Volume 2013, Issue 1 138068

Research Article

Open Access

Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

Zhi-Han Zhao,

Corresponding Author

Zhi-Han Zhao

[email protected]

Department of Mathematics and Information Engineering, Sanming University, Sanming, Fujian 365004, China

Search for more papers by this author

Yong-Kui Chang,

Yong-Kui Chang

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China lzjtu.edu.cn

Search for more papers by this author

Juan J. Nieto,

Juan J. Nieto

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Search for more papers by this author

Zhi-Han Zhao,

Corresponding Author

Zhi-Han Zhao

[email protected]

Department of Mathematics and Information Engineering, Sanming University, Sanming, Fujian 365004, China

Search for more papers by this author

Yong-Kui Chang,

Yong-Kui Chang

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China lzjtu.edu.cn

Search for more papers by this author

Juan J. Nieto,

Juan J. Nieto

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Search for more papers by this author

First published: 15 December 2013

https://doi.org/10.1155/2013/138068

Citations: 3

Academic Editor: Dumitru Baleanu

Share a link

Email
Wechat
Bluesky

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

()

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

()

Note that L²(ℙ, ℍ) is a Hilbert space equipped with the norm

()

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

. 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

()

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 , 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

, 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

()

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

()

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 , there exist a subsequence {s_n} _n∈ℕ and a stochastic process y : ℝ → L²(ℙ, ℍ) such that

()

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 , there exists a subsequence {s_n} _n∈ℕ and a function such that

()

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

()

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 ɛ > 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 < ɛ 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 .

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 .

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

()

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

()

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:

()

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
()
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,
()
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,
()
for all t ∈ ℝ⁺.
(H4)
We have
()
where and .
(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
()
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

()

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

()

where

and

First we prove that G(t) ∈ AA(ℝ; L²(ℙ, ℍ)). Let be an arbitrary sequence of real numbers. Since g ∈ AA(ℝ; L²(ℙ, ℍ)), there exists a subsequence {s_n} _n∈ℕ of such that for a certain stochastic process

()

hold for each t ∈ ℝ. Now, let

for each σ ∈ ℝ. Note that

is also a Brownian motion and has the same distribution as W. Moreover, if we let

, then by making a change of variables σ = s − s_n to get (see the equation (10.6.6) in [34])

()

and, hence, using the Ito’s isometry property of stochastic integral, we have the following estimations

()

Then by (20), we obtain that

for each t ∈ ℝ. In a similar way, we can show that

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 ɛ > 0, there exists a constant T > 0 such that E∥h(s)∥² ≤ ɛ and for all s ≥ T. Then, for all t ≥ 2T, we obtain

()

This inequality proves the assertion since ɛ 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

()

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 ɛ > 0 such that for each u₀∈B_ɛ(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

()

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

()

Let 0 < λ < 1. We affirm that the assertion holds for ɛ = λr. In fact, let u₀ ∈ B_ɛ(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

()

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

()

That is,

()

Then, one has

()

For any x, y ∈ 𝔻 and t ≥ 0, we have

()

Thus, we get

()

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

()

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

()

Therefore

()

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:

()

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

()

Now, we can define the functions f, g : ℝ⁺ × L²(ℙ, ℍ) → L²(ℙ, ℍ) by

()

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.

References

1 Cuesta E., Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete and Continuous Dynamical Systems Series A. (2007) no. supplement, 277–285, MR2409222, ZBL1163.45306.
Google Scholar
2 Cuevas C. and de Souza J. C., S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations, Applied Mathematics Letters. (2009) 22, no. 6, 865–870, https://doi.org/10.1016/j.aml.2008.07.013, MR2523596.
10.1016/j.aml.2008.07.013
Web of Science® Google Scholar
3 Agarwal R. P., Lakshmikantham V., and Nieto J. J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods & Applications. (2010) 72, no. 6, 2859–2862, https://doi.org/10.1016/j.na.2009.11.029, MR2580143, ZBL1188.34005.
10.1016/j.na.2009.11.029
Web of Science® Google Scholar
4 Anh V. V. and Mcvinish R., Fractional differential equations driven by Lévy noise, Journal of Applied Mathematics and Stochastic Analysis. (2003) 16, no. 2, 97–119, https://doi.org/10.1155/S1048953303000078, MR1989577, ZBL1042.60034.
10.1155/S1048953303000078
Google Scholar
5 Diethelm K. and Ford N. J., Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications. (2002) 265, no. 2, 229–248, https://doi.org/10.1006/jmaa.2000.7194, MR1876137, ZBL1014.34003.
10.1006/jmaa.2000.7194
Web of Science® Google Scholar
6 El-Sayed A. M. A., Nonlinear functional-differential equations of arbitrary orders, Nonlinear Analysis: Theory, Methods & Applications. (1998) 33, no. 2, 181–186, https://doi.org/10.1016/S0362-546X(97)00525-7, MR1621105, ZBL0934.34055.
10.1016/S0362-546X(97)00525-7
Web of Science® Google Scholar
7 Gorenflo R. and Mainardi F., Fractional calculus: integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics, 1997, 378, Springer, New York, NY, USA, 223–276, MR1611585.
10.1007/978-3-7091-2664-6_5
Google Scholar
8 Henry B. I. and Wearne S. L., Existence of Turing instabilities in a two-species fractional reaction-diffusion system, SIAM Journal on Applied Mathematics. (2002) 62, no. 3, 870–887, https://doi.org/10.1137/S0036139900375227, MR1897726.
10.1137/S0036139900375227
Web of Science® Google Scholar
9 Hilfer H., Applications of Fractional Calculus in Physics, 2000, World Scientific, River Edge, NJ, USA, https://doi.org/10.1142/9789812817747, MR1890104.
10.1142/3779
Google Scholar
10 Lakshmikantham V. and Vatsala A. S., Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications. (2008) 69, no. 8, 2677–2682, https://doi.org/10.1016/j.na.2007.08.042, MR2446361, ZBL1161.34001.
10.1016/j.na.2007.08.042
Web of Science® Google Scholar
11 Miller K. S. and Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, 1993, John Wiley & Sons, New York, NY, USA, A Wiley-Interscience Publication, MR1219954.
Google Scholar
12 N′Guérékata G. M., Sur les solutions presqu′automorphes d′équations différentielles abstraites, Les Annales des Sciences Mathématiques du Québec. (1981) 5, no. 1, 69–79, MR626577, ZBL0503.34035.
Google Scholar
13 Bugajewski D. and N′Guérékata G. M., On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces, Nonlinear Analysis: Theory, Methods & Applications. (2004) 59, no. 8, 1333–1345, https://doi.org/10.1016/j.na.2003.08.012, MR2101648, ZBL1071.34055.
10.1016/j.na.2003.08.012
Web of Science® Google Scholar
14 Diagana T., Hernández E. M., and dos Santos J. P. C., Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations, Nonlinear Analysis: Theory, Methods & Applications. (2009) 71, no. 1-2, 248–257, https://doi.org/10.1016/j.na.2008.10.046, MR2518032, ZBL1172.45002.
10.1016/j.na.2008.10.046
Web of Science® Google Scholar
15 Ding H.-S., Xiao T.-J., and Liang J., Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, Journal of Mathematical Analysis and Applications. (2008) 338, no. 1, 141–151, https://doi.org/10.1016/j.jmaa.2007.05.014, MR2386405, ZBL1142.45005.
10.1016/j.jmaa.2007.05.014
Web of Science® Google Scholar
16 dos Santos J. P. C. and Cuevas C., Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Applied Mathematics Letters. (2010) 23, no. 9, 960–965, https://doi.org/10.1016/j.aml.2010.04.016, MR2659119, ZBL1198.45014.
10.1016/j.aml.2010.04.016
Web of Science® Google Scholar
17 Chang Y.-K., Zhao Z.-H., and N′Guérékata G. M., A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations, Nonlinear Analysis: Theory, Methods & Applications. (2011) 74, no. 6, 2210–2219, https://doi.org/10.1016/j.na.2010.11.025, MR2781750, ZBL1217.60043.
10.1016/j.na.2010.11.025
Web of Science® Google Scholar
18 Chang Y.-K., Zhao Z.-H., and N′Guérékata G. M., Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Computers & Mathematics with Applications. (2011) 61, no. 2, 384–391, https://doi.org/10.1016/j.camwa.2010.11.014, MR2754146, ZBL1211.60025.
10.1016/j.camwa.2010.11.014
Web of Science® Google Scholar
19 Chang Y.-K., Zhao Z.-H., and N′Guérékata G. M., Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space, Advances in Difference Equations. (2011) 2011, article 9, MR2820283, ZBL1266.60101.
Google Scholar
20 Chang Y.-K., Zhao Z.-H., N′Guérékata G. M., and Ma R., Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations, Nonlinear Analysis: Real World Applications. (2011) 12, no. 2, 1130–1139, https://doi.org/10.1016/j.nonrwa.2010.09.007, MR2736199, ZBL1209.60034.
10.1016/j.nonrwa.2010.09.007
Web of Science® Google Scholar
21 Cao J., Yang Q., and Huang Z., Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations, Stochastics. (2011) 83, no. 3, 259–275, https://doi.org/10.1080/17442508.2010.533375, MR2810592, ZBL1221.60078.
10.1080/17442508.2010.533375
Google Scholar
22 Fu M. and Liu Z., Square-mean almost automorphic solutions for some stochastic differential equations, Proceedings of the American Mathematical Society. (2010) 138, no. 10, 3689–3701, https://doi.org/10.1090/S0002-9939-10-10377-3, MR2661567, ZBL1202.60109.
10.1090/S0002-9939-10-10377-3
Web of Science® Google Scholar
23 Fu M., Almost automorphic solutions for nonautonomous stochastic differential equations, Journal of Mathematical Analysis and Applications. (2012) 393, no. 1, 231–238, https://doi.org/10.1016/j.jmaa.2012.04.017, MR2921664, ZBL1244.60056.
10.1016/j.jmaa.2012.04.017
Web of Science® Google Scholar
24 Fu M. and Chen F., Almost automorphic solutions for some stochastic differential equations, Nonlinear Analysis: Theory, Methods & Applications. (2013) 80, 66–75, https://doi.org/10.1016/j.na.2012.12.005, MR3010755, ZBL1268.60090.
10.1016/j.na.2012.12.005
Web of Science® Google Scholar
25 Chen Z. and Lin W., Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, Journal of Functional Analysis. (2011) 261, no. 1, 69–89, https://doi.org/10.1016/j.jfa.2011.03.005, MR2785893, ZBL1233.60030.
10.1016/j.jfa.2011.03.005
Web of Science® Google Scholar
26 N′Guerekata G. M., Almost Automorphic and almost Periodic Functions in Abstract Spaces, 2001, Kluwer Academic/Plenum Publishers, New York, NY, USA, MR1880351.
10.1007/978-1-4757-4482-8
Web of Science® Google Scholar
27 N′Guérékata G. M., Topics in Almost Automorphy, 2005, Springer, New York, NY, USA, MR2107829.
Google Scholar
28 Zhao Z.-H., Chang Y.-K., and Nieto J. J., Square-mean asymptotically almost automorphic process and its application to stochastic integro-differential equations, Dynamic Systems and Applications. (2013) 22, no. 2-3, 269–284, MR3100203.
Web of Science® Google Scholar
29 Agarwal R. P., de Andrade B., and Cuevas C., Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Analysis: Real World Applications. (2010) 11, no. 5, 3532–3554, https://doi.org/10.1016/j.nonrwa.2010.01.002, MR2683811, ZBL1248.34004.
10.1016/j.nonrwa.2010.01.002
Web of Science® Google Scholar
30 Cuevas C. and Lizama C., Almost automorphic solutions to a class of semilinear fractional differential equations, Applied Mathematics Letters. (2008) 21, no. 12, 1315–1319, https://doi.org/10.1016/j.aml.2008.02.001, MR2464387, ZBL1192.34006.
10.1016/j.aml.2008.02.001
Web of Science® Google Scholar
31 Fattorini O., Second Order Differential Equations in Banach Spaces, 1985, North-Holland, Amsterdam, The Netherland.
Google Scholar
32 Haase M., The Functional Calculus for Sectorial Operators, 2006, 169, Birkhäuser, Basel, Switzerland, https://doi.org/10.1007/3-7643-7698-8, MR2244037.
10.1007/3-7643-7698-8
Google Scholar
33 Prüss J., Evolutionary Integral Equations and Applications, 1993, 87, Birkhäuser, Basel, Switzerland, https://doi.org/10.1007/978-3-0348-8570-6, MR1238939.
10.1007/978-3-0348-8570-6
Google Scholar
34 Kuo H.-H., Introduction to Stochastic Integration, 2006, Springer, New York, NY, USA, MR2180429.
Google Scholar

Citing Literature

All articles

Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

Abstract

1. Introduction

2. Preliminaries

2.1. Sectorial Linear Operators

2.2. Square-Mean Asymptotically Almost Automorphic Processes

3. Main Results

4. An Example

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

Abstract

1. Introduction

2. Preliminaries

2.1. Sectorial Linear Operators

2.2. Square-Mean Asymptotically Almost Automorphic Processes

3. Main Results

4. An Example

Acknowledgments

References

Citing Literature

References

Related

Information