Existence of Mild Solutions to Delay Diffusion Equations with Hilfer Fractional Derivative

: Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions.


Introduction
Differential equations with fractional derivatives play a pivotal role across a wide spectrum of fields such as the natural sciences, engineering, biological sciences, finance, and physics, capturing the interest of numerous scholars [1][2][3][4][5][6][7].The phenomenon of time delay, a ubiquitous occurrence in the fabric of real-life scenarios, has spurred a significant body of research.In recent years, growing interest has been observed in exploring the characteristics of solutions to delay differential equations, particularly focusing on the existence and stability of solutions involving the Caputo fractional derivative, as highlighted in studies [8][9][10].
The introduction of n delay terms increases the complexity when studying the existence of solutions.To address this problem, we need to consider the specific continuous function space . Because of the unboundedness and continuity of solutions to equations containing Hilfer fractional derivatives at zero, we examine the initial value of the Hilfer fractional diffusion equation with delay in the form of t (µ−1)(1−ν) φ(x,t) Γ(µ(1−ν)+ν) on the interval [−r, 0), and introduce a new solution operator to ensure the meaningfulness of the studied equation's solutions at zero.
In the main results, we initially assume the compactness of the analytic semigroup and relax the continuity condition for the function f required by reference [11], demanding instead that f be continuous in other variables for almost all of the time variables.Additionally, we assume that the norm of f is governed by the L Based on these assumptions, we employ the Leray-Schauder fixed-point theorem to demonstrate the existence of mild solutions.On this basis, we do not impose compactness on the analytic semigroup and we stipulate that the measure of f is controlled by the measures of delayed terms.Then, we utilize a non-compact measure approach to further prove the existence of mild solutions.Lastly, we assume that the norms of f satisfy a Lipschitz condition in another space Y.Following this, we apply the Banach contraction mapping principle to prove the existence and uniqueness of mild solutions.However, the interval µ ∈ [0, 1] in [11] is not applicable in this study because the proof of the strong continuity of R µ,ν (t) = t (1−µ)(1−ν) S µ,ν (t) necessitates the condition that µ is not zero.For simplicity, this paper restricts µ to the interval (0, 1).Moreover, Theorem 3.3 in the Gu and  Trujillo [15] represents a special case of Theorem 2 in this paper when f has no time delays.
The structure of this manuscript is articulated as follows.Section 2 delineates the requisite space and norm pertinent to this study, alongside a review of some foundational results.Subsequently, it proceeds to articulate the solution operator for Equation (4).Section 3 leverages fixed-point theorems, under specified conditions, to establish the existence of a solution for Equation (4).An illustrative example that underscores the derived outcomes is presented in Section 4. Concluding the discourse, Section 5 offers an all-encompassing recapitulation of the paper's content.

Preliminaries
Let X = L 2 (Ω) be a Banach space, where the norm is ∥ • ∥.The space of all continuous functions maps J into X is denoted by C J = C(J, X), where J is an interval and J ⊂ R. For arbitrary y ∈ C J and closed interval J, we define the norm ∥ y ∥ ∞ = sup t∈J ∥ y(t) ∥.The Lebesgue measurable functions ω : J → R with 1 ≤ p ≤ ∞ construct a Banach space, which is written as L p (J, R). Define Then, with the norm ∥ • ∥ Y , Y becomes a Banach space.

Definition 2 ([16]
).The fractional integral of the real function f is defined by where p > 0, and Γ(•) denotes the Gamma function.

Definition 3 ([17]
).The definition of the generalized fractional Riemann-Liouville derivative with order µ, ν ∈ (0, 1) and lower limit a is provided the right-hand side is well-defined.

Lemma 1 ([19]
).Let S 1 , S 2 ⊂ X, and S 1 , S 2 be bounded.Furthermore, let c be a real number.The noncompactness measure possesses the following properties , where coS 1 represents the convex closure of S 1 .
Then, we have where λ(W(t)) is the Lebesgue integral on J.

Definition 5 ([23]
).Let S ⊂ X be nonempty.If for any bounded set B ⊂ S and continuous mapping T : S → X , there exists a constant k ∈ [0, 1) such that [24]) and (Au)(t)x = Au(x, t).Then, A generates an analytic semigroup {Q(t)} t≥0 on X.Without losing generality, we assume that {Q(t)} t≥0 is a uniformly bounded linear operator.Thus, there exists M ≥ 1, such that Set , then (1) can be formulated in an abstract Cauchy problem form as where The equivalent integral equation for Equation ( 4) is given by Similar to [15], we obtain the following result.Lemma 6.If integral Equation ( 5) holds, then we have where S µ,ν (t The wright function M ν (θ), where ν ∈ (0, 1) and θ ∈ C, is defined as the infinite series .
This function satisfies the integral equality Definition 6.We define the mild solution of Equation ( 4) as a function u ∈ C (0,b] for which u satisfies

Lemma 7 ([15]
).For any t > 0, by the continuity of Q(t), we know that P ν (t) is continuous according to the uniform operator topology.
Subsequently, we have where B(•, •) represents the Beta function (see [26]).Therefore, we obtain That is, by the arbitrariness of t 0 , R µ,ν (t)(t > 0) is continuous in the uniform operator topology.

Main Results
In order to obtain the existence of mild solutions for Equation (4), we provide some assumptions.
(H 1 ) (H 4 ) For any bounded, equicontinuous and countable sets for any Proof.In view of Lemma 8, for t ∈ (0, b], it follows that For v ∈ B k 0 , according to (H 2 ), and u k t (k = 0, 1, •••, n) is continuous in t, and we have .
By applying (H 3 ) and Hölder inequality, for t ∈ (0, b], we derive We obtain by Lemma 8 and (9) that Consequently Firstly, we demonstrate that T is a completely continuous operator. Assume Considering that by the dominated convergence theorem, we have Thus, we know that T is continuous.Subsequently, we prove that Tv, v ∈ B k 0 is relatively compact.It suffices to demonstrate that Tv, v ∈ B k 0 is uniformly bounded and equicontinuous, and (Tv According to (H 3 ), there is a constant k 0 > 0, such that For t ∈ [0, b], by ( H 3 ) and Lemma 8, we obtain where It is obvious that I 1 → 0 as t 1 → t 2 by Remark 1.By condition (H 3 ), it can be deduced that lim t 1 →t 2 I 2 = 0. Noting that and ds exists, then by the dominated convergence theorem, we obtain then, it can be deduced that lim t 1 →t 2 I 3 = 0.For sufficiently small ε > 0, we have where By Lemma 7, we know that I 41 → 0 as t 1 → t 2 .Similarly, one can establish the proof for I 2 → 0 and I 3 → 0 , then, we have I 42 → 0 and I 43 → 0 as ε → 0 .Therefore, which means that Tv, v ∈ B k 0 is equicontinuous.Now, we need to establish that for any t ∈ [−r, b], (Tv)(t), v ∈ B k 0 is relatively compact in X.

Conclusions
Compared with other fractional derivatives, such as the Riemann-Liouville derivative and Caputo derivative, the Hilfer fractional derivative is built on a new theoretical foundation of fractional calculus, which provides a more complete and unified definition to better describe the behavior of complex systems.Therefore, studying the existence of solutions to the Hilfer fractional delay diffusion equation contributes to a deeper understanding of the behavior of such equations and provides accurate mathematical models for solving practical problems.This endeavor holds significant academic significance in optimizing engineering designs, predicting outcomes, and controlling natural systems.The main focus of this article is to investigate the existence of solutions to delay diffusion equations with Hilfer fractional derivatives.Under the assumption that the analytic semigroup is compact or non-compact, the Leray-Schauder fixed-point theorem and non-compactness measure method were employed to prove the existence of mild solutions, while Banach contraction mapping principle was utilized to establish the uniqueness of mild solutions.Moving forward, we aim to further explore the regularity and stability of mild solution to delay diffusion equations with the Hilfer fractional derivative.
), which is equivalent to T has a fixed point on C [−r,b] .Lemma 10 ([23]).Let B ⊂ X be a bounded closed convex set and that the operator T : B → B is k-set-contractive.Then, T has a fixed point in B.

Theorem 3 .
operator.Thus, we know from Lemma 10 that the Cauchy Equation (4) has a mild solution.□ Under assumptions (H 5 ), Equation (4) has a unique mild solution provided Mb (1−µ)(1−ν) t) = Au(t) + f (t, u(t), u(t − τ)), t ∈ (0, Under the assumptions ( H 2 ), ( H 3 ), ( H 4 ), Equation (4) has a mild solution if Similar to Theorem 1, T : B k 0 → B k 0 is continuous, and Tv, v ∈ B k 0 is uniformly bounded and equicontinuous.Let G = coT B k 0 .Then, it is simple to demonstrate that T maps G into itself and G ⊂ B k 0 is equicontinuous.By Lemma 2, we have that for any B ⊂ G, there exists a countable set B 0 11, T has a fixed point.That is, Equation (4) has a mild solution.□ Theorem 2.