Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion

This article is devoted to the study of the existence and uniqueness of mild solution to a class of Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion. Our results are obtained by using fractional calculus, stochastic analysis, and the fixed-point technique. Moreover, an example is provided to illustrate the application of the obtained abstract results.


Introduction
Fractional calculus has gained considerable popularity during the past decades since it has been recognized as one of the best tools to model physical systems possessing longterm memory and long-range spatial interactions, which also plays an important role in diverse areas of science and engineering, as well as other applied sciences. As a result of the intensive development of fractional calculus, there has been a significant breakthrough in theoretical analysis and applications of fractional differential equations in the literature [8,14,15,17,23]. In recent decades, the existence and uniqueness of mild solutions, as well as controllability for fractional differential equations with Caputo fractional derivative, have attracted much attention. We can refer to [1-3, 11, 13, 16, 18-20, 24, 25] and the references therein.
It is worth mentioning that by applying Laplace transform and probability density functions, Zhou et al. [26] gave a suitable concept of mild solutions for a class of Riemann-Liouville fractional evolution equations with nonlocal conditions. Li et al. [10] considered the Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives by using the α-resolvent operator. Yang and Wang [22] investigated the approximate controllability of Riemann-Liouville fractional differential inclusions. For more details about the existence of mild solutions of Riemann-Liouville fractional differential equations, one can refer to [12,20]. On the other hand, Ahmed and El-Borai [1] studied Hilfer fractional stochastic integro-differential equations. We can refer to [5][6][7] for more details about the existence of mild solutions and approximate controllability of Hilfer fractional differential equations and inclusions.
As an generalization of Brownian motion, fractional Brownian motion has received a lot of attention in the recent years, which as we know is a Gaussian process with selfsimilarity and stationary increments, as well as long-range dependence properties. When Hurst parameter H ∈ (0, 1 2 ), fractional Brownian motion is neither a semimartingale nor a Markov process. Hence, the classical stochastic analysis techniques are invalid to study it. Recently, the research of stochastic differential equations driven by fractional Brownian motion has been investigated by many authors, see [4,9,21,28] and the references therein.
However, we would like to emphasize that it is natural and also important to study the existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion since it has not yet been sufficiently studied in contrast with the integer-order case. In this paper we are concerned with the following Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions driven by both Wiener process and fractional Brownian motion: where L D α t denotes the Riemann-Liouville fractional derivative in time defined for 1 2 < α < 1, I α is the temporal Riemann-Liouville fractional integral operator of order α; x(·) takes values in a separable Hilbert space X, A is the infinitesimal generator of an analytic semigroup {S(t)} t≥0 on a real separable Hilbert space X with inner product ·, · and norm · . Assume K to be another separable Hilbert space with inner product ·, · and norm · K . Let L(K, X) denote the space of all bounded linear operators from K to X, h : J × X → X and F : J × X → L(K, X) be functions satisfying some specific assumptions given in (H 2 )-(H 3 ); {ω(t)} t≥0 is a given K -valued Wiener process with a finite trace nuclear covariance operator Q > 0 defined on the filtered complete probability space ( , F, P); B H is a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1). The initial data x 0 is an F 0 -measurable, stochastic process independent of the Wiener process ω and fBm B H (t) with finite second moment.
A brief outline of this paper is given as follows. In Sect. 2, we recall some notations and preliminaries about fractional Brownian motion and fractional calculus. Using the fixed point theorem, Sect. 3 establishes the existence of mild solutions for system (1.1). In Sect. 4, we will give an example to illustrate the application of the obtained abstract results. Conclusions and discussions are given in the final section.

Preliminaries
In this section, we first present some notations, definitions, and preliminary facts.
We first recall some basic facts about fBm and the Wiener integral with respect to fBm. Let ( , F, P) be a complete probability space. We choose a time interval J = [0, b] with arbitrary fixed horizon b, assume {B H (t), t ∈ J} to be a one-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). The definition implies that B H is a continuous and centered Gaussian process with covariance function In what follows, we always assume 1 2 < H < 1. Consider the square integrable kernel We will denote by H the reproducing kernel Hilbert space of the fBm. In fact, H is the closure of the linear space of indicator functions {I [0,t] , t ∈ [0, T]} with respect to the scalar product The mapping I [0,t] → B H (t) can be extended to an isometry between H and the first Wiener chaos, and we denote by B H (ϕ) the image of ϕ under this isometry.
Set K * H be the linear operator from H to L 2 ([0, b]) defined by Recall that the operator K * H is an isometry between H and L 2 ([0, b]). Let the process ω = ω(t), t ∈ [0, b] be defined by Then ω is a Wiener process, while B H has the integral representation We recall that for ϕ, ψ on [0, b], their scalar product in H is given by The embedding relationship is as follows [21]: Assume that Y is a real separable Hilbert space. Suppose that there exists a complete orthogonal system {e n } ∞ n=1 in Y . Let Q ∈ L(Y , Y ) be an operator with finite trace Tr Q = ∞ n=1 λ n < ∞ (λ n ≥ 0) such that Qe n = λ n e n . By using the covariance operator Q, the infinite-dimensional fBM on Y can be defined as where B H n (t) are one-dimensional standard fractional Brownian motions mutually independent on ( , F, P). Assume that the space L 0 2 : and that the space L 0 2 equipped with the inner product ϕ, We next denote by L 2 ( , X) the collection of all strongly-measurable, square-integrable, X-valued random variables. Obviously, L 2 ( , X) is a Banach space equipped with norm To define the mild solution of system (1.1), we also need to consider the Banach space Throughout this paper, we assume that 0 ∈ ρ(A) where ρ(A) denotes the resolvent set of A. Then it is possible for us to define the fractional power A η as a closed linear operator on its domain D(A η ) for 0 < η ≤ 1 (see [25]). For an analytic semigroup {S(t)} t≥0 , the following properties hold: (i) There exists M ≥ 1 such that (ii) For ∀η ∈ (0, 1], there exists a positive constant C η such that For further convenience, set which involves the Wright-type function connected with the following one-sided stable probability function:

Lemma 2.2 ([23])
The operators S α and P α have the following properties: (i) For any fixed t ≥ 0, S α (t) and P α (t) are linear and bounded operators, i.e., for any x ∈ X, 25,27]) For any t > 0 and 0 ≤ γ < 1, there exists a positive constant C γ such that .

Lemma 2.5
If Proof Referring to [25], we omit the proof here.

Main results
In this section, we present and prove the existence of mild solutions for system (1.1). To develop our results, we first give the concept of mild solution for system (1.1).

Definition 3.1 An F t -adapted and measurable stochastic process
is integrable and the following integral equation is verified: For our readers' convenience, we first introduce some notations: To establish the main results, we require the following hypotheses: : X → X is continuous for each t ∈ J, and for each x ∈ X, the function h(·, x) : J → X is strongly measurable; (3b) there exist constants β ∈ (0, 1) and L > 0 such that h ∈ D(A β ) and for any We define the operator : C α (J, X) → C α (J, X) as follows: According to assumption (H 3 ) and Lemma 2.3, we obtain For the last term of the mild solution, from assumption (H 5 ), we have the estimate In the following, we set to present our first existence result for system (1.1).
Then it is obvious that B q is a bounded, closed, convex set in C α (J, X). We demonstrate the proof in six steps.
Step 1. We shall show that there exists a constant r = r(a) such that (B r ) ⊂ B r .
In fact, if this claim is not true, then for each positive constant r there exists some x (r) ∈ B r such that (x (r) ) / ∈ B r , i.e., Dividing both sides by r and letting r → ∞ yields This contradicts the assumption of Theorem 3.1, which implies that there exists r such that maps B r into itself.
Step 2. We show that 1 is a contraction on B r .
For any x, y ∈ B r , we derive Thus, 1 is a contraction by assumption of Theorem 3.1.
To prove this assertion, we subdivide Step 3 into three claims. Claim 1. 2

maps bounded sets into uniformly bounded sets in C α (J, X).
We only need to show that there exists constant > 0 such that for each 2 x, x ∈ B r , 2 x C α ≤ holds. In fact, for each t ∈ J , by using Hölder's inequality, we have Consequently, for each ρ ∈ 2 x, we have For 0 < t 1 < t 2 ≤ b, the strong continuity of {P α (t) : t ≥ 0} implies that there exists a constant δ > 0 such that |t 2t 1 | < δ and P α (t 1 ) -P α (t 2 ) < τ . In addition, note that c = α-1 1-α 1 , then for ∀x ∈ B r , this yields that where Thus I 2 tends to zero as t 2t 1 → 0. Similarly, we can get that I 3 tends to zero as t 2t 1 → 0.
Therefore, the relationship of E and { 2 x : x ∈ B r } implies that 2 is equicontinuous on Let 0 < t ≤ b be fixed. Then for ∀λ ∈ (0, t) and ∀δ > 0, x ∈ B r , define an operator From the compactness of S(λ α δ), λ α δ > 0, we obtain that for ∀λ ∈ (0, t) and ∀δ > 0, the set The right-hand side of the above inequality tends to zero as λ, δ → 0. So we can deduce that ( 2 x)(t) -( λ,δ 2 x)(t) C α → 0 as λ, δ → 0 + which enables us to claim that there are relatively compact sets arbitrarily close to the set We deduce, from Claims 1-3 and the Arzola-Ascoli theorem, that 2 is a completely continuous map. Using Krasnoselskii's fixed point theorem, we claim that the operator equation x = 1 x + 2 x has a fixed point on B r which is a mild solution for system (1.1). The proof is complete.
To give our last existence theorem, we require the following hypotheses: For convenience, let To end this section, we shall proceed with our last existence and uniqueness theorem for system (1.1) based on the Banach contraction principle. Proof Define operator as in Theorem 3.1. Then by similar arguments employed in Theorem 3.1, we can get that operator maps B r into itself, where B r is defined as in Theorem 3.1.
Moreover, we have Hence, the Banach contraction principle implies that has a unique fixed point in B r which is a mild solution for system (1.1). This completes the proof.

An example
As an application, we present an example to illustrate our results. Consider the following fractional stochastic evolution equation driven by both fBm and Wiener process: where L D 3 4 denotes the Riemann-Liouville fractional derivative of order 3 4 , I 1 4 is a Riemann-Liouville integral of order 1 4 , n is a positive integer, 0 < t 0 < t 1 < · · · < t n < b, z ∈ [0, π].
To write the above system (4.1) into the abstract form of (1.1), we choose the space X = L 2 [0, π]. Next we define an operator A by Av = v with the domain D(A) = {v ∈ X : v, v absolutely continuous, v ∈ X, v(0) = v(π) = 0}. Then A is a generator of a strongly continuous semigroup {S(t) t≥0 } which, as we know, is compact, analytic, and self-adjoint.
With the above choices of A, F, h, σ , system (4.1) can be seen as an abstract form of system (1.1). On the other hand, suppose that the assumptions of Theorem 3.1 hold. Thus, using Theorem 3.1, we claim that system (4.1) admits a mild solution on [0, b] under the above additional assumptions.

Conclusion
In this paper, we have considered a class of Riemann-Liouville fractional stochastic evolution equations. In particular, the fixed point technique, fractional calculus, and stochastic analysis were used for achieving the sufficient conditions to ensure the existence of mild solutions of Riemann-Liouville fractional stochastic evolution equations driven by Wiener process and fBm. We last gave an illustrative example. Our future work will address the explosive solutions of fractional Navier-Stokes stochastic differential equations driven by fBm and multiplicative noise.