science and engineering (see, e.g., [3–11] and references therein).

The concept of asymptotically almost automorphic functions was firstly introduced by N’Guérékata in [12]. Since then these functions have become of great interest to several mathematicians and gained lots of developments and applications, we refer the reader to [13–16] and the references listed therein.

Recently, the existence of almost automorphic and pseudo almost automorphic solutions to some stochastic differential equations has been considered in many publications such as [17–27] and the references therein. In a very recent paper [28], the authors introduced a new notation of square-mean asymptotically almost automorphic stochastic processes including a composition theorem. However, to the best of our knowledge, the existence of square-mean asymptotically almost automorphic mild solutions to the problem (1) is an untreated topic. Therefore, motivated by the works [16, 28], the main purpose of this paper is to investigate the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the problem (1). Then, we present an example as an application of our main results.

The rest of this paper is organized as follows. In Section 2, we recall some basic definitions and facts which will be used throughout this paper. In Section 3, we prove some existence results of square-mean asymptotically almost automorphic mild solutions to the problem (1). Finally, we give an example as an application of our abstract results.

2. Preliminaries

In this section, we introduce some basic definitions, notations, and preliminary facts which will be used in the sequel. For more details on this section, we refer the reader to [28–30].

Throughout the paper, (ℍ,‖⋅‖,⟨⋅,⋅⟩) stands for a real separable Hilbert space. (Ω,ℱ,ℙ) denotes a complete 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) Note that 𝐿²(ℙ,ℍ) is a Hilbert space equipped with the norm ‖𝑥‖₂∶=_Ω‖𝑥‖²𝑑ℙ^1/2,foreach𝑥∈𝐿²(ℙ,ℍ). (3) We denote by 𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)) the collection of all bounded continuous stochastic processes 𝜑 from ℝ⁺ into 𝐿²(ℙ,ℍ) such that lim_𝑡→+∞𝐸‖𝜑(𝑡)‖²=0. It is then easy to check that 𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)) is a Banach space when it is endowed with the norm ‖𝜑‖_𝐶0∶=sup_𝑡∈ℝ+‖𝜑(𝑡)‖₂. Similarly, 𝐶₀(ℝ⁺×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) stands for the space of the continuous stochastic processes 𝑓∶ℝ⁺×𝐿²(ℙ,ℍ)→𝐿²(ℙ,ℍ) such that lim_𝑡→+∞𝐸𝑓(𝑡,𝑥)²=0 (4) uniformly for 𝑥∈𝐾, where 𝐾⊂𝐿²(ℙ,ℍ) is any bounded subset. Additionally, 𝑊(𝑡) will be 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}.

2.1. Sectorial Linear Operators

A closed and linear operator 𝐴 is said to be sectorial of type 𝜛 and angle 𝜃 if there exist 0<𝜃<𝜋/2,𝑀>0, and 𝜛∈ℝ such that its resolvent exists outside the sector 𝜛+𝑆_𝜃∶={𝜛+𝜆∶𝜆∈ℂ,|arg(−𝜆)|<𝜃} and ‖(𝜆−𝐴)⁻¹‖≤𝑀/|𝜆−𝜛|, 𝜆∉𝜛+𝑆_𝜃.

Definition 1 (see [2]). Let 𝐴 be a closed and linear operator with domain 𝐷(𝐴) defined on a Banach space 𝕏. We call 𝐴 the generator of a solution operator if there exist 𝜛∈ℝ and a strongly continuous function 𝑆_𝛼∶ℝ⁺→ℒ(𝕏) such that {𝜆^𝛼∶Re(𝜆)>𝜛}⊂𝜌(𝐴) and 𝜆^𝛼−1(𝜆^𝛼−𝐴)⁻¹𝑥=∫^∞₀𝑒^−𝜆𝑡𝑆_𝛼(𝑡)𝑥𝑑𝑡, 𝑅𝑒(𝜆)>𝜛,𝑥∈𝕏. In this case, 𝑆_𝛼(⋅) is called the solution operator generated by 𝐴.

We note that if 𝐴 is sectorial of type 𝜛 with 0≤𝜃<𝜋(1−𝛼/2), then 𝐴 is the generator of a solution operator given by 𝑆_𝛼(𝑡)∶=(1/2𝜋𝑖)∫_𝛾𝑒^𝜆𝑡𝜆^𝛼−1(𝜆^𝛼−𝐴)⁻¹𝑑𝜆,𝑡≥0, where 𝛾 is a suitable path lying outside the sector 𝜛+𝑆_𝜃. Recently, Cuesta in [1] proved that if 𝐴 is a sectorial operator of type 𝜛<0 for some 𝑀>0 and 0≤𝜃<𝜋(1−𝛼/2), then there exists a constant 𝐶>0 such that 𝑆_𝛼(𝑡)≤𝐶𝑀1+|𝜛|𝑡^𝛼,𝑡≥0. (5)

Remark 2. Note that 𝑆_𝛼(𝑡) is, in fact, integrable. For more details on the solution family 𝑆_𝛼(𝑡) and related issues, we refer the reader to [31–33].

2.2. Square-Mean Asymptotically Almost Automorphic Processes

We recall some basic facts for a symptotically almost automorphic processes which will be used in the sequel.

Definition 3 (see [22]). A stochastic process 𝑥∶ℝ→𝐿²(ℙ,ℍ) is said to be stochastically continuous if lim_𝑡→𝑠𝐸‖𝑥(𝑡)−𝑥(𝑠)‖²=0. (6)

Definition 4 (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 {𝑠^′_𝑛}_𝑛∈ℕ, there exist 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 (7) hold for each 𝑡∈ℝ. The collection of all square-mean almost automorphic stochastic processes 𝑥∶ℝ→𝐿²(ℙ,ℍ) is denoted by 𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)).

Definition 5 (see [17]). 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 exists 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 (8) for each 𝑡∈ℝ and each 𝑥∈𝕂. Denote by 𝐴𝐴(ℝ×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) the set of all such functions.

Lemma 6 (see [22]). (𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)),‖⋅‖_∞) is a Banach space equipped with the norm ‖𝑥‖_∞∶=sup_𝑡∈ℝ‖𝑥(𝑡)‖₂=sup_𝑡∈ℝ𝐸‖𝑥(𝑡)‖^21/2, (9) for 𝑥∈𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)).

Lemma 7 (see [17]). 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.

Definition 8 (see [25]). A stochastically continuous process 𝑓∶ℝ⁺→𝐿²(ℙ,ℍ) is said to be square-mean asymptotically almost automorphic if it can be decomposed as 𝑓=𝑔+ℎ, where 𝑔∈𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)) and ℎ∈𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)). Denote by 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) the collection of all the square-mean asymptotically almost automorphic processes 𝑓∶ℝ⁺→𝐿²(ℙ,ℍ).

Definition 9 (see [28]). 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 asymptotically almost automorphic if it can be decomposed as 𝑓=𝑔+ℎ, where 𝑔∈𝐴𝐴(ℝ×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) and ℎ∈𝐶₀(ℝ⁺×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)). Denote by 𝐴𝐴𝐴(ℝ⁺×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) the set of all such functions.

Lemma 10 (see [28]). If 𝑓, 𝑓₁, and 𝑓₂ are all square-mean asymptotically almost automorphic stochastic processes, then the following hold true: (I)𝑓₁+𝑓₂ is square-mean asymptotically almost automorphic; (II)𝜆𝑓 is square-mean asymptotically almost automorphic for any scalar 𝜆; (III)there exists a constant 𝑀>0 such that sup_𝑡∈ℝ+𝐸‖𝑓(𝑡)‖²≤𝑀.

Lemma 11 (see [28]). Suppose that 𝑓∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) admits a decomposition 𝑓=𝑔+ℎ, where 𝑔∈𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)) and ℎ∈𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)). Then {𝑔(𝑡)∶𝑡∈ℝ}⊂{𝑓(𝑡)∶𝑡∈ℝ⁺}.

Corollary 12 (see [28]). The decomposition of a square-mean asymptotically almost automorphic process is unique.

Lemma 13 (see [28]). 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) is a Banach space when it is equipped with the norm 𝑓_{𝐴𝐴𝐴(ℝ+;𝐿2(ℙ,ℍ))}∶=sup_𝑡∈ℝ𝑔(𝑡)₂+sup_𝑡∈ℝ+‖ℎ(𝑡)‖₂, (10) where 𝑓=𝑔+ℎ∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) with 𝑔∈𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)), ℎ∈𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)).

Lemma 14 (see [28]). 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) is a Banach space with the norm 𝑓_∞∶=sup_𝑡∈ℝ+𝑓(𝑡)₂=sup_𝑡∈ℝ+𝐸𝑓(𝑡)^21/2. (11)

Remark 15 (see [28]). In view of the previous lemmas it is clear that the two norms are equivalent in 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)).

Lemma 16 (see [28]). Let 𝑓∈𝐴𝐴(ℝ×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) and let 𝑓(𝑡,𝑥) be uniformly continuous in any bounded subset 𝕂⊂𝐿²(ℙ,ℍ) uniformly for 𝑡∈ℝ⁺. Then 𝑓(𝑡,𝑥) is uniformly continuous in any bounded subset 𝕂⊂𝐿²(ℙ,ℍ) uniformly for 𝑡∈ℝ.

Lemma 17 (see [28]). Let 𝑓∈𝐴𝐴𝐴(ℝ⁺×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) and suppose that 𝑓(𝑡,𝑥) is uniformly continuous in any bounded subset 𝕂⊂𝐿²(ℙ,ℍ) uniformly for 𝑡∈ℝ⁺. If 𝑢(𝑡)∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)), then 𝑓(⋅,𝑢(⋅))∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)).

We now give the following concept of mild solution of (1).

Definition 18. Let 𝑆_𝛼(𝑡) be an integrable solution operator on 𝐿²(ℙ,ℍ) with generator 𝐴. An ℱ_𝑡-adapted stochastic process 𝑥∶[0,+∞)→𝐿²(ℙ,ℍ) is called a mild solution of the problem (1) if 𝑥(0)=𝑢₀ is ℱ₀-measurable and 𝑥(𝑡) satisfies the corresponding stochastic integral equation: 𝑥(𝑡)=𝑆_𝛼(𝑡)𝑢₀−𝑓0,𝑢₀((𝑡++𝑓𝑡,𝑥))_𝑡0𝑆_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑊(𝑠),𝑡≥0. (12)

3. Main Results

In this section, we establish the existence of square-mean asymptotically almost automorphic mild solutions to the problem (1). For that, we need the following technical results.

First, we list the following basic assumptions.(H1)The operator 𝐴 is a sectorial operator of type 𝜛<0 for some 𝑀>0 and 0≤𝜃<𝜋(1−𝛼/2), and then there exists  𝐶>0 such that 𝑆_𝛼(𝑡)≤𝐶𝑀1+|𝜛|𝑡^𝛼,𝑡≥0, (13) where 𝑆_𝛼(𝑡) is the solution operator generated by 𝐴.(H2)The function 𝑓∈𝐴𝐴𝐴(ℝ⁺×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) and there exists a continuous and nondecreasing function 𝐿_𝑓∶[0,+∞)→[0,+∞) such that for each 𝑟≥0 and for all 𝐸‖𝑥‖²≤𝑟, 𝐸‖𝑦‖²≤𝑟, 𝐸𝑓(𝑡,𝑥)−𝑓𝑡,𝑦²≤𝐿_𝑓(𝑟)𝐸𝑥−𝑦² (14) for all 𝑡∈ℝ⁺.(H3)The function 𝑔∈𝐴𝐴𝐴(ℝ⁺×𝐿²(ℙ,ℍ);𝐿²(ℙ,ℍ)) and there exists a continuous and nondecreasing function 𝐿_𝑔∶[0,+∞)→[0,+∞) such that for each 𝑟≥0 and for all 𝐸‖𝑥‖²≤𝑟, 𝐸‖𝑦‖²≤𝑟, 𝐸𝑔(𝑡,𝑥)−𝑔𝑡,𝑦²≤𝐿_𝑔(𝑟)𝐸𝑥−𝑦² (15) for all 𝑡∈ℝ⁺.(H4)We have sup_𝑟>0𝑟6(𝐶𝑀)²−𝐿_𝑓(𝑟)𝑟(𝐶𝑀)²−|𝜛|^−1/𝛼(1−1/𝛼)𝜋𝐿_𝑔(𝑟)𝑟()>1𝛼sin𝜋/𝛼−𝜆𝑟1+(𝐶𝑀)²𝑀_𝑓+|𝜛|^−1/𝛼(1−1/𝛼)𝜋()𝑀𝛼sin𝜋/𝛼^𝑔, (16) where 𝑀_𝑓=sup_𝑡∈ℝ+𝐸‖𝑓(𝑡,𝑥(𝑡))‖² and 𝑀_𝑔=sup_𝑡∈ℝ+𝐸‖𝑔(𝑡,𝑥(𝑡))‖².(H5)The operator 𝐴 is a sectorial operator of type 𝜛 with 0≤𝜃<𝜋(1−𝛼/2), and there exists 𝜙(⋅)∈𝐿¹(ℝ⁺) such that 𝑆_𝛼(𝑡)²≤𝜙(𝑡)∀𝑡≥0,lim_𝑡→+∞𝜙(𝑡)=0, (17) where 𝑆_𝛼(𝑡) is the solution operator generated by 𝐴.

Lemma 19. Suppose that assumption (H1) holds and let 𝑓∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). If 𝐹 is the function defined by 𝐹(𝑡)∶=^𝑡₀𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)𝑑𝑊(𝑠),𝑡≥0, (18) then 𝐹∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)).

Proof. Since 𝑓∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)), we have by definition that 𝑓=𝑔+ℎ, where 𝑔∈𝐴𝐴(𝑅;𝐿²(𝑃,𝐻)) and ℎ∈𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)). Then 𝐹(𝑡)=^𝑡₀𝑆_𝛼()𝑔(𝑠)(𝑠)+𝑡−𝑠𝑑𝑊_𝑡0𝑆_𝛼()ℎ(𝑠)(𝑠)=𝑡−𝑠𝑑𝑊_𝑡−∞𝑆_𝛼()𝑔(𝑠)(𝑠)−𝑡−𝑠𝑑𝑊_0−∞𝑆_𝛼()𝑔(𝑠)(𝑠)+𝑡−𝑠𝑑𝑊_𝑡0𝑆_𝛼()ℎ(𝑠)(𝑠)(𝑡)(𝑡),𝑡−𝑠𝑑𝑊=𝐺+𝐻 (19) where 𝐺(𝑡)=∫^𝑡_−∞𝑆_𝛼(𝑡−𝑠)𝑔(𝑠)𝑑𝑊(𝑠) and 𝐻(𝑡)=−∫⁰_−∞𝑆_𝛼(𝑡−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)+∫^𝑡₀𝑆_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑊(𝑠).
First we prove that 𝐺(𝑡)∈𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)). Let {𝑠^′_𝑛}_𝑛∈ℕ be an arbitrary sequence of real numbers. Since 𝑔∈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 𝑔lim_𝑛→∞𝐸𝑔𝑡+𝑠_𝑛−𝑔(𝑡)²=0,lim_𝑛→∞𝐸𝑔𝑡−𝑠_𝑛−𝑔(𝑡)²=0 (20) hold for each 𝑡∈ℝ. Now, let 𝑊(𝜎)∶=𝑊(𝜎+𝑠_𝑛)−𝑊(𝑠_𝑛) for each 𝜎∈ℝ. Note that 𝑊 is also a Brownian motion and has the same distribution as 𝑊. Moreover, if we let 𝐺(𝑡)=∫^𝑡_−∞𝑆_𝛼(𝑡−𝑠)𝑔(𝑠)𝑑𝑊(𝑠), then by making a change of variables 𝜎=𝑠−𝑠_𝑛 to get (see the equation (10.6.6) in [34]) 𝐸𝐺𝑡+𝑠_𝑛−𝐺(𝑡)²=𝐸^{𝑡+𝑠𝑛}_−∞𝑆_𝛼𝑡+𝑠_𝑛𝑔(𝑠)(𝑠)−−𝑠𝑑𝑊_𝑡−∞𝑆_𝛼(𝑡−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)²=𝐸^𝑡_−∞𝑆_𝛼(𝑡−𝜎)𝑔𝜎+𝑠_𝑛−𝑔(𝜎)𝑑𝑊(𝜎)² (21) and, hence, using the Ito’s isometry property of stochastic integral, we have the following estimations 𝐸𝐺𝑡+𝑠_𝑛−𝐺(𝑡)²≤𝐸^𝑡_−∞𝑆_𝛼(𝑡−𝜎)𝑔𝜎+𝑠_𝑛−𝑔(𝜎)²≤𝑑𝜎_𝑡−∞𝑆_𝛼(𝑡−𝜎)²𝐸𝑔𝜎+𝑠_𝑛−𝑔(𝜎)²𝑑𝜎≤sup_𝑡∈ℝ𝐸𝑔𝑡+𝑠_𝑛−𝑔(𝑡)^2𝑡_−∞𝐶𝑀1+|𝜛|(𝑡−𝜎)^𝛼2≤()𝑑𝜎𝐶𝑀²sup_𝑡∈ℝ𝐸𝑔𝑡+𝑠_𝑛−𝑔(𝑡)^2∞₀1(1+|𝜛|𝑠^𝛼)²≤()𝑑𝑠𝐶𝑀²|𝜛|^−1/𝛼(1−1/𝛼)𝜋𝛼sin(𝜋/𝛼)sup_𝑡∈ℝ𝐸𝑔𝑡+𝑠_𝑛−𝑔(𝑡)². (22) Then by (20), we obtain that lim_𝑛→∞𝐸‖𝐺(𝑡+𝑠_𝑛)−𝐺(𝑡)‖²=0 for each 𝑡∈ℝ. In a similar way, we can show that lim_𝑛→∞𝐸‖𝐺(𝑡−𝑠_𝑛)−𝐺(𝑡)‖²=0 for each 𝑡∈ℝ. Thus we conclude that (⋅)∈𝐴𝐴(ℝ;𝐿²(ℙ,ℍ)).
Next, let us show that 𝐻(⋅)∈𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)). Since ℎ∈𝐶₀(ℝ⁺;𝐿²(ℙ,ℍ)) and 1/(1+|𝜛|𝑠^𝛼)² is integrable in [0,+∞), for any sufficiently small 𝜀>0, there exists a constant 𝑇>0 such that 𝐸‖ℎ(𝑠)‖²≤𝜀 and ∫^∞_𝑇(1/(1+|𝜛|𝑠^𝛼)²)𝑑𝑠≤𝜀 for all 𝑠≥𝑇. Then, for all 𝑡≥2𝑇, we obtain 𝐸‖𝐻(𝑡)‖²=𝐸₀^𝑡/2𝑆_𝛼()ℎ(𝑠)(𝑠)+𝑡−𝑠𝑑𝑊_𝑡𝑡/2𝑆_𝛼()ℎ(𝑠)(𝑠)−𝑡−𝑠𝑑𝑊_0−∞𝑆_𝛼(𝑡−𝑠)𝑔(𝑠)𝑑𝑊(𝑠)²≤3𝐸₀^𝑡/2𝑆_𝛼(𝑡−𝑠)ℎ(𝑠)²𝑑𝑠+3𝐸^𝑡_𝑡/2𝑆_𝛼(𝑡−𝑠)ℎ(𝑠)²𝑑𝑠+3𝐸⁰_−∞𝑆_𝛼(𝑡−𝑠)𝑔(𝑠)²𝑑𝑠≤3₀^𝑡/2𝐶𝑀1+|𝜛|(𝑡−𝑠)^𝛼2𝐸‖ℎ(𝑠)‖²𝑑𝑠+3^𝑡_𝑡/2𝐶𝑀1+|𝜛|(𝑡−𝑠)^𝛼2𝐸‖ℎ(𝑠)‖²𝑑𝑠+3⁰_−∞𝐶𝑀1+|𝜛|(𝑡−𝑠)^𝛼2𝐸𝑔(𝑠)²()𝑑𝑠≤3𝐶𝑀²sup_𝑡∈ℝ+𝐸‖ℎ(𝑡)‖^2𝑡_𝑡/21(1+|𝜛|𝑠^𝛼)²()𝑑𝑠+3𝐶𝑀²𝜀₀^𝑡/21(1+|𝜛|𝑠^𝛼)²()𝑑𝑠+3𝐶𝑀²sup_𝑡∈ℝ𝐸𝑔(𝑡)^2∞_𝑡1(1+|𝜛|𝑠^𝛼)²()𝑑𝑠≤3𝐶𝑀²sup_𝑡∈ℝ+𝐸‖ℎ(𝑡)‖^2∞_𝑇1(1+|𝜛|𝑠^𝛼)²()𝑑𝑠+3𝐶𝑀²𝜀^∞₀1(1+|𝜛|𝑠^𝛼)²()𝑑𝑠+3𝐶𝑀²sup_𝑡∈ℝ𝐸𝑔(𝑡)^2∞_𝑇1(1+|𝜛|𝑠^𝛼)²()𝑑𝑠≤3𝐶𝑀²𝜀sup_𝑡∈ℝ+𝐸‖ℎ(𝑡)‖²+|𝜛|^−1/𝛼(1−1/𝛼)𝜋𝛼sin(𝜋/𝛼)+sup_𝑡∈ℝ𝐸𝑔(𝑡)². (23) This inequality proves the assertion since 𝜀 is arbitrary. Recalling that 𝐹(𝑡)=𝐺(𝑡)+𝐻(𝑡) for all 𝑡≥0, we get 𝐹(𝑡)∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). The proof is completed.

It is easy to see that, by arguments similar to those in the proof of Lemma 19, we have the following result.

Lemma 20. Suppose that assumption (H5) holds and let 𝑓∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). If 𝐹 is the function defined by 𝐹(𝑡)∶=^𝑡₀𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)𝑑𝑊(𝑠),𝑡≥0, (24) then 𝐹∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)).

Now, we are ready to establish our main results.

Theorem 21. Assume that (H1)–(H4) hold. Then there exists 𝜀>0 such that for each 𝑢₀∈𝐵_𝜀(0,𝐿²(ℙ,ℍ)) there exists a unique square-mean asymptotically almost automorphic mild solution 𝑥(⋅,𝑢₀) of the problem (1) on [0,∞) such that 𝑥(0,𝑢₀)=𝑢₀.

Proof. We define a nonlinear operator Υ by (Υ𝑥)(𝑡)=𝑆_𝛼(𝑡)𝑢₀−𝑓0,𝑢₀((𝑡++𝑓𝑡,𝑥))_𝑡0𝑆_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑊(𝑠)𝑡≥0. (25)
First we prove that Υ(𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)))⊆𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). Given 𝑥∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)), from the properties of {𝑆_𝛼(𝑡)}_𝑡≥0, 𝑓, and 𝑔, we infer that Υ𝑥 is well defined and continuous. Since 𝑥(𝑡) is bounded, we can choose a bounded subset 𝕂 of 𝐿²(ℙ,ℍ) such that 𝑥(𝑡)∈𝕂 for all 𝑡∈ℝ⁺. It follows from (H2) and (H3) that both 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) are uniformly continuous on the bounded subset 𝕂 uniformly for 𝑡∈ℝ⁺. Moreover, from Lemmas 17 and 19 and taking into account (H1), it follows that Υ𝑥∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)).
Now, by (H4), there exists a constant 𝑟>0 such that 𝑟6(𝐶𝑀)²−𝐿_𝑓(𝑟)𝑟(𝐶𝑀)²−|𝜛|^−1/𝛼(1−1/𝛼)𝜋𝐿_𝑔(𝑟)𝑟()>1𝛼sin𝜋/𝛼−𝜆𝑟1+(𝐶𝑀)²𝑀_𝑓+|𝜛|^−1/𝛼(1−1/𝛼)𝜋()𝑀𝛼sin𝜋/𝛼^𝑔. (26) Let 0<𝜆<1. We affirm that the assertion holds for 𝜀=𝜆𝑟. In fact, let 𝑢₀∈𝐵_𝜀(0,𝐿²(ℙ,ℍ)). Define the space 𝔻={𝑥∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ))∶𝑥(0)=𝑢₀,𝐸‖𝑥(𝑡)‖²≤𝑟,𝑡≥0}. Then 𝔻 is a closed subspace of 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). We claim that Υ𝔻⊆𝔻. If 𝑥∈𝔻 and 𝑡∈ℝ⁺, we get 𝐸‖(Υ𝑥)(𝑡)‖²≤3𝐸𝑆_𝛼(𝑡)𝑢₀−𝑓0,𝑢₀²+3𝐸𝑓(𝑡,𝑥(𝑡))²+3𝐸^𝑡₀𝑆_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑊(𝑠)²≤6(𝐶𝑀)²𝐸𝑢₀²+𝐸𝑓0,𝑢₀²+3𝐸𝑓(𝑡,𝑥(𝑡))−𝑓(𝑡,0)+𝑓(𝑡,0)²+3𝐸^𝑡₀𝑆_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))²()𝑑𝑠≤6𝐶𝑀²𝜆𝑟+6(𝐶𝑀)²𝐸𝑓0,𝑢₀²+6𝐿_𝑓(𝑟)𝑟+6sup_𝑡∈ℝ+𝐸𝑓(𝑡,0)²+6(𝐶𝑀)^2𝑡₀11+|𝜛|(𝑡−𝑠)^𝛼2𝐿_𝑔(𝑟)+𝑟𝑑𝑠_𝑡011+|𝜛|(𝑡−𝑠)^𝛼2𝐸𝑔(𝑠,0)²()𝑑𝑠≤6𝐶𝑀²𝜆𝑟+6(𝐶𝑀)²𝑀_𝑓+6𝐿_𝑓(𝑟)𝑟+6𝑀_𝑓+6(𝐶𝑀)²|𝜛|^−1/𝛼(1−1/𝛼)𝜋()𝐿𝛼sin𝜋/𝛼^𝑔(𝑟)𝑟+𝑀_𝑔, (27) which from (26) implies that 𝐸‖(Υ𝑥)(𝑡)‖²≤𝑟 for all 𝑡≥0, and so that Υ𝔻⊆𝔻.
Next, to complete the proof, we need to show that Υ(⋅) is a contraction from 𝔻 into 𝔻. By (26), we know that 𝑟6(𝐶𝑀)²−𝐿_𝑓(𝑟)𝑟(𝐶𝑀)²−|𝜛|^−1/𝛼(1−1/𝛼)𝜋𝐿_𝑔(𝑟)𝑟𝛼sin(𝜋/𝛼)−𝜆𝑟>0. (28) That is, 𝑟6(𝐶𝑀)²>𝐿_𝑓(𝑟)𝑟(𝐶𝑀)²+|𝜛|^−1/𝛼(1−1/𝛼)𝜋𝐿_𝑔(𝑟)𝑟𝛼sin(𝜋/𝛼)+𝜆𝑟. (29) Then, one has 6𝐿_𝑓(𝑟)+6(𝐶𝑀)²|𝜛|^−1/𝛼(1−1/𝛼)𝜋()𝐿𝛼sin𝜋/𝛼^𝑔(𝑟)+6𝜆(𝐶𝑀)²<1. (30) For any 𝑥,𝑦∈𝔻 and 𝑡≥0, we have 𝐸(Υ𝑥)(𝑡)−Υ𝑦(𝑡)²≤2𝐸𝑓(𝑡,𝑥(𝑡))−𝑓𝑡,𝑦(𝑡)²+2𝐸^𝑡₀𝑆_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))−𝑔𝑠,𝑦(𝑠)𝑑𝑊(𝑠)²≤2𝐿_𝑓(𝑟)sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)²+2(𝐶𝑀)^2𝑡₀11+|𝜛|(𝑡−𝑠)^𝛼2×𝐸𝑔(𝑠,𝑥(𝑠))−𝑔𝑠,𝑦(𝑠)²𝑑𝑠≤2𝐿_𝑓(𝑟)sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)²+2(𝐶𝑀)²|𝜛|^−1/𝛼(1−1/𝛼)𝜋()𝐿𝛼sin𝜋/𝛼^𝑔(𝑟)×sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)²≤2𝐿_𝑓(𝑟)+2(𝐶𝑀)²|𝜛|^−1/𝛼(1−1/𝛼)𝜋()𝐿𝛼sin𝜋/𝛼^𝑔(𝑟)×sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)². (31) Thus, we get Υ𝑥−Υ𝑦_∞=sup_𝑡∈ℝ+𝐸(Υ𝑥)(𝑡)−Υ𝑦(𝑡)^21/2≤2𝐿_𝑓(𝑟)+2(𝐶𝑀)²|𝜛|^−1/𝛼(1−1/𝛼)𝜋()𝐿𝛼sin𝜋/𝛼^𝑔(𝑟)𝑥−𝑦_∞. (32) It follows from (30) that Υ is a contraction mapping on 𝔻. So by the Banach contraction mapping principle, we draw a conclusion that there exists a unique fixed point 𝑥(⋅) for Υ in 𝔻. It is clear that 𝑥 is a square-mean asymptotically almost automorphic mild solution of (1). The proof is complete.

The next result is proved using the similar steps as in the proof of the previous result, so we omit the details.

Theorem 22. Assume that (H1)–(H3) hold. If 𝐿_𝑓(𝑟)≡𝐿_𝑓 and 𝐿_𝑔(𝑟)≡𝐿_𝑔 for all 𝑟≥0 and 𝐿_𝑓+(𝐶𝑀)²|𝜛|^−1/𝛼(1−1/𝛼)𝜋𝐿_𝑔/𝛼sin(𝜋/𝛼)<1/2, then for every 𝑢₀∈𝐿²(ℙ,ℍ), there exists a unique square-mean asymptotically almost automorphic mild solution 𝑥(⋅,𝑢₀) of the problem (1) on [0,∞) such that 𝑥(0,𝑢₀)=𝑢₀.

Theorem 23. Suppose that assumptions (H2), (H3), and (H5) hold. If 𝐿_𝑓(𝑟)≡𝐿_𝑓 and 𝐿_𝑔(𝑟)≡𝐿_𝑔 for all 𝑟≥0 and 𝐿_𝑓+𝐿_𝑔‖𝜙‖₁<1/2, then for every 𝑢₀∈𝐿²(ℙ,ℍ), there exists a unique square-mean asymptotically almost automorphic mild solution 𝑥(⋅,𝑢₀) of the problem (1) on [0,∞) such that 𝑥(0,𝑢₀)=𝑢₀.

Proof. Consider the nonlinear operator Υ given by (Υ𝑥)(𝑡)=𝑆_𝛼(𝑡)𝑢₀−𝑓0,𝑢₀((𝑡,++𝑓𝑡,𝑥))_𝑡0𝑆_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑊(𝑠),𝑡≥0. (33)
First we prove that Υ maps 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) into itself. Given 𝑥∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)), from the properties of {𝑆_𝛼(𝑡)}_𝑡≥0, 𝑓 and 𝑔, we infer that Υ𝑥 is well defined and continuous. Since 𝑥(𝑡) is bounded, we can choose a bounded subset 𝕂 of 𝐿²(ℙ,ℍ) such that 𝑥(𝑡)∈𝕂 for all 𝑡∈ℝ⁺. It follows from conditions (H2) and (H3) that both 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) are uniformly continuous on the bounded subset 𝕂 uniformly for 𝑡∈ℝ⁺. Moreover, from Lemmas 17 and 20 and taking into account (H5), it follows that Υ𝑥∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)).
Next we prove that Υ is a contraction mapping from 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) into itself. Note that we have already proved Υ(𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)))⊆𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). Moreover, for any 𝑥,𝑦∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) and 𝑡≥0, we have 𝐸(Υ𝑥)(𝑡)−Υ𝑦(𝑡)²≤2𝐸𝑓(𝑡,𝑥(𝑡))−𝑓𝑡,𝑦(𝑡)²+2𝐸^𝑡₀𝑆_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))−𝑔𝑠,𝑦(𝑠)𝑑𝑊(𝑠)²≤2𝐿_𝑓sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)²+2^𝑡₀𝜙(𝑡−𝑠)𝐸𝑔(𝑠,𝑥(𝑠))−𝑔𝑠,𝑦(𝑠)²𝑑𝑠≤2𝐿_𝑓sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)²+2𝐿_𝑔sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)^2∞₀𝜙(𝑠)≤𝑑𝑠2𝐿_𝑓+2𝐿_𝑔𝜙₁sup_𝑡∈ℝ+𝐸𝑥(𝑡)−𝑦(𝑡)². (34) Therefore Υ𝑥−Υ𝑦_∞=sup_𝑡∈ℝ+𝐸(Υ𝑥)(𝑡)−Υ𝑦(𝑡)^21/2≤2𝐿_𝑓+2𝐿_𝑔𝜙₁𝑥−𝑦_∞. (35) That is, Υ is a contraction mapping on 𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). By the Banach contraction mapping principle, Υ has a unique fixed point 𝑥(𝑡)∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)). It is clear that 𝑥 is a square-mean asymptotically almost automorphic mild solution of (1). The proof is then complete.

4. An Example

In this section, we apply the results obtained previously to investigate the existence of square-mean asymptotically almost automorphic mild solutions for the following partial stochastic fractional differential system: 𝜕𝑢()−𝜕𝑡𝑡,𝜉^𝑡_−∞𝑎(𝑡)𝑎₁(𝑡−𝑠)𝑢(𝑠,𝜉)𝑑𝑠=𝐽_𝑡^𝛼−1𝜕²𝜕𝜉²−𝜈𝑢(𝑡,𝜉)−^𝑡_−∞𝑎(𝑡)𝑎₁()𝑢()+𝑡−𝑠𝑠,𝜉𝑑𝑠_𝑡−∞𝑎(𝑡)𝑎₂()𝑢()(𝑡),𝑢()()𝑢()𝑡−𝑠𝑠,𝜉𝑑𝑠𝑑𝑊𝑡,0=𝑢𝑡,𝜋=0,𝑡≥0,0,𝜉=𝑢₀(𝜉),𝜉∈𝐼=[0,𝜋], (36) where 𝑎∈𝐴𝐴𝐴(ℝ⁺;ℝ), 𝑎₁,𝑎₂∶ℝ⁺→ℝ are continuous functions, and 𝜈>0 is a fixed constant.

Let ℍ=𝐿²([0,𝜋]) with the norm ‖⋅‖ and 𝐴∶𝐷(𝐴)⊆ℍ→ℍ be the operator defined by 𝐴𝑥=𝑥^′′−𝜈𝑥 domain 𝐷(𝐴)={𝑥∈ℍ∶𝑥^′′∈ℍ,𝑥(0)=𝑥(𝜋)=0}. It is well known that Δ𝑥=𝑥^′′ is the infinitesimal generator of an analytic semigroup {𝑆(𝑡)}_𝑡≥0 on ℍ. Furthermore, 𝐴 is sectorial of type 𝜛=−𝜈<0.

In the sequel, we assume 𝐿_𝑓=‖𝑎‖_∞⁰_−∞𝑎₁(−𝑠)²𝑑𝑠^1/2𝐿<∞,_𝑔=‖𝑎‖_∞⁰_−∞𝑎₂(−𝑠)²𝑑𝑠^1/2<∞. (37) Now, we can define the functions 𝑓,𝑔∶ℝ⁺×𝐿²(ℙ,ℍ)→𝐿²(ℙ,ℍ) by 𝑓(𝑡,𝑥)(𝜉)=𝑎(𝑡)⁰_−∞𝑎₁()𝑥()𝑔(𝜉)(𝑡)−𝑠𝑠,𝜉𝑑𝑠,𝑡,𝑥)(=𝑎⁰_−∞𝑎₂()𝑥()𝐽−𝑠𝑠,𝜉𝑑𝑠,_𝑡𝛼−1ℎ(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠, (38) which permits to transform the system (36) into the abstract system (1). Moreover, it is not difficult to see that 𝑓,𝑔 are continuous and Lipschitz in the second variable with Lipschitz constants 𝐿_𝑓 and 𝐿_𝑔, respectively.

The next result is a consequence of Theorem 22.

Theorem 24. Under the previous assumptions, (36) has a unique mild solution 𝑥∈𝐴𝐴𝐴(ℝ⁺;𝐿²(ℙ,ℍ)) whenever 𝐿_𝑓 and 𝐿_𝑔 are small enough.

Acknowledgments

The first author was supported by Research Fund for Young Teachers of Sanming University (B2011071Q). The second author was supported by NSF of China (11361032), Program for New Century Excellent Talents in University (NCET-10-0022), and NSF of Gansu Province of China (1107RJZA091). And the third author was partially supported by Ministerio de Economia y Competitividad (Spain), project MTM2010-15314, and cofinanced by the European Community fund FEDER.