Existence of Mild Solutions for Impulsive Fractional Stochastic Differential Inclusions with State-Dependent Delay

We study the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with state-dependent delay. Sufficient conditions for the existence of solutions are derived by using the nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan. An example is given to illustrate the theory.


Introduction
During the past two decades, fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, and engineering [1][2][3]. There has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical senses. One can see the monographs of Kilbas et al. [2], Miller and Ross [4], Podlubny [5], and Lakshmikantham et al. [6] and the survey of Agarwal et al. [7,8].
To study the theory of abstract differential equations with fractional derivatives in infinite dimensional spaces, the first step is how to introduce new concepts of mild solutions. A pioneering work has been reported by El-Borai [9,10]. Very recently, Hernández et al. [11] showed that some recent papers of fractional differential equations in Banach spaces were incorrect and used another approach to treat abstract equations with fractional derivatives based on the welldeveloped theory of resolvent operators for integral equations. Moreover, Wang and Zhou [12], Zhou and Jiao [13] also introduced a suitable definition of mild solutions based on Laplace transform and probability density functions.
On the other hand, the theory of impulsive differential equations or inclusions has become an active area of investigation due to its applications in fields such as mechanics, electrical engineering, medicine, biology, and ecology. One can refer to [14,15] and the references therein. Recently, the problems of existence of solutions and controllability of impulsive differential equations and differential inclusions have been extensively studied [16,17]. Benedetti in [18] proved an existence result for impulsive functional differential inclusions in Banach spaces. Obukhovskii and Yao [19] considered local and global existence results for semilinear functional differential inclusions with infinite delay and impulse characteristics in a Banach space. Some existence results were obtained for certain classes of functional differential equations and inclusions in Banach spaces under assumption that the linear part generates an compact semigroup (see, e.g., [20][21][22]). The existence results of impulsive differential equations and inclusions have been generalized to stochastic differential equations with impulsive conditions [23,24] and for stochastic impulsive differential inclusions [25][26][27].
We would like to mention that the impulsive effects also widely exist in fractional stochastic differential systems [28][29][30], and it is important and necessary to discuss the qualitative properties for stochastic fractional equations with impulsive perturbations with state-dependent delay. However, to the authors' knowledge, no result has been reported on the existence problem of impulsive fractional stochastic differential inclusions with state-dependent delay and the aim of this paper is to fill this gap.
Motivated by this consideration, in this paper we will discuss the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with statedependent delay in Hilbert spaces. Specifically, sufficient 2 Chinese Journal of Mathematics conditions for the existence are given by means of the nonlinear alternative of Leray-Schauder type for multivalued maps due to O'Regan.

Preliminaries and Basic Properties
In this section, we provide definitions, lemmas, and notations necessary to establish our main results. Throughout this paper, we use the following notations. Let (Ω, F, P) be a complete probability space equipped with a normal filtration F , ∈ = [0, ] satisfying the usual conditions (i.e., right continuous and F 0 containing all P-null sets). We consider two real separable Hilbert spaces H, K with inner product (⋅, ⋅) H , (⋅, ⋅) K and norm ‖ ⋅ ‖ H , ‖ ⋅ ‖ K . Let = ( ) ≥0 be a -Wiener process defined on (Ω, F , P) with the linear bounded covariance operator such that Tr( ) < ∞. Assume that there exists a complete orthonormal system { } ≥1 in K, a bounded sequence of nonnegative real numbers { } such that = , = 1, 2, . . ., and a sequence { } ≥1 of independent Brownian motions such that and F = F , where F is the sigma algebra generated by { ( ), 0 ≤ ≤ }. Let (K, H) denote the space of all bounded linear operators from K to H equipped with the usual operator norm ‖ ⋅ ‖ (K,H) . For ∈ (K, H) we define If ‖ ‖ 2 < ∞, then is called a -Hilbert-Schmidt operator. Let (K, H) denote the space of all -Hilbert-Schmidt operators . The completion (K, H) of (K, H) with respect to the topology is induced by the norm ‖ ⋅ ‖ where ‖ ‖ 2 = ( , ) is a Hilbert space with the above norm topology. Let 2 (Ω, H) be a Banach space of all strongly measurable, square integrable, H-valued random variables equipped with the norm ‖ (⋅)‖ 2 = (E‖ (⋅, )‖ 2 ), where E(⋅) denote the expectation with respect to the measure P. Let C( , 2 (Ω, H)) be the Banach space of all continuous maps from to 2 (Ω, H) satisfying the condition sup 0≤ ≤ E‖ ( )‖ 2 < ∞. Let 0 2 (Ω, H) denote the family of all F 0 -measurable, H-valued random variables (0).
Recall the following known definitions. For more details see [2]. Definition 1. The fractional integral of order with the lower limit 0 for a function is defined as provided the right-hand side is pointwise defined on [0, ∞), where Γ is the gamma function.

Definition 2.
Riemann-Liouville derivative of order with lower limit zero for a function : [0, ∞) → R can be written as If ( ) ∈ [0, ∞), then Obviously, the Caputo derivative of a constant is equal to zero. The Laplace transform of the caputo derivative of order > 0 is given as Chinese Journal of Mathematics 3 Definition 4 (see [31]). A closed and linear operator is said to be sectorial if there are constants ∈ R, ∈ [ /2, ], > 0, such that the following two conditions are satisfied: Definition 5 (see [30]). Let be a closed and linear operator with the domain ( ) defined in a Banach space . Let ( ) be the resolvent set of . We say that is the generator of an -resolvent family if there exist ≥ 0 and a strongly continuous function : R + → ( ), where ( ) is a Banach space of all bounded linear operators from to and the corresponding norm is denoted by ‖ ⋅ ‖, such that { : Re > } ⊂ ( ) and where ( ) is called the -resolvent family generated by .

Definition 6.
Let be an -resolvent operator family on Banach space with generator . Then, the following assertions hold: Definition 7 (see [30]). Let be a closed and linear operator with the domain ( ) defined in a Banach space and > 0. We say that is the generator of a solution operator if there exist ≥ 0 and a strongly continuous function where ( ) is called the solution operator generated by .
The next result is a consequence of the phase space axioms. The reader can refer to [34] for the proof.
In what follows, we use the notations P(H) for the family of all nonempty subsets of H and denote P (H) = { ∈ P (H) : is closed} , Now, we briefly introduce some facts on multivalued analysis. For details, one can see [35].
If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph; that is, Definition 10 (see [35]). Let : H → P , (H) be a multivalued map. Then is called a multivalued contraction if there exists a constant ∈ (0, 1) such that for each , ∈ H The constant is called a contraction constant of .
Next, we mention the statement of a nonlinear alternative of Leray-Schauder type for multivalued maps due to O'Regan.

The Mild Solution and Existence
Before stating and proving the main result, we present the definition of the mild solution to the system (3) . . , , the restriction of (⋅) to the interval ( , +1 ] ( = 0, 1, . . . , ) is continuous, and there exists ∈ S Σ, such that ( ) satisfies the following integral equation: . . . The following result on the operator ( ) appeared and is proved in [31].
Theorem 13. If ∈ (0, 1) and ∈ A ( 0 , 0 ) is a sectorial operator, then for any ∈ H and > 0, one has where is a constant depending only on and .
In order to establish the results, we first assume that the function is continuous from × B into (−∞, ] and we impose the following additional hypotheses. (H1) If ∈ (0, 1) and ∈ A ( 0 , 0 ) is a sectorial operator, then for ∈ H and > 0, for more details, see [31]) . (ii) → ( , ) is upper semicontinuous (u.s.c.) for almost all ∈ , and for each fixed ∈ B, the set S Σ, of selections of Σ is nonempty.
(H4) There exists a positive integrable function ] uniformly in ∈ for a nonnegative constant Λ, where (H5) The function : × B → H is continuous and there exists > 0 such that  The following lemma is required for the main result. The reader can refer to [37,38] for the lemma and to [32] for more details about the proof.
where ∈ S Σ, = { ∈ 2 ( (K, H)) : ( ) ∈ Σ( , ( , ) ) for a.e. ∈ } and : (−∞, 0] → H such that 0 = and = on . We shall show that Φ has a fixed point, which is then a mild solution for the problem (3). To this end we show that Φ satisfies all the conditions of Lemma 11. For the sake of convenience, we divide the proof into several steps.
Step 3. We show that the operator Φ is condensing. Let Φ 1 : → P(BPC) and Φ 2 : → P(BPC) be defined by We first show that Φ 1 is a contraction while Φ 2 is a completely continuous operator.
As a consequence of the above Claims 1-3, we conclude that Φ is a condensing map. All of the conditions of Lemma 11 are satisfied; we deduce that Φ has a fixed point in BPC which is a mild solution of the problem (3).

An Example
To apply our abstract results, we consider the following impulsive fractional stochastic partial differential inclusions with state-dependent delay of the form where ( ) is a standard cylindrical Wiener process in H defined on a stochastic space (Ω, F, {F }, P); is the Caputo fractional derivative of order 0 < < 1; is continuous; and 0 < 1 < 2 < ⋅ ⋅ ⋅ < < are prefixed numbers. Let It follows from the above expressions that ( ( )) ≥0 is a uniformly bounded compact semigroup, so that ( , ) = ( − ) −1 is a compact operator for all in the resolvent set of . ≥ 0, ≥ 1 and let : (−∞, ] → R be a nonnegative measurable function which satisfies the conditions (H-5) and (H-6) in the terminology of Hino et al. [34]. Briefly, this means that is locally integrable and there is a non-negative, locally bounded function ℎ on (−∞, 0] such that ( + ) ≤ ℎ( ) ( ) for all ≤ 0 and ∈ (−∞, − ) \ , where ⊆ (−∞, − ) is a set whose Lebesgue's measure is zero. Let , for ≥ 0 (see [34]).