The Existence of Mild Solutions for Hilfer Fractional Stochastic Evolution Equations with Order µ ∈ ( 1, 2 )

: In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order µ ∈ ( 1,2 ) and type ν ∈ [ 0,1 ] . We prove the existence of mild solutions of Hilfer fractional stochastic evolution equations when the semigroup is compact as well as noncompact. Our approach is based on the Schauder ﬁxed point theorem, the Ascoli–Arzelà theorem and the Kuratowski measure of noncompactness. An example is also provided, to demonstrate the efﬁcacy of this method.

Random disturbances are commonplace in many real-life scenarios.Stochastic differential equations can solve option pricing problems in the field of economics, and can be used for escape and jump problems of Brownian particles in the field of physics, among other applications.As a result, stochastic differential equations are widely employed in diverse scientific fields, including economics, chemistry, physics and social sciences.The existence, uniqueness, stability and controllability of solutions for fractional stochastic differential equations are very important.To this end, researchers have established mild solutions for various classes of stochastic evolution equations.Specifically, Refs.[9][10][11] demonstrated the existence of mild solutions for a class of Caputo fractional stochastic evolution equations with order µ ∈ (0, 1).Refs. [12,13] established the existence of mild solutions for a class of stochastic evolution equations with Riemann-Liouville fractional derivatives.Additionally, refs.[14,15] proved the existence of mild solutions for a class of stochastic evolution equations with Hilfer fractional derivatives H D µ,ν 0 + , where µ ∈ (0, 1), ν ∈ [0, 1].
To provide a clear structure for this article, we have divided it into several sections.In Section 2, we introduce some basic facts that are needed for our analysis.In Section 3, we prove some lemmas, and make certain assumptions.In Section 4, we verify the existence problem of Equation ( 1) based on the proved lemma and some new methods.In Section 5, we provide an example, to verify the validity of our results.Finally, a summary of this thesis is provided in Section 6.

Preliminaries
Denoting L 2 (Ω, H) as a Banach space of all strongly measurable, square-integrable and H-valued random variables, the norm y( L(K, H) denote the space of all bounded linear operators from K into H, with the norm • .We assume that there exists a complete orthonormal basis {e k } k≥1 in K. Denote By Proposition 2.9 in [16], provided that ψ(t) ∈ L(K, H), and that ψ(t) is measurable with respect to F t for t ∈ [0, b], and satisfies then we have the following property: For convenience of calculation in this paper, we introduce the function g α (•), which is defined as where α > 0, and gamma function Γ(•) satisfies αΓ(α) = Γ(α + 1).In case α = 0, we denote g 0 (t) = δ(t); the Dirac measure is concentrated at the origin.
Definition 1 (see [2]).The Riemann-Liouville fractional integral is defined as follows: where * is the convolution.Definition 2 (see [2]).The Riemann-Liouville fractional derivative is defined as follows: in particular, its Laplace transform is as follows: Definition 3 (see [2]).The Caputo fractional derivative is defined as follows: where the function y(t) is n − 1 times continuously differentiable, and is absolutely continuous.
Definition 4 (see [1]).The Hilfer fractional derivative is defined as follows: Remark 2. The focus of this article is on the case where 0 < ν < 1; however, we note that when ν = 1 or ν = 0, the conclusions in this article still apply.
where {C(t)} t∈R is a strongly continuous cosine family in X.
The infinitesimal generator of a strongly continuous cosine family {C(t)} t∈R is the operator A: X → X, defined by where D(A) = {y ∈ X : C(t)y is a twice continuously differentiable function with respect to t}.
Lemma 2 (see [19]).Strongly continuous cosine family {C(t)} t∈R satisfying ||C(t)|| X ≤ M 0 e ω|t| in X, for all t ≥ 0 and some ω ≥ 0, M 0 ≥ 1, and A being the infinitesimal generator of {C(t)} t∈R .Then, for Reλ > ω, λ 2 ∈ ρ(A) and λR(λ In this paper, because A is the infinitesimal generator of a strongly continuous cosine family of uniformly bounded linear operators {C(t)} t≥0 in H, there exists a constant M ≥ 1, such that C(t) L(H) ≤ M for t ≥ 0. Lemma 3. The problem Equation (1) is considered in the corresponding integral form, as follows: Proof.When 0 < t ≤ b, it follows from Definitions 2 and 4 that We can deduce, based on Definition 5, that Thus, the operators I µ 0 + act simultaneously on both sides of Equation (1), by using ( 5) and ( 6), and we can deduce that Proof.From Lemma 3, and using convolution calculation, we can obtain Assuming Reλ > 0, we denote the Laplace transform by L, and then By using the Laplace transform on Equation ( 8), we obtain By Lemma 2 and µ = 2p, we obtain In Appendix B of [18], the one-sided stable probability is denoted as follows: and its Laplace transform is Meanwhile, we can obtain where M p (ϑ) refers to the Wright function defined in Definition 6.Thus, we have Similarly, we can derive

and as the Laplace inverse transform
, according to Equation ( 3) and lim Then, by combining the aforementioned arguments with the uniqueness theorem of Laplace transform, Equation ( 7) can be derived.The proof is completed.Definition 8. y: (0, b] → H is an F t -adapted stochastic process; it is said to be the mild solution of the Cauchy problem (1) Lemma 5.For any y ∈ H, the following inequality is true: In addition, Q p (t) is uniformly continuous: that is, for any t 2 , t 1 ≥ 0, Proof.As C(t) ≤ M for any t ≥ 0, we can obtain Thus, by Definition 6 and Lemma 4, we can derive We will now demonstrate the uniform continuity of operator Q p (t) for t 2 > t 1 ≥ 0: Thus, the proof is concluded.
Lemma 6.If Equation ( 4) holds for any t > 0 and y ∈ H, the following formula is true: Proof.As d dt S(t p ϑ)y = pt p−1 ϑC(t p ϑ)y for t > 0 and y ∈ H, it can be calculated that Thus, we obtain By Definition 6 and Lemma 5, we can obtain In conclusion, the proof is finished.
Lemma 7 (A generalized Gronwall inequality; see [20]).Suppose that (i) γ > 0, 0 < T ≤ ∞, (ii) non-negative function a(t) and u(t) are locally integrable on 0 ≤ t < T, and (iii) continuous function g(t) is a non-negative, non-decreasing and bounded, 0 ≤ t < T. If ) is relatively compact if and only if the following conditions hold: Lemma 9 (Schauder fixed point theorem; see [3]).Let B be a closed, covex and nonempty subset of a Banach space X.Let Ψ : B → B be a continuous mapping, such that ΨB is a relatively compact subset of X.Then, Ψ has at least one fixed point in B.

Some Lemmas
To demonstrate the main outcome of this paper, the following assumptions are necessary: (A 1 ): For any 0 < t ≤ b, we assume that f (t, •) is Lebesgue measurable; for each y ∈ H, we assume that f (•, y) is continuous; (A 2 ): For any 0 < t ≤ b, we assume that h(t, •) is F t -measurable, and (A 4 ): There exists a constant, l > 0, and a bounded set, D ⊂ H, such that where ∨ means the maximum of the two-for example, if a > b, then a ∨ b = a.Now, we introduce another space: , it is clear that the space is a Banach space.
Define mapping Φ : where According to (A 3 ) above, there is a positive constant R, such that ) Let Evidently, B R and BR are convex, nonempty and closed subsets of Next, we prove several lemmas that are relevant to our main result.
Proof.As part of our analysis, we aim to prove that U is equicontinuous; therefore, we need to show that lim . This proof can be divided into the following two steps: Step I: u : Because C(0) = I and Definition 6, we can obtain Similarly, we obtain Due to the known relations between the Caputo fractional derivative and the Riemann-Liouville fractional derivative, we obtain By using Equations ( 12)-( 15), as well as Lemmas 4 and 6, we can establish the following result: .
By using the equality mentioned above, and the C r inequality, we can obtain the following result, when t 1 = 0 and t 2 ∈ (0, b]: For any 0 < t 1 < t 2 ≤ b, we can apply the C r inequality, to obtain the following result: where By employing the C r inequality, we obtain We can observe that I 11 → 0 as t 2 → t 1 .According to Equation ( 15), we can derive By using Lemma 6, we can derive the following result: Noting that then, by Lebesgue's dominated convergence theorem and Lemma 6, we derive This implies that Thus, by I 11 → 0 and I 12 → 0 as t 2 → t 1 , we derive Using similar methods for I 1 → 0 as t 2 → t 1 , we can obtain the following result: Thus, we obtain After conducting the aforementioned analysis, we can conclude that the set {u : Step II: {u : When t 1 = 0, 0 < t 2 ≤ b, according to Lemma 5, Equation ( 2), (A 3 ) and Hölder s inequality, we obtain where Next, we prove J 2 → 0 as t 2 → t 1 , according to C r inequality and Lemma 5, obtaining where We can deduce that lim Then, Moreover, Lemma 6 and Equation (2) imply that Hence, J 2 → 0 as t 2 → t 1 .Thus, we can prove J 1 → 0 as t 2 → t 1 in a way similar to J 2 → 0. Consequently, According to the above analysis, lim As y(t) = t α−2 z(t), t ∈ (0, b], according to (A 1 ) and (A 2 ), we obtain By employing (A 3 ), we can obtain we can use the Lebesgue dominated convergence theorem to derive Similarly, we obtain Hence, Ψ is continuous.

Main Results
Theorem 1. Suppose that (A 1 )-(A 3 ) are satisfied, and that {S(t)} t>0 is compact, then there exists at least one mild solution to problem (1) in BR .
Proof.Evidently, if the operator Ψ has a fixed point, z ∈ B R is equivalent to Equation (1), and there exists a mild solution y ∈ BR , where z(t) = t 2−α y(t).Hence, we only need to prove that the operator Ψ has a fixed point in B R .According to Lemmas 11 and 12, we know that Ψ is continuous, and that ΨB R ⊂ B R .In order to prove that Ψ is a completely continuous operator, we need to prove that Ψ(B R ) is a relatively compact set.From Lemma 10, the set U = {u : u(t) = (Ψz)(t)), z ∈ B R } is equicontinuous.According to the Ascoli-Arzelà theorem, we only need to prove that U(t) = {u(t) : (Ω, H), we only need to prove that the set U(t) is relatively compact in L 2 (Ω, H) for t > 0.
Similarly, we can obtain d 1 → 0. Thus, U(t) is also relatively compact in L 2 (Ω, H); therefore, by employing the Schauder fixed point theorem, we can deduce that Ψ has at least one fixed point z * ∈ B R .Let y * = t α−2 z * for t ∈ (0, b].Thus, The proof is completed. Theorem 2. Suppose that (A 1 )-(A 4 ) are satisfied, and that {S(t)} t>0 is noncompact, then there exists at least one mild solution to the problem of Equation (1) in BR .
m=0 is relatively compact.By Lemma 10, we can deduce that the set V is equicontinuous.According to the Ascoli-Arzelà theorem, we only need to prove that V According to Lemmas 1 and 5 and (A 5 ), we obtain For any y 1 , y 2 ∈ H, by employing Lemma 5 and Equation (2), we can derive Thus, according to Equation ( 19), (A 4 ) and [14], we obtain .
The above estimates yield .
In addition, we obtain According to Lemma 7, we can also obtain χ(V(t)) = 0; therefore, V(t) is relatively compact.According to the Ascoli-Arzelà theorem, V is relatively compact.Thus, there exists y * ∈ B R , such that lim Let y * (t) = t 2−α z * (t).Thus, y * ∈ BR is a mild solution to Equation (1).

Conclusions
By employing the Ascoli-Arzelà theorem and novel techniques, this paper has explored the existence of mild solutions for Hilfer fractional stochastic evolution equations with order 1 < µ < 2 and type 0 ≤ ν ≤ 1.Our proof demonstrates the theorem of the existence of mild solutions in both compact and noncompact cases: specifically, the satisfaction of the Lipschitz condition is not required for f (t, •) and h(t, •).The techniques presented in this paper are suitable for investigating the existence of solutions for non-autonomous evolution equations, fractional evolution equations with instantaneous/non-instantaneous impulses and fractional neutral functional evolution equations.We refer readers to the relevant papers [21,22].