Existence Results of Mild Solutions for the Fractional Stochastic Evolution Equations of Sobolev Type

: In this paper, by utilizing the resolvent operator theory, the stochastic analysis method and Picard type iterative technique, we ﬁrst investigate the existence as well as the uniqueness of mild solutions for a class of α ∈ ( 1, 2 ) -order Riemann–Liouville fractional stochastic evolution equations of Sobolev type in abstract spaces. Then the symmetrical technique is used to deal with the α ∈ ( 1, 2 ) -order Caputo fractional stochastic evolution equations of Sobolev type in abstract spaces. Two examples are given as applications to the obtained results.


Introduction
Since fractional differential equations can describe many problems in the fields of physical, biological and chemical and so on, some properties of solutions for the fractional differential equations have been considered by many authors, see [1][2][3][4][5][6][7][8]. In [2], when the nonlinearity satisfies non-Lipschitz conditions, Wang studied the existence of mild solutions of α ∈ (0, 1)-order fractional stochastic evolution equations with Caputo derivative in abstract spaces. Li et al. [3] obtained the existence as well as the uniqueness of weak solutions and strong solutions of an inhomogeneous Cauchy problem of order α ∈ (1, 2) involving Riemann-Liouville fractional derivatives via the technique of fractional resolvent.
Sobolev type (fractional) differential equation arises in various areas of physical problems, see [4,5], hence it has been investigated by researchers recently, see [4][5][6][7]. Feckan et al. [5] proved the controllability results for α ∈ (0, 1)-order fractional functional evolution equations of Sobolev type in abstract spaces. By virtue of the characteristic solution operators, they obtained the exact controllability results via Schauder fixed point theorem. In [8], by using the characterizations of compact resolvent families, Ponce investigated the Cauchy problem for a class of fractional evolution equations. Furthermore, the stochastic perturbation is unavoidable in the natural systems. Therefore, it is important to consider stochastic effects in studying fractional differential systems. Recently, in [6], by means of the operator semigroup theory, fractional calculus and stochastic analysis technique, Benchaabane et al. established a group of sufficient conditions to guarantee the existence as well as the uniqueness of solutions for the α ∈ (0, 1)-order fractional stochastic evolution equations of Sobolev type. As far as we know, the existence as well as the uniqueness of mild solutions for the Sobolev type fractional stochastic evolution equations of order α ∈ (1, 2) have not been extensively discussed yet.
In the present work, we consider the existence as well as the uniqueness of mild solutions for two classes of the initial value problems (IVPs) of fractional stochastic equations of Sobolev type in a Hilbert space X where 1 < α < 2 and b > 0 are constants, L D α t and C D α t denote, respectively, the α-order fractional derivative operators of Riemann-Liouville and Caputo, A : D(A) ⊂ X → X is a densely defined and closed linear operator in X, S : D(S) ⊂ X → X is also a closed linear operator in X, ξ and η are X-valued random variable, f , Σ, W and g 2−α will be specified later.
In the previous works, see [5,6], the authors often make the following assumptions on A and S when they investigate the Sobolev type differential equations.
(i) D(S) ⊂ D(A) and S is bijective; (ii) S has the compact and bounded inverse S −1 .
In this situation, −AS −1 generates a semigroup T(t) := e −AS −1 t for t ≥ 0 and S : D(S) ⊂ X → X may be bounded.
In this paper, without assuming (i) and (ii) on A and S as well as any compactness conditions on f and Σ, we investigate the existence as well as the uniqueness of mild solutions of the IVPs (1) and (2). More precisely, we first present the concept of (α, α − 1)-resolvent family and (α, 1)-resolvent family generated by the pair (A, S). With the help of (α, α − 1)-resolvent family and (α, 1)-resolvent family and Laplace transform, the correct definitions of mild solutions of the IVPs (1) and (2) are presented. Under some essential conditions on f and Σ, we study the existence as well as the uniqueness of mild solutions of the IVPs (1) and (2) by virtue of the iteration technique of Picard type. We have to emphasize that we do not assume the compactness of the (α, α − 1)-resolvent family and the (α, 1)-resolvent family in our main results.

Preliminaries
In this part, we first recall some definitions of fractional calculus. The definition of the fractional resolvent family is also given in this section. By using the fractional resolvent family and Laplace transform, the concepts of mild solution of the IVPs (1) and (2) are introduced, and an inequality is given in Lemma 1.
Denote by (Ω, F , {F t } t≥0 , P ) the complete probability space involving a filtration {F t } t≥0 , which satisfies the usual conditions. On (Ω, F , {F t } t≥0 , P ), {W(t)} t≥0 is a Q-wiener process with values in X, where Q is a bounded linear covariance operator and trQ < +∞. Let k ≥ 0 be a bounded sequence and {e k } k≥1 a complete orthonormal system of X satisfying Qe k = k e k for k = 1, 2, · · · . Let {β k } k≥1 be independent Brownian motions satisfying Further, let F t be the σ-algebra generated by {W(θ) : 0 ≤ θ ≤ t}. Put L 0 2 := L 2 (Q 1 2 , Y). Then L 0 2 is a real separable Hilbert space endowed with ϕ 2 the set of strongly F -measurable random variables with values in X. Then L p (F , X) is a Banach space satisfying E x p ≤ +∞. Let C(I, L p (F , X)) be the Banach space of all continuous maps from I to L p (F , X) satisfying the condition sup t∈I E x(t) p < +∞.
Let C p b ⊂ C(I, L p (F , X)) be the closed subspace of C(I, L p (F , X)), which consist of F t -adapted and measurable processes u(t). Put (3) ) is a Banach space. ξ and η in the IVPs (1) and (2) are F 0 -measurable and X-valued random variable independent of W.
Firstly, we recall a group of concepts of fractional calculus, see [9,10] for more details. For every We define the finite convolution of the functions f and g by

Definition 3.
For α > 0, the α-order Caputo fractional derivative of all u ∈ L 1 (I) is defined by If u ∈ C m (R + ), for α ∈ (m − 1, m), the α-order Caputo fractional derivative is defined by Hence from [10], since g α (λ) = λ −α for any α > 0, by Remark 1 and the properties of the Laplace transform, we have and In the following, we establish the concept of the fractional resolvent family which is a basic concept in our main results, see [4,8] for more details. Let {Π(t)} t≥0 be a strongly continuous family of B(X). If there exist M ≥ 1 and ω > 0 satisfying According to the Definition 5 of [8], we present the following definition.
Then the pair (A, S) generates an (α, Consequently, for all u ∈ X, we have Without loss of generality, we put M := sup Applying the Laplace transform to the first equation of the IVP (1), by virtue of Su(λ) = S u(λ) and (4), we have Together this fact with (7) and (8), we obtain by the properties of Laplace transform, we get Thus, based on the above discussion, the mild solution of the IVP (1) is defined below.

Definition 6.
A stochastic process u ∈ C(I, L p (F , X)) is called a mild solution of the IVP (1) if it satisfies the integral Equation (9).
Similarly, we can define the mild solution of the IVP (2) by applying (5).

Definition 7.
A stochastic process u ∈ C(I, L p (F , X)) is called a mild solution of the IVP (2) if it satisfies the integral equation where {C S α,1 (t)} t≥0 is the (α, 1)-resolvent family generated by (A, S), and At last, we recall an inequality, which cites from the Proposition 1.9 of [11][12][13][14].
where L Σ > 0 is a constant involving p and b.

Main Results
In this part, by utilizing the iteration technique of Picard type, we will prove the existence as well as the uniqueness of mild solutions of the IVPs (1) and (2). To this end, the following assumptions are needed. Hypothesis 1 (H1). f : I × X → X and Σ : I × X → L 0 2 are continuous functions and there is a function Φ : Hypothesis 2 (H2). The function Φ : I × R + → R + satisfies the assumptions: (i) For every u ∈ [0, ∞), Φ(·, u) is locally integrable. (ii) For every t ∈ I, Φ(t, ·) is nondecreasing and continuous. (iii) For all C 1 > 0, C 2 ≥ 0, the equation has a global solution on I.

Remark 3.
If u(t) is a global solution on I of the IVP of the first-order ordinary differential equation where C 1 > 0, C 2 ≥ 0 are constants, then the assumption Hypothesis 2 (H2)(iii) holds.
We will use Picard type approximate technique to prove our main results. For this purpose, we define the sequence of stochastic process {u n } n≥0 as follows: where Lemma 2. Let (A, S) be the pair which generates an (α, α − 1)-resolvent family {C S α,α−1 (t)} t≥0 of type (M, ω). If the Hypothesis 1 (H1) and Hypothesis 2 (H2) hold, the sequence {u n } n≥0 is well-defined. Moreover, there is a constant C > 0 satisfying sup Proof of Lemma 2. By (11), we have By applying Hölder inequality and Lemma 1, we get Together these facts with the monotonicity of Φ, by (3), we conclude that where In view of Hypothesis 2 (H2)(iii), the solution u(·) of the integral equation global exists on I. In the following, we prove u n C p t ≤ u(t) for all t ∈ I, n ≥ 0 by utilizing the induction method. Indeed, Let u n C p t ≤ u(t) for all t ∈ I, n ≥ 0. By means of (13) and (14), we obtain This implies that (12) holds with C := u(b) and the sequence {u n } n≥0 is well-defined.  (11), for any m ≥ n ≥ 0, we have

By the monotonicity of Ψ and (3), we can obtain
where τ : By (15), we have Since {φ n (t)} n≥0 is monotone and uniformly bounded due to Lemma 2, we know that there exists a function φ(t) satisfying lim n→∞ φ n (t) = φ(t), ∀t ∈ I.
Taking n → ∞ in the inequality (16), by the continuity of Ψ and dominated convergence theorem, we deduce that Hence, φ(t) ≡ 0 for all t ∈ I in view of Hypothesis 4 (H4).
Then taking n → ∞ in the second equality of (11), by the continuity of f , Σ and dominated convergence theorem, we can obtain Therefore, the IVP (1) has a mild solution u * belongs to C p b due to Definition 6. Next, we prove the uniqueness. Let the IVP (1) have mild solutions u * and v * . By a similar method as above, we obtain ≡ 0 for all t ∈ I. Thus, u * ≡ v * and the proof is completed.
For the IVP (2), by (10), we define the sequence of stochastic process {v n } n≥1 by By utilizing similar techniques as in the proof of Lemma 2 and Theorem 1, the following conclusions are obtained.   [4,8].

Remark 5.
In Theorems 1 and 2, we do not assume the existence, boundedness and compactness of S −1 , which are essential assumptions of [5,6]. So, the operator S in the IVP (1) and (2) may be unbounded. Therefore, our results improve the ones of [5,6].

Conclusions
In the present work, we investigate the existence as well as the uniqueness of mild solutions for the IVPs of Sobolev type fractional evolution equations involving α ∈ (1, 2)-order Riemann-Liouville or Caputo fractional derivatives. By using the stochastic analysis method, the Laplace transform and the fractional resolvent family, we first present the concept of mild solutions to the concerned problems.
Then the existence as well as the uniqueness theorems are proved by using an iteration technique of the Picard type. At the end of this paper, two examples are provided as applications of the abstract results.
Funding: The research is partially supported by the NNSF of China (No. 11701457).