Some inequalities on multi-functions for applying in the fractional Caputo–Hadamard jerk inclusion system

Results reported in this paper establish the existence of solutions for a class of generalized fractional inclusions based on the Caputo–Hadamard jerk system. Under some inequalities between multi-functions and with the help of special contractions and admissible maps, we investigate the existence criteria. Fixed points and end points are key roles in this manuscript, and the approximate property for end points helps us to derive the desired result for existence theory. An example is prepared to demonstrate the consistency and correctness of analytical findings.


Introduction
With the presentation of new analytical results in recent years, the power of fractional calculus in describing processes and modeling physical events and engineering tools has become clear to everyone. In most published papers we are able to observe different generalized fractional modelings of standard equations in which the Caputo or Riemann-Liouville derivatives or their extensions have been utilized in fractional differential equations (FDEs) and fractional differential inclusions (FDIs) such as pantograph inclusion [1], hybrid thermostat inclusion [2], q-differential inclusion on time scale [3], Langevin inclusion [4], and higher order fractional differential inequalities [5]. One can find many published works on various applications of fractional calculus in different fields of science (see, for example, [6][7][8][9][10][11][12][13][14][15][16]).
The authors in [5] showed how fractional differential inequalities can be useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. The nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative has been obtained [5]. On the other hand in [20] the authors analyzed the properties of fractional operators with fixed memory length in the context of Laplace transform of the Riemann-Liouville fractional integral and derivative with fixed memory length [20] on the fractional differential equation These facts could be used to better explain the motivation behind the present study [20]. Jleli et al. studied the wave inequality with a Hardy potential where is the exterior of the unit ball in R N , (N ≥ 2), p > 1, and under the inhomogeneous boundary condition α ∂y ∂x (t, x) + βy(t, x) ≥ w(x) on (0, ∞) × ∂ , where α, β ≥ 0 and (α, β) = (0, 0) [21]. The Caputo-Hadamard derivation operator [22] is another extension of the above operators that many researchers got help from it in their modelings. For instance, we can find the applications of this generalized operator in modeling of the Sturm-Liouville-Langevin problem [23], investigation of the combination synchronization of a Caputo-Hadamard system [24], description of an uncertain BVP' [25], studying the proportional Langevin BVP [26], etc.
Our main novelty in this work is to use the Caputo-Hadamard operator for generalizing the standard jerk problem in the form of a fractional inclusion problem. In fact, a jerk system is a simple form of a nonlinear ODE of third order depicted by where, in mechanics, the nonlinear mapping F(·, ·, ·) is equivalent to the 1st-derivative of acceleration. For this reason, it is introduced as a jerk [27,28]. The mathematical analysis of this generalized system is our main purpose in this work. To do this, we decided to utilize a new family of multi-functions belonging to φadmissibles and φ-ψ-contractions for proving theorems based on fixed point methods. Also, those multi-functions that have approximate property for their end points play a fundamental role in our analysis. These items present the novelty and contribution of our work in this regard, because most researchers get help from standard fixed point techniques in their papers. For example, the Leray-Schauder, the Banach principle, Krasnoselskii, degree principle, Schaefer are the most famous of them, and they are applied in more papers including the generalized proportional equation by Das et al. in [29], impulsive implicit problem by Ali et al. in [30], nonlinear φ-Hilfer problem on compact domain by Mottaghi et al. in [31], multi-term multi-strip coupled system by Ahmad et al. in [32], ψ-Hilfer system of coupled Langevin equations by Sudsutad et al. in [33], sequential RL-Hadamard-Caputo problem by Ntouyas et al. in [34], sequential post-quantum integrodifference problem by Soontharanon et al. in [35] and Samei in [36][37][38], Neumann symmetric Hahn problem by Dumrongpokaphan et al. in [39], etc.
By virtue of the idea of a standard jerk equation and extending it to the generalized fractional Caputo-Hadamard settings, we here introduce and study new existence methods based on some special multi-functions to guarantee the existence of solution for the extended fractional jerk inclusion problem illustrated as in which ι 1 , ι 2 , ι 3 ∈ (0, 1] and CH D p 1 + displays the derivative operator in the sense of Caputo-Hadamard subject to p ∈ {ι 1 , ι 2 , ι 3 } and also t ∈ I := [1, e] and η ∈ (1, e). In addition to these, we have considered the operator G : I × R 3 → P(R) as a multi-function in which P(R) illustrates all nonempty subsets of R.
This research is conducted as follows. Section 2 is fundamental and necessary in its nature since it collects definitions and required results. Section 3 is divided into two parts: one is in relation to the existence criterion via fixed points and the second is in relation to the existence criterion via end points. In fact, in Sect. 3.1, some inequalities between multifunctions and contractions and admissible functions play the role to prove the desired results via fixed point notion. Accordingly, Sect. 3.2 is devoted to proving similar results via end points and approximate property for end points. Section 4 discusses an example for simulating and analyzing the results numerically. Section 5 completes our research via conclusions.

Preliminaries
Here, we shall review some primitive and fundamental concepts in the direction of used approaches and techniques in the present study. As you will observe, these notions and properties are utilized throughout the paper. The readers can find more details in [22,40,41]. 40,41]) Let q ≥ 0. Then the Hadamard fractional q th -integral of a continuous function y : (a, ∞) → R of order q is formulated by H I 0 a + y(t) = y(t) and in which n -1 < q < n and δ = t d dt . Note that, for q = n ∈ N, we have From here onwards, we denote the abbreviations HF-integral and CHF-derivative for the above fractional operators. To find other information on the CHF-operators, we direct the interested readers to [22]. ([22, 40, 41]) Let q, p ∈ R + . Then:
In what follows we give a brief introduction to some special function spaces and multivalued operators. We assume (A, · ) as a normed space. We mean by P CL (A), P BN (A), P CP (A), and P CV (A) the category of all closed, bounded, compact, and convex sets, respectively, belonging to A.

Definition 2.6 ([42]) The (Pompeiu-Hausdorff ) metric, displayed by
is introduced as in which ρ is a metric of A and In the next step, we recall a specific family of multi-functions introduced by Amini-Harandi [42] in 2010 which we utilize in our proofs.

Definition 2.8 ([42]) Let
A be a metric space and G be a multi-valued operator on it.
(2) G admits the AEP-property (approximate end point property) whenever Later, in 2013, Mohammadi, Rezapour, and Shahzad [43] provided another family of multi-functions based on two operators ψ and φ which is a generalized structure of a similar notion pertinent to single-valued operators given by Samet et al. [44] in 2012.

Definition 2.9 ([43]) Let
be a family of all increasing mappings ψ : (3) A admits the property (C φ ) if for each {y n } n≥1 ⊂ A with y n → y and φ(y n , y n+1 ) ≥ 1, To follow the required arguments on the existence of a solution for the Caputo-Hadamard fractional jerk problem (CHF-jerk problem) (1), we begin this section by introducing a Banach space as follows: for all y ∈ A.

Existence results via fixed-points and end points
Now, in the next proposition, the solution's structure for the supposed CHF-jerk problem (1) is exhibited in the format of an integral equation.

, e) and T ∈ C(I, R). Then the solution of the linear CHF-jerk problem
is obtained as where Proof Let y satisfy the linear CHF-jerk problem (2). In view of the semi-group property for HF-integrals given in Lemma 2.3, since ι 1 ∈ (0, 1], so by utilizing the HF-integral of order ι 1 , we get where c 0 ∈ R. Again, utilizing the HF-integral of order ι 2 ∈ (0, 1] to both sides of (5), we get where c 1 ∈ R. At last, utilizing the HF-integral of order ι 3 ∈ (0, 1] to both sides of (6), the general series solution of (2) can be derived by where c 2 ∈ R. To obtain the values c i (i = 0, 1, 2), we first consider the third boundary condition and (5), and so the coefficient c 0 is obtained as In the sequel, the second boundary condition and the obtained value for c 0 in (8) yield Finally, (8) and (9) and the first boundary condition give At this moment, we insert the value of the coefficients c i , by (8)-(10), into (7) and obtain showing that y satisfies (3) and F 1 (t), F 2 (t) are continuous functions represented in (4). This ends the proof.

Fixed-point and jerk model (1)
In this part, we define the solution to the CHF-jerk problem (1).

Definition 3.2
The function y ∈ C(I, A) is named the solution to the supposed CHF-jerk problem (1) whenever it fulfills the given BCs and ∃g ∈ L 1 (I) s.t.
for almost all t ∈ I and ∀t ∈ I. For each y ∈ A, we specify selections of G as In the sequel, define the multi-function K : A → P(A) by for which By making use of the following theorem relying on some inequalities between special multi-functions such as φ-ψ-contractions and φ-admissible, we establish the first criterion guaranteeing the existence of solution for the CHF-jerk problem (1). (1) G is φ-admissible and φ-ψ-contraction; (2) φ(y 0 , y 1 ) ≥ 1 for some y 0 ∈ A and y 1 ∈ Gy 0 ; (3) A involves the (C φ )-property.
Proof Definitely, the fixed point of the mapping K : A → P(A) is a solution of the CHFjerk problem (1). Note that S G,y is nonempty. Indeed, the multifunction is both measurable and closed-valued for any y ∈ A, so S G,y = ∅. Firstly, we will claim that K(y) ⊆ A is closed ∀y ∈ A. As for, take a sequence {y n } n≥1 in K(y) such that y n → y as n → ∞. For each n ≥ 1, there is g n ∈ S G,y such that e r ι 1 +ι 2 +ι 3 -1 g n (r) dr r for all t ∈ I. Since the multifunction G has compact values, there is indeed a subsequence of {g n } n≥1 (following the same notation) that converges to some g ∈ L 1 (I). Subsequently, g ∈ S G,y and y n (t) → y(t) dr r for all t ∈ I. As a result, we can deduce that y ∈ K(y) and K is closed-valued. The boundedness of K(y) is obvious from the compactness of multifunction G. Next, we prove that K is a φ-ψ-contraction. To do this, we regard φ : A 2 → R ≥0 by φ(y,ȳ) = 1 whenever and φ(y,ȳ) = 0 otherwise, where y,ȳ ∈ A. Consider y,ȳ ∈ A and 1 ∈ K(ȳ) and choose g 1 ∈ S G,ȳ such that for all t ∈ I. By making use of (17), we get Now, consider a mapping U : I → P(A) defined by for any t ∈ I. Since g 1 and are measurable, so the multivalued function is also measurable. Now, suppose so that we have t r ι 1 +ι 2 +ι 3 -1 e r ι 1 +ι 2 +ι 3 -1 for any t ∈ I. Then we get the following inequalities as a result.
Then G admits one and exactly one end point iff G contains the AEP-property. Then the CHF-jerk problem (1) has a solution.
Proof We want to establish that the multifunction K : A → P(A) possesses an end point. Initially, we claim that K(y) is closed ∀y ∈ A. As the multifunction is both measurable and closed-valued for any y ∈ A, so the G has a measurable selection and S G,y = ∅. By using the same procedure as that given in Theorem 3.5, it can be easily deduced that K(y) is closed-valued. Also, the compactness of G ensures the boundedness of K(y). Next, assume that y,y ∈ A and 1 ∈ K(y) and choose g 1 ∈ S G,y such that for all t ∈ I. Also, for all y,y ∈ A and t ∈ I, we have There exists We give a mapping : for any t ∈ I. Because g 1 and is too. Take s.t. for all t ∈ I we get . Define 2 ∈ K(y) by e r ι 1 +ι 2 +ι 3 -1 e r ι 1 -1 g 2 (r) dr r for any t ∈ I. Using the same techniques that were employed in the proof of Theorem 3.5, we get that ( 1 + 2 + 3 )ψ y -y = ψ y -y .
Hypothesis (H 10 ) gives the approximate property for the end points of K . Hence, due to Theorem 3.6, ∃y * ∈ A s.t. K(y * ) = {y * }. As a result, y * is a solution of the CHF-jerk problem (1).

Example
We give an example for simulating and analyzing the results numerically.
where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ One can see the results of F 1 (t), F 2 (t) for t ∈ [1, e] in Table 1 and can see a graphical representation of them in Fig. 1. As the multi-function K possesses an approximate end point  1 Graphical representation of F 1 (t) and F 2 (t) for t ∈ I property, hence by using Theorem 3.7, the supposed CHF-jerk problem (19) admits a solution.

Conclusion
In this research work, a generalization of the standard jerk equation in the context of the Caputo-Hadamard differential inclusion (1) was provided, in which we used some inequalities and important properties of multi-valued functions in the framework of the special contractions and admissible mappings. We extracted existence properties of solutions of the mentioned inclusion (1) by applying two different notions of fixed points and end points in functional analysis. This type of the Caputo-Hadamard structure for a jerk problem is a newly-defined FBVP, and we tried to establish our results based on some new non-routine techniques of fixed point and end point theories. With the help of an example, we described our method numerically and graphically. Due to the importance of jerk in the modern physics, it is necessary that we continue our study on the extended models of such physical structures and investigate other qualitative properties of them.