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Existence and Stability of Square-Mean S-Asymptotically Periodic Solutions to a Fractional Stochastic Diffusion Equation with Fractional Brownian Motion
In this paper, a generalized Gronwall inequality is demonstrated, playing an important role in the study of fractional differential equations. In addition, with the fixed-point theorem and the properties of Mittag–Leffler functions, some results of the existence as well as asymptotic stability of square-mean S-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion are obtained. In the end, an example of numerical simulation is given to illustrate the effectiveness of our theory results.
Originated in 1695, fractional calculus has been widely applied in physics, chemistry, economics, biology, and other fields. Recent decades have witnessed the rapid development of fractional calculus, with the emergence of many related researches [1–9]. The dynamic behavior of some complex processes in reality can be explained by fractional differential equations. For example, anomalous diffusion phenomena can be described with fractional diffusion equations. Compared with the traditional diffusion equations (first order), with fractional diffusion equations, subdiffusion or supdiffusion phenomenon can be described when its order is between 0 and 1 or between 1 and 2, respectively.
In addition, the diffusion phenomenon in real life is often affected by random factors, promoting the generation of fractional stochastic diffusion equations. However, this research has not aroused much concern until recent years. In , a class of nonautonomous fractional stochastic reaction-diffusion equations was studied, obtaining the regularity of random attractors. The Galerkin method was applied by Wang  to investigate the existence of tempered pullback random attractors for nonautonomous fractional reaction-diffusion equations with multiplicative noise. Chen  studied the stochastic time-fractional diffusion equations with multiplicative white noise, obtaining the Hölder continuity of the solution. Peng and Huang  established the existence of mild solutions for a nonlocal backward problem for fractional stochastic diffusion equations.
We also notice that some researches focus on the stability of solutions to fractional stochastic differential equations of order . Li and Wang  studied the existence and asymptotic behavior of solutions to fractional stochastic delay evolution equations with integral term and Wiener process by using fractional resolvent operator theory and the Schauder fixed-point theorem. Mathiyalagan and Balachandran  studied the finite-time stochastic stability of fractional-order singular systems with time delay and white noise utilizing the Gronwall approach and stochastic analysis technique. In , applying the Laplace transform method, the authors obtained the existence, uniqueness, and Hyers–Ulam stability of solutions to a class of linear fractional differential equations involving Mittag–Leffler kernel. The resolvent operator technique and contraction mapping principle were used in  to study the existence and uniqueness of mild solution to fractional neutral stochastic integrodifferential equations involving impulses driven by fractional Brownian motion (FBM), and a new impulsive-integral inequality was used to obtain the exponential stability for these equations. Moreover, the existence and asymptotic stability in the -th moment of mild solutions to a class of fractional stochastic partial differential equations with Wiener process was investigated by Zhang et al. . Because the form of the equations in this paper is different from those in the above studies, the methods to prove the stability in these studies cannot be directly applied to this paper.
Inspired by the above researches, in order to study the stability and periodicity of anomalous diffusion phenomena affected by random factors, we consider the fractional stochastic diffusion equation involving Dirichlet boundary conditions:where denotes the Caputo fractional derivative, and is a bounded open domain, whose boundary is sufficiently smooth. Functions and satisfy some appropriate conditions, with the initial data for , is the fractional Brownian motion (FBM), and . In addition,and the functions satisfywhere represents a constant, and satisfies
Firstly, a new generalized Gronwall inequality is given. Then, we obtain the existence and uniqueness of the S-asymptotically periodic solutions for problem (1) based on the characteristics of Mittag–Leffler functions, Hölder inequality, and the inequality for fractional stochastic integral with FBM and Banach’s fixed-point theorem. In addition, with the help of the generalized Gronwall inequality, some conditions are given to ensure the asymptotic stability of the S-asymptotically periodic solutions for problem (1). We notice that the generalized Gronwall inequality in  cannot be applied to Theorem 2 because the estimation obtained by that method is not stable.
Compared with previous research results, the innovations of this paper include the following: (1) the equations studied contain both fractional differential operators and FBM. It is worth mentioning that the standard Brownian motion, without long memory, cannot represent all types of noise. A good long-term memory noise could be described by FBM of Hurst parameter . For instance, the continuous disturbance and long-term dependence in the financial market model can be considered as a kind of FBM , with the impact of nuclear waste on the environment being seen as FBM in ecological models. Other research studies on FBM can be referred to [22–30]. (2) Stability of the S-asymptotically periodic solutions is studied by means of a new generalized Gronwall inequality.
The paper is organized as follows: the readers are allowed to review Section 2 for the necessary basic knowledge, followed by some results of the existence and uniqueness of S-asymptotically periodic solutions in Section 3. Subsequently, the asymptotic stability of S-asymptotically periodic solutions is studied in Section 4, with a numerical simulation example in Section 5.
For the sake of convenience in writing, throughout this paper, by , we mean . denotes a complete filtered probability space, and are two separable Hilbert spaces. The space of bounded linear operators from into is written as . For convenience, the same notation is applied to denote norms in and ; is applied to denote the inner produce of and . Moreover, is the space of all strongly measurable and square-integrable -valued random variables under the Banach norm .
A stochastic process is called stochastically bounded if called stochastically continuous if for all , and called square-mean S-asymptotically -periodic if where is a constant.
Denote by the space of all stochastically bounded and continuous processes from into and its normwhere . Then, is a Banach space.
We use to denote the space of square-mean S-asymptotically -periodic stochastic process from into . Then, is a Banach space with the sup norm and is a linear closed subspace of .
In the following, we introduce the definition and properties of FBM. We denote by a two-sided one-dimensional FBM . Then, is a continuous-centered Gaussian process, whose variance function is
In addition, if is a Wiener process, thenfor , where , with denoting the beta function.
Let be an operator with and for constants and a complete orthonormal basis in . The infinite dimensional FBM on can be expressed bywhere is the covariance operator and are two-sided one-dimensional FBMs, which are mutually independent on .
Let be the collection of all Q-Hilbert–Schmidt operators , where is a Hilbert–Schmidt operator, and the norm is
For convenience, set . The space is a separable Hilbert space whose inner product . Then, we define the stochastic integral of with regard to bywhere is the deterministic function with values in .
Now, we recall the Mittag–Leffler function and the probability density functions which play important roles in fractional differential equations .
We notice that the Mittag–Leffler function is a generalization of exponential function , which is .
Set , , and . Let , and we define by
We suppose that generates an exponentially stable -semigroup satisfyingwhere and are constants.
Thus, (1) can be transformed intowhere is the Caputo-fractional derivative. Later, in the paper, .
Remark 1. We see that (13) is much easy to be verified. For example, letThen, has eigenvalues , whose normalized eigenvectors , generates an analytic, compact, and exponentially stable semigroup , and
Proof. The proof of is as follows. In fact, for , in view of with , we haveIn order to get the stability of the square-mean S-asymptotically -periodic solution, we need the following generalized Gronwall inequality for fractional differential equations.
Lemma 3. Let , , be two constants. If a continuous function satisfiesthen
Remark 3. Compared with the generalized Gronwall inequality in , does not have to be a nondecreasing function, and is not necessarily nonnegative. This is very important to prove the stability of the solution.
Next, we give a result which is very useful for the estimations of fractional stochastic integral with FBM.
3. Existence Uniqueness of Square-Mean S-Asymptotically -Periodic Solutions
Lemma 5. Let and if satisfies.
For in every bounded subset of is S-asymptotically -periodic in . Moreover, there exists a positive constant such thatfor and .
Proof. Step 1: for , we prove thatFirstly, we haveThen,In view of Lemma 2, it is obvious thatBy combining Hölder inequality with and using Lemma 2, we haveThe last formula and Lemma 1 yield thatSince implies then for , there exists such that whenever . Then, we haveDue to Lemma 1, it is obvious that implies thatUsing a strategy similar to the one in the proof of (36), we getThen, .
Step 2: For , we prove that is stochastically bounded and continuous.
On the one hand, for a given , we haveLemma 2 (1) implies that . Arguing similarly as in (33), we see thatwhich means that .For an arbitrary sequence of real numbers with as for , we havewhich is due to . Hence,for every sufficiently large. In view ofthen it follows from Lebesgue’s dominated convergence theorem thatAdditionally, according to the arbitrariness of , we have thatwhich gives . Then, we know that is stochastically continuous.
On the other hand,which implies that is stochastically bounded.
By Steps 1-2, we obtain .
Lemma 6. Let andIf satisfies.
For in every bounded subset of , is stochastically bounded and continuous in . Furthermore, for and , there exists such that for and .
There exists a positive constant such thatfor and .
Proof. Step 1: for , we prove thatLet for each . Then, it is easy to find that is identically distributed like . Next, for , we see thatThen,Moreover, , Lemmas 2 and 4 yield thatThen,and , , and Lemma 4 imply thatwhere can be gotten similarly to (53). Moreover, without loss of generality, we may suppose for , , for , owing to :where is the beta function. Thus, . Therefore, .
Step 2: For , we prove that is stochastically bounded and continuous.
For a given number and , we getIt follows from and Lemmas 2 and 4 that