The Existence Results of Solutions to the Nonlinear Coupled System of Hilfer Fractional Differential Equations and Inclusions

: This paper is dedicated to studying the existence results of solutions to the nonlinear coupled system of Hilfer fractional differential equations and inclusions, with multi-strip and multi-point mixed boundary conditions. Through tools such as the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems, along with Leray-Schauder degree theory, the main results can be obtained. The Hilfer fractional differential system has practical implications for specific physical phenomena. Examples are provided to clarify the application of our main results.


Introduction
Classical calculus began with Newton's invention of forward flow in 1665, while fractional calculus was born in 1695 when Leibniz and L'Hospital discussed the significance of 1  2 derivatives, see [1].It can be said that fractional and integer calculus have almost the same research history span.However, fractional calculus has only been revitalized in the last few decades, thanks to the development of various fields that promote research in fractional calculus.The new study of fractional calculus can also offer novel ideas for solving challenging problems in various fields, see [2][3][4][5][6][7].
Although Riemann-Liouville and Caputo fractional derivatives are considered valuable tools for modeling many real-world problems, R. Hilfer found that the traditional fractional derivatives of Riemann-Liouville and Caputo could not meet the requirements for solving new problems during the study of fractional time evolution [8].Therefore, in order to separate fractional integrals, a generalized definition of fractional derivatives is proposed based on the Riemann-Liouville integral by R. Hilfer, which is H D α,β u(t) = I β(n−α) D n I (1−β)(n−α) u(t), where n − 1 < α < n, 0 ≤ β ≤ 1, D = d dt .Many authors later called this definition as the Hilfer fractional derivative.The reader is referred to references [9] for the distinction between the same order α but with different values of β.
Initial value and boundary value problems involving the Hilfer fractional derivative have attracted a lot of research.In [10], K.M. Furati discussed the existence of solutions to a Hilfer fractional differential equation for the following initial value problem.Moreover, the stability of the solution to a weighted Cauchy-type problem is also analyzed.
where H D α,β is the Hilfer fractional derivative of order α, and type β.
In [11], K. Dhawan investigated the coupled Hilfer fractional differential equations with nonlocal conditions.By applying the Leray-alternative Schauder's and the Contraction principle, the author proved the existence and uniqueness of the solution.Furthermore, the Ulam stability of the solution was discussed for the defined problem.
In [12], B. Ahmad studied the nonlinear generalized coupled fractional differential equations accompanied by nonlocal coupled multipoint Riemann-Stieltjes and generalized fractional integral boundary conditions using the Leray-Schauder alternative and Banach contraction mapping principle.
It is worth noting that J. Pradeesh considered the existence of a mild solution for the following Hilfer fractional stochastic differential system in [13].
where H D v,µ 0 + denotes the Hilfer fractional derivative of order µ and type v such that 1 < µ < 2 and 0 ≤ v ≤ 1, while I 2−γ 0 + is the Riemann-Liouville integral operator with order (2 − γ), where γ = µ + v(2 − µ).A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous cosine family C(t) t≥0 of uniformly bounded linear operators and z(•) takes values in a separable Hilbert space H with inner product ⟨•, •⟩ and norm ∥ • ∥.The authors employed fractional calculus, multivalued analysis, sine and cosine operators, and Bohnenblust-Karlin's fixed point theorem to investigate the existence of a mild solution for the Hilfer fractional stochastic differential system.
However, there are very few articles that have studied Hilfer fractional differential equations and inclusions simultaneously.Inspired by the aforementioned works, we are initiating a study on the existence of solutions to a coupled system of Hilfer fractional differential equations and inclusions subject to the coupled fractional integral and discrete mixed boundary conditions where H D α k ,β , H D γ k ,β denote the Hilfer fractional derivative of order α k , γ k respectively and parameter β, I l ki is the Riemann-Liouville fractional integral of order l ki , 1 In the research of fractional time evolution, a significant challenge arises when generalizing traditional equations of motion, as it involves deciding whether to utilize the Riemann-Liouville fractional derivative or the Caputo fractional derivative.R. Hilfer introduced the Hilfer fractional derivative during the study.When β = 0, the Hilfer system can be simplified to the Riemann-Liouville system studied by authors such as X.Zhao, A. Guezane-Lakoud, T. Jankowski, and others.For more details, refer to [14][15][16].When β = 1, the Hilfer system can be reduced to the Caputo system discussed by investigators like Y. Zi, A. Alsaedi, J. Xie, B. Ahmad, and others.References can be found in [17][18][19][20].We learned that models of fractional diffusion equations with Hilfer fractional derivatives are used in the context of glass relaxation and aquifer problems [21].Also, fractional reaction-diffusion equations and space-time fractional diffusion equations involving the Hilfer fractional derivative are studied in [22,23].And more importantly, differential inclusion, which can handle uncertainty problems, has been applied to dynamical systems and stochastic processes, such as control problems and sweeping processes [24,25].Therefore, our studies on the nonlinear coupled Hilfer fractional differential inclusions have great practical applications in physical phenomena.
In the coupled Hilfer fractional differential system, there are u(t), v(t) that mutually influence each other.The nonlinear terms, f 1 and f 2 , consist of u(t), v(t), H D γ 1 ,β u(t), H D γ 2 ,β v(t), where γ 1 , γ 2 are less than α 1 , α 2 .The inclusion of H D γ 1 ,β u(t), H D γ 2 ,β v(t) in f 1 , f 2 enhances the model's capability to address real-world problems The coupled Hilfer fractional differential inclusions have the same structure.Moreover, the nonlocal boundary conditions consist of the Riemann-Liouville fractional integral and numerous discrete points.The value of the unknown function u(t) at the right endpoint t = 1 is equal to the sum of the values of the Riemann-Liouville fractional integral of the unknown function v(t) on the subinterval [0, ξ i ](i = 1, 2, . . ., m) and the discrete values of the unknown function v(t) at η j (j = 1, 2, . . ., n).
In addition, there are now many well-established methods for studying fractional differential equations and inclusions, such as Guo-Krasnoselskii's fixed-point theorem, the Banach contraction mapping principle, and monotone iteration techniques.We employed the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems, along with Leray-Schauder degree theory, to explore the existence of solutions for the Hilfer fractional differential equations and inclusions, respectively.
In fact, fractional derivatives have been greatly developed and applied, leading to the emergence of several mature definitions such as the Riemann-Liouville fractional derivative, the Hadamard fractional derivative, the Caputo-Katugampola fractional derivative, the Katugampola fractional derivative, and others.In contrast, the derivative under Hilfer's definition requires more research efforts to promote its development.Whether transferring mature research techniques to Hilfer or developing new technical methods, the work is meaningful.Differential inclusions can be regarded as a collection of differential equations and inequalities.Moreover, this paper examines the coupled system of Hilfer differential equations and inclusions, which is of great significance for practical applications.
The rest of the paper is structured as follows: In Section 2, some fundamental concepts of fractional calculus and lemmas are presented.Section 3 is dedicated to presenting the main results, which are illustrated through examples.Section 4 contains a summary of previous work and future prospects.

Preliminaries
In this section, we present some basic definitions, lemmas, and auxiliary results for the proof that will be utilized in the next section.
Lemma 12 (Nonlinear alternative of Leray-Schauder type [29]).Let X be a Banach space, Ω be a closed convex subset of X, and U be an open subset of Ω with 0 ∈ U. Suppose that F : U → P cp,cv (Ω) is an upper semicontinuous compact map.Then either (1) F has a fixed point in U, or (2) there is ∂U and λ ∈ (0, 1) such that x ∈ λF(x).
For each z ∈ C(J), we defined the set of selections of F by For convenience, we denote In the forthcoming analysis, we always need to make the following assumptions: where l 1 , l 2 are defined by (12).
Subject to BVP ( 5) and ( 7), we consider a corresponding linear differential system as follows and establish the expression of the corresponding Green's functions.
with boundary conditions (5) has an integral representation we obtain c 12 = c 22 = 0, and we get From the remaining conditions in (5), it can be inferred that Further, we can reduce (18) to Combining ( 23) and ( 24), it can be seen that where l k (k = 1, 2) is defined by (12).From ( 22) and ( 25), we have This completes the proof of the lemma.Moreover, according to Lemma 4, and fractional order derivative of solution (11) can be expressed as 2−θ 2 (J), then u, v satisfie ( 5) and ( 7) if and only if u, v satisfies (30) and (31).
Proof.We define the operator T : C → C by where Also, according to ( 28) and ( 29), it is easy to see that We will show that the operator T : C → C is completely continuous.
For any (u, v) ∈ Ω, we get Similarly, we obtain that and Hence, for (u, v) ∈ Ω, T 1 , T 2 is uniformly bounded.Thus it follows from the above inequalities that the set TΩ is uniformly bounded.
For any (u, v) ∈ Ω and t 1 , t 2 ∈ J such that t 1 < t 2 , we have and Therefore the set T 1 Ω is equicontinous for all (u, v) ∈ Ω.Similarly, we can get the set T 2 Ω is equicontinous for all (u, v) ∈ Ω.As a consequence, the set TΩ is equicontinous for all (u, v) ∈ Ω.By applying the Arzelá-Ascoli theorem, the set TΩ is relative compact which implies that the operator T is completely continuous.
Lastly, we shall show that the set (u, v).For any t ∈ J, we have Then, we get which imply that (83) Thus, we obtain where , which shows that the set ξ is bounded.Therefore, by applying Lemma 11, the operator T has at least one fixed point.Therefore, we deduce that problem ( 5) and ( 7) has at least one solution on J.
The proof is completed.

The Existence Results of the Coupled Hilfer Fractional Differential Inclusions
X is a real (or complex) separable Banach space with a norm ∥ • ∥, defined by ∥u∥ = sup t∈J |u(t)|, P (X) is the family of all nonempty subsets of X.For a normed space (X, ∥ • ∥), let Y be a subset of X.We denote are given multivalued maps.When F, G are convex valued, to complete our result we need the following assumtions: and for u, v, w, z ∈ X and a.e.t ∈ J.
Lemma 15. [28] Let X be a Banach space.Let F : J × X 4 → P cp,cv (X) be an L 1 -Carathéodory multivalued map and T be a linear continuous mapping from L 1 (J, X) to C(J, X).Then the operator is a closed graph operator in C(J, X) × C(J, X).
Proof.For each (u, v) ∈ C, define the sets of selections of F, G by and Define the multivalued operators N 1 : and where Consider the continuous operator N : C → P (C) defined by Clearly, the fixed points of N are solutions of the system ( 6) and ( 7).
Then there exist f i ∈ S F,(u,v) , g i ∈ S G,(u,v) (i = 1, 2) such that for any t ∈ J, t = 1, 2, we have Let 0 ≤ λ ≤ 1.Then, for any t ∈ J, we have and Since F and G are convex valued, we infer that S F,(u,v) and S G,(u,v) are convex.Obviously, Step 2. N maps bounded sets into bounded sets in C.
Let r > 0, B r = {(u, v) ∈ C : ∥(u, v)∥ C ≤ r} be a bounded subset of C, (h 1 , h 2 ) ∈ N(u, v) and (u, v) ∈ B r .Then there exist f ∈ S F,(u,v) and g ∈ S G,(u,v) such that for any t ∈ J, We have which yields which yields Thus In a similar manner, we have and Hence we have Step 3. N maps bounded sets into equicontinuous sets in C. Let B r be a bounded set of C as in step 2. Let 0 ≤ t 1 ≤ t 2 ≤ 1 and (u, v) ∈ B r .
Analogously, we can obtain Therefore, the operator N(u, v) is equicontinuous.By the Arzelá-Ascoli theorem, we infer that the operator N(u, v) is completely continuous.
Step 4. N has a closed graph.
Step 5. A priori bounds on solutions.Let (u, v) ∈ λN(u, v) for some λ ∈ (0, 1).Then there exist f ∈ S F,(u,v) and g ∈ S G,(u,v) such that for all t ∈ J, With the same arguments as in Step 2 of our proof, for each (u, v) ∈ C, we obtain (124) Now we set U = {(u, v) ∈ C : ∥(u, v)∥ C < K}.Clearly, U is an open subset of C and (0, 0) ∈ U.As a consequence of Steps 1-4, together with the Arzelá-Ascoli theorem, we can conclude that N : U → P cp,cv (C 1 ) × P cp,cv (C 2 ) is upper semicontinuous and completely continuous.From the choice of U, there is no (u, v) ∈ ∂U such that (u, v) ∈ λN(u, v) for some λ ∈ (0, 1).Therefore, by Lemma 12, we deduce that N has a fixed point (u, v) ∈ U, which is a solution of the coupled system ( 6) and (7).
This completes the proof.